piezoelectric transducers modeling and...

PIEZOELECTRIC TRANSDUCERS MODELING AND

CHARACTERIZATION

August 2004

SERIES OF BOOKS RECOMENDED BY MPI

Keywords: Ultrasonics, Piezoelectric, Transducers, Converters Resonance, Impedance, Active and Reactive Power, Optimal Operating Regimes, Converter Quality Parameters.

Every engineer working in high power ultrasonics with piezoelectric transducers and ultrasonic power supplies would need to know precise answers on topics presented in this book. Instead of spending years of work and enormous amounts of money in establishing proper R&D ultrasonic lab, it is much faster and easier to consult this book which presents synthesis of author’s results during more than 25 years in highly professional R&D environment.

PIEZOELECTRIC TRANSDUCERS MODELING AND CHARACTERIZATION

Author: Miodrag Prokic Published 2004 in Switzerland by MPI 266 pages, Copyright © by MPI All international distribution rights exclusively reserved for MPI Book can be ordered from:

MP Interconsulting Marais 36 Phone/Fax: +41- (0)-32-9314045 2400, Le Locle email: [email protected] http://[email protected] http://mastersonic.com

http://www.mpi-ultrasonics.com/


mailto:[email protected]

CONTENTS: PIEZOELECTRIC TRANSDUCERS

MODELING AND CHARACTERIZATION

1. Piezoelectric transducers modeling2. Converters measurements and characterization3.0 Loading-general3.1 Loading and Losses3.2 Water loading3.3 Connecting boosters and sonotrodes4.0 Impedance combinations and resonances4.1 Inductive Compensation4.2 Piezoelectric Converters Operating High Power4.3 Acoustically Sensitive Zones 4.4 Impedance and Power Matching in practice5. Power performances6. Assembling, Quality and tolerances6.1 Quality Spec-examples7. Ultrasonic Hammer Transducer

ANNEX: MMM Technology

AnalogiesMMM-basicsAnalytic Signal and Power presentation

LITERATURE (see the CD)

1. Piezoelectric Converters Modeling and Characterization

Here are pictures of products that are related to modeling and characterization techniques presented in this paper. All of such devices and components can be measured, modeled and characterized using (more or les) the same methodology and the same terminology.

Converters modeling Miodrag Prokic 27.12.03 Le Locle - Switzerland

1-2

There are many possibilities to present and analyze equivalent models of piezoelectric converters. The analysis presented here will primarily cower modeling of high-power piezoelectric, Langevin, sandwich-converters (applicable in ultrasonic cleaning, plastic welding, sonochemistry and other power industrial applications), and the same analysis can easily be extended to cover much wider field of different piezoelectric transducers and sensors (but this was not the main objective of this paper). For electrical engineering needs (as for instance: when optimizing ultrasonic power supplies, in order to deliver maximal ultrasonic power to a mechanical load) we need sufficiently simple and practical (lumped parameter), equivalent models, expressed only using electrical (and easy measurable or quantifiable) parameters (like resistance, capacitances, inductances, voltages and currents). Of course, in such models we should (at least) qualitatively know which particular components are representing purely electrical nature of the converter, and which components are representing mechanical or acoustical nature of the converter, as well as to know how to represent mechanical load. For here described purpose, the best lumped parameter equivalent circuits that are fitting a typical piezoelectric-converter impedance (the couple of series and parallel resonance of an isolated vibration mode) are shown to be Butterworth-Van Dyke (BVD) and/or its electrical dual-circuit developed by Redwood (booth of them derived by simplifying the Mason equivalent circuit and/or making the best piezoelectric impedance modeling based on experimental results and electromechanical analogies). In this paper the two of mutually equivalent (above mentioned, and later slightly modified), dual electrical models will be used to present a piezoelectric converter operating in its series and/or parallel resonance, Fig. 1. Above described objective has been extremely simplified after electronics industry developed Network Impedance (Gain-Phase) Analyzers (such as HP 4194A and similar instruments). Practically, for the purpose of modeling, it is necessary to select one single converter's operating mode (to select a frequency window which captures only the mode of interest, or the single couple of series and parallel resonance belonging to that mode) and let Impedance Analyzer to perform electrical impedance measurements by producing sweeping frequency signal in the selected frequency interval (and by measuring voltage and current passing on the converter connected to the input of Impedance Analyzer). The next step (implemented in Impedance Analyzers) is to compare the measured impedance parameters with theoretically known converter model (lumped parameters model, already programmed as an modeling option inside of Impedance Analyzer), and to calculate model parameters (practically performing the best curve fitting that places measured impedance values into theoretical impedance model). This way, in a few (button pressing) steps we are able to get numerical values of all (R, L, C) electrical components relevant (only) for selected converter mode and selected frequency range (and this is in most cases the most important for different engineering purposes, such as: optimizing ultrasonic power supplies, realizing optimal resonant frequency and output power control, optimizing converters quality…). We are also able to compare (using modern impedance analyzers) how close are measured impedance values (of a real converter), and values resulting from impedance curve fitting process. In cases of well-designed converters (and converters with sufficiently high mechanical quality factor) we are able to get almost 100% correct modeling in a selected frequency window (meaning that all measured and calculated R, L and C, lumped model parameters, are numerically almost 100% correct). The objectives of this paper are: 1. To explain the most important (and simplified, practical and easy quantifiable), electrical

lumped-parameters equivalent circuits, suitable to represent piezoelectric converter in its series and/or parallel resonance, for purposes such as converters characterization, qualification and optimization, for different electrical design purposes, as well as to explain qualitatively converter models regarding higher frequency harmonics, and


1-3

models when converter transforms mechanical input into electrical output (operating as a receiver or sensor).

2. To establish the very general concept of mechanical loading of piezoelectric converters

(where mechanical load is presented in normalized form using the mechanical-load units comparable to internal resistance of the converter-driving electric circuit, or ultrasonic power supply).

3. To analyze the optimal power transfer of piezoelectric converters (when transforming

electrical input into mechanical output), operating in series and parallel resonance, and to explain mechanical loading process and losses in both situations.

im

Cos um C1

Cop

L1 C2 L2 R2⇔

R1

Fig. 1 Piezoelectric Converter (One-Port), Dual, BVD Models valid in the close vicinity of an isolated couple of converters series and parallel resonances

The Fig. 1 presents two of the most widely used lumped-parameters piezoelectric converter impedance models (mutually equivalent, BVD = Butterworth-Van Dyke, dual-circuits models), valid for isolated couple of series and parallel resonances (of a non-loaded). In fact, on the fig Fig. 1 are presented the simplest models applicable for relatively high mechanical quality factor piezoelectric converters, where thermal dissipative elements in piezoceramics could be neglected. The more general models (again mutually equivalent), representing real piezoelectric (non-loaded) converters with dissipative dielectric losses and internal resistive electrode-elements (Rop (=) Leakage AC and DC resistance, Ros1 ≈ Ros2 (=) Dielectric resistive loss of piezoceramics) in piezoceramics are presented on the Fig. 2. For high quality piezoceramics Rop, is in the range of 10 MΩ - 50 MΩ, and Ros1, Ros2 are in the range between 50 Ω and 100 Ω, measured at 1 kHz, low signal (and can be calculated from piezoceramics tanδ value, or using HP 4194A, and similar Impedance Analyzers). In most of cases of high quality piezoceramics we can neglect Rop as too high resistance, and Ros1, Ros2, as too low values, but we should also know that dielectric and resistive losses are becoming several times higher when converter is driven high power, in series or parallel resonance, comparing them to low signal measurements.


1-4

im Ros2

R2

Cos C1

Cop Rop

L1 um Rop C2 L2

Ros1(≈ MΩ) R1(≈ Ω)

Fig. 2 BVD Piezoelectric Co The other dissipative power losses (branch and come from converter jpiezoceramics and metal parts, hysteresis-related losses (internal mmodels from the Fig. 2 can be schelectric-elements symbolic prescapacitances together with their belinstance: For any electric combinatione resistance we shall introduce between one inductance and one rewe can always find exact circuit traparallel connection). Doing this waygiven on the Fig. 3, and applicable on the Fig. 4.

im

C1

Rop

L*1

C*op

Fig. 3 Simplified BV(L* and C* are presenting real in

integrated, dissipative eleme

Converters modeling Miodrag

⇔

(≈ kΩ)

nverter Models with dissipative elements

R1 and R2) are belonging to the mechanical circuit oint losses, from planar friction losses between from mounting elements and from material echanical damping in all converter parts). The

ematically simplified if we introduce abbreviated enting dissipative (real) inductances and onging resistances, using only one symbol, as for on (or connection) between one capacitance and the symbol C*, and for any electric combination sistance we shall introduce the symbol L* (since

nsformations between two elements in serial and , models presented on Fig. 2 will be simplified as circuit equivalents (used in Fig. 3) are presented

C2

um

C*os

Rop

L*2

⇔

D Piezoelectric Converter Models ductances and capacitances with internally nts: Fully equivalent to models on Fig. 2)

Prokic 27.12.03 Le Locle - Switzerland

1-5

Cs

Rs

Rp Cp C*

⇔ ⇔

Ls

Rs

Rp Lp L*

⇔

R R Q R R Q

C C Q C C QQ C R Q R C f

p s s s p p

p s s s p p

s s s p p p

= + = +

= =

= = =

( / ) , / ( / )

π

/ , ,, / ,

1 1 1 1

1 2

2 2

2 2

ω ω ω

R R

L LQ L

p s

p s

s s

=

=

=

(

(

1

1ω

Fig. 4 Circuit Equivalents & Simplifications (exp

The other elements on the Fig. 2 are: Cos ≈ Cop (=) piezoceramics, C1/2, L1/2 (=) motional mass and stmechanical oscillating circuit/s (see Fig. 7 to find apprbetween all model parameters). We could also add interminals the cable (and winding) resistance, since electrodes, soldered or bonded (electrical) joints, anparameters). The influence of an external acoustic load on the conon the Fig. 5, by introducing loading resistances RL1 amuch simplified equivalent of the real converter loadinRL1 and RL2, sometimes should be treated as comgeneral case).

im

C*⇔ C1

iL1 Rop

RopC*op

L*1 RL1

Fig. 5 Alternative BVD Models of Loaded Pi

Converters modeling Miodrag Prokic 27.12.0

Q R

Q LR Q R

s s

s s

s p

+ =

+

=

) ,

/ ) ,/ ,

1

2

2

laining mo

Clamped, stiffness elemoximate ma series to anevery real d a cable (

verters’ mond RL2, as tg (in reality plex impeda

C2

um

os

L

ezoelectric

3 Le L

⇔

R Q

L QL

p p

p p

p p

+

= +

/ ( )

/ ( // ,

1

1 12

2

2

ω f=

)ω π

dels from Fig. 3)

atic capacitance/s of ents of converter’s thematical relations y of input converter

converter has input presently neglected

deling is presented he closest and very loading resistances nces as the most

iL2

*2 RL2

Converters

ocle - Switzerland

1-6

Based on equivalent electric circuits presented on Fig. 4, we can easily place parallel-loading resistances from Fig. 5 in series with inductances, just by calculating new equivalent frequency-dependant elements-values. In literature regarding the same problematic it is very usual to see that left-side piezoelectric converter-model from Fig. 5 has loading resistance in series with motional inductance and capacitance, and for the model on the right side of the Fig. 5 is usual that loading resistance is found in parallel with motional inductive and capacitive circuit elements (but using Electric Circuit Theory we can easily play with any of parallel or series elements combination, as presented on the Fig. 4). It is also clear that loading nature or load-resistance would change, depending how and where we place it (in situations, like in series connection/s with motional inductance, load resistance would increase with load-increase (starting from very low value), and in case of placing it in parallel with motional inductance (as presented on Fig. 5), load resistance would decrease with load-increase (starting from very high value)). In all above given converter models (Figs. 1,2,3,5), we can recognize motional current im and motional voltage um as the most important mechanical-output power/amplitude controlling parameters of piezoelectric converters in series and parallel resonance. When converter is operating in series resonance, in order to control its output power and/or amplitude we should control its motional current im, and in the regime of parallel resonance, output power and/or amplitude are directly proportional to the motional voltage um. More precisely, when we compare two operating regimes of the same converter, when converter is producing the same output power (in series and/or parallel resonance), we can say that converter operating in series resonance is able to deliver to its load high output force (or high pressure) and relatively low velocity, and when operating in parallel resonance it is able to deliver high output velocity and relatively low force (knowing that output converter power is the product between velocity and force delivered on its front emitting surface). Here we are using the electromechanical analogy system: (CURRENT ⇔ FORCE) & (VOLTAGE ⇔ VELOCITY). When we are talking about converter’s series-resonance frequency zone, this is the case of motional Current-Force resonance (where converter’s impedance has low values), and when we are talking about converter’s parallel-resonance frequency zone, this is the case of motional Voltage-Velocity resonance (where converter has high impedance values). Automatically, if we realize by electrical means high motional current (current resonance, equal to series resonance), the converter will produce high motional force (it will operate in a force resonance). Also, if we realize by electrical means high motional voltage (voltage resonance, equal to parallel resonance), the converter will produce high motional velocity (it will operate in a velocity resonance). All above conclusions, for the time being, are based only on the analogy (CURRENT ⇔ FORCE) & (VOLTAGE ⇔ VELOCITY), and later on, some more (experimental) supporting facts will be presented. It is also important to underline which circuit-elements (in all above found circuits, Figs. 1,2,3,5) are representing purely electrical elements of a piezoelectric converter, and which elements are only given as functional (and analog) electrical equivalents of converter’s mechanical parts and its mechanical properties (including loading elements), see Fig.6.


1-7

It is very important to know that mechanical converter-loading, presented on Figs. 5 & 6, is equally and coincidently influencing changes, both in series and parallel converter impedance, basically reducing equivalent mechanical quality factor/s of a loaded converter (or coincidently increasing its series resonant-impedance and decreasing parallel resonant-impedance). This is the principal reason why (in this paper) an isolated couple of series and parallel resonances is treated as the same, single and unique oscillating-mode that can be driven in its current or voltage resonance, and produce force or velocity-dominant mechanical output. It is also shown to be possible to drive an ultrasonic converter high power, extremely efficiently, in any frequency (continuously) between its series and parallel resonance (when special converter reactive impedance-compensation is used). Since there is certain frequency shift between each couple of series and parallel resonances, and since most of today’s converters (made by big players in ultrasonic industry worldwide) operate either in series or parallel resonance, in literature regarding converters modeling and converters measurements, many authors are talking about different vibration modes or different harmonics.


1-8

C2

0s

RL1

⇔

RL2 R0p

C1

L*1

C*op

Mecele

Purely electrical elements

⇓⇑

⇔

*

Current Transformer

Fig. 6 Alternative BVD Models of Loseparation of purely electrica

Down: The same models with transformer-se

Also in different literature regarding pievery similar equivalent circuits (as prevthem developed from Mason model), wpaper are not present or have different all mutually equivalent piezoelectric mpresent 100% dual (equivalent) electric celements, and presenting the same (eland/or AC electrical currents and vo(regardless frequency).

Converters modeling Miodrag Pro

C*

R0p

Purely electricalelements

⇓

VoTran

aded Piezoelectril and purely mechparation between electr

zoelectric converteiously presented ohere some of circtopology. Here is odels (Figs. 1,2,3ircuit-structures, h

ectrical) impedanceltages, connected

kic 27.12.03

L*2

Mechanical elements

hanical ments

⇑

*

ltage sformer

c Converter with block anical elements

ical and mechanical circuits

rs modeling, we can find n Figs. 1,2,3,5 & 6, all of uit elements found in this accepted the strategy that ,5 & 6) should mutually aving the same number of s for the same input DC

to their input terminals

Le Locle - Switzerland

1-9

ωp

ωs

L2

Cos

ωs

ωp

Cop

C1L1

⇔ ⇓

C2 ⇓

C*os L*2

L*1 C1 ⇔ C2

C*op

ω ω π

ω ω π

sos

osop

os

pop

os

opos

os

op

op

os op opop

os

osos

fL C L C C

C C ff f

CC

CC

fL C

LC C

C C

C C CC C

CC C

ff

C C C C C Cf

f fL C C

L C

LLC

C C LL CC

= = = =+

=−

≈

= = = =

+

=+ +

=

= − = +−

=

= + =

1 11 1 2 2

12

2

22

12

1

2

2 22 2

11

2

2

2 1

1

2

2

1 2 11

2

22

12

1 1

2

12

12 2

1 1

2 1 1

2 1 1

1

( ), ,

, ,

, ( ) ,

( ) , (

,

( ) ,

− =+

= ⋅CC

kC

C CCC

ff

op

os op

os) , .2 1

1 2

12

22

Fig. 7 Simplified Evolution of BVD, Equivalent, Dual Circuits

Following this strategy, it was necessary to introduce certain electric elements that are not found in other literature sources regarding the same problematic in order to satisfy circuits symmetry and DC and AC current/s and voltage/s balance, when applying Kirchoff’s, Ohm’s and other Circuit Theory Laws. Practically, as presented in a very simplified and condensed form on Fig. 7, we can easily see that all of above discussed equivalent electric circuits (Figs.1,2,3,5,6) are based (or established) on the evolution of mutually equivalent (or dual) electric circuits (known in modern filter design), combined with circuit equivalents from Fig. 4. Different relations between dual-model elements found on Fig. 7 are also applicable (as the best approximation for sufficiently high quality factor circuits) to all models presented on Figs. 1, 2, 3, 5, 6 and to other piezoelectric transducer models based on Mason model. In order to give the full picture of the modeling strategy presented in this paper, it would be interesting to explain how model of a (single) piezoelectric converter transforms when we connect a horn or booster to it (of course, added horn or booster should have almost the same resonant frequency as a converter). Practically we should know where in the basic converter model we place one more equivalent


1-10

circuit-model presenting added horn. The most important background related to this situation is to know that added horn also presents one mechanical resonant structure that can be replaced with an electrical (equivalent) resonant circuit. In most of the literature regarding converters modeling, added horns are treated as converter loads, but here we shall (primarily) treat them as added resonant boxes, resonant circuits, or added filter circuits (all of them operating at the same resonance). Starting from evolution-equivalent circuits presented on Fig. 7, and from basic converter models presented on Fig. 3 we can illustrate what means “converter+horn” modeling with new step-by-step circuit-evolution models given in Fig. 8. We know that regardless how many boosters and horns we add to a converter (all of them having the same resonant frequency); -finally, after making impedance measurements (with HP 4194A, for instance) we get principally the same models as models on Fig. 3 (just particular circuit parameters and mechanical quality factor are changed). Of course this is a kind of over-simplified statement, since added sonotrodes with complex shapes can create new resonant frequencies, but here we accept (in advance) to limit our observations only to a well isolated and separated couple of series and parallel resonances. What is happening (Fig. 8) is that by adding horns to a converter we create higher-orders electrical and mechanical filter-circuits that can be again (going backwards and applying Electric Circuits Theory) transformed into basic converter models presented on Figs. 1,2,3,5,6. We also see that in simplified circuit evolution chart on Fig. 8, we deal with strongly mutually-coupled resonant-circuits (either the same current or the same voltage are coupling such circuits), operating on the same resonant frequency, and that we can place the Load where it would be physically (loading should again be treated on the same way as presented on Figs. 5 & 6). The internal losses of added horns are also not neglected, since we can find dissipative elements in added-horn-related resonant circuits.

Cs C*os C1

L*s

C*op C*op C2 L*2⇒ ⇔

L*s1

Cs1

Added horn L*1

L1 = Ls +Ls1C1 = (CsCs1)/(Cs + Cs1) CsLs = Cs1Ls1 = C1L1

Fig. 8 Evolution of “Converter-Horn” BVD, Dual Circuits

We also know (from many years of experience in ultrasonic engineering) that when we add the horn, booster or some other (well designed) sonotrode to a converter, new, resulting mechanical quality factor of the combination “converter+horn” will


1-11

become much higher than it was in the case of a single converter, meaning that mechanical resonant circuits (“converter+horn”) should be mutually connected in a connection that creates higher mechanical quality factors. This is in agreement with filter theory knowledge (regarding connecting multiple filter blocks in series (or sometimes parallel, or more complex) connection, having the same resonance/s), and also strongly supports the evolution chart on Figs. 8 & 9. We can also create another simplified circuit evolution chart (see Fig. 9), equivalent to one of Fig. 8, in order to explain “converter+horn” modeling (when converter and horn operate on the same resonant frequency), starting from a dual circuit model where motional capacitance and inductance are in parallel connection. It is also important to underline that motional inductances of “converter+horn” combination, both given on Fig. 8 and Fig. 9 are strongly (acoustically, or mechanically) coupled, oscillating on the same resonant frequency, what makes presented circuit transformations easier to understand. From the point of view of Electric Circuit Theory, a strong “converter+horn” mechanical coupling on Figs. 8 & 9 can also be presented in the convenient equivalent form of coupled inductances (L*s coupled with L*s1 and L*p coupled with L*p1), or in the form with ideal transformer/s coupling (where for Fig. 8, L*s is the primary transformer section and L*s1 and Cs1 are in the secondary transformer section, or where for Fig. 9, L*p is in the primary transformer section and L*p1 and Cp1 are in the secondary transformer section).

L*p1Cp1L*p

Added horn

⇒

C*os

C1C*os

C*op

L*2

⇔ Cp C2 L*1

L2 =CpL

Fig. 9 Evolution of “Con It remains also to explain converter hframe of already established lumpedFig. 10-b are presented different cconnection of basic resonant-circuit present (simplified) direct resistive loa The single-resonant-mode loading equivalent circuit transformations) in

Converters modeling Miodrag P

C2 = Cp +Cp1 (LpLp1)/(Lp + Lp1) p = Cp1Lp1 = C2L2

verter-Horn” BVD, Dual Circuits

armonics (higher frequency resonances) in the -parameters modeling. On the Fig. 10-a and ircuit possibilities using series and parallel

configurations. The cases on Fig. 10-a also ding on the first resonant mode.

(Fig. 10-a) can also be presented (using the form of inductive transformer coupling,

rokic 27.12.03 Le Locle - Switzerland

1-12

where RL can be placed on the secondary side of the transformer, which primary side is the converter motional inductance, or we can make some other transformer and/or inductive-coupled (equivalent circuits) combinations in series with motional inductances. Most of above mentioned “inductive and transformer coupling options” we can implement in a certain (most convenient) form of equivalent circuit models to all of loading situations presented on Figs. 10-b until 10-e.

* * *

Higher Resonant Modes & Harmonics

C*op

* RL

⇑ R

Fig

The mcase previoloadin(1), (2 For inconneoperaload (mechais the sonotr(4) pre

Con

⇓

*

* * L

*

C*os

. 10-a Dual-Circuits, Converter Modeling Including Harmonics, valid for acoustically or mechanically non-loaded converters

uch more general case of converter (dual) models with harmonics (than the on Fig 10-a), including different loading situations (which are covering all usly discussed models), is presented on the Fig. 10-b. All possible mechanical g situations (Fig. 10-b), are presented by (dual) load-impedances marked with ), (3) and (4).

stance, if we take only the upper converter model (Fig. 10-b), when load (1) is cted to mechanical output terminals (short-circuit connection), converter is ted in idle, or no-load condition, and it is called a piezoelectric resonator. If the 2) is connected to mechanical output terminals, this is the case of resistive nical loading. If the load (3) is connected to mechanical output terminals, this case of a simple mechanical resonator similar to a single-resonant-frequency ode or booster, or some other simple mechanical oscillating system. The load sents the most general case of arbitrary and complex mechanical impedance.

verters modeling Miodrag Prokic 27.12.03 Le Locle - Switzerland

1-13

For the lower converter model (Fig. 10-b) we again have the same (analogue and dual-models) situation, as already explained for the upper model. The only exception is that when load (1) is connected to mechanical output terminals (in fact nothing, or open-contacts are connected), converter is operated in idle, or no-load condition, and it is called a piezoelectric resonator. The loads (2), (3) and (4) have the same meaning as already explained for the upper model (but presented as dual circuits).

*

RLS*

C*op

⇓⇑ C*os

Fig. 10-b Dual-Circuit

Loads (here m In most of cases of poperating resonant modassuming that mechanvariable (load-dependaloading models presentloading-models presentequivalent (dual electricequivalent circuits (andresistive acoustic load (resistance (RLs1,2 or Rconverter equivalent cirhave only series load have proportional inccircuit options where

Converters modeling

ZLS* * *

* *

*

(1) (2) (3) (4)

ZLPRLP

*

s, Converter Modeling Including Harmonics and Darked by electric impedances (1), (2), (3) and (4))

ractical interest (in ultrasonic technology), we use e, and operate converter in its series or parallel res

ical (or acoustic) load can be approximated or replnt) resistance. For such simplified situations, insteaded on Figs. 5, 6 … 10-b, we can standardize the used on the Fig. 10-c. All of models on Fig. 10-c are circuits), and can be mutually transformed using the relations) presented on Fig. 7 and Fig. 4. Effecon Fig. 10-c) is presented either by series or parallel

Lp1,2) placed (appropriately) in the motional partcuit/s. If we apply the circuit options (Fig. 10-c) wresistances, RLs1,2, by the acoustic load increase wrease of series load resistances. Also, if we ap we have only parallel load resistances, RLp1,2,

Miodrag Prokic 27.12.03 Le Locle - Switz

*

)
(1) (2) (3) (4
Different LoadingPossibilities

Electrical InputTerminals

ifferent

a single onance, aced by of using e of the mutually generic tively, a variable of the here we

e shall ply the

by the

erland

1-14

acoustic load increase we shall have proportional reduction of parallel load resistances. Consequently series load resistances RLs1,2 should be directly proportional to the acoustic impedance ρvs (= density x sound speed x surface), and parallel load resistances RLp1,2 should be directly proportional to the acoustic admittance or mobility = 1/ρvs (of the media in contact with a piezoelectric converter). This situation is very important step in understanding and modeling of piezoelectric-converters loading, since (based on models on Fig. 10-c) we can establish generally valid platform for different analyses of optimal power transfer, impedance matching situations, reactive impedance compensation, mechanical output control etc.

RLS1 RLp1

RLS2 RLp2

L*1sC1

C*op

C1

L*1pC*opRop Rop

L*2s

L*2p

C*os

C2 C2

C*os

Rop Rop

⇔

⇔

⇓⇑ ⇓⇑

Electrical Input

Parallel Mechanical

Load

Electrical Input

Series Mechanical

Load

Fig. 10-c Single-Mode, Series and Parallel, Resistive Load-Impedance Models (Developed using equivalent circuits from Fig. 4)

Using similar equivalent-circuit idea, like presented on the lower part of the Fig. 3, we can also make transformer-separation between every converter and its load (applicable, for instance, to Fig. 10-b, Fig. 10-c…), by introducing voltage or current transformers (which will not change equivalent input impedance/s of the models where such separation would be applied). Every piezoelectric converter is in the same time able to detect mechanical vibrations, serving as an accelerometer, microphone, receiver or sensor (producing an electrical output signal, proportional to its mechanical excitation). If we again consider only simplified converter models (without harmonics; Figs. 1, 2, 3, 5 and 6), we can present converter’s mechanical excitation by dual models given on Fig. 10-d.


1-15

Later on, if we would like to include harmonics into Fig. 10-d, we can simply use the modeling structures presented on Fig. 10-b and keep the mechanical excitation in the same place/s as presented on the Fig. 10-d. It is important to have a feeling how external mechanical excitation influences converter’s operation (and modeling) because in many situation of interest (like ultrasonic cleaning, welding, liquid processing…), when we drive a converter electrically in order to produce mechanical output, in the same time its load can produce (or reflect) mechanical excitation and generate additional charges, currents and voltages inside of a converter’s structure. When piezoelectric converter is used only as the sensor of mechanical excitation, we also need to know what are its most convenient electrical models in order to qualify and quantify sensor parameters and sensor output signals. Models presented on the Fig. 10-d are also important if we would like to operate a piezoelectric converter as an (lock-in) active vibration source able in the same time to detect external mechanical excitation in the well-selected frequency zone (or using piezoelectric converters and sensors in interactive and impedance sensitive operating regimes).

Low Frequency Sensor Operation (resonant mode elements neglected)

Sensor Operating in Resonance Area (including low frequency domain and load)

C

Ca

C1 C*op

1C*op Charge

Source Qa

Va Rop C*c R C*

ZLLo

⇑⇓

Voltage Source ua

V S⇑⇓

ZLPLoad

C*os

Rop C*c

V QC C C

QC Ca

a

op c

a

os

≅+ +

≅+1

Fig. 10-d Converter as a Receiver of M Piezoelectric converters when operating as sensorsconstant mechanical excitation (external force), ge

Converters modeling Miodrag Prokic 27.12

L*

op c

Sad

oltage ource ua

C2

C*os

L*2 Rop C*c

c

echanical Excitation

are also able to detect nerating certain consta

.03 Le Locle - Swit

Cable pacitance

Cable apacitance

Converter Elements

Va

Charge Source

Qa

Sensor Elements& Excitation

static and nt charge

zerland

1-16

and/or voltage, because of the presence of permanent (frozen) electrical field inside of the crystal structure of piezoelectric materials (created during production of that piezoelectric material by separating internal positive and negative electric charges in the form of oriented electrical dipoles). When external (static) force is applied on a piezoelectric sensor, internally separated electrical charges (belonging to polarized piezoelectric crystal structure) will slightly change their equilibrium position and release certain amount of free electrical charge (proportional to the applied force), which will appear on the external sensor electrodes. In cases when there is only a static, or low frequency mechanical excitation applied to a piezoelectric sensor, we can safely apply two sensor models presented on the left side of Fig. 10-d, and treat charge source Qa and voltage source ua as static electric sources. The same situation becomes more interesting when external mechanical excitation is originated from a dynamical, high frequency vibration source, which is in the same frequency range where sensor has its resonant frequencies, because this type of excitation usually has very low, or close to zero average-level of amplitudes (regarding its average force or average velocity). Sensor will still remain able to react on constant external force, and in the same time be able to detect external dynamic excitation. For such situations we can create models where static and dynamic (or variable) excitation sources are separated, as presented on Fig. 10-e (or we could also separate them in the same positions where they are already placed on Fig. 10-d). Dynamic external excitation (Fig. 10-e) is introduced in the form of inductive coupled voltage sources with motional inductances.


1-17

C1

L*1

C*op

Rop

C*c

Qas

C*os

C2L*2 Rop

uasC*c

⇑ ⇓

Cable Capacitance

Static Charge, Qas , and Static Voltage, uas, Sources

ZG

Dynamic External Vibration Sources uG , ZG

uG

ZG

uG

Fig. 10-e Converter as a Receiver of Mechanical Excitation where static and dynamic excitation are separated

Using Circuit Theory, we could create many of equivalent models, similar to models presented on Fig. 10-d and Fig. 10-e, “playing” with equivalent and/or dual circuits, with transformations between voltage and current/charge generators, etc. Let us go back to converter models with isolated or single resonant mode presented on Fig. 1 until Fig. 6. It is obvious that we can drive piezoelectric converter in any of its characteristic resonant frequencies (series or parallel) exercising different advantages and disadvantages, depending on application and on converter’s operating regime. Also, it is very important to know that in any of the resonant-frequency zones (in series and/or parallel resonance) we could create either maximal converter amplitude, or maximal velocity or maximal force, depending how we control the resonant output circuit and compensate the piezoelectric converter, and what we prefer to maximize as the mechanical output. For instance, in series resonance zone we shall have maximal motional current and maximal motional force mutually in phase (presenting natural, series mechanical resonance), but maximal velocity and maximal displacement (in the same zone) would be realized on two other resonant frequencies (and all of them would be relatively close to each other). The same is valid for parallel resonance zone where we shall have maximal motional voltage and maximal motional velocity mutually in phase (presenting natural, parallel mechanical resonance), but maximal force and maximal amplitude (in the same zone) would be realized on two other resonant frequencies (and again, all of them would be relatively


1-18

close to each other). When we create electrical resonance on the converter input terminals, maximizing either converter’s input-current or input-voltage, such electrical resonance/s will not present in the same time the best (particular) mechanical resonance (or saying differently, converter’s input-current resonance will be only close to converter’s series mechanical-resonance (but not identical to it), and converter’s input-voltage resonance will be close to converter’s parallel-mechanical resonance, too). Such finesses, regarding differences between electrical and mechanical resonance operating areas usually pass unrecognized by majority of people active in ultrasonic design and engineering. In this paper the frequency zone of converter series resonance and minimal impedance will be simply characterized by frequency f1 (or expressed as the frequency interval [f1]), and frequency zone of converter parallel resonance and maximal impedance will be characterized by frequency f2 (or expressed as the frequency interval [f2]), since for high mechanical-quality-factor power converters, all electrical and mechanical resonances in the zone [f1] are very close to each other, and the same is valid for zone [f2]. The more detailed analyze of the piezoelectric converter impedance (of an isolated resonant mode: Figs. 1- 6) will show that the typical impedance-frequency curve (as well as other impedance parameters in the function of frequency) are similar as presented on the Fig. 11-a. In order to give precise mathematical expressions for all possible converter resonances (valid for single resonance; -models from Figs. 1,2,3,5,6), let us present converter complex impedance as two electrical parts in series connection, or as a connection between resistive impedance Re and reactive impedance Xe (Fig. 11-a).

Z =R jX R X e Z e ZXR

e e e ej X

Rj X

R j

e

e

e

e

e

e+ = + ⋅ = ⋅ = ⋅

=

2 2arctan arctan

,

arctan .

θ

θ

e

It will also be very much useful to present reactive parts of converter’s motional impedances (see dual electrical circuits: Figs. 1,2,3,5,6 & Fig. 11-a) as X1 and X2,

X X X LC

X X X LL C

L C

L C

1 1 1 11

2 2 22

22 2

1

1

= + = − =

= =−

=

ωω

ωω

( motional inductance & capacitance in series) ,

( motional inductance & capacitance inparallel)


1-19

IMPEDANCE

Z = ⎢Z ⎢e j θ = = Re + jXe

PHASE θ = arctg(Xe/Re)

Fig. 11-a Impedance Functions vs. Frequency


1-20

The meaning of characteristic resonant frequencies (on Fig. 11-a) is as follows:

RESONANCES CONDITIONS EQUATION fm – frequency where absolute value of impedance reaches its minimum

Z Z= min. f frm s= −( )1

2

2α

fs – series motional current-force, mechanical resonant frequency

X X X

LC

L C1 1 1

11

1 0

= + =

= − =ωω

fL Cs =1

2 1 1π

fr – series electrical, input-current resonance, where phase pass zero line

θ = =arctan ,XR

Xe

ee0 0= f f

rr s= +( )12

2α

fa – parallel, electrical, input-voltage resonance, where phase pass zero line

θ = =arctan ,XR

Xe

ee0 0= f f

ra s= +−( )1 12

2α

fp – parallel motional voltage-velocity, mechanical resonant frequency

X LL C2

22

2 21=

−→ ∞

ωω

fL Cp =1

2 2 2π

fn – frequency where absolute value of impedance reaches its maximum

Z Z= max. f frn s= +

+( )1 12

2α

Abbreviated symbols rCC

ff f

rQ

C Rf L C

op

m

op

s

= =−

= =1

12

22

12

1

1

1 12, α

π

A piezoelectric converter when operating at fs, presents the source of high motional force and relatively low motional velocity (comparing it with the same converter operating in fp and producing the same power). If we operate the same converter slightly increasing its operating frequency towards fr, the converter would start decreasing its motional force and increasing its motional velocity. Since all of the frequencies, fm, fs and fr are mutually very close (for high mechanical quality-factor converters), in most of analyzes here we shall replace all of them with f1 ≈ (fm< fs< fr). Something similar (but in much wider frequency range) would also happen if we start operating converter (high power) from fs, decreasing its operating frequency towards lower frequencies. Also, a piezoelectric converter when operating at fp, presents the source of high motional velocity and relatively low motional force (comparing it with the same converter operating in fs and producing the same power). If we operate the same converter slightly decreasing its operating frequency towards fa, the converter would start decreasing its motional velocity and increasing its motional force. Since all of the frequencies, fa, fp and fn are mutually very close (for high mechanical quality-factor converters), in most of analyzes here we shall replace all of them with f2 ≈ (fa< fp< fn). Operating in the vicinity of f2 towards higher frequencies is not beneficial


1-21

regarding producing high power, since the amount of accumulated elastomechanical (potential) energy in that frequency area is on the very low level. The converter output mechanical power is equal to the product between its motional velocity and motional force, or to the product between its motional voltage and motional current. An optimal power transfer (from electrical to oscillatory, mechanical power) is achieved when motional current and motional voltage are mutually in phase, producing only the active or real output power (and consequently converter should operate either in fs or in fp). Some ultrasonic companies are also operating their converters in the frequency area between f1 and f2, this way realizing the output power regulation by moving the impedance-frequency operating point (or effectively controlling the phase difference between motional current and motional voltage). The same phase-impedance regulation concept (moving the impedance-frequency operating point) can also be realized for the frequency area in close vicinity to f1 and left from f1, towards lower frequencies. The simplest circuit replacements (equivalent circuits) for complex impedance of a piezoelectric converter, in the function of operating frequency interval/s, are presented on Fig. 11-b. For the most of design needs regarding electronics where piezoelectric converters and sensors are involved, it is very important to know the parameters of the circuits presented on Fig. 11-b in order to select the proper converter or sensor regime. Obviously the piezoelectric converter impedance is complex and changing its character (depending on frequency), being dominantly capacitive, or dominantly inductive and/or certain combination of capacitive, resistive and inductive elements. On the Fig. 11-c are presented different aspects of converter impedance vs. frequency that can be measured using Network Impedance Analyzer (programmed to detect only particular impedance element, such as input converter capacitance, or input inductance or input resistance).


1-22

⇑⇓

Fig. 11-b Simplified Piezoelectric-Impedance Models vs. Frequency


1-23

Z

Z = Re + jXe

Z = Z ⎢e j θ Z = (Re

2 + Xe

2) θ= arctg(Xe/Re)

Z, θ, Re, Ls/p, Cs/p, Xe, Rs/p

Fig. 11-c Different Converter-Impedance Elements vs. Frequenc

Based on Mobility system of electromechanical analogies (Current = Force= Velocity), electrical equivalent circuits and particular electrical impedancpresented on Fig. 11-b and Fig. 11-c can be directly transformed into corremechanical equivalents, as given on Fig. 11-d.

Converters modeling Miodrag Prokic 27.12.03 Le Locle - Swit

Logarithmic (dB)Scale

Linear Scale

Z = Re + jXe

y

, Voltage e curves sponding

zerland

1-24

Fig. 11-d Simplified Piezoelectric-Impedance Electrical and Mechanical Models

vs. Frequency (based on Mobility type of Analogies). In ultrasonic engineering, regarding (isolated) resonant operating regimes of piezoelectric converters (and sensors), the most interesting is to determine the simplest and sufficiently applicable equivalent circuit models in order to realize


1-25

optimal converter driving, optimal power transfer, the best signal reception and proper electrical and mechanical converter impedance matching. Let us take the most widely used converter (dual) models presented on Fig. 1 (used later on, too). If we now imagine that converter is operating precisely in its series resonance, the corresponding motional reactive-impedance part would be equal zero,

X LC1 1

1

1 0= − =ωω

. In case if converter would operate exactly in its parallel

resonance, the corresponding motional reactive-impedance part would disappear

becoming extremely high impedance, X LL C2

22

2 21=

−→ ∞

ωω

. Now, for resonant

operating regimes of sufficiently high mechanical quality factor converters (where we can apply all reasonable approximations and circuit simplifications, as already explained in this chapter), the models presented on Fig. 1 can be even more simplified, as presented on Fig. 11-d.

Cop

Cos

R1 R2

Part of purely electrical circuit

Part of mechanical/motional circuit

Converter in

Parallel Resonance f∈[f2] , X2 →∞

Converter in Series Resonance

f∈[f1], X1 = 0

Fig. 11-d Approximated Converter Models in Series and Parallel Resonance in free, no-load conditions

If we apply equivalent circuit transformations as presented on Fig. 4, it will be possible to additionally transform both models from Fig. 11-d into two more of mutually corresponding series and parallel connection/s between equivalent resistance/s and capacitance/s. It is also important to underline that two circuit models presented on Fig. 11-d are no more mutually equivalent (as their predecessors), since each of them is previously simplified for different operating frequency. In order to maximize real (or active) output power of a piezoelectric converter operating in one of its resonances it would be necessary to neutralize the capacitive components belonging to circuits on Fig. 11-d, by connecting externally one inductance in parallel or series connection (respectively). In reality, when we analyze the optimal power conversion, we should upgrade models on Fig. 11-d with other circuit-elements visible on Figs. 2,3,5,6…that are presently neglected. We


1-26

should also know that for non-loaded piezoelectric converters, in most cases of practical interests (Fig. 11-d, Figs. 1,2,3,5,6…), R1 is in the range between 5 Ω and 100 Ω, and R2 is in the range between 10 kΩ and 100 kΩ (or higher than 100 kΩ). Coincidently with converter load increase we can find that, R1 would also proportionally increase (typically until approx. 10 times higher value), and R2 would proportionally decrease (typically until approx. 10 times lower value). In certain circuit configurations, piezoelectric converter can be conveniently compensated (by adding external inductive and capacitive components to its input electric terminals) to become pure resistive impedance in relatively large frequency intervals, and still be able to operate high power inside of such frequency intervals.


2-1

2. Converters Measurements and Characterization

Let us now start using converters modeling (based on previously established equivalent models) for the purpose of converters characterization, optimization and defining converters quality parameters (using only electrical measurements data). For the purpose of illustrating measured and calculated converters’ quality parameters, in the table T 1.1 are given (low-signal) measurement data of BRANSON’s converter, model 502/932R, max. 3000 Watts, 20 kHz (see also Fig. 12 and Fig. 13 with impedance-phase curves of the same, non-loaded and fully-loaded 502/932R converter), measured using HP 4194A Network Impedance Analyzer. The converter example chosen here is neither the best nor the worst case of 502/932 BRANSON converters, and it is only taken for the purpose of having numerical illustration/s following converters modeling presented in this paper. Converter loading (regarding data in T 1.1) is realized by placing its front emitting surface on a thick rubber-foam full with water (with wooden-plate backing), and by pressing the converter towards its load (applying a force of about 20 kg, on the converter housing). Practically, this kind of load simulation is measured by HP 4194A under low current and voltage signals (sweeping frequency signal of 1 v, rms, produced internally in HP 4194A). In reality loading measurements should be made with operating converter driven high power, but from long experience in this field, we know that it is sufficiently representative to make low-signal measurements (similar to above explained situation) in order to get useful results that would be in the error-limits of max. 10%, compared to a situation if measurements were made operating converter full power. Here we shall follow the strategy to explain qualitative changes during converters (mechanical or acoustical) loading, and until standardized loading situations are not specified, loading would stay application-dependant and descriptively explained process.

502/932R converter


2-2

T 1.1 (BRANSON 502/932R, 3000 Watts, 20 kHz converter; measurement data)

Assembled Converter In Series Resonance In Parallel Resonance

C nFC nFL mHR Zf HzQ

op

m

=

==

= =

==

18 14 04617 5344 6 4 66

18900453 73

1

1

1

1

01

. ,. ,.

. , .

.

minΩ Ω

C nFC nFL HR K Zf HzQ

os

m

K

===

= =

==

22 05101 53570 5094 20 96

209121256 7

2

2

2

2

02

. ,. ,.

. ,

.

max

µΩ Ω

ZZ

RR

max

min

.1000 1000

20 542

1

≅ ≅

Non-Loaded (modest quality) Converter Fig. 12 Coupling factor of the fully assembled, converter (PZT8, k33=0.64)

k Energystored in mechanical formTotal input energy

Mechanical energy stored by (assembled) converterMechanical energy that can be stored (only) in piezoceramics

c2 = =

+≅ ⋅ ≅ ≅

≅ ≅

CC C

CC

ff

k

kk

op

osc

c

1

1 2

12

22

2

332

2

018 04243

042430640

044

. , .

( ..

) .≅ .

(the ideal converter design would be when ) ( / )k kc 332 1≅

C nFC nFL mHR Zf HzQ

op

mL

=

==

= =

==

18 783 8019 3757 574 56 24

1852539 205

1

1

1

1

1

. ,. ,.. , .

.

minΩ Ω

C nFC nFL HR K Zf HzQ

os

mL

K

===

= =

==

22 5896 74631 231 50 1 879

2038718 63

2

2

2

2

2

. ,. ,.

. , .

.

max

µΩ Ω

ZZ

RR

max

min

.1000 1000

0 032

1

≅ ≅

Fully-Loaded (modest quality) Converter (still well operational and able to deliver maximal output power), Fig. 13

k Energystoredinmechanical formTotal input energy


c2 = =

+≅ ⋅ ≅ ≅

≅ ≅

CC C

CC

ff

k

kk

op

osc

c

1

1 2

12

22

2

332

2

018 0425

04250640

0441

. , .

( ..

) .≅ .



2-3

Assembled Converter In Series Resonance In Parallel Resonance

C nC nFL mRZf HzQ

p

m

0

1

1

1

1

01

17 324013 904418 3335187

19818815

1139

=

===

=

==

. ,. ,.

. ,. ,

,min

Ω

Ω

F

H

C nC nL HR KZ Kf HzQ

s

m

0

2

2

2

2

02

21 22897 08423595 6216

19120828

2485

FF

====

=

==

. ,. ,

. ,,

,max

µΩ

Ω

ZZ

RR

max

min1000 10001062

1

≅ ≅

Non-Loaded, High Quality 502/932R Converter (well selected, just for comparison with modest quality converter)

Coupling factor of the fully assembled, converter (PZT8, k33=0.64)

k Energystoredinmechanical formTotal input energy


(The ideal converterdesign would be when (k .

c2

c

= =+

≅ ⋅ ≅ ≅

≅ ≅

≅

CC C

CC

ff

k

kk

k

op

osc

c

1

1 2

12

22

2

332

2

332

01811 04256

042560640

0442

1

. , . ,

( ..

) .=

/ ) )

,

Only

piezoceramics In Series Resonance In Parallel Resonance PZT-8, Vernitron – Morgan-Matroc, USA

n = 6 piezoceramic rings: ≅ φ50 x φ20 x 5 mm d33 = 245 x 10-12 [v/m] ± 10% , k33 = 0.640,

Cinp.(1 kHz) = 19.55 nF, tan δ (1 kHz) = 0.000279, Rs(1 kHz) = 56 Ω, Rp(1 kHz) = 52 MΩ

C nC pL mRZf HzQ

op

m

=

===

=

==

2 68554336 55269 616412 741

12 872432880

1129

1

1

1

1

10

..

.

.min. .

Ω

Ω

FFH

C nC nFL HR kZf HQ

os

m

F

z

====

=

==

3 022124 26857 9260 045

256 86134880

1385

2

2

2

2

20

....

max. .

µΩ

ZZ

RR

max

min

.1000 1000

20 182

1

≅ ≅

Parameters of Non-Loaded, single, PZT8 piezoceramic ring/s, used for assembling BRANSON 502/932R (first, natural, radial resonance mode/s)

k k f ff

CC C

CC

ffc eff

op

os2 2 22

12

22

1

1 2

12

22 01110 01110 03332= =

−≅

+≅ ⋅ ≅ ≅. . , . . .

(numerical values relevant only for the first radial mode)


2-4

Fig. 12 Impedance curve of non-loaded BRANSON 502, 932R converter


2-5

Fig. 13 Impedance curve of fully-loaded BRANSON 502, 932R converter


2-6

Converters Quality Parameters

The most important quality quantifications of a high power ultrasonic converter (operating in certain selected resonant mode) are its mechanical quality factors (related and found only for the mechanical oscillatory circuit-part/s of the equivalent converter models, Fig. 6), equal to,

Q EEm

s

d

= =2 2π πEnergy stored in the transducer during one full periode

Energy dissipated in the transducer during one full periode, (1.1)

and can be expressed (similar as in Electric Circuit Theory) for series resonance and for non-loaded piezoelectric converter (hanging in air) as,

Q ZR R

LC

for BRANSONmocm

101

1 1

1

1

1 452 73= = =( . 502 / 932R, from T 1.1) , (1.2)

and for parallel resonance (for non-loaded piezoelectric converter, hanging in air) as,

Q RZ

R CL

for BRANSONmocm

22

022

2

2

1256 7= = =( . 502 / 932R, from T 1.1) . (1.3)

When we are talking about converters’ mechanical quality factors, this is usually related to non-loaded (fully free) converters, hanging in air, and in all above given expressions (as well as in other parts of this paper) every indexing with “o” indicates this situation (“o” = non-loaded). Index “m” indicates that we are talking about mechanical-circuit related parameter/s, and indexing with “1” and “2” is related to series and parallel resonances. Since the same parameters can be found for loaded converters, indexing “L” is reserved for loading and/or load-influenced parameters. A well operating and highly efficient (high power) converter, operating in a continuous regime should be designed to have as higher as possible mechanical quality factors (either being non-loaded, or fully loaded), and in this paper we shall mostly address only such kind of converters (what will significantly simplify different mathematical expressions and formulas related to such converters). In (1.2) and (1.3) we can see two of unusual mechanical parameters (not found in literature), which are here addressed as “characteristic mechanical impedances of series and parallel resonance, of non-loaded converters”, defined as,

Z LC f C

for BRANSON Z

ZLC f C C

f ff f

ZC

C Cf f

f fZ for BRANSON

cm cm

cmp

cmp

cm

011

1 1 102

022

2 1 0 1

22

12

1 201

1

0 1

22

12

1 2

02

12

2081745

12

74 96

= = = >>

= =+

⋅−

= ⋅+

⋅−

=

π

π

( . , )

( )( . , ) .

Ω

Ω

502 / 932R

502 / 932R

(1.4)


2-7

Another couple of unusual mechanical parameters (invented in this paper for the purpose of converters characterization and comparison), are “characteristic average, axial wave-velocities of non-loaded half-wavelength converters, operating in series and/or parallel resonance”, defined as,

01cm01 01 01 cm02

021 1 1 1

cm01

cm02 02 02 cm01 02 012 2 2 2

cm02

f1 Hv f 2H 2Hf v ,f2 L C L C

m(v 4611.6 ,H 122mm ,for BRANSON 502/932R, from T 1.1) ,s

1 Hv f 2H 2Hf v , (f f )2 L C L C

m(v 5102.528 , H 122mm ,for BRANs

= λ ⋅ = ⋅ = = =π π

⎡ ⎤= =⎢ ⎥⎣ ⎦

= λ ⋅ = ⋅ = = > >π π

⎡ ⎤= =⎢ ⎥⎣ ⎦SON 502/932R, from T 1.1) ,

H ( ) Effective Axial Transducer Length .2λ

= =

(1.5)

In all previously given expressions, it is obvious that we are talking about series and parallel resonant impedance characteristics, where series and parallel resonant frequencies can be approximated (in the case of high quality converters) as,

fL C

f f Z Z

fL C

fCC

f f Z Z

s

pp

011 1

0 02 01 0

022 2

011

00 01 02 0

12

12

1

= ≅ < =

= ≅ + ≅ > =

−

−

π

π

, ( , ) ,

( ,

min.

max.

series resonance

series resonance ) . (1.6)

Also, the absolute values of converter’s impedance in its series and parallel resonance (for non-loaded converter) are, Z Z Z R Series resonance f

Z Z Z R Parallel resonance fs

p

01 0 0 1 01

02 0 0 2 02

= = ≈

= = ≈

−

−

min.

max.

, ( , ) ,

( , ) . (1.7)

High quality converters usually have very low series impedance ( Z Z Z Rs01 0 0 1= = ≈−min. ≈ 10 Ω range, and lower values), and very high parallel impedance ( Z Z Z Rp02 0 0 2= = ≈−max. ≈ 100 KΩ range, and higher values). Now, after introducing definitions and symbolic related to most important converter “static” parameters, we can address the other very-important, dynamic (loading) parameters of high power piezoelectric converters. Converters loading process is changing its electrical impedance on the way that (under loading) minimal series impedance is increasing, and that maximal parallel impedance is decreasing (coincidently). Both mechanical quality factors ((1.2) & (1.3)) of series and parallel resonance mechanical-circuits are also dropping down coincidently following mechanical load increase.


2-8

It will be shown that the mechanical quality-factors ratio, between the state of non-loaded and fully-loaded converter, is one of the best measure of converter’s dynamic (loading) performances (see T 1.2). Let us establish the following convention and symbolic regarding converters mechanical quality factors (in non-loaded and loaded conditions):

Q ZR R

LC

Z Rfor BRANSON

Q Q Fully loadedconverterQuality Factor at f Z R

for BRANSON

mocm

mL mo

L L L

101

01 01

01

01

01 01

1 1

1 1 1

1

452 73

39 205

= = = −

≈

=

R

S|||

T|||

U

V|||

W|||

<< = −

≈

=

RS|T|

UV|W|

( )

,( .

,

( ),

( .

Non loaded converter

Quality Factor at f502 / 932R, T 1.1)

502 / 932R, T 1.1)

01

(1.8)

Q RZ

R CL

Z Rfor BRANSON

Q Q Fully loadedconverterQuality Factor at f Z R

for BRANSON

mocm

mL mo

L L L

202

0202

02

02

02 02

2 2

2 2 2

1256 7

18 63

= = = −

≈

=

R

S|||

T|||

U

V|||

W|||

<< = −

≈

=

RS|T|

UV|W|

( )

,( .

,

( ),

( .

Non loaded converter

Quality Factor at f502 / 932R, T.1.1)

502 / 932R, T 1.1)

02

Under fully-loaded converter’s quality factors, (1.8), we shall understand that converter is measured when heavily (maximally) loaded, but being still well operational and able to deliver (safely) 100% of its maximal operating power. This time we shall not describe what means heavy-loaded converter, since this is an application-dependent situation, but generally, certain loading situations could be standardized, such as process of immersing the front emitting face of the converter in water, or contacting and pressing converter’s front emitting face to some other material (until it properly operates full-power). In reality, we should first test the chosen maximal load situation, by operating converter full-power (to see where is maximal loading level, when converter delivers its maximal output power), and then we should keep the same loading (the same kind of contact between converter and its load), and make low-signal impedance measurements using Impedance Analyzer (this time converter is only connected to Impedance Analyzer). Of course, the best would be to have an impedance analyzer able to operate in high power conditions, measuring high voltage and high current values, and to avoid load simulation, until all loading standards and correlations between Low-signal and High-signal measurements become well known (but such instruments are still not available). Using symbolic from (1.8) we can formulate the following, most important dynamic loading parameters of ultrasonic high power converters, related separately to converters’ efficiency (η in %) in series and parallel resonance operation/s:


2-9

QQ

ZZ

RR

and f

for BRANSONQ

Qx Q

Q

mo

mL

L LL

mL

mo

mL

mo

1

1

1

01

1

011

11

1

1

1

1157 12 07 12 52

100 1

≅ ≅

= ≅ ≅

≈−

= − ≅

R

S|||

T|||

U

V|||

W|||

,

. . .

% [ ]%

measured at series resonance / s f

( 502 / 932R, T 1.1)Q ( 91.36% for BRANSON 502 / 932R)

01

mo1η

QQ

ZZ

RR

and f

for BRANSONQ

Qx Q

Q

mo

mL L LL

mL

mo

mL

mo

2

2

02

2

02

22

22

2

2

2

67 45 5109 62 60

100 1

≅ ≅

= ≅ ≅

≈−

= − =

R

S|||

T|||

U

V|||

W|||

,

. . .

% [ ]%

measured at parallel resonance / s f

( 502 / 932R, T 1.1)Q ( 98.52% for BRANSON 502 / 932R)

02

mo2η

(1.9)

Practically, the higher the ratio between non-loaded and fully-loaded mechanical quality factors (1.9) is, the higher efficiency η, and better dynamic (loading) converter performances are (respecting that fully-loaded converter is still well operational and able to deliver its maximal power). The ratio/s (1.9) also present the measure/s of converters capacity to accumulate and exchange potential elastomechanical energy with input electric energy (when converter operates in resonance and when electrical and elastomechanical energy are mutually transforming, producing converter oscillations). In the close relation with converters dynamic performances are relations presenting resonant frequency shift (or frequency deviation) from non-loaded until fully-loaded situation, expressed as,

f ff f

ff

ff

for BRANSON

f ff f

ff

ff

ff

ff

ff

for BRANSON

L L L

L

L L LC

C

L

L L

L

2 1

02 01 0

0

1 2

01 02 0 0

0

0

0

18622012

0 92545

1 02313 0 1011 0 0957

−−

= =

= =

RS||

T||

UV||

W||

++

= = =

= = =

RS||

T||

UV||

W||

∆∆

∆∆

∆ ∆

( )

.

( )

. ; . , . ,

resonant frequency intervals ratio

( 502 / 932R, T 1.1)

central frequencies ratio

( 502 / 932R)

(1.10) High quality converters should have frequency intervals and central frequencies ratios (1.10) close to 1 (or saying differently, as this ratio is closer to 1, the converter is more stable and performing better). Regarding frequency stability between loaded and non-loaded converter states we can also calculate the central frequency-shift ratios on the following way,


2-10

f ff f

ff

for BRANSON

f ff f

ff

for BRANSON

f f f f f f f f f

C LC

L L

C

L

C LC C

C LC L L C C LC

0

2 1

0

02 01 0

0 01 02 1 2 0

4501862

0 242

4502012

0 2236

0 5 0 5 0 5

−−

= = =

−−

= = =

= + = + = +

∆∆∆∆

( 502 / 932R, T 1.1)

( 502 / 932R, T 1.1)

.

. ,

. ( ) , . ( ), . ( ) .

(1.11)

High quality converters should have all central frequency shifts (1.11) as lower as possible. Relations given in (1.10) and (1.11) are also useful for designing ultrasonic power supplies and circuits for automatic, PLL resonant frequency tracking. Since in certain operating regimes converters can operate between series and parallel resonant frequency, and in any case series and parallel resonance (of the same converter) are mutually coupled and dependant (both, minimal and maximal converter’s impedances are coincidently changing under loading), we can also define “mixed-mode, static and low signal quality parameters”, formulating a kind of effective parameters, practically creating average value/s between every quality parameter defined for series and for parallel resonance (see the last column in T 1.2). The table T 1.2 is presenting an overview of converters quality parameters (as already introduced above), and all numerical values in T 1.2 are calculated for Branson 502/932R converter (see measured impedance-phase curves and model parameters presented on Fig. 12 & Fig. 13 and in T 1.1). There is another important quality parameter of a piezoelectric converter, dominantly related to its mechanical design, and to elastomechanical material properties of all components and parts of that converter, known as Converter’s Electromechanical Coupling Factor, kc. If we take converter/s from T 1.1 (either modest or high quality, non-loaded and/or loaded, one), we shall see that effective Coupling Factor in all mentioned cases remains almost unchanged,

k Energystored in mechanical formTotal input energyc

2 = = =−

≅+

≅ ⋅ ≅

≅

k f ff

CC C

CC

ff

k

effop

os

c

. . ,

.

2 22

12

22

1

1 2

12

22 018

04243. (1.12)

If we now compare the Converter Coupling factor kc with the Coupling Factor, k33 of the single and load-free piezoceramics (that is the part of the same converter), we shall see how good (or optimal) the mechanical design of that converter is,

( ) ( ..

) .kk

c

33

2 2042430640

044≅ ≅Mechanical energy stored by (assembled) converter

Mechanical energy that can be stored (only) in piezoceramics.≅ (1.13)

since the ideal mechanical converter design would be when converter is not changing the initial Coupling Factor of its piezoceramics, ( / , meaning that elastomechanical properties of converter metal parts would perfectly match such properties of piezoceramics.

)k kc 332 1≅


2-11

T 1.2 (All numerical values in this table are related to BRANSON 502/932R converter: see T 1.1 & Figs. 12,13)

Parameter ⇓

Series Resonance ⇓

Parallel Resonance ⇓

Mixed-mode ⇓

Mechanical Quality Factors

⇒

(non-loaded converter)

QZR

RLC

mocm

101

01

01

01

01

1

453 73

= =

= =

= .

QR

Z

R CL

mocm

202

02

0202

02

1256 7

= =

= =

= .

Q Q Q

RR

ZZ

RR

LL

CC

RR

C CC

f ff f

m mo mo

cm

cm

p

*

.

0 1 2

02

01

01

02

02

01

01

02

02

01

02

01

0 1

1

01 02

022

012

755 12

= ⋅ =

= ⋅ =

= ⋅

+

−

=

=

Mechanical Quality Factors

⇒

(fully-loaded converter)

Q mL1 39 205= . QmL 2 18 63= . Q Q QmL mL mL*.= ⋅

=1 2

27 0257=

Ratio of Mechanical

Quality Factors ⇒

&

Efficiency

(non-loaded): (fully-loaded

converter)

QQ

Z RZ R

RR

L CL C

ZZ

RR

mo

mL

cm L

cmL

L L

L

L L

1

1

01 1

1 01

1

01

01 1

1 01

1

01

1

01

1157 12 0712 52

=

= ≅

≅ ≅

= ≅≅( . .

. )≅

η1

1

1

100

91 36

≈−

≅

Qmo1 QQ

mL

mo

. %

QQ

Z RZ R

RR

L CL C

ZZ

RR

mo

mL

cmL

cm L

L

L

L

L L

2

2

2 02

02 2

02

2

2 02

02 2

02

2

02

2

67 45 510962 8

=

= ≅

≅ ≅

= ≅≅( . .

. )≅

η 2

2

2

100

98 52

≈−

≅

Qmo2 QQ

mL

mo

. %

QQ

QQ

QQ

Z ZZ Z

R RR R

R RR R

L LL L

C CC C

ZZ

ZZ

RR

RR

m

mL

m

mL

m

mL

cm cmL

cmL cm

L

L

L

L

L

L

L

L

L

L

L

L

01

1

02

2

0

01 2

1 02

1 02

2 01

1 02

2 01

01 2

1 02

1 02

2 01

1

2

02

01

1

2

02

01

27 94 24 83 28 04

= =

=

≅ ≅

= ≅ ≅

**

( . . . )

Characteristic Mechanical

Impedance ⇒


ZLC

f C

cm011

1

1 1

122081 745

= =

= =

=π

. Ω

Z L

Ccm022

2

74 96

= =

= . Ω

Z Z Z

LC

LC

R R QQ

cm cm cm

m

m

*

.

0 01 0

1

1

2

2

01 0201

02

395 029

= =

= ⋅ =

= ⋅ =

= Ω

2

Characteristic Mechanical

Impedance ⇒

(fully-loaded converter)

Z LCcm1

1

1

2257 73

= =

= . Ω

Z L

Ccm22

2

80 77

= =

= . Ω

Z Z Zcm cm cm*

.= =

=1 2

427 033 Ω


2-12

Characteristic Axial Wave Velocity ⇒


v f

HL C

HL C

Hf

ms

H mm

cm01 01

1 1

1 101

2 12

2

4611 6

122

= ⋅ =

= ⋅ =

= =

= LNMOQP

=

λ

π

π

. ,

v f

HL C

HL C

Hf

ms

H mm

cm02 02

2 2

2 202

2 12

2

5102 528

122

= ⋅ =

⋅ =

= =

= LNMOQP

=

λ

π

π

. ,

v v v

Hf HL C L C

ms

H mm

cm cm cm

C

*

*

. ,

0 01 02

1 1 2 2

2

4850 86

122

= =

=

= LNMOQP

=

π

Characteristic Axial Wave Velocity ⇒

(fully-loaded

converter)

v fms

H mm

cm1 1 12

4520 1

122

Hf= ⋅ =

= LNMOQP

=

λ

. ,

v f Hfms

H mm

cm2 2 22

4974 43

122

= ⋅ = =

= LNMOQP

=

λ

. ,

v v vHf

ms

H mm

cm cm cm

C

**

. ,

= =

=

= LNMOQP

=

1 2

2

474173

122

Resonant Frequency ⇒ (non-loaded converter)

fL C

f

Hz

s011 1

01

218900

= ≅

=

π f

L Cf

Hz

p022 2

01

220912

= ≅

=

π f f f

HzC*

.0 01 02

19880 56= =

=

Resonant Frequency ⇒ (fully-loaded

converter)

fL C

f

Hz

s11 1

12

18525

= ≅

=

π f

L Cf

Hz

p22 2

12

20387

= ≅

=

π f f f

HzC*

.= =

=1 2

19433 71

Converter’s Resonant

Frequency Stability

Parameters

f ff f

ff

f ff f

ff

ff

ff

ff

f ff f

ff

f ff f

ff

f f f Hzf f f Hzf f

L L L L L LC

C

L

L

L

C LC

L L

C

L

C LC C

C

LC L L

C C

2 1

02 01 0

1 2

01 02 0 0

0

0

0

2 1

0

02 01 0

0 01 02

1 2

0

18622012

0 92545 102313

0 1011 0 0957

4501862

0 242 4502012

0 2236

0 5 199060 5 194560 5

−−

= = =++

= = =

= =

−−

= = =−−

= = =

= + == + =

= +

∆∆

∆ ∆

∆∆

∆∆

. , . ;

. , .

. , . ,

. ( ) ,

. ( ) ,. ( f HzLC ) .= 19681

Coupling factor of the assembled

converter from T 1.1

(PZT8, k33=0.64)

k Energystored in mechanical formTotal input energyMechanical energy stored by (assembled) converter

Mechanical energy that can be stored (only) in piezoceramics

c2 = =

+≅ ⋅ ≅ ≅

≅ ≅

CC C

CC

ff

k

kk

op

osc

c

1

1 2

12

22

33

2 2

018 04243

042430640

044

. , .

( ) ( ..

) .≅ .


The other important quality parameters of power ultrasonic converters Static & Low-signal parameters (converter is operating in air = no-load)

C at kHz High enough nFinp. ( )1 → Total input capacitance E max. N / m 2→ Piezoceramics Young modulus


2-13

Q of assembled, non- loaded converterQ of single, non- loaded piezoceramics max.mo1,2

mo1,2

((

≥

≥

))→

Relations between Mechanical Quality Factors of assembled, non-loaded converter and Quality factor of piezoceramics

(that is the part of the same converter)

Q* Q Q max.,( 1000 , for instance)

mo mo1 mo2= ⋅ →

≥ Effective mechanical quality factor

(here invented parameter)

Q * Q * Q max.,( 1000 , for instance)

emo mo e= ⋅ →

≥ Effective electromechanical quality factor (here

invented parameter)

tan minimumδ ( )at kHzQe

1 1= → Dielectric loss factor

Qe = Dielectric quality factor

d High enough, 10 mV

CN33

12→ ⋅LNMOQP

− = Piezoelectric charge constant

Z1000 Z

R1000 R

max.

( 100, for

max.

min.

2

⋅≅

⋅→

≥1

,

example) Maximum to minimum impedance ratio

T High enough (max.> 300) , Cc → ° Curie temperature ∆

∆f f f

ff

= − →

⋅ ≥

( )

. %

2 1

100 10

max.

for ex : Frequency gap between series and parallel resonant frequency


3.0-1

3.0 Loading of piezoelectric transducers

Field(s) of Application: In applications such as Ultrasonic Welding, single operating, well-defined, resonant frequency transducers are usually used (operating often on 20, 40 and sometimes around 100 kHz and higher). In recent time, some new transducer designs can be driven on sweeping frequency intervals (applied to a single transducer). In Sonochemistry and Ultrasonic Cleaning we use single or multiple ultrasonic transducers (operating in parallel), with single resonant frequency, two operating frequencies, multi-frequency regime, and all of the previously mentioned options combined with frequency sweeping. Frequency sweeping is related to the vicinity of the best operating (central) resonant frequency of transducer group. Frequency sweeping can also be applied in a low frequency (PWM, ON-OFF) group modulation (producing pulse-repetitive ultrasonic train, sometimes-called digital modulation). Also, multi-frequency concept is used in Sonochemistry and Ultrasonic Cleaning when we can drive a single transducer on its ground (basic, natural) frequency and on several higher frequency harmonics (jumping from one frequency to another, without changing transducer/s). Real time and fast automatic resonant (or optimal operating frequency) control/tuning of ultrasonic transducers is one of the most important tasks in producing (useful) ultrasonic energy for different technological applications, because in every application we should realize/find/control: 1° The best operating frequency regime in order to stimulate only desirable vibrating modes. 2° To deliver a maximum of real or active power to the load (in a given/found operating frequency domain/s). 3° To keep ultrasonic transducers in a pulse-by-pulse, real time, safe operating area regarding all critical overload/overpower situations, or to protect them against: overvoltage, overcurrent, overheating, etc. All of the previously mentioned (control and protecting) aspects are so interconnected, that none of them can be realized independently, without the other two. All of them also have two levels of control and internal structure: a) Up to a certain (first) level, with the design and hardware, we try to insure/incorporate the most important

controls and protecting, (automatic) functions. b) At the second level we include certain logic and decision-making algorithm (software) which takes care of real-

time and dynamic changes and interconnections between them. It is necessary to have in mind that in certain applications (such as ultrasonic welding), operating and loading regime of ultrasonic transducer changes drastically in relatively short time intervals, starting from a very regular and no-load situation (which is easy to control), going to a full-load situation, which changes all parameters of ultrasonic system (impedance parameters, resonant frequencies…). In a no-load and/or low power operation, ultrasonic system behaves as a typically linear system; however, in high power operation the system becomes more and more non-linear (depending on the applied mechanical load). The presence of dynamic and fast changing, transient situations is creating the absolute need to have one frequency auto tuning control block, which will always keep ultrasonic drive (generator) in its best operating regime (tracking the best operating frequency). The meaning of mechanical loading of ultrasonic transducers: Mechanical loading of the transducer means realizing contact/coupling of the transducer with a fluid, solid or some other media (in order to transfer ultrasonic vibrations into loading media). All mechanical parameters/properties (of the load media) regarding such contact area (during energy transfer) are important, such as: contact surface, pressure, sound velocity, temperature, density, mechanical impedance, … Mechanical load (similar to electrical load) can have resistive or frictional character (as an active load), can be reactive/imaginary impedance (such as masses and springs are), or it can be presented as a complex mechanical impedance (any combination of masses, springs and frictional elements). In fact, direct mechanical analogue to electric impedance is the value that is called Mobility in mechanics, but this will not influence further explanation. Instead of measuring complex mechanical impedance (or mobility) of an ultrasonic transducer, we can easily find its complex electrical impedance (and later on, make important conclusions regarding mechanical impedance). Mechanical loading of a piezoceramic transducer is transforming its starting impedance characteristic (in a no-load situation in air) into similar new impedance that has lower mechanical quality factor in characteristic resonant area/s. There are many

M. Prokic MPI, Marais 36, 2400, Le Locle, Switzerland, E-mail: [email protected]

3.0-2

electrical impedance meters and network impedance analyzers to determine/measure full (electrical) impedance-phase-frequency characteristic/s of certain ultrasonic transducers on a low sinus-sweeping signal (up to 5 V rms.). However, the basic problem is in the fact that impedance-phase-frequency characteristics of the same transducer are not the same when transducer is driven on higher voltages (say 200 Volts/mm on piezoceramics). Also, impedance-phase-frequency characteristics of one transducer are dependent on transducer’s (body) operating temperature, as well as on its mechanical loading. It is necessary to mention that measuring electrical Impedance-Phase-Frequency characteristic of one ultrasonic transducer immediately gives almost full qualitative picture about its mechanical Impedance-Phase-Frequency characteristic (by applying a certain system of electromechanical analogies). We should not forget that ultrasonic, piezoelectric transducer is almost equally good as a source/emitter of ultrasonic vibrations and as a receiver of such externally present vibrations. While it is emitting vibrations, the transducer is receiving its own reflected (and other) waves/vibrations and different mechanical excitation from its loading environment. It is not easy to organize such impedance measurements (when transducer is driven full power) due to high voltages and high currents during high power driving under variable mechanical loading. Since we know that the transducer driven full power (high voltages) will not considerably change its resonant points (not more than ±5% from previous value), we rely on low signal impedance measurements (because we do not have any better and quicker option). Also, power measurements of input electrical power into transducer, measured directly on its input electrical terminals (in a high-power loading situation) are not a simple task, because we should measure RMS active and reactive power in a very wide frequency band in order to be sure what is really happening. During those measurements we should not forget that we have principal power delivered on a natural resonant frequency (or band) of one transducer, as well as power components on many of its higher and lower frequency harmonics. There are only a few available electrical power meters able to perform such selective and complex measurements (say on voltages up to 5000 Volts, currents up to 100 Amps, and frequencies up to 1 MHz, just for measuring transducers that are operating below 100 kHz). Optimal driving of ultrasonic transducers: For optimal transducer efficiency, the best situation is if/when transducer is driven in one of its mechanical resonant frequencies, delivering high active power (and very low reactive power) to the loading media. Since usually resonant frequency of loaded transducer is not stable (because of dynamical change of many mechanical, electrical and temperature parameters), a PLL resonant frequency (in real-time) tracking system has to be applied. When we drive transducer on its resonant frequency, we are sure that the transducer presents dominantly resistive load. That means that maximum power is delivered from ultrasonic power supply (or ultrasonic generator) to the transducer and later on to its mechanical load. If we have a reactive power on the transducer, this can present a problem for transducer and ultrasonic generator and cause overheating, or the ultrasonic energy may not be transferred (efficiently) to its mechanical load. Usually, the presence of reactive power means that this part of power is going back to its source. The next condition that is necessary to satisfy (for optimal power transfer) is the impedance matching between ultrasonic generator and ultrasonic transducer, as well as between ultrasonic transducer and loading media. If optimal resonant frequency control is realized, but impedance/s matching is/are not optimal, this will again cause transducer and generator overheating, or ultrasonic energy won’t be transferred (efficiently) to its mechanical load. Impedance matching is an extremely important objective for realizing a maximum efficiency of an ultrasonic transducer (for good impedance matching it is necessary to adjust ferrite transformer ratio and inductive compensation of piezoelectric transducer, operating on a properly controlled resonant frequency). Output (vibration) amplitude adjustments, using boosters or amplitude amplifiers (or attenuators) usually adjust mechanical impedance matching conditions. Recently, some ultrasonic companies (Herman, for instance) used only electrical adjustments of output mechanical amplitude (for mechanical load matching), avoiding any use of static mechanical amplitude transformers such as boosters (this way, ultrasonic configuration becomes much shorter and much more load-adaptable/flexible, but its electric control becomes more complex). By the way, we can say that previously given conditions for optimal power transfer are equally valid for any situation/system where we have energy/power source and its load (To understand this problem easily, the best will be to apply some of the convenient systems of electromechanical analogies). It is important to know that Impedance-Phase-Frequency characteristics of one transducer (measured on a low sinus-sweeping signal) are giving indicative and important information for basic quality parameters of one transducer, but not sufficient information for high power loaded conditions of the same transducer. Every new loading situation should be rigorously tested, measured and optimized to produce optimal ultrasonic effects in a certain mechanical load. It is also very important to know that safe operating limits of heavy-loaded ultrasonic transducers have to be controlled/guaranteed/maintained by hardware and software of ultrasonic generator. The usual limits are maximal operating temperature, maximal-operating voltage, maximal operating current, maximal operating power, operating frequency band, and maximum acceptable stress. All of the previously mentioned parameters should be


3.0-3

controlled by means of convenient sensors, and protected/limited in real time by means of special protecting components and special software/logical instructions in the control circuits of ultrasonic generator. A mechanism of very fast overpower/overload protection should be intrinsically incorporated/included in every ultrasonic generator for technologically complex tasks. Operating/resonant frequency regulation should work in parallel with overpower/overload protection. Also, power regulation and control (within safe operating limits) is an additional system, which should be synchronized with operating frequency control in order to isolate and select only desirable resonances that are producing desirable mechanical output. Electronically, we can organize extremely fast signal processing and controls (several orders of magnitude faster than the mechanical system, such as ultrasonic transducer, is able to handle/accept). The problem appears when we drive ultrasonic configuration that has high mechanical quality factor and therefore long response time, which is when mechanical inertia of ultrasonic configuration becomes a limiting factor. Also, complex mechanical shapes of the elements of ultrasonic configuration are creating many frequency harmonics, and low frequency (amplitude) modulation of ultrasonic system influencing system instability that should be permanently monitored and controlled. We cannot go against physic and mechanical limitations of a complex mechanical system (such as ultrasonic transducer and its surrounding elements are), but in order to keep ultrasonic transducer in a stable (and most preferable) regime we should have absolute control over all transducer loading factors and its vital functions (current, voltage, frequency…). This is very important in case of applications like ultrasonic welding, where ultrasonic system is permanently commuting between no-load and full load situation. In a traditional concept of ultrasonic welding control we can often find that no-load situation is followed by the absence of frequency and power control (because system is not operational), and when start (switch-on) signal is produced, ultrasonic generator initiates all frequency and power controls. Some more modern ultrasonic generators memorize the last (and the best found) operating frequency (from the previous operating stage), and if control system is unable to find the proper operating frequency, the previously memorized frequency is taken as the new operating frequency. Usually this is sufficiently good for periodically repetitive technological operations of ultrasonic welding, but this situation is still far from the optimal power and frequency control. In fact, the best operating regime tuning/tracking/control should mean a 100% system control during the totality of ON and OFF regime, or during full-load and no-load conditions. Previously described situation can be guaranteed when Power-Off (=) no-load situation is programmed to be (also) one transducer-operating regime which consumes very low power compared to Power-ON (=) full-load situation. This way, transducer is always operational and we can always have the necessary information for controlling all transducer parameters. Response time of permanently controlled/driven ultrasonic transducers can be significantly faster than in the case when we start tracking and control from the beginning of new Power-ON period.

When transducers are driven full power, it happens in the process of harmonic oscillation, so input electrical energy is permanently transformed to mechanical oscillations. What happens when we stop or break the electrical input to the transducer? - The generator no longer drives the transducer, and/or they effectively separate. The transducer still continues to oscillate certain time, because of its elastomechanical properties, relatively high electro-mechanical Q-factor, and residual potential (mechanical) energy. Of course, the simplest analogy for an ultrasonic transducer is a certain combination of Spring-Mass oscillating system. Any piezoelectric or magnetostrictive transducer is a very good energy transformer. It means that if the input is electrical, the transducer will react by giving mechanical output; but, if the active, electrical input is absent (generator is not giving any driving signal to the transducer) and the transducer is still mechanically oscillating (for a certain time), residual electrical back-output will be (simultaneously) generated. It will go back to the ultrasonic generator through the transducer’s electrical terminals (which are permanently connected to the US generator output). Usually, this residual transducer response is a kind of reactive electrical power, sometimes dangerous to ultrasonic generator and to the power and frequency control. It will not be synchronized with the next generator driving train, or it could damage generator’s output switching components. Most existing ultrasonic generator designs do not take into account this residual (accumulated) and reversed power. In practice, we find different protection circuits (on the output transistors) to suppress self-generated transients. Obviously, this is not a satisfactory solution. The best would be never to leave the transducers in free-running oscillations (without the input electrical drive, or with “open” input-electrical terminals on the primary transformer side). Also, it is necessary to give certain time to the transducers for the electrical discharging of their accumulated elasto-mechanic energy.

Resonant frequency control under load: Frequency control of high power ultrasonic converters (piezoelectric transducers) under mechanical loading conditions is a very complex situation. The problem is in the following: when the transducer is operating in air, its resonant frequency control is easily realizable because the transducer has equivalent circuit (in the vicinity of this frequency) which is similar to some (resonant) configuration of oscillating R-L-C circuits. When the transducer is


3.0-4

under heavy mechanical load (in contact with some other mass, liquid, plastic under welding…), its equivalent electrical circuit loses (the previous) typical oscillating configuration of R-L-C circuit and becomes much more closer to some (parallel or series) combination of R and C. Using the impedance-phase-network analyzer (for transducer characterization), we can still recognize the typical impedance phase characteristic of piezo transducer. However, it is considerably modified, degraded, deformed, shifted to a lower frequency range, and its phase characteristic goes below zero-phase line (meaning the transducer becomes dominantly capacitive under very heavy mechanical loading). If we do not have the transducer phase characteristic that is crossing zero line (between negative and positive values, or from capacitive to inductive character of impedance) we cannot find its resonant frequency (there is no resonance), because electrically we do not see which one is the best mechanical resonant frequency. Active and Reactive Power and Optimal Operating Frequency: The most important thing is to understand that ultrasonic transducers that are used for ultrasonic equipment (piezoelectric or sometimes magnetostrictive) have complex electrical impedance and strong coupling between their electrical inputs and relevant mechanical structure (to understand this we have to discuss all relevant electromechanical, equivalent models of transducers, but not at this time). This is the reason why parallel or serial (inductive for piezoelectric, or capacitive for magnetostrictive transducers) compensation has to be applied on the transducer, to make the transducer closer to resistive (active-real) electrical impedance in the operating frequency range. The reactive compensation is often combined with electrical filtering of the output, transducers driving signals. Universal reactive compensation of transducers is not possible, meaning that the transducers can be tuned as resistive impedance only within certain frequencies (or at maximum in band-limited frequency intervals). Most designers think that this is enough (good electrical compensation of the transducers), but, in fact, this is only the necessary first step. This time we are coming to the necessity of making the difference between electrical resonant frequency and mechanical resonant frequency of an ultrasonic converter. In air (non-loaded) conditions, both electrical and mechanical resonant frequencies of one transducer are in the same frequency point/s and are well and precisely defined. However, under mechanical loading this is not always correct (sometimes it is approximately correct, or it can be the question of appearance of some different frequencies, or of something else like very complicated impedance characteristic). From the mechanical point of view, there is still (under heavy mechanical load) one optimal mechanical resonant frequency, but somehow it is covered (screened, shielded, mixed) by other dominant electrical parameters, and by surrounding electrical impedances belonging to ultrasonic generator. To better understand this phenomenon, we can imagine that we start driving one ultrasonic transducer (under heavy loading conditions), using forced (variable frequency), high power sinus generator, without taking into account any PLL, or automatic resonant frequency tuning. Manually (and visually) we can find an operating frequency producing high power ultrasonic (mechanical) vibrations on the transducer. As we know, heavy loaded transducer presents kind of dominantly capacitive electrical impedance (R-C), but it is still able to produce visible ultrasonic/mechanical output (and we know that we cannot find any electrical pure resonant frequency in it, because there is no such frequency). In fact, what we see, and what we can measure is how much of active and reactive power circulates from ultrasonic generator to piezoelectric transducer (and back from transducer to generator). When we say that we can see/detect a kind of strong ultrasonic activity, it means most probably that we are transferring significant amount of active/real electrical power to the transducer, and that much smaller amount of reactive/imaginary power is present, but we cannot be absolutely sure that such loaded transducer has proper resonant frequency (it could still be dominantly capacitive type of impedance, or some other complex impedance). In fact, in any situation, the best we can achieve is to maximize active/real power transfer, and to minimize reactive/imaginary power circulation (between ultrasonic generator and piezoelectric ultrasonic transducer). If/when our (manually controlled) sinus generator produces/supplies low electrical power, the efficiency of loaded ultrasonic conversion is also very low, because there is a lot of reactive power circulating inside of loaded transducer (and back to the generator). Here is the most interesting part of this situation: if we intentionally increase the electrical power that drives the loaded transducer (keeping manually its best operating frequency, or maximizing real/active power transfer), the transducer becomes more and more electro-acoustically efficient, producing more and more mechanical output, and less and less reactive power. Also, thermal dissipation (on the transducer) percentage-wise (compared to the total input energy) becomes lower. What is really happening: under heavy mechanical loading and high power electrical driving (on the manually/visually found, best operating frequency, when real power reaches its maximum) the transducer is again recreating/regaining (or reconstructing) its typical piezoelectric impedance-phase


3.0-5

characteristic which, now, has new phase characteristic passing zero line, again (like in real, oscillatory R-L-C circuits). Somehow, high mechanical strain and elasto-mechanical properties of total mechanical system (under high power driving) are accumulating enough (electrical and mechanical) potential energy, and the system is again coming back, mechanically decoupling itself from its load (for instance from liquid) and/or starting to present typical R-L-C structure that is easy for any PLL resonant frequency control (having, again, real/recognizable resonant frequency). Of course, loaded ultrasonic transducer (optimally) driven by high power will have some other resonant frequency, different than the frequency when it was driven by low power, and also different than its resonant frequency (or frequencies) in non-loaded conditions (in air), because resonant frequency is moving/changing according to time-dependant loading situation (in the range of ±5% around previously found resonant frequency). To better understand the importance of active power maximization, we know that when we have optimal power transfer (from the energy source to its load), the current and the voltage time-dependant shapes/functions (on the load) have to be in phase. This means that in this situation electrical load is behaving as pure resistive, or active load. (Electrically reactive loads are capacitive and inductive impedances). The next condition (for optimal power transfer) is that load impedance has to be equal to the internal impedance of its energy source (meaning the generator). In mechanical systems, this situation is analogous or equivalent to the previously explained electrical situation, but this time force and velocity time-dependant shapes/functions (on the mechanical load) have to be in phase, which means that in such situations mechanical load is behaving as pure (mechanically) resistive, or active load. Active mechanical loads are basically frictional loads (and mechanically reactive loads/impedances are masses and springs in any combination). We usually do not know/see exactly (and clearly) if we are producing active mechanical power, but by following/monitoring/controlling electrical power, we know that when we succeed in producing/transferring certain amount of active electrical power to one ultrasonic transducer, that corresponds, at the same time, to one directly proportional amount of active mechanical power (dissipated in mechanical load). Delivering active power to some load usually means producing heat on active/resistive elements of this load. We also know that productivity, efficiency and quality of ultrasonic action (in Sonochemistry, plastic welding, ultrasonic cleaning…) strongly and directly depend on how much active mechanical power we are able to transfer to a certain mechanical load (say to a liquid or plastic, or something else). When we have visually strong ultrasonic activity, but without transferring significant amount of active power to the load, we can only be confused in thinking (feeling) that our ultrasonic system is operating well, but in reality, we do not have big efficiency of such system. Users and engineers working in/with ultrasonic cleaning know this situation well. Sometimes, we can see very strong ultrasonic waving in one ultrasonic cleaner (on its liquid surface), but there is no ultrasonic activity and cleaning effects are missing. In conclusion, it is correct to say that: active electrical power & active mechanical power, for an electromechanical system where we transfer electrical energy to the mechanical load. Another conclusion is that we also need to install convenient mechanical/acoustical/ultrasonic sensors which are able to detect, follow, monitor and/or measure resulting ultrasonic/acoustical/mechanical activity (in real-time) on the mechanical load, in order to be 100% sure that we are transferring active mechanical power to certain mechanical load, and to be able to have a closed feedback loop for automatic (mechanical, ultrasonic) power regulation. For instance, in liquids (Sonochemistry and ultrasonic cleaning applications), the appearance of cavitation is the principal sign of producing active ultrasonic power. To control this we need sensors of ultrasonic cavitation. Also, we know that the last step in any energy chain (during electromechanical energy transfer) is heat energy. By supplying electrical resistive load with electrical energy we produce heat. The same is valid for supplying mechanical resistive/frictional load with ultrasonic energy, when the last step in this process is again heat energy (but, again, force and velocity wave shapes of delivered ultrasonic waves have to be in phase, measured on its load). From the previous commentary we can conclude that the best sensors for measuring active/resistive ultrasonic energy transfer in liquids are real-time, very fast responding temperature sensors (or some extremely sensitive thermocouples, and/or thermopiles). There can be a practical problem (for resonant frequency tracking) if we start driving certain transducer full power, under load, if we are not sure that we know its best operating mechanical resonant frequency (because we can destroy the transducer and output transistors if we start with a wrong frequency). In real life, every well designed PLL starts with a kind of low power sweep frequency test (say giving 10% of total power to the transducer), around its known best operating frequency taking/accepting one frequency interval that is given in advance. When the best operating resonant frequency is confirmed/found, PLL system tracks this frequency, and at the same time the power regulation (PWM) increases output power (of ultrasonic generator) to the desired maximum. Of course, when the transducer is in air (mechanically non-loaded), previous explanation is readily applicable because we can easily find its best resonant frequency, and later on we can start gradually increasing the power on the transducer. If starting and operating situation is with already heavy loaded transducer (which can be represented by


3.0-6

dominantly R-C impedance), the problem is much more serious, because we should know how to recognize (automatically) the optimal mechanical resonant frequency (without the possibility of using phase characteristic that is crossing zero line). There are some tricks, which may help us realize such control. Of course, before driving one transducer in automatic PLL regime, we should know its impedance-phase vs. frequency properties (and limits) in non-loaded and fully loaded situations. In order to master previous complexity of driving ultrasonic transducers (and to explain this situation) we should know all possible and necessary equivalent (electrical) circuits of non loaded and loaded ultrasonic transducers, where we can see/discuss/adjust different methods of possible PLL control/s. Since ultrasonic transducer is always driven by using ultrasonic generator which has output ferrite transformer, inductive compensation and other filtering elements, it is necessary to know the relevant (and equivalent) impedance-phase characteristics in all of such situations in order to take the most convenient and proper current and voltage signals for PLL. All previous comments are relevant when driving (input) signal is either sinusoidal or square shaped, but always with a (symmetrical, internal) duty cycle of 1:1 (Ton: Toff = 1:1), meaning being a regular sinusoidal or square shaped wave train. There is a special interest in finding a way/method/circuit capable of driving ultrasonic transducers directly using high power (and high ultrasonic frequency), PWM electrical (input) signals, because of the enormous advantages of PWM regulating philosophy. Applying special filtering networks in front of an ultrasonic transducer can be very useful when we want to drive ultrasonic transducer with PWM signals. Influence of External Mechanical Excitation: One of the biggest problems for PLL frequency tracking is when ultrasonic (piezoelectric) transducer under mechanical load, driven by ultrasonic generator, produces mechanical oscillations, but also receives mechanical response from its environment (receiving reflected waves). Sometimes, received mechanical signals are so strong, irregular and strangely shaped that equivalent impedance characteristic of loaded transducer becomes very variable, losing any controllable (typical impedance) shape. It looks like all the parameters of equivalent electrical circuit of loaded transducer are becoming non-linear, variable and like transients signals. There is no PLL good enough to track the resonant frequency of such transducer, but luckily, we can introduce certain filtering configuration in the (electrical) front of transducer and make this situation much more convenient and controllable (meaning that external mechanical influences can be attenuated/minimized). Sometimes loaded ultrasonic transducer (in high power operation) behaves as multi-resonant electrical and mechanical impedance, with its entire equivalent-model parameters variable and irregular. Optimal driving of such transducer, either on constant or sweeping frequency, becomes uncontrollable without applying a kind of filtering and attenuation of external vibrations and signals received by the same transducer. In fact, the transducer produces/emits vibrations and at the same time receives its own vibrations, reflected from the load. There is a relatively simple protection against such situation by adding a parallel capacitance to the output piezoelectric transducer. Added capacitance should be of the same order as input capacitance of the transducer. This way, ultrasonic generator (frequency control circuit) will be able to continue controlling such transducer, because parallel added capacitance cannot be changed by transducer parameters variation. In case of large band frequency sweeping, we can also add to the input transducer terminals certain serial resistive impedance (or some additional L-R-C filtering network). This way we avoid overloading the transducer by smoothly passing trough its critical impedance-frequency points (present along the sweeping interval). In any situation we can combine some successful, useful and convenient PLL procedures with a real/active power maximizing procedure incorporated in an automatic, closed feedback loop regulation (of course, trying in the same time to minimize the reactive/imaginary power). Objectives and new R&D tasks: Traditional ultrasonic equipment exploits mainly single resonant frequency sources, but it becomes increasingly important to introduce/use different levels of frequency and amplitude modulating signals, as well as low frequency (ON – OFF) group PWM digital-modulation in low and high frequency domains. Several modulation levels and techniques could be applied to maximize the power and frequency range delivered to heavily loaded ultrasonic transducers (and, this way, many of the above-mentioned loading problems could be avoided or handled in a more efficient way). Discussing such situations can be a subject of a special chapter. As we can see from the previous explanation, it is not so simple and easy to explain all the aspects of future ultrasonic generators, but it is already a big advantage to understand the basic principles and principal targets. The question is: how is it possible that a number of producers in the ultrasonic field are working/ producing


3.0-7

relatively good ultrasonic equipment, if we take into account all previous design problems and tasks. The answer is that probably almost none of them has neither in practice nor in R&D all of the previously mentioned activities. Possibly, at least some of them understand the existence of such problems, and during a long experimental practice (or sometimes by luck and copying from the others), they have found and optimized some particular situations (designs), which are operating satisfactorily in certain conditions. Each proposal, without the exact knowledge of what has to be done, is an expensive R&D and technological excursion. When possible, it is reasonable to avoid this kind of adventure. It is sure that all traditional applications that can tolerate old design philosophy are already obsolete; for the new generation of Ultrasonic Generators, we have to take a new approach in R&D and strategic planning. CONCLUSION: Digital-Phase-Frequency-Locked-Loop, combined with a convenient frequency counting /searching algorithm, and real/active power tracking, with a programmable mono-frequency, multi-frequency and frequency sweeping concept/s can be included/combined in one FPGA microchip. Of course, resonant frequency control has to be simultaneously linked to (automatic) output power regulation and multilevel overload protection. This could be a powerful (multifunction) controller for today’s technologies of different ultrasonic processing and similar applications. Today, the most demanding digital PLL (resonant frequency) control of industrial, high power ultrasonic transducers (operating between 20 and 100 kHz) is connected to the following set of parameters/characteristics: 1. Master clock 100 MHz, in order to derive the frequency resolution better than 0.5 Hz. 2. Frequency band for automatic PLL regulation (∈) [foper. – 10% foper. to foper. + 10% foper.]. 3. Searching for sgn (PHI) i.e. abs (PHI) is the best way to identify a singular resonant system, which is capable

of oscillating. 4. Over-damped/overloaded systems are too broad for any accurate identification. 5. A/D capability and embedded micro-controller conditions for different calculations and synchronous output

power regulation are necessary.


3.0-8

PLL FLOW CHART (example)

Find..and/or..Set:Power(=)Low

PLL

)L(PLL

)0()L(

)0(.res

)L(.res

)0(.res

)f(,)f(,)f(

,)f(,f,f,f

∆∆∆

∆

Take Inputs:

.inp.inptt i,u,i,u

Calculate / Find:

t

tt.inpt i

uZ,, =θθ

IF:[ ][ ].maxZ

and0

t

t

=

=θ

Than: f 1.res f=

NO

Find:

2.res

t

ff.maxZ

=

=

Find:

3res

inp

ff,0

=

=θ

)fff(31f 321res ++=

Memorize last found f1

YES

( )fchange0:If

)fff(31f0:If

ff0:If

t

321.rest

1.rest

∆⇒>

++=⇒<

=⇒=

θ

θ

θ

fres. Set: Full power

0:If t <θ

YES

NO

( )restartand

fchange0t

∆⇒>θ

Set New

fres., ∆f…


3.1-1

3.1 Converters Loading and Associated Losses Let us try to understand converters loading process, converters internal losses and the meaning of the mechanical quality factors ratio (referring to numerical results for Branson converter 502/932R; -see T 1.1 and T 1.2). We know that mechanical quality factor presents the ratio of stored and dissipated energy per one oscillation of a converter,

Q EEm

s

d

= =2 2π πEnergy stored in the transducer during one full periode

Energy dissipated in the transducer during one full periode

Q EE

Q EE

Q EE

Q EE

QQ

EE

EE

EE

QQ

x QQ

ms

dm

s

d

mLsL

dLmL

sL

dL

m

mL

s

sL

dL

d

dL

d

mL

mo

mL

mo

0101

0102

02

02

11

12

2

2

01

1

01

1

1

01

1

01

11

1

1

1

2 453 73 2 1256 70

2 39 205 2 18 63

1157

100 1

= = = =

= = = =

= ≅ =

≈−

= − ≅

π π

π π

η

. , . ,

. , .

.

% [ ]%

( ) ,

Q ( 91.36% )mo1

QQ

EE

EE

EE

QQ

x QQ

QQ

QQ

QQ

EE

EE

EE

EE

EE

EE

m

mL

s

sL

dL

d

dL

d

mL

mo

mL

mo

m

mL

m

mL

m

mL

s

sL

s

sL

dL

d

dL

d

dL

d

dL

d

02

2

02

2

2

02

2

02

22

2

2

2

0 01

1

02

2

01

1

02

2

1

01

2

02

1

01

2

02

67 45

100 1

27 93

= ≅ =

≈−

= − ≅

= = ≅ =

( )

Q ( 98.52% )

( )

mo2

. ,

% [ ]%

**

.

η (1.12)

For instance, in case of BRANSON 502/932R converter (data given in T 1.1, Figs. 12 & 13), we know (based on measurements) that average power dissipation in series resonance (in air = non-loaded converter) is P Wd01 60 65≈ ÷ , and in parallel resonance Pd02 10 32≈ W÷ , while converter is producing the same output amplitude of 20 µm-pp (in both resonant regimes). We can also notice that converter’s efficiency is higher in parallel resonance than in series resonance (98.52% > 91.36%, for BRANSON 503/932R and for many other power converters, too). Thermal power dissipation of non-loaded 502/932R converter was measured using Clarke-Hess power meter connected just before converter (keeping converters current and driving voltage in phase). The converter output amplitude in series resonance, was stabilized to 20 micrometers (peak-to-peak), the voltage on the converter necessary to produce that amplitude was found to be in the range of 21.6 V, rms, and the converter input current was 2.98 A, rms. Of course, the input current into converter was many times higher comparing it to the converter current when operating in parallel resonance. Measured total power consumption (in series resonance) was in the range of 60 to 65 Watts (and this is the thermal dissipation, since converter was operating load-free in air). In order to avoid high internal power dissipation, when converter is operating non-loaded in series resonance, most of

Converters modeling Miodrag Prokic Le Locle - Switzerland

3.1-2

modern designs of ultrasonic power supplies are implementing PWM, load-current control, reducing automatically converter’s current, and shifting a little bit converter operating frequency towards higher impedance, when converter is not delivering power to its load. Series resonance regimes of piezoelectric converters should be permanently loaded in order to produce minimal or acceptable internal power dissipation. We also know that series resonance is current-resonance (high current & limited voltage), and that parallel resonance is voltage-resonance (high voltage & limited current). Converter operating in series resonance is able to produce high output force (or pressure) and relatively low output velocity (proportional to its high driving, motional-current and relatively low input-voltage), and converter operating in parallel resonance is able to produce high output velocity and relatively low output force (proportional to its high driving, motional-voltage and relatively low input-current consumption). Here we should pay attention on the fact/s that motional current in series resonance is a bit smaller than converter input current, and that motional voltage in parallel resonance is also smaller than input converter voltage, since equivalent, motional converter circuit/s is/are separated from converter input by the presence of reactive, capacitive elements Cop and Cos. For parallel-resonance dissipation-measurements Branson is testing converters the same way as in series resonance (the input converter electrical-field, directly on piezoceramics, is in the range between 186 and 300 V-rms/mm, and the motional converter voltage is stabilized on 930 V-rms for piezoceramics thick 5 millimeters = 930/5 = 186 V-rms/mm), while converter is producing 20 micrometers, peak-to-peak amplitude (keeping the same amplitude as in the case of series resonance measurements). The best what was possible to find in parallel resonance regime (but seldom and for well selected, excellent quality converters), regarding 502/932R, non-loaded converters operating in air, was between 5 and 10 Watts of total internal power dissipation, or total power consumption, and in larger number of measurements the consumption was between 15 and 32 Watts. For standard quality of 502/932R converters Branson is specifying allowed power dissipation (for parallel resonance of non-loaded converter) to be between 10 and 20 Watts. Since when we make a ratio between two mechanical quality factors, we shall get the ratio of corresponding energies, and this is in the same time the ratio between corresponding powers, consequently, based on (1.12), we can roughly estimate how much would be the (worst case scenario of) power dissipation, P a , of fully-loaded (and still acceptable quality: T 1.1, Figs. 12 & 13) converter in its series and parallel resonance,

nd PdL dL1 2

P PP P W

P P P W

dL d

dL d

dL dL dL

1 01

2 02

1 2

1157 60 1157 694 2067 45 10 67 45 674 5

684 28

≅ ⋅ = W⋅ =≅ ⋅ = ⋅ =

= =

. . .. . .

* . .

(1.13)

As we can see from (1.13), estimated thermal power dissipation in the case of fully-loaded high-power converter delivering its maximal output power (here is selected one of modest but still acceptable-quality, BRANSON 502/932R converters, T 1.1, Figs. 12 & 13, which is also able to perform much better in cases of well selected


3.1-3

converters), would be too high (close to 700 Watts), meaning that we should try to design and/or select only converters that already have extremely low power dissipation in non-loaded conditions. The second conclusion is that we should not operate high-power converters in long-time, continuous (full-power) regime/s; -it is always better to have certain pulse-repetitive, ON-OFF regime (for instance 1:1), allowing to a converter’s forced-cooling eliminating excessive heat. Until present we were only taking opposite extremes in analyzing converters performances, such as non-loaded and fully-loaded situations. Loading process between no-load and maximal-load (when converter still operates well) is continuously influencing converter’s performances, and for the purpose of better converters characterization it would be meaningful to create the functional diagram (based on measurements), as presented in the table, T 1.3 (numerically illustrated again for the same BRANSON 502/932R converter, Figs. 12, 13 & 14, T 1.1). T 1.3 Loading of BRANSON 502/932R converter (data from T 1.1, Figs. 12 & 13)

Effective Mechanical

Quality Factors

Ratio, (1.11)

Non-Loaded Converter

( ) Q QmL x m* *− = 0

Load increase

( Q Q Qm mL x* * *0 mL< <− )

Fully-Loaded Converter

( Q Q )mL x mL* *− = = max.

α = −QQ

mL x

mo

**

α = 1 1 > α > 0.0358 α = 0 0358.

Of course, the table T 1.3 can be extended to have separate values for series and parallel resonance, or some other loading function can be created (tracing, under loading: output converter amplitude, velocity, force, different model parameters…). Until present, under converter loading (in this paper) we are only simulating real load conditions (by immersing converter in water, or some other similar method), and making low-signal impedance measurements. In reality, for exact and correct converters characterization we need a kind of instrument similar to Network Impedance Analyzer (for instance similar to HP 4194A), able to perform frequency sweeping, power, impedance, and spectral measurements, using high voltage and high current signals (still not available in one compact and sophisticated instrument). In order to simplify understanding of internal power losses, we can go back to a basic knowledge from elementary electro-technique. Power losses inside of a converter are losses on its equivalent resistive components. When converter is not loaded, the most significant resistive components able to dissipate power are: in series resonance R1 (at f1), and in parallel resonance R2 (at f2). For Branson non-loaded converter presented on Figs. 12 & 13, T 1.1, R1 = 4.6 Ω and R2 = 94.20 kΩ (f1 = 18900 Hz, f2 = 20912 Hz). Since experimentally we already know that for reaching converter amplitude of 20 µm-pp in series-resonance, we need 21.6 V-rms and 2.98 A-rms at the converter input terminals, we can (approximately) calculate the power internally dissipated in the converter as ≈ R1 x I2 = 4.6 x 2.982 = 41 W, that is close to the experimentally measured range of 60 – 65 W. Also, we know (based on measurements) that for reaching converter amplitude of 20 µm-pp in parallel-resonance, we need approximately 1000 V-rms at the converter input terminals, and


3.1-4

we can (approximately) calculate the power internally dissipated in the converter as ≈ U2 / R2 = 10002 / 94200 = 10.62 W, that is also close to the experimentally measured values. When the same converter is fully-loaded, R1L = 57.574 Ω and R2L = 1.50 kΩ (f1 = 18525 Hz, f2 = 20387 Hz), the same approximate calculation would produce results similar to results already predicted in (1.13), as for instance: PdL1 ≈ R1L x I2 ≈ 57.574 x 2.982 = 511 W, and PdL2 ≈ U2 / R2L ≈ 10002 / 1500 = 667 W. In order to present another view of the same situation (regarding power losses), let us consider an ultrasonic converter as simple combination/s of equivalent capacitive, inductive and resistive elements. If we program an Impedance Analyzer to measure converter (in its resonant area) as a series connection between an equivalent resistance and an equivalent inductance, and if we again take the same Branson converter (Figs. 12 & 13, T 1.1), we shall get inductance-resistance plots presented on Figs. 14.1 – 14.2.

Non-loaded converter

Ls

Rs

Capacitive Inductive Capacitive

f1: Rs= 5.597 Ω, Ls= 5.69 µH, f2: Rs= 98.17 kΩ, Ls= 338.288 mH

Fig. 14.1 Non-loaded converter as the series Inductance-Resistance

Fully loaded converter

Ls Rs


f1: Rs= 74 Ω, Ls= 13.34 µH, f2: Rs= 1.378 kΩ, Ls= 105.224 mH

Fig. 14.2 Fully-loaded converter as the series Inductance-Resistance


3.1-5

If we again program an Impedance Analyzer to measure converter (in its resonant area) as a parallel connection between an equivalent resistance and an equivalent inductance, and if we again take the same Branson converter (Figs. 12 & 13, T 1.1), we shall get inductance-resistance plots presented on Figs. 14.3 – 14.4.


LpRp


f1: Rp= 5.46 Ω, Lp= 359.132 µH, f2: Rp= 96.9 kΩ, Lp= 488.246 mH

Fig. 14.3 Non-loaded converter as the parallel Inductance-Resistance

Fully-loaded converter

Lp Rp


f1: Rp= 76.97 Ω, Lp= 76.615 mH, f2: Rp= 1.437 kΩ, Lp= 542.2 mH

Fig. 14.4 Fully-loaded converter as the parallel Inductance-Resistance If we now program an Impedance Analyzer to measure converter (in its resonant area) as a series connection between an equivalent resistance and an equivalent capacitance, for the same Branson converter (Figs. 12 & 13, T 1.1), we shall get capacitance-resistance plots presented on Figs. 14.5 – 14.6.


3.1-6

Cs

Cs


Rs


f1: Rs = 4.68 Ω, Cs = 4.17 µF, f2: Rs = 95.33 kΩ, Cs= 563 pF

Fig. 14.5 Non-loaded converter as the series Capacitance-Resistance

Cs CsRs



f1: Rs = 73.55 Ω, Cs = 4.061 µF, f2: Rs = 1.54 kΩ, Cs= 204.492 nF

Fig. 14.6 Fully-loaded converter as the series Capacitance-Resistance If we now program an Impedance Analyzer to measure converter (in its resonant area) as a parallel connection between an equivalent resistance and an equivalent capacitance, for the same Branson converter (Figs. 12 & 13, T 1.1), we shall get capacitance-resistance plots presented on Figs. 14.7 – 14.8.


3.1-7

Rp

Cp CpNon-loaded converter


f1: Rp= 4.549 Ω, Cp= 957.582 nF, f2: Rp= 97.12 kΩ, Cp= 11.59 pF

Fig. 14.7 Non-loaded converter as the parallel Capacitance-Resistance


Cp Rp

Cp


f1: Rp= 76 Ω, Cp= 8 nF, f2: Rp= 1.45 kΩ, Cp= 65 pF

Fig. 14.8 Fully-loaded converter as the parallel Capacitance-Resistance In all of here presented Inductance, Capacitance vs. Resistance plots Figs. 14.1 – 14.8) we can find that equivalent non-loaded converter resistance in its series resonance is always in the range between 4.549Ω and 5.597Ω, and in fully loaded situation between 73.55Ω and 76.97Ω (similar to model parameters from T 1.1). Also in parallel resonance, resistance of non-loaded converter is in the range between 95.33kΩ and 98.17kΩ, and in fully loaded situation between 1.3kΩ and 1.54kΩ (again close to model parameters from T 1.1). Simply considering that in series resonance converter current is producing power dissipation on its series resistance, and that in parallel resonance converter voltage is responsible for power dissipation on its parallel resistance, we shall again get similar results as already estimated in (1.13), as for instance: Pdo1 ≈ R1 x I2 ≈ 45 W, PdL1 ≈ R1L x I2 ≈ 668 W, Pdo2 ≈ U2 / R2 ≈ 10 W, and PdL2 ≈ U2 / R2L ≈ 704 W.


3.1-8

The meaning of here presented analysis of converter power losses is to give us estimation tools to qualify converters (in order to optimize them), and to (roughly) quantify their losses, using low-signal impedance measurements, while having enough prediction-security to say that what we are predicting is approximately and sufficiently valid (and applicable) to conditions when converter is operating full-power (driven either with high input current or high input voltage, depending on its operating resonant regime). Another platform for analyzing (or understanding) converter power losses in resonance is to measure its effective tangent delta, or dissipation factor (in resonant operating area), as presented on Figs. 14.9 – 14.10.

Non-loaded

Cp

CpD = tan δ

D D


f1 = 18900 Hz, D = 7.182; f2 = 20912 Hz, D = 5.552

Fig. 14.9 Non-loaded converter, Dissipation Factor

Fully-loaded

Cp

Cp

D = tan δ D D

Capa

f1 =

Fig. 14

Converters model

citive Inductive Capacitive

18675 Hz, D = 98.39; f2 = 20462 Hz, D = 73.64

.10 Fully-loaded converter, Dissipation Factor

ing Miodrag Prokic Le Locle - Switzerland

3.1-9

From Figs. 14.9 -14.10 we can see that dissipation factor in series resonance is always a little bit higher than in parallel resonance (7.182 > 5.552; 98.39 > 73.64), but they are approximately close to each other, too. We can also see that the ratio between fully-loaded converter’s dissipation factor/s to non-loaded converter’s dissipation factor/s, in this example (for Branson converter Figs. 12 & 13, T 1.1), is close to 13 (98.39/7.182 = 12.58, 73.64/5.552 = 13.264), meaning that converter internal losses would also rise approximately 13 times, comparing non-loaded and fully-loaded situation (similar to results of mechanical quality ratio/s found in (1.12)). Another important conclusion is that maximal power performances (and efficiency) of a piezoelectric converter are approximately the same, either converter is operating in its series or parallel resonance, but converter loading nature and electrical driving configurations are very much different regarding series and parallel resonance. Also, converter mechanical output in series resonance presents dominant force or pressure source, and converter output in parallel resonance presents dominant velocity source. We can also see that dissipation factors are reaching maximal values in converter’s electrical resonant points (Impedance zero-phase points), meaning that in real operating conditions, in order to avoid high internal dissipation, we should operate converter either in its motional (mechanical) series resonance, or in its motional (mechanical) parallel resonance, or in the frequency zone between them (also in the frequency interval towards lower frequencies). In order to analyze converter losses and thermal dissipation it is important to have convenient converter models that are applicable for low and high signal driving conditions. There is a kind of understanding in modern ultrasonic technology that simplified, lumped parameters, piezo-converter models (described in the first chapter) are only well applicable for very low signal measurements and converter characterization (for instance, when we use Network Impedance Analyzer that produces 1 V sinusoidal frequency sweeping signal). Since still we are not able to make full-scale impedance measurements when converter is operating full power (on high driving voltages and high input currents), because for such situations we do not have high-power Network Impedance Analyzers, based on particular measurements in high power and heavy loading driving conditions, we can conclude that low-signal lumped parameters models are not applicable. Generally, this is not correct. If converter is operating in piezoceramics’ safe operating area, and in any of its linear regimes, or steady, stationary and stable oscillatory (sinusoidal) regimes, and if converter is not overloaded, models are well applicable (for low and high driving signals). Of course, under higher voltage (or high power) driving we can have certain resonant frequency shifts, which are never higher than 10% comparing them to low signal converter parameters. In Branson case (converter analyzed here: 502/932R), since converters are sufficiently well optimized, resonant frequency shifts (in high power driving conditions) are in the range of 1% or less (depending on piezoceramics choice, central bolt choice, static pressure on ceramics, surface finishing...). Also, if we know what are essential signal-feedback values for controlling mechanical output parameters of converters (motional voltage and motional current), and if we know how to extract such signals from our power circuits, we would conclude that lumped parameters models are very much applicable for low and high signal driving conditions, as long we stay in the safe operating limits of


3.1-10

piezoceramics (for instance not to apply more than 200V-rms. per millimeter of piezoceramic thickness, directly on piezoceramics, etc.). It is another problem (regarding modeling) when we go to converter overload situations. Most of lumped parameter models are significantly evolving and degrading because of load contribution/s. Some of the lumped circuit model parameters will significantly degenerate, and a few of them will stay relatively stable under heavy loading conditions, and we should profit exploiting this knowledge in order to design proper external matching electronics. It would be too easy to conclude that in high power (usually under overloading) our simple converter models are not valid. In reality, this could be conditionally correct, because our simple models (under heavy loading) are progressively evolving and becoming even simpler models (becoming dominantly a combination of R and C components), or saying differently, our models are still valid, but our knowledge of models evolution under loading is most probably missing. In the next part of this chapter we shall analyze some experimental loading situations in order to give deeper insight in this situation.


3.2-1

3.2 Model Parameters Evolution Under Water Loading Let us now take as an example, the water loading of the particular bell-shaped titanium sonotrode connected to titanium booster (1:1), and to Sonics & Materials ultrasonic converter type CV-154 (similar to Branson high power converters), where we shall trace the changes in converter impedance parameters caused by load increase.

T 1.4.1 Piezoelectric Converter Impedance Parameters in Series Resonance (CV-154) Water

Loading in %

Cop [nF]

C1[pF]

L1[mH]

R1

[Ω] f1

[kHz] Qm1[1]

α (1.11) T 1.3

LC

k1

1

Ω

(1.4) 0.00 14.6487 158.360 402.506 14.960 19.934 3370 1 50.415 0.80 14.7020 164.165 389.809 17.820 19.895 2734 0.8113 48.720 8.33 14.8980 167.179 386.398 19.536 19.802 2461 0.7303 48.078 25.0 15.1125 160.505 404.183 22.760 19.760 2205 0.6543 50.186 36.0 15.1504 146.329 447.355 95.050 19.673 582 0.1727 55.319 42.0 15.5491 107.887 617.514 164.83 19.506 459 0.1362 75.657

T 1.4.2 Piezoelectric Converter Impedance Parameters in Parallel Resonance (CV-154)

Water Loading

in %

Cos [nF]

C2[µF]

L2[µH]

R2[kΩ]

f2[kHz]

Qm2[1]

α (1.11) T 1.3

LC

2

2

Ω

(1.4) 0.00 14.80706 1.38004 45.6939 29.880 20.042 5193 1 5.7539 0.80 14.86617 1.37674 45.9680 20.729 20.005 3587 0.6907 5.7789 8.33 14.87437 1.36744 46.7808 17.650 19.912 3018 0.5812 5.8482 25.0 15.27301 1.39914 45.8797 12.780 19.864 2232 0.4298 5.7258 36.0 15.29673 1.31706 49.2255 2.1500 19.763 352 0.0678 6.1079 42.0 15.65699 1.25125 52.9027 1.2000 19.559 184 0.03543 6.5217

T 1.4.3 Coupling factor (kc = k ≅ k’ ≅ k’’) under loading (for CV-154)

Water Loading in %

k CC Cop

'=+1

1

k CC

ff

os' '= ⋅2

12

22

k k' ' ' 0 5. ( ' ' ' )k k+

0.00 0.103416 0.103027 0.1032213 0.1032215 0.80 0.100850 0.103342 0.1020884 0.1020960 8.33 0.105342 0.104903 0.1051223 0.1051225 25.0 0.102514 0.103932 0.1032206 0.1032230 36.0 0.097806 0.107279 0.1024330 0.1025425 42.0 0.083010 0.111559 0.0962316 0.0972845

The loading level will be quantified as the percentage of the bell-shaped-sonotrode axial-length immersed in water. All low-signal impedance measurements data are related to models presented on Figs. 1-6 and summarized in the tables T 1.4.1, T 1.4.2 and T 1.4.3. Impedance-Phase measurements-data from T 1.4.1 and T 1.4.2 can be found on the Figs. 15.1 – 15.6, and curve-fitting plots of all model parameters (from T 1.4.1 and T 1.4.2) are given on Figs. 16.1 –16.6.


3.2-2

Fig. 15.1 Non-loaded Converter Impedance (CV-154 + Booster + Sonotrode)

Fig. 15.2 Water-loaded Converter Impedance (Loading = 0.80%)

Fig. 15.3 Water-loaded Converter Impedance (Loading = 8.33%)


3.2-3

Fig. 15.4 Water-loaded Converter Impedance (Loading = 25%)




3.2-4

Loading by Immersion in waterexpressed in % of immersed sonotrode length

0 10 20 30 40 50

Mod

el P

aram

eter

s: C

op, C

1, L

1, R

1, f1

0

100

200

300

400

500

600

700

Load % vs CopLoad % vs C1Load % vs L1Load % vs R1Load % vs f1

L1 [mH]

C1 [pF]R1 [Ohm]

f1 [kHz]

Cop [nF]

Fig. 16.1 Series Resonance, Model Parameters from T 1.4.1 vs Load %

Piezoelectric Converter Impedance ParametersIn Parallel Resonance (CV-154)

Loading by Immersion in Waterexpressed in % of immersed sonotrode length

0 10 20 30 40 50

Mod

el P

aram

eter

s: C

os, C

2, L

2, R

2, f2

0

10

20

30

40

50

60

Load % vs CosLoad % vs C2Load % vs L2Load % vs R2Load % vs f2

L2 [uH]

R2 [kOhm]

f2 [kHz]

Cos [nF]

C2 [uF]

Fig. 16.2 Model Parameters from T 1.4.2 vs Load %


3.2-5

Piezoelectric Converter, Motional Impedance Parameters (CV-154)


0 10 20 30 40 50

Mot

iona

l Mod

el P

aram

eter

s: C

1, L

1, C

2, L

2

0

100

200

300

400

500

600

700

Load % vs C1Load % vs L1Load % vs C2Load % vs L2

L1 [uH]

C1 [pF]

L2 [uH]

C2 [uF]

Fig. 16.3 Motional Model Parameters from T 1.4.1 & T 4.1.2 vs Load %

Piezoelectric Converter Impedance Parameters (CV-154)


0 10 20 30 40 50

Mod

el P

aram

eter

s: C

op, C

os, C

1, C

2, f1

, f2

0

20

40

60

80

100

120

140

160

180

Load % vs CopLoad % vs C1Load % vs f1Load % vs CosLoad % vs C2Load % vs f2

C1 [pF]

f1, f2 [kHz]

C2 [uF]

Cop, Cos [nF]

Fig. 16.4 Capacitive Parameters from T 1.4.1 & T 1.4.2 vs Load %

For the regular and acceptable loading interval in this example (and generally) is taken the situation when f1 and f2 and mechanical coupling factor k (from T 1.4.3) under loading are remaining very stable (almost constant), and when electrical


3.2-6

complex impedance-phase-angle between f1 and f2 remains positive. We can also see from Figs. 16.1 – 16.6 that the most-stable model parameters (vs Water-Load) are capacitances Cop and Cos. Converter loading would become unacceptably too high (or start going to overloading) when f1, f2, Cop, Cos and mechanical coupling factor k start changing drastically (irregularly and in a non-linear fashion), compared to values under low and moderate loading. Motional capacitance and inductance C2 and L2 are also sufficiently stable parameters (vs Water-Load) in comparison with Cop and Cos, but motional capacitance and inductance C1 and L1 are obviously much less stable (vs Water-Load). We can also see that resistive motional model-parameters R1 and R2 are behaving very much non-linear in the function of water-loading. The reason of relative “instability” of motional parameters C1 and L1 is in the character of a loading process associated to series and/or parallel resonance, or saying differently, the loading interval of a converter operating in parallel resonance is much wider (and acoustically much different) than the loading interval of a converter operating in series resonance. From mechanical quality factors changes in the function of water loading (Fig. 16.5) we can conclude that for low and moderate loads, it is more convenient to operate converter in parallel resonance (since Qm2 > Qm1), and for heavy mechanical-load situations, series resonance becomes more favorable operating regime (since Qm1 > Qm2). This example will only serve to initiate possible ideas and conclusions regarding future and more rigorous analyses regarding power conversion in piezoelectric converters. For instance, we can conclude that certain converter parameters are remaining almost unchanged under load and use them in realizing converter reactive (inductive) compensation, optimal power transfer, resonant frequency control, choice of series or parallel resonance, etc.


3.2-7

Loading by Im m ersion in W aterexpressed in % of im m ersed sonotrode length

0 10 20 30 40 50

Mec

hani

cal Q

ualit

y Fa

ctor

s

0

1000

2000

3000

4000

5000

6000

Load % vs Q m 1Load % vs Q m 2

Q m 1

Q m 2

Characteristic Mechanical Impedances (see eqs. (1.4) and T 1.2, Fig. 16.6), are also interesting for qualification of a piezoelectric converter loading process. For instance, on the Fig. 16.6 we can find that Characteristic Mechanical Impedance in parallel resonance, Zcm2, remains relatively stable (constant) for wide loading interval that is not the case with Characteristic Mechanical Impedance of series resonance, Zcm1. This is only confirming that the nature of piezoelectric converter loading is very much different between converter operating in series and parallel resonance.

Qm1

Qm2

Fig. 16.5 Mechanical Quality Factors from T 1.4.1 & T 1.4.2 vs Load %


3.2-8


0 10 20 30 40 50Cha

ract

eris

tic M

echa

nica

l Im

peda

nces

: Zcm

1 , Z

cm

0

10

20

30

40

50

60

70

80

Load % vs Zcm1 Load % vs Zcm2

Zcm1 [kOhm]

Zcm2 [Ohm]

Fig. 16.6 Characteristic Mechanical Impedances from T 1.4.1 & T 1.4.2 vs Load %


3.3-1

3.3 Evolution of Model Parameters When Connecting Boosters and Sonotrodes

Here are pictures of typical converter-booster-sonotrode configurations.

Converter

Booster

Sonotrode

Different Boosters and Sonotrodes


3.3-2

In literature, it is habitual to find that adding boosters and sonotrodes to a converter is addressed as the case of converter loading. Here we can simply name the same situation as: “assembling” a complex transducer structure (see Figs. 8, 9, 10). By adding new resonant elements to a converter we are simply creating a chain of mechanical filters (or chain of strongly coupled resonators), where every new resonant element is contributing by increasing the total mechanical quality factor of the complex transducer structure (if all resonant elements are mutually tuned to operate in the same frequency range). This fact is also supported by the measurement results presented in T 1.5.1 – T 1.5.4, given below. We should also make the difference between converters low frequency (1 kHz), input, static, clamped capacitance (Cinp.), and model parameters Cop and Cos, valid (or fitted) for certain high frequency resonant mode (for an isolated couple of resonances in f1 and f2). Usually, the low frequency (1 kHz) input capacitance of a converter will always stay the same (constant), regardless how many boosters and sonotrodes we attach to the same converter, but particular model parameters Cop and Cos would increase significantly when adding boosters and sonotrodes to a converter (of course we should add only compatible boosters and sonotrodes, operating in the same frequency range as the basic converter). If there is a case of a single converter (without attached boosters and sonotrodes), its low frequency input capacitance would be approximately equal to model parameter clamped capacitances, Cinp.(1kHz) ≤ Cos ≅ Cop + C1, but when we start attaching boosters and sonotrodes to it, Cop and Cos would start increasing significantly (with every added booster or horn), comparing them to the low frequency input capacitance (Cinp. at 1 kHz) that will not change. For the same example as already introduced by T 1.4.1, T 1.4.2 and Figs. 15.1 – 16.6, which presents the combination of: Conics & Materials converter (CV-154) + Titanium Booster (1:1) + Bell Sonotrode, we can trace (in no-load conditions) the changes of Cop and Cos, starting from the situation when (1°) there is only the single converter (CV-154), Fig. 17.1, then (2°) Converter + Booster, Fig. 17.2, and finally (3°) Converter + Booster + Bell-Sonotrode (Fig. 15.1). The measurement data for cases (1°), (2°) and (3°) are summarized in the tables T 1.5.1 and T 1.5.2.


3.3-3

Fig. 17.1 Non-loaded Converter Impedance (single CV-154)

Fig. 17.2 Non-loaded Converter Impedance (CV-154 + Booster)

It is important to notice that if we create a chain of ultrasonic resonators connected axially to each other, such as: converter + booster + sonotrode(1) + sonotrode(2) +… the resulting impedance-phase characteristic, after adding every new element, will become more and more narrow and sharp, making the difference between parallel and series resonance/s gradually smaller, and creating gradually higher mechanical


3.3-4

quality factors, as presented on the Fig. 17.3. The best frequency matching criteria in such situations (between all resonators in the chain) is to keep the central frequency (a middle point between any couple of series and parallel resonant frequency), f f f f f f f0 11 21 12 22 13 230 5 0 5 0 5≈ + ≈ + ≈ + ≈. ( ) . ( ) . ( ) Constant. approximately constant. The resonant frequency control (PLL) also becomes gradually more and more difficult because the difference between parallel and series resonant frequency becomes increasingly smaller and mechanical quality factors are significantly increasing.

f f f f f f ff f f f f f Q Q Qm m m

0 11 21 12 22 13 23

21 11 22 12 23 13 3 2 1

0 5 0 5 0 5≈ + ≈ + ≈ +− > − > − > >

. ( ) . ( ) . ( ),,

Fig. 17.3 Impedance Evolution of Non-loaded Converter

(1)- only converter, (2)- converter + booster, (3)- converter + booster + sonotrode


3.3-5

Generally (see the picture below, Fig. 17.3-1), in ultrasonic engineering, depending of chosen resonant operating regime, we can make frequency-matching between certain converter and sonotrode taking as an objective that new configuration (converter-sonotrode) would operate either in series 1Z , f1, or in parallel 2Z , f2

resonant frequency of the single converter Z , or as the optimal-case scenario

between series and parallel resonant frequency (case 3Z ) , as already presented on Fig. 17.3.

Fig. 17.3-1 Different possibilities for converter-sonotrode frequency-matching


3.3-6

T 1.5.1 Piezoelectric Converter Impedance Parameters in Series Resonance (CV-154…)

No-Load C(1kHz) =11.12nF

Cop[nF]

C1[pF]

L1[mH]

R1

[Ω] f1

[kHz] Qm1[1]

∆f [Hz]

LC

k1

1

Ω

equat. (1.4) (1°) 10.72 1422.24 46.324 8.34 19.625 684 1250 5.7071 (2°) 12.61 271.099 237.741 9.877 19.825 2998 210 29.6134 (3°) 14.6487 158.360 402.506 14.960 19.934 3370 108 50.415

T 1.5.2 Piezoelectric Converter Impedance Parameters in Parallel Resonance (CV-154…)


Cos[nF]

C2

[µF] L2

[µH] R2

[kΩ] f2

[kHz] Qm2[1]

∆f [Hz]

LC

2

2

Ω

equat. (1.4) (1°) 12.142 0.092797 626.764 102.46 20.875 1247 1250 82.1836 (2°) 12.88 0.637556 98.9538 97.24 20.035 7805 210 12.4582 (3°) 14.80706 1.38004 45.6939 29.880 20.042 5193 108 5.7539

From results in T 1.5.1 and T 1.5.2 we can easily calculate that (in this specific configuration; -converter CV-154) Cop is increasing for 31% and Cos is increasing for 19.8% (comparing single converter and Converter+Booster+Bell-Horn). Of course, Cop and Cos will not endlessly increase if we continue adding boosters and sonotrodes to a converter (the experimentally known limit seems to be less than +50%). For the same example we can also see that C1 is decreasing, L1 increasing, Zcm1 increasing, C2 increasing, L2 decreasing, Zcm2 decreasing, and that in almost all cases mechanical quality factors are increasing (while difference between parallel and series resonant frequencies is decreasing, also when comparing single converter and Converter+Booster+Bell-Horn). The input, low frequency, clamped (and static) capacitance of the same single converter (CV-154) is Cinp.(1kHz) = 11.12 nF, and it is not changing when we add Booster + Bell-Horn to the converter. The above-described situation (regarding mainstream tendency of model parameters evolution) is typical and generally valid for all complex converter-booster-sonotrode structures. For instance, if we again take Branson 502/932R converter (data for single converter are given in T 1.1, T 1.2, Figs. 12 & 13) and summarize the same situation as already presented in T 1.5.1 and T 1.5.2 (starting from single converter (1°), then Converter + Booster (2°), and finally Converter + Booster + Large Sonotrode (3°)), we will get results presented in T 1.5.3 and T 1.5.4 (which are qualitatively equal or similar to results from T 1.5.1 and T 1.5.2).

T 1.5.3 Piezoelectric Converter Impedance Parameters in Series Resonance (502/932R…)


Cop[nF]

C1[pF]

L1[mH]

R1[Ω]

f1[kHz]

Qm1[1]

∆f [Hz]

LC

k1

1

Ω

equat. (1.4)

(1°) 18.1 4046 17.534 4.6 18.900 454 2012 2.0817 (2°) 26.6025 958.442 66.0934 4.724 20.000 1758 350 8.304 (3°) 26.5929 63.8892 981.240 34.4779 20.101 3594 24 123.9291


3.3-7

T 1.5.4 Piezoelectric Converter Impedance Parameters in Parallel Resonance (502/932…)


Cos[nF]

C2

[µF] L2

[µH] R2

[kΩ] f2

[kHz] Qm2[1]

∆f [Hz]

LC

2

2

Ω

equat. (1.4) (1°) 22.05 0.10153 570.50 94.20 20.912 1257 2012 80.77 (2°) 27.56094 0.82906 73.7318 29.2549 20.350 3102 350 9.430 (3°) 26.65678 16.8820 3.70444 5.68529 20.125 12137 24 0.468437

In reality, looks logical that the low frequency input capacitance Cinp.(1 kHz), and model-parameter capacitance Cos should always stay constant, mutually equal, static capacitance of a converter. The reason why we are getting the difference (between Cinp. and Cos), when making impedance measurements with Network Impedance Analyzer, is just in fact that we measure Cinp in one discrete frequency point (at 1 kHz), and Cos is found after best curve fitting, respecting piezoelectric converter models, on a limited frequency interval that captures only the close vicinity of a series and parallel resonant frequency couple f1 and f2. Since we give the limited data-input to the Network Impedance Analyzer (taking into account the frequency interval only a bit larger than ∆f = f2 – f1), neglecting enormously large interval of impedance values between very low frequency (say 1 kHz) and first resonant mode (and also neglecting frequencies higher than f2), results of best curve fitting are creating the value of Cos that seems different (higher) than Cinp. Since in resonant operating conditions we usually care only about converter characteristics in the close vicinity of certain resonance, it is obvious that we need to know the model parameters Cos and Cop valid for the same operating frequency interval. This is also confirming that lumped parameters models, we usually use for describing piezoelectric impedance, are only locally valid (for a close vicinity of a well-isolated, single resonant mode). In ultrasonic applications, when making inductive converter compensation (to eliminate Cos or Cop), we can often notice that calculated inductance in reality is different than the value we really need to apply (to make converter to become resistive load), because in converter resonant area (for frequencies lower than f1 and higher than f2), effective converter (total input) capacitance is very mach frequency-dependant. For instance, in ultrasonic cleaning applications, when many single converters are operating as a group (in parallel connection), series compensating-inductance is often significantly lower than calculated-value based on Cinp. or Cos (because cleaning converters, as a group, usually operate in the area just below series resonance, where effective converters capacitance is significantly higher than Cinp.(1 kHz)). In order to graphically visualize this situation (regarding capacitive and inductive character of a piezoelectric converter vs. frequency), on the Fig. 18 are qualitatively improvised equivalent converter capacitance and inductance in the area of a selected resonant mode (only showing their typical tendency, without any precise and quantitative significance). It is also known that single piezoelectric converters can easily be forced to operate high power on frequencies lower than f1, since in this area converter has significantly high effective capacitance (higher than Cinp.(1 kHz)), but the same is not valid for frequency area higher than f2, since in that frequency interval converter has relatively low effective capacitance (much lower than Cinp.(1 kHz), which is increasing slowly and exponentially towards Cinp.). Obviously we can conclude that sufficiently high effective converter capacitance is enabling converter to accumulate proportionally high potential energy, necessary for continuous and high power oscillating regime. Analogous logic (regarding inductive


3.3-8

potential energy accumulation) should be applicable to converters operation below f2 (between f1 and f2). Differently formulating the same conclusion we can say that natural and highly efficient converters oscillating regimes (for producing high power) are in frequency areas left from f2 (or between f1 and f2: LMAX.), and left from f1 (towards lower frequencies: CMAX.). The same situation (regarding best operating-power frequency areas) would be only slightly changed when we make piezoelectric converter inductive compensation/s, but again it will be valid that best operating areas will stay below converter’s parallel resonance, between series and parallel resonance, and below series resonance, in the frequency areas where converter’s effective capacitance and/or effective inductance are significantly elevated.

Cinp.(1 kHz)

Frequency f

1 1

Fig. 18 Piezoele

In order to make a piezreal power), we shoucompensate it by addiconverter’s static capacdominant, static convert We know that all piezoea large frequency intervnumber of limited frequcomplex impedance chathe normalized impeda(later on) with a norma(dominant) static capaci

Zj C j fC

ZZ fC Z

= = ⇒

= ⋅ =

1 12

12

12

0 0

0 0 0

ω π

π π

Converters modelin

fL C

fL CMIN MAX MAX MIN

1 22 2≅ ≅

π π,

ctric Converter Effective Capacitance & Inductance

oelectric converter operating as a resistive load (to produce ld know the dominant static converter capacitance and ng certain series or parallel inductance that will neutralize itance. Here we shall explain the procedure how we can find er capacitance (valid for wide frequency interval).

lectric converters as electrical components (characterized in al) are dominantly capacitive components, except in a certain ency intervals where converters could have inductive and/or racter (in converters resonance zones). Let us first establish

nce concept for an ideal capacitor Co in order to compare it lized impedance of the converter that would have the same tance (as being an ideal capacitor):

ZfC

Zf C

Const f const

f C Z f fff

= = = =

⋅= ⇒

12

12

10

00 0

0

0 0 0 0

0

π π, .,

( / )( )

.,

g Miodrag Prokic Le Locle - Switzerland

3.3-9

⇒

⇒

⇒

log(Z

Z) = -log(2 f C Z ) - log( f

f) ,

Y = log(ZZ

) , X = log( ff

) , A = -log(2 f C Z ) = Const.

Y + X = A

00 0 0

0

0 00 0 0

π

π

From the equation of line, Y + X = A, we can see that normalized impedance of an

ideal capacitor Co, Y = log(Z

Z)

0

, can be presented in the function of a normalized

frequency, X = log( ff

)0

, similar to the curve on the left side of Fig. 19.

If we now take a piezoelectric converter (which should have the same dominant capacitance Co), and using Impedance Analyzer measure its normalized impedance,

Y = log(Z

Z)

0

, in the function of normalized frequency, X = log( ff

)0

, (setting

horizontal and vertical axes, X and Y, to have logarithmic divisions), we shall get the Y vs X curve, as presented on the right side of Fig. 19.

Fig. 19

When using Nthe value of thnormalized pieasymptotic lineThis way, wecapacitance C

C = 12 f C Z0

0 0π

Converter

0

Normalized Impedance ofPiezoelectric C

etwork Impedance Analyzere constant A = -log(2 f C0 0πzoelectric impedance curve that would cross the X an

shall be in a position to 0 that should satisfy the equa

0 ⋅10A .

s modeling Miodrag

0

an Ideal Capacitor (left), and of a onverter (right)

, we should be able to find experimentally Z )0 , by simple linear extrapolation of the (right side of Fig. 19), passing the middle d Y axis in two points: (0, A) and (A, 0). calculate the dominant (static) converter tion:

Prokic Le Locle - Switzerland

3.3-10

If we make the comparison between Co, low frequency capacitance Cinp.(1 kHz) and model capacitance Cos of the same piezoelectric converter, we shall again find that all of them are only approximately equal, Co ≅ Cinp.(1 kHz) ≅ Cos . As the conclusion regarding static converter capacitances, we can say that when converter is operating on a very low frequency or in a constant driving-voltage regime, the most important should be Cinp.(1 kHz), and when converter is operating in a very large frequency interval, then Co becomes important static capacitance parameter. In other situations when we drive converter in a single resonance point, then Cos should be considered as the most important static capacitance.


3.3-11

Impedance Changes in Case of Converters Bonded to Thin-Walls Vessels In many applications like ultrasonic cleaning and ultrasonic liquid processing, ultrasonic converters are bonded or glued to (thin walls) cleaning or reactor vessels. For the purpose of electrical and mechanical impedance matching and optimal power conversion (from an ultrasonic power supply, or ultrasonic generator to ultrasonic converters) it is important to know the range of possible impedance changes, comparing the impedance-characteristics of non-loaded (free in air) converters, and characteristics of the same converters after bonding and after water loading. Generally valid situation (see Fig. 20) is that converter after bonding or gluing to a cleaning vessel has a bit lower series and parallel resonant frequencies (than non-loaded converter), and that every additional loading is reducing converter’s maximal (parallel-resonance) impedance, and increasing minimal (series-resonance) impedance. Fig. 20 presents (qualitatively) a typical loading situation regarding impedance changes, when single piezoelectric converter is: in the case (0), non-loaded and non-bonded, in air; -the case (1), when the same converter is bonded to an empty and thin-walls vessel or ultrasonic cleaning tank; -and case (2), when the same vessel (or ultrasonic cleaning tank) is full with water. There are many bonding and gluing techniques, and many different adhesives for fixing piezoelectric converters to ultrasonic cleaning vessels or liquid processing reactors. Depending on bonding technology and used adhesives, piezoelectric converter would change its impedance parameters in the frames given numerically in T 1.6 (of course, all data in T 1.6 are presenting approximate values, typical for ultrasonic cleaning converters).

Single non-loaded converter (0)

Converter after bonding (1)

Bonded and water-loaded converter (2)

[ f1 ] [ f2 ]

= (0) = Z(0)= non-loaded converter in air

= (1) = Z(1)= bonded, non-loaded converter

= (2) = Z(2)= bonded, water-loaded converter

Fig. 20 Piezoelectric Converter Impedance after Bonding & Water Loading


3.3-12

T 1.6 Typical impedance limits in cases of bonding & water loading (see Fig. 20) Z min , f1 Z max , f 2

ZZ

ff

Z Z

(1)min

(0)min

1(1)

1(0)

(0)min

(1)min

∈ ≈

= ⇒ ≤ ≤

( , ) , .5 15 0 96

15 75 225Ω Ω Ω

ZZ

ff

Z Z

(0)max

(1)max

2(1)

2(0)

(0)max

(1)max

∈ ≈

= ⇒ ≤ ≤

( , ) , .

.

4 9 0 94

60 6 67 15k kΩ Ω kΩ

ZZ

ff

Z Z

(2)min

(1)min

1(2)

1(1)

(1)min

(2)min

∈ ≈

= ⇒ ≤ ≤

( , ) , ( . , . )2 7 0 96 0 99

150 300 1050Ω Ω Ω

ZZ

ff

Z Z

(1)max

(2)max

2(2)

2(1)

(1)max

(2)max

∈ ≈

= ⇒ ≤ ≤

( , ) , ( . , . )

.

2 6 0 98 105

10 1 67 5k kΩ Ω kΩ

10 100 0 92 0 95

15 150 1500

≤ ≤ ≈

= ⇒ ≤ ≤

ZZ

ff

Z Z

(2)min

(0)min

1(2)

1(0)

(0)min

(2)min

, ( . , . )

if Ω Ω Ω

6 60 0 92 0 99

60 1 10

≤ ≤ ≈

= ⇒ ≤ ≤

ZZ

ff

Z Z

(0)max

(2)max

2(2)

2(0)

(0)max

(2)max

, ( . , . )

if k k kΩ Ω Ω

Ultrasonic cleaning and liquid processing systems are usually driven by arrays of many similar converters, connected electrically in parallel, and glued to the same surface. If all converters in parallel connection are identical, or very similar to each other, we can approximately estimate the resulting impedances of certain converters group, just by dividing all impedance values valid for a single transducer (found in Fig. 20 and T 1.6) by the number of transducers in a group (like having identical resistances in parallel connection), and doing this way we shall stay in the error limit of ±10%. When we know the absolute limits of impedance variations of a converter group in parallel connection, we shall be able to design optimal power supplies for them, to calculate optimal driving current and voltages, to calculate transformer ratio for optimal load-impedance matching (to ultrasonic generator), and to implement proper power control strategy. In all other cases, when converters are mutually dissimilar, we should experimentally and by (Impedance-Phase) measurements determine equivalent impedance parameters and frequency areas of high converters efficiency.


4.0-1

4.0 Converter Electric Impedance Modifications and Mechanical Resonances

In this chapter we shall describe several experiments where converter electrical impedance is externally changed by adding to it in series and/or parallel connection passive electric components (adding resistances, capacitances and inductances). The resulting converter resonances are measured using electrical means (Impedance Network Analyzer), and mechanical means (mechanical vibrating platform with variable frequency, and wide-frequency-band mechanical amplitude sensor). For this situation it was chosen a very simple cylindrical, bolted, Langevin, sandwich converter (back mass: stainless steel, front mass: aluminum 7075, and 2 of PZT8 piezoceramic rings in the middle area). The choice of the converter was made to satisfy the criteria that its first resonance-couple (natural series and parallel resonance) is sufficiently isolated on the frequency axis (or that all higher harmonics are placed very far from the first resonance couple), and to have regular and typical piezoelectric impedance shape (see Fig. 1). In many situations of practical interest, when driving piezoelectric converters, ultrasonic generators or electric power supply units are used, where different R, L and C elements are effectively placed in parallel or series connection to a converter (for the purpose of impedance and power matching, and for converter reactive impedance compensation). Consequently, it is clear that in any case converter impedance characteristics and resonant operating zones are affected by the presence of surrounding electronics (as well as by converters mechanical loading conditions). In order to establish the correlation between converter electrical and mechanical resonant zones, converter was placed (vertically or axially) on the mechanical vibrating platform, VP (made for measuring axial, resonant, half-wave lengths of ultrasonic sonotrodes). This way, converter was considered as a solid cylinder (rod), and it was possible to measure its half-wave mechanical resonances (λ/2 – resonant frequencies). On the opposite converter face, the wide-band (piezoelectric) displacement-sensor was placed (touching converter vertically, only in one point), in order to detect the λ/2 – axial resonances of such (composite) solid rod. The vibrating platform VP was equipped with variable frequency (sinusoidal signal-shape) generator and frequency meter, and this frequency (produced by VP) was considered as a converter mechanical frequency fm (see Fig. 2, d)). Electrical converter impedance was (first) externally modified (by adding different electrical components to converter’s input terminals) and resulting converter (equivalent) impedance was measured with Impedance Network Analyzer, and then the same converter (including all external electrical modifications) was moved to vibration platform VP, and resulting mechanical λ/2 – axial-resonances were measured. This way (for a non-loaded converter) it was possible to establish relations between typically electrical resonances (measured only using electrical means), and typically mechanical resonances (measured using mechanical means). Mechanical vibrating platform VP (used here) was able to measure very precisely the resonant frequency fm and to give sufficiently good information about mechanical amplitude of an oscillating λ/2 –rod, on its opposite side (in relative units from 1 to 10). When resonant λ/2-mechanical frequency was found, the amplitude sensor S was generating a high electrical signal.


4.0-2

Fig. 1 Typical Non-loaded Converter Impedance-Phase Characteristics The next, more profound benefit of such impedance modifications is to become able to compare the same converter-characteristics electrically modified (with passive electric components connected externally), and the converter characteristics under different mechanical loading situations (without any external, electrical impedance modification). This way, indirectly, we can conclude that certain types of mechanical loads are modifying converter equivalent impedance on the similar way as connecting resistance or capacitance or inductance… to the same (non-loaded) converter. Consequently, we can get a feeling about modeling of different mechanical loads (or we can give the answers to what means mechanical resistive, capacitive, inductive and/or some other complex loading, applying analogies).


4.0-3

Fig. 2 Electrical and Mechanical, Non-loaded Converter Characteristics a) & b) Typical Electric Impedance-Phase Curves c) λ/2 – axial mechanical resonance with shortened input terminals d) λ/2 – axial mechanical resonance with open input terminals

On the Fig. 2 are presented (in a simplified, summary form) almost the same facts found on Fig. 1 (such as electrical impedance and phase, Fig. 1, a) & b)), and two more measurements when the same converter is placed on the vibrating platform VP (Fig. 2, c) & d)). Fig. 2, c) presents converter mechanical amplitude and mechanical


4.0-4

resonant frequency when its input electrical terminals are in a short circuit: switch = ON. Fig. 2, d) presents converter mechanical amplitude and mechanical resonant frequency when its input electrical terminals are open: switch = OFF. We can immediately conclude that series electrical resonance corresponds to the high-current driven converter, when electrical driving circuit has very low internal impedance (input terminals in a short circuit), and that parallel electrical resonance corresponds to the high-voltage driven converter, when electrical driving circuit has very high internal impedance (input terminals open). We can also (indirectly) conclude that if a converter should dominantly operate under heavy mechanical loading, it should be operated in its series resonance, and if a converter should dominantly operate under low and moderate mechanical loading conditions, it should be operated in its parallel resonance. We can also find out that electrical and mechanical series and/or parallel resonance/s are (approximately) in the same frequency zone/s (almost mutually equal for high mechanical quality-factor converters). Since parallel resonance is higher than series resonance, and because, when making mechanical resonance measurements, we do not change converter’s axial length (it is always the same converter), it looks that a short-circuited converter has lower effective (or resulting) sound speed than open-terminals converter (because in the case of here described mechanical measurements, using VP, we treat converter only as a solid, elastic rod). Let us now generalize the measurements presented on the Fig. 2, by connecting in parallel to the same converter input-terminals a variable resistance (Rp ∈ 10 Ω until 500 kΩ), as shown on the Fig. 3. When parallel resistance is very high, this corresponds to open converter terminals (Fig. 2, d)), and when parallel resistance is very low, this corresponds to converter terminals in a short circuit (Fig. 2, c)). Fig. 3 presents electrical Impedance-Phase characteristics (measured by Network Impedance Analyzer) when parallel resistance is taking different values. We can easily notice (on the Fig. 3) that parallel resistance will not change series and parallel converter resonant frequencies. The much more interesting situation is to find out what is happening with mechanical resonant frequencies when we use mechanical vibrating platform VP (as previously explained), Fig. 4. Fig. 4 a) & b) presents in a simplified form the same electric Impedance-Phase curves, as already seen on the Fig. 3. Fig. 4, c) is just underlining where are electrical resonant frequencies (in order to compare them with mechanical resonant frequencies), Fig. 4, d) presents mechanical resonances measured on the VP, and Fig.4, e) presents positions and relative amplitudes of λ/2–axial-mechanical resonances. The basic and most important conclusions relevant for Fig. 4 are practically (almost) the same as all conclusions relevant for Fig. 2.


4.0-5

Fig. 3 Converter Impedance-Phase Characteristics with Parallel Resistance


4.0-6

Fig. 4 Converter Characteristics with Parallel Resistance

a) & b) Electric Impedance-Phase Curves c) Electrical resonances d) Mechanical resonances e) λ/2 – axial-mechanical resonances

We can now connect variable resistance (Rs ∈ 10 Ω until converter input terminals, and trace the equivalent impresented on the Fig. 5. When series resistance is very low, corresponds to a free converter from the Fig. 1. If series

Converters modeling Miodrag Prokic Le L

(see Fig. 3)

100 kΩ) in series with pedance changes, as this situation electrically resistance increases,

ocle - Switzerland

4.0-7

Fig. 5 Converter Impedance-Phase Characteristics with Series Resistance

converter series resonance will simply disappear, becoming weaker and weaker (by amplitude), and moving a little bit towards parallel resonance, until completely becomes undetectable for sufficiently high values of Rs. It only remains a well detectable converter’s parallel resonance (for high values of Rs). On the Fig. 6, a), b) & c) is presented the same situation in a simplified summary form, in order to make more clear conclusions that can be drawn from the Fig. 5. When we place the same converter (see Fig. 5 d) & e)) on the mechanical vibrating platform VP (as before), we shall have exactly the same mechanical-situation (λ/2 –


4.0-8

axial mechanical resonance with converter’s open input terminals) as already explained for the Fig. 2, d).

Fig. 6 Converter Characteristics with Series Resistance (see Fig. 5)



4.0-9

The next example presented on the Fig. 7 and Fig. 8, is a variable capacitance connected in parallel to the same converter (Cp ∈ 0.1 nF until 100 nF). When parallel capacitance Cp is very low (close to zero), this corresponds (electrically and mechanically) to open converter terminals (see Fig. 2, d)), and when parallel capacitance is very high (much higher than converter static capacitance), this situation corresponds (only) mechanically to converter terminals in a short circuit (see Fig. 2, c)). Fig. 7 and Fig. 8, a) & b) present electrical Impedance-Phase characteristics (measured by Network Impedance Analyzer) when parallel capacitance is taking different values. We can easily notice (on the Fig. 7) that parallel capacitance will not change (electrically or mechanically) converter series resonant frequency. Electrically, increase of parallel capacitance is only reducing and destroying natural parallel resonance of the converter, making converter to become dominantly a capacitive component, and moving its parallel resonance (electrically and mechanically) towards converter series resonance (see also Fig. 8, where the same situation is simplified and presented in a summary form, convenient for comparison between electrical and mechanical resonance measurements, as shown before). Fig. 8, c) is just qualitatively underlining where are electrical resonant frequencies (in order to compare them with mechanical resonant frequencies), Fig. 8, d) gives a rough idea about evolution of mechanical resonances measured on the VP, and Fig.8, e) presents positions and relative amplitudes of λ/2–axial-mechanical resonances. Effectively, piezoelectric converter with parallel capacitance becomes (using only electrical means) variable-sound-speed solid-rod (in the frequency area between its series and parallel resonance), since resulting sound speed will depend on connected capacitance (see Fig. 8, e)). We can easily imagine producing different semi-active ultrasonic tools such as boosters, sonotrodes and solid-wave-guides with possibility for easy external sound speed tuning and resonance adjustment (installing one piezoceramic, sandwiched layer inside of such tools, and adding variable parallel capacitance to it). Also a number of electromechanical band-limited filters, selective acoustic receivers, and frequency and amplitude modulators can be designed based on variable-sound-speed solid-rod (operating as λ/2–axial-mechanical resonator).


4.0-10

Fig. 7 Converter Impedance-Phase Characteristics with Parallel Capacitance


4.0-11

Fig. 8 Converter Characteristics with Parallel Capacitance (see Fig. 7)



4.0-12

The next example presented on the Fig. 9 and Fig. 10, is a variable capacitance connected in series to the same converter (Cs ∈ 0.1 nF until 10 nF). When series capacitance Cs is very high (much higher than converter static capacitance), this corresponds (electrically and mechanically) to an open converter terminals (see Fig. 1 and Fig. 2, d)), and when series capacitance is decreasing towards very low values, series converter resonance is mowing closer to parallel resonance and progressively disappearing (see Fig. 9 and Fig. 10, a), b) and c)). Fig. 9 and Fig. 10, a) & b) present electrical Impedance-Phase characteristics (measured by Network Impedance Analyzer) when series capacitance is taking different values. We can easily notice (on the Fig. 9 and Fig. 10) that series capacitance will not change (electrically or mechanically) converter parallel resonant frequency. Electrically, a decrease of series capacitance is only reducing and destroying natural series resonance of the converter. Fig. 10, c) is just qualitatively underlining where are electrical resonant frequencies (in order to compare them with mechanical resonant frequencies), Fig. 10, d) shows where converter will resonate mechanically measured on the VP, and Fig. 10, e) presents relative mechanical amplitude of converter’s λ/2–axial-mechanical resonance. Here described situation with series capacitance is showing us how (in certain applications) we can eliminate undesirable series resonant frequencies and make a piezoelectric converter operating only on its (high impedance) parallel resonance, where converter becomes extremely sensitive as a sensor of mechanical vibrations, as a hydrophone, or sensitive microphone, since mechanical and electrical parallel resonance will stay unchanged, and only series resonance would disappear.


4.0-13

Fig. 9 Converter Impedance-Phase Characteristics with Series Capacitance


4.0-14

Fig. 10 Converter Characteristics with Series Capacitance (see Fig. 9)

a) & b) Electric Impedance-Phase Curves c) Electrical resonances d) Mechanical resonance e) λ/2 – axial-mechanical resonance


4.0-15

The next example presented on the Fig. 11 and Fig. 12, is a variable inductance connected in parallel to the same converter (Lp ∈ 60 µ until 35 mH). When parallel inductance Lp is very high, this approximately corresponds (electrically and mechanically, but only in oscillatory operating conditions) to open converter terminals (see Fig. 1 and Fig. 2, d)), except that booth series and parallel electrical and mechanical resonances would be easy detectable (and operational) by electrical and/or mechanical means. When parallel inductance is very low, this situation approximately corresponds (only mechanically) to converter terminals in a short circuit (see Fig. 2, c)). Based only on electrical impedance measurements (Fig. 11 and Fig. 12, a) & b)), we can also notice that variable parallel inductance generates two (frequency and inductance dependant) high impedance zones (lower and higher parallel resonance zones), and keeps relatively stable converter’s series resonance. Fig. 11 and Fig. 12, a) & b) present electrical Impedance-Phase characteristics (measured by Network Impedance Analyzer) when parallel inductance is taking different values. We can easily notice (on the Fig. 11) that parallel inductance will change (electrically and mechanically) booth of converter’s parallel resonant frequencies. Electrically, starting from very high parallel inductance and then decreasing will shift lower (first) parallel resonance of the converter towards higher frequencies (until its series resonance), and it will also shift the higher (second) parallel resonance towards higher frequencies, while keeping series resonance in a stable position. Fig. 12, c) is just qualitatively underlining the evolution of electrical resonant frequencies (in order to compare them with mechanical resonant frequencies). Fig. 12, d) gives a rough idea about evolution of mechanical resonances measured on the VP, and Fig.12, e) presents positions and relative amplitudes of λ/2–axial-mechanical resonances. Effectively, piezoelectric converter with parallel inductance becomes (using only electrical means) variable-sound-speed solid-rod, since resulting sound speed (and converter’s parallel resonances) will depend on connected inductance (see Fig. 12, e)). What is very interesting is that electrically we can notice existence of movable, lower and higher parallel-resonant-zone (see Fig. 11, Fig. 12, a), b) and c)), but in reality, after making measurements using vibrating platform VP, we will find that converter operating as λ/2–axial-mechanical resonator is inefficient in the higher (or second) parallel-resonant-zone, since it will have relatively low mechanical amplitudes in such resonant states (see Fig. 12, e)). Contrary, the lower (movable) parallel resonant zone will be mechanically fully efficient in a large frequency interval, starting from series resonance towards lower frequencies, since λ/2–axial-mechanical resonant amplitudes will be relatively high (Fig. 12, e)). Again, similar to parallel capacitance option (Fig. 7 and Fig. 8), we can easily imagine making different semi-active ultrasonic tools such as boosters, sonotrodes and solid-wave-guides with possibility for easy external sound speed tuning and resonance adjustment (installing one piezoceramic, sandwiched layer inside of such tools, and adding variable parallel inductance to it). Also a number of electromechanical filters, selective acoustic receivers, and frequency and amplitude modulators can be designed based on variable-sound-speed solid-rod (operating as λ/2–axial-mechanical resonator).


4.0-16

Fig. 11 Converter Impedance-Phase Characteristics with Parallel Inductance


4.0-17

Fig. 12 Converter Characteristics with Parallel Inductance (see Fig. 11) a) & b) Electric Impedance-Phase Curves c) Electrical resonances d) Mechanical resonances e) λ/2 – axial-mechanical resonances


4.0-18

The next example presented on the Fig. 13 and Fig. 14, is a variable inductance connected in series to the same converter (Ls ∈ 10 µH until 10 mH). When series inductance Ls is very low (or zero), this corresponds (electrically and mechanically) to an open single converter terminals (see Fig. 1 and Fig. 2, d)). Based only on electrical impedance measurements (Fig. 13 and Fig. 14, a) & b)), we can also notice that variable series inductance generates two (frequency and inductance dependant) low impedance zones (lower and higher series resonance zones), and keeps relatively stable converter’s parallel resonance. When series inductance start increasing its value, electrical, lower series converter resonance starts mowing towards lower frequencies, and higher series resonance also moves towards converter’s parallel resonance or effectively also decreasing (see Fig. 13 and Fig. 14, a), b) and c)). Fig. 13 and Fig. 14, a) & b) present electrical Impedance-Phase characteristics (measured by Network Impedance Analyzer), when series inductance is taking different values. We can easily notice (on the Fig. 13 and Fig. 14) that series inductance will not change (electrically or mechanically) converter’s parallel resonant frequency. Fig. 14, c) is just qualitatively underlining where are electrical resonant frequencies (in order to compare them with mechanical resonant frequencies), Fig. 14, d) shows where converter will resonate mechanically measured on the VP, and Fig. 14, e) presents relative mechanical amplitude of converter’s λ/2–axial-mechanical resonance. Now, based on all above described experimental situations, we can generalize the following conclusions:

a) By connecting in series any of passive R, L and C components to a single piezoelectric converter we shall be able to modify, and/or move, or attenuate series resonance zone/s of the converter (impedance minimum zone/s), while its parallel resonance zone remains in a stable position.

b) By connecting in parallel any of passive R, L and C components to a

single piezoelectric converter we shall be able to modify, and/or move, or attenuate parallel resonance zone/s of the converter (impedance maximum zone/s), while its series resonance zone remains in a stable position.

c) In any case of practical interest, piezoelectric converter can be

efficiently driven in its low frequency area, below its series resonant frequency, then in certain situations the frequency interval between converter’s series and parallel resonant frequencies can also be efficiently activated (including operations in series and parallel resonances), and in a remaining frequency area, higher than converter’s parallel resonant frequency, converter becomes electromechanically inefficient (regardless of possible presence of different electrical resonant zones).


4.0-19

Fig. 13 Converter Impedance-Phase Characteristics with Series Inductance


4.0-20

Fig. 14 Converter Characteristics with Series Inductance (see Fig. 13) a) & b) Electric Impedance-Phase Curves c) Electrical resonances d) Mechanical resonance e) λ/2 – axial-mechanical resonance


4.1-1

4.1 Inductive Compensation and Sensitivity of Piezoelectric Converters

On the Fig. 4.1-1 are presented series and parallel inductive compensations of a piezoelectric converter (Ls = series inductance, Lp = parallel inductance). All circuits on the left side of the Fig. 4.1-1 are mutually equivalent, and the same is valid for the circuits on the right side (but there is no equivalence between left and right side circuits). We can also notice on the Fig. 4.1-1 that most significant elements responsible for creating different resonant zones are encircled in order to visualize the appearance of three different resonant frequencies in each case.

⇓

⇓

⇓⇑

⇓⇑

Z Ls

CopfSH

Ls

Cos

Z

p

fPH

f2Cos

fSL fPL

a)

Ls

Fig. 4.1-1 Series, a), and Parallel, b), Inductive Com

Series inductive compensation is naturally enabling converter topower in its parallel mechanical-resonance fp ≈ f2, ancompensation is naturally enabling converter to produce reamechanical-resonance fs ≈ f1. One of the problems for designeboth, series and parallel inductive compensations are creatingresonant zones. For instance, when we use series inductstimulate parallel mechanical-resonance f2, we also create, leftlow-impedance resonant zones with frequencies fSL and fSH (re3.1. Also, when we use parallel inductive compensationmechanical-resonance f1, we also create, left and right from f1resonant zones with frequencies fPL and fPH (respectively), Fig.that piezoelectric converter can also operate in all resonant zon

Converters modeling Miodrag Prokic Le Lo

⇓⇑

L

Cop

Lp

f1

⇓⇑

Series Inductive Compensation
Parallel Inductive Compensation
Lpb)

pensation

produce real (active) d parallel inductive l power in its series rs in this field is that two more side-band ive compensation to and right from f2, the spectively), Fig. 4.1-

to stimulate series , the high-impedance 4.1-3.2. It is the fact es created by series

cle - Switzerland

4.1-2

and/or parallel inductive compensation (in fPL, f1, fPH, fSL, f2, fSH), but not all of them are equally efficient (regarding power conversion). The objective in optimal inductive compensation (in most of cases) is to make f1 to be in the middle point between fPL, and fPH, and f2 to be in the middle point between fSL, and fSH. Another objective (in most of cases) is to create maximal possible difference between fPL, and fPH, and between fSL, and fSH, in order to make a safe operating frequency band larger (placed symmetrically around chosen resonant frequency: see Fig. 4.1-3.1 and Fig. 4.1-3.2). It is also easy, by changing the value of series or parallel inductance, to create unsymmetrical positioning of fxL, and fxH around series or parallel resonance point/s. This option can be useful in cases when we intend to apply series inductive compensation in order to drive converter in its series mechanical resonant frequency (by making fSL ≈ f1), or when we intend to apply parallel inductive compensation and drive converter in its parallel mechanical resonant frequency (by making fPH ≈ f2). In reality, by adding external inductances (in series or in parallel to a converter), we are modifying the effective sound speed inside the structure of certain converter, effectively modifying its piezo-properties. In most of literature explaining inductive compensation of piezoelectric converters and sensors (operating in resonance), above described situation is oversimplified suggesting that for operating in parallel mechanical resonance we should apply

series inductance that satisfies the expression: fL Cs os

21

2=

π, and in cases when

operating in series mechanical resonant frequency we should apply parallel

inductance that satisfies the expression: fL Cp op

11

2=

π. In reality, calculating the

necessary inductance values that way, we are making certain errors (for instance, actual and the best experimentally found inductance, for single converter compensation, could be approximately +10% higher than calculated values, but this statement cannot be generalized for all piezoelectric converters). Since the differences between calculated and experimentally found inductance values are not too high, for the sake of simplicity, we are using above mentioned expressions for calculating inductive compensation, and later on, experimentally we should make necessary corrections. Let us give a couple of practical examples regarding series inductive compensation,

A) When transducer will operate in its parallel mechanical resonance fm ≅ f2, and B) When transducer will operate in equivalent electrical series resonance fm = f’SL ≤ f1

Without entering into any formula-development, we can present the table of best empirically-found and approximated results regarding how to calculate series inductance (see the following picture and table).


4.1-3

Resonant Regime Values of series inductive compensation Case A)

Series inductive compensation ( ) HighLL sAs == ⇒Transducer is operating in its parallel mechanical, or voltage resonance: . const.ff 2m ≅=

∆f 6∆f∆f ∆f 3∆f∆f

HL

HL

≈+≈≅

02

12

0212

20

22

2s

022

2 Cf4π 1

Cff4π 1

)2f∆f

(1Cf4π

1L

Cf4π 1

<

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

≈

≈−

≈

<

Case B) Series inductive compensation

( LowLLL sAsBs =≈=21

) ⇒

Transducer is operating in equivalent series or current resonance:

dependentLoadff(f 1'sLm −=≤= )

∆f 6∆f∆f

∆f∆f)f (∆'H

'L

H'L

≈+

<≈ '

02

12

0212

20

22

2s

022

2 Cf8π 1

Cff8π 1

)2f∆f

(1Cf8π

1L

Cf8π 1

<

⎪⎪

⎭

⎪⎪

⎬

⎫

⎪⎪

⎩

⎪⎪

⎨

⎧

≈

≈−

≈

<


4.1-4

We can also give a couple of practical examples regarding parallel inductive compensation,

C) When transducer will operate in its series mechanical resonance fm f≅ 1, and D) When transducer will operate in equivalent electrical parallel resonance fm = f’PH ≥ f2

Without entering into any formula-development, we can present the table of best empirically-found and approximated results regarding how to calculate parallel inductance (see the following picture and table).

Resonant Regime Value of parallel inductive compensation Case C)

Parallel inductive compensation ( ) LowLL pCp == ⇒Transducer is operating in its series mechanical, or current resonance: . const.ff 1m ≅=

∆f 6∆f∆f ∆f 3∆f∆f

HL

HL

≈+≈≅

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−≈<

)f

f 2∆(1Cf4π

1L

Cf4π 1

10

21

2p

02

12


4.1-5

Case D) Series inductive compensation ( ) ⇒ High2LLL pCpDp =≈=Transducer is operating in equivalent parallel or voltage resonance:

dependentLoadff(f 2'pLm −=≥= )

∆f 6∆f∆f

∆f∆f)f (∆'H

'L

L'H

≈+

<≈ '

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−≈<

)f

f 2∆(1Cf2π

1L

Cf2π 1

10

21

2p

02

12

Only resonant regimes in mechanical parallel and series resonances (case A) and case C)) are able to keep stable and almost-constant operating frequency, f2 or f1). Most of ultrasonic plastic welding applications are covered by cases A) and C), combined with a convenient PLL resonance-control circuit (but there are some other regimes where combined series + parallel inductive compensation is very successfully used, as kind of high power filtering). We can notice that changing series or parallel (compensating) inductances in very large intervals is not significantly changing f2 or f1, but it is changing very much all values for fL and fH. When operating in electrically-created, equivalent series and/or parallel resonance/s (case B) and case D)), mechanical resonant frequency is very much load-dependent (meaning moderately-variable), since all electrical and mechanical parameters are slightly changing in the function of the acoustic load (load current and voltage are changing, equivalent impedance is changing, ferrite core properties are current dependent…). Cases B) and D) are very good for ultrasonic cleaning, Sonochemistry and other liquid processing applications, as well as for wide band and multifrequency, large sweeping operating regimes. Transducers operating in regimes described by B) and D) are often producing extraordinary effects, which can not be produced by regimes A) and C). Because of security and circuit-simplicity reasons (such as short circuit protection, load current limiting etc.), the most-widely used is series inductive compensation described by cases A) and B). If the objective is to make an ideal series and/or parallel inductive compensation applied on the same piezoelectric converter (of course, not in the same time), in order to get the widest possible bandwidth and symmetrical distribution of Low and High electrical resonant points (cases A and C), we will notice that (experimentally found) series and parallel inductances are mutually different, and that always a parallel inductance will be higher than series inductance, while relevant and mutually corresponding frequency intervals (between characteristic resonant points) would stay approximately equal. Just to get a rough idea about what means such Ideal and Symmetrical piezoelectric impedance matching in both resonances see the picture below.


4.1-6

The purpose of making inductive compensation of piezoelectric converters is not only to enable them operating efficiently on a stable (constant) resonant frequency. Applying proper inductive compensation (or combined parallel and series inductive compensation) we can also make converter operating efficiently in a relatively large frequency interval (for instance in the interval between series and parallel resonant frequency), maximize active (or real) output converter-power, and control particular mechanical output/s such as: output oscillatory speed, acceleration and amplitude. In cases when making acoustically well-matched emitter-receiver transducers (or coupled sensors), we will try to realize that pHsHpLsL ff,ff ≅≅ (see later Fig. 4.1-3.3). Let us take as an example a single (non-loaded) piezoceramic, PZT8 ring. All equivalent-models data (regarding its first, radial resonant mode) are given in T. 4.1-1 (measured using Network Impedance Analyzer HP4194A). For such (and any other) piezoelectric converter we can find many different equivalent-models, depending how large frequency interval we consider as relevant for our modeling.


4.1-7

T. 4.1-1 Piezoelectric Converter Models (Ultrasonic Welding Piezoceramics) Piezoceramic Ring:Vernitron, PZT8, 45 / 89, First Radial ResonanceParameters

∅ ∅2 0 8 0 2" . " .x x "

f Hzf HzZ

Z kQQ

m

m

1

2

10

20

3288034880

12 8724

256 86111291385

==

=

=

==

min. .

max. .

Ω

Ω

A

ZA

C pFHL m

R

1

1

1

336 81269 556512 6210

===

... Ω

A

ZA

C pL mR M

p

p

1

1

1

336 81269 556516 8762

=

==

... Ω

FH

B

⇔

ZB

C nL HR k

2

2

2

24 2667857 904260 045

===

...

µΩ

F

B

C nL HR m

s

s

2

2

2

24 2667857 903142 611

===

...

µΩ

F

C

⇔ ZB

C nFC pFL mR

op =

===

2 68554336 55269 616412 741

1

1

1

..

.

. ΩH

C

C C CnF

C nL HR k

os op= +

F

=

====

1

2

2

2

3 022124 26857 9260 045

....µ

Ω

⇔Z Z


4.1-8

We can now, experimentally, make the perfect inductive matching (inductive compensation) using the same piezoceramic ring (or piezoelectric converter) presented in T. 4.1-1, and find the best values for series and parallel compensating inductances. Let us create the equivalent circuit, Fig. 4.1-2, a) that corresponds to series inductive compensation (see Fig. 4.1-1, a) and Fig. 4.1-3.1), and equivalent circuit, Fig. 4.1-2, b) that corresponds to parallel inductive compensation (see Fig. 4.1-1, b), and Fig. 4.1-3.2). The series and parallel compensating inductances are experimentally found by satisfying the criteria of optimal inductive matching, creating large and symmetrically shaped impedance curve around parallel and/or series resonant frequency (or making that: L f f f f L f f f fs SH SL p PH PL⇒ − ≅ − ⇒ − ≅ −( ) , (2 2 1 1 ) , see Fig. 4.1-3.1 and Fig. 4.1-3.2). All measurements and model data are also presented on the Fig. 4.1-2.

Ls CosC nos = 3 0221. F

i z = iinp.iinp.

a) b)

uinp. u z um

C2 L2 R2

iinp. i z

i m iC iL

uinp.= u z

L1 C1 R1

Lp Cop

Fig. 4.1-2 Series, a), and Parallel, b), Inductive Comp

Equivalent circuits for the converter from T. 4. We shall, first, analyze interesting aspects of series inductive cconverter (from T. 4.1-1) operates in its parallel mechanical resona), Fig. 4.1-2, a) and Fig. 4.1-3.1). The input driving-signal involtage uinp. and mechanical output is represented by motional vo

Converters modeling Miodrag Prokic Le Loc

C nL HR kL mHf Hf Hf kf k

s

SL

SH

========

24 26857 9260 0457 31

328803488028 6540 975

2

2

2

1

2

...

.

.

.

µΩ

F

zz

HzHz

C nop = 2 68554. F
C pL mRL mHf Hf Hf kf kH
p

PL

PH

====

====

336 55269 616412 7418 7

328803488026 62539 3

1

1

1

1

2

...

.

.

.

Ω

FH

zzHzz

ensation 1-1.

ompensation, when ance f2 (Fig. 4.1-1, this case is input ltage um, effectively

le - Switzerland

4.1-9

representing converter’s output velocity (Fig. 4.1-2, a)). The objective of optimal power and frequency control (and best inductive matching) in this case is to realize linear dependence between uinp. and um, or to make them mutually equal and in phase (see Fig. 4.1-3.1, e) and f)), in the largest possible frequency interval (∆f f fSH= − ≅ ( ) .f f kHzSL2 6 15( )2 − ≈ ,∆f f f0 2 1= −( ) = 2kHz , f fSH SL− = 2∆f ≈ 6 0∆f≅ 12 3. kHz ), while having parallel mechanical resonance f2 in the middle point of that interval ( f f fSH SL2 0 5≈ +. ( ) ). This way, by controlling the input voltage we are effectively controlling the output velocity. Of course, in order to control the input voltage uinp. we need to detect converter motional voltage um, and to use it as the feedback information for automatic, closed-loop regulation. In high power ultrasonics world is more commonly found the explanation that when keeping um constant we are keeping the converter output (displacement) amplitude constant. In reality, since (usually) high power converters operate on a constant resonant frequency (f2 = const., PLL controlled), this is the same because if peak velocity-amplitude is vp = const., then the peak converter displacement will be ap = vp / 2πf2 = Const. Now, we can analyze interesting aspects of parallel inductive compensation, when converter (from T. 4.1-1) operates in its series mechanical resonance f1 (Fig. 4.1-1, a), Fig. 4.1-2, b) and Fig. 4.1-3.2). The input driving-signal in this case is input current iinp. and mechanical output is represented by motional current im, effectively representing converter’s output force (Fig. 4.1-2, b)). The objective of optimal power and frequency control (and best inductive matching) in this case is to realize linear dependence between iinp. and im, or to make them mutually equal and in phase (see Fig. 4.1-3.2, e) and f)), in the largest possible frequency interval (∆f f fPH= − ≈( )1

, ( ) .f f kHzPL1 6 3− ≈ ∆f f f0 2 1= −( ) = 2kHz , f f f f kHPH PL z− = ≈ ≅2 6 12 60∆ ∆ . ), while having series mechanical resonance f1 in the middle point of that interval ( ). This way, by controlling the input current we are effectively controlling the output force. Of course, in order to control the input current if f fPH PL1 0 5≈ +. ( )

inp. we need to detect converter motional current im, and to use it as the feedback information for automatic, closed-loop regulation. In high power ultrasonics world is more commonly found the explanation that when keeping im constant we are keeping the converter output (displacement) amplitude constant. In reality, since (usually) high power converters operate on a constant resonant frequency (f1 = const., PLL controlled), this is the same because if peak force-amplitude is Fp = const., then the peak converter displacement will be ap = Fp / (2πf1)2 = Const.. All Impedance-Phase and Transfer function plots on Fig. 4.1-3.1 and Fig. 4.1-3.2 are realized exactly for the same converter from T. 4.1-1, using models and data given on Fig. 4.1-2, a) and b). Measurements (and inductive tuning) are originally made using Network Impedance Analyzer HP 4194A, and in order to present them in a very systematic and summary form, all model data and component values are introduced in the corresponding P-Spice (and Orcad) models, resulting with Fig. 4.1-3.1 and Fig. 4.1-3.2. Doing this way (using P-Spice models) we are able to reconstruct electromechanical transfer functions (Fig. 4.1-3.1, e), f) and Fig. 4.1-3.2, e), f)), because otherwise we can not directly measure such transfer functions. We also see that in the area of series and/or parallel resonant frequency, transfer functions are almost flat (Atr. (=) Gain (=) 1 (=) 0 dB), and phase of the transfer


4.1-10

function/s is equal zero, in a large frequency interval (almost between fL and fH), what makes motional voltage and motional current to be ideal parameters for mechanical output control (or for converter output power control).

[dB]

acFM

Z

ui

z

z

=

a)

f1 f2

[ ]°

θ z b)
[dB] Z
ui

eq.

inp.

inp.

=

c)

fSL f1 f2 fSH

[ ° ] θ eq.

d)

[dB] A

uu

tr.

m

inp.

e)

[ ° ] θ tr.

f)

Fig. 4.1-3.1 Series Inducti(corresponds to conve

) Converter Impedance vs. F) Equivalent Input Impedancerequency, e) Motional Transotional Transfer Function vs.

Converters modeling

fSL f1 f2 fSH

ve Compensation Impedance-Phase Curves rter data from T. 4.1-1 and Fig. 4.1-2, a)) requency, b) Converter Phase vs. Frequency, vs. Frequency, d) Equivalent Input Phase vs. fer Function vs. Frequency, f) Phase of the Frequency.

Miodrag Prokic Le Locle - Switzerland

4.1-11

[dB]

acFM

Z

ui

z

z

=

a)

[ ]°

θ z

b)

[dB]

Z

ui

eq.

inp.

inp.

=

c)

fPL f1 f2 fPH

[ ]°

θ eq.

d)

[dB]
A
ii

tr.

m

inp.

= e)

fPL

[ ]°

θ tr.

f)

Fig. 4.1-3.2 Parallel Inductive Compen

(corresponds to converter data fro) Converter Impedance vs. Frequency, b) Equivalent Input Impedance vs. Frequerequency, e) Motional Transfer Functiootional Transfer Function vs. Frequency.

Converters modeling Miodrag Prok

f1 f2

f1 f2 fPH

sation Impedance-Phase Curves m T. 4.1-1and Fig. 4.1-2, b)) ) Converter Phase vs. Frequency, ncy, d) Equivalent Input Phase vs. n vs. Frequency, f) Phase of the

ic Le Locle - Switzerland

4.1-12

Piezoelectric Converters Operating as Sensors From flat and zero-phase transfer-functions (Fig. 4.1-3.1, e), f) and Fig. 4.1-3.2, e), f)), we also see that efficient and large frequency sweeping (between f a ) can be applied when converter operates in any of its resonance area. This is particularly interesting situation for acoustic or ultrasonic sensors design, since we can extend operating sensor/s bandwidth, booth in the transmitting and receiving mode. When we design well-matched sensors (transmitter-receiver couple), we usually search to find series and parallel inductive compensation that makes:

nd fL H

( ) ( ) ,f f f fSH SL PH PL− ≅ − ( )f f fSL PL L≅ ,= ( )f fSH PH Hf , ( ) ( )f f f fH L− ≅ −1 2≅ = . One of excellent examples regarding piezoelectric transmitter-receiver matching and tuning (in air), Fig. 4.1-3.3, can be found in the catalogue of Massa Products Corporation, company specialized in producing different piezoelectric sensors and transducers (Massa Products Corporation, 280 Lincoln Street, Hingham, MA 02043, USA).

f1 f2

a) fL fH

b) fL fH

f1, f2

d) c) fL fH

Fig. 4.1-3.3 Series and Parallel Inductive Compensation Applied for Tuning Transmitter-Receiver Sensor-Couple TR-89/B, Produced by Massa

a) Transmitting Sensor Response with Series Tuning b) Receiving Response with Parallel Tuning c) Equivalent Impedance Curves (untuned sensor, series-tuned sensor, and parallel or shunt-tuned sensor) d) Sensor’s Directional Response


4.1-13

Series Compensation

Z

dB

a)

Z

ui

dB

b)

PdB

B

A

c)

Arrow direction → (=) Series resistance increase

Fig. 4.1-3.4 Transmitter Tuning with Series Compensation a) Evolution of Equivalent Converter Impedance in Series with a Resistor. b) Evolution of Equivalent Converter Impedance in Series with a Resistor and

Inductance. c) Evolutionary Transmitting Sensors Response (or pressure generated in front of

a sensor): -under a), only sensor and series resistance (=) case A; -and under b), sensor, series resistance and series inductance (=) case B.


4.1-14

Usually, when we use an acoustic (piezoelectric) sensor as a transmitter, we apply series inductive compensation, Fig. 4.1-3.4, since in that case the sound field pressure in front of such sensor is maximized (why this is the case, it will be explained). We can notice that in case when we have only a sensor in series connection with a variable resistance (Fig. 4.1-3.4, a)), by increasing resistance we are practically eliminating or attenuating sensor series resonance, and only parallel resonance area remains active (see Fig. 4.1-3.4, c), case A). In case if we have series resistance and series inductance connected to a sensor (Fig. 4.1-3.4, b)), we shall get similar equivalent impedance as presented on Fig. 4.1-3.1, c), and with resistance increase, again, only sensor parallel resonance area will remain active. Now, if we connect such sensor-transmitter (first with series resistance = Case A = untuned sensor, and then with series resistance and series inductance = Case B = tuned sensor) to a constant rms. voltage-source with variable frequency sinusoidal signal, (Fig. 4.1-3.4, c)), and if we measure the sound pressure in front of the sensor (in function of frequency), we shall see that pressure peaks are always produced in zones where equivalent impedance (sensor impedance + resistor, or sensor + resistor + inductance) is minimal, or in all impedance zones where a kind of series resonance is present. All series resonance zones (or zones of minimal impedance: in , when sensor is non-tuned, and in f f when sensor is series inductance tuned) are high current consuming, because sensor current increases with equivalent-impedance decrease, and by analogy CURRENT-FORCE, we can conclude that sensor in series resonance will be a high (or dominant) force-source (or high pressure source, since pressure is equal to force per surface area).

f1 SL SH,

Obviously, applying VOLTAGE-VELOCITY analogy, we can also conclude that piezoelectric sensors or converters driven in zones of high equivalent impedances, or in zones of parallel resonances ( f f ), are presenting dominant velocity-sources.

fPL PH, ,2

On the Fig. 4.1-3.4, c) we can see that series-resistance increase will produce in case A), when we have only a sensor with series resistance, progressive diminishing of output pressure and frequency shift (of maximal pressure) toward sensor parallel resonant-frequency f2, and in case B), when we have sensor, series resistance and series inductance, certain diminishing of output pressure, and pressure-curve flattening will be experienced (applicable in cases when we need to create a more-linear transmitter response-function: see also Fig. 4.1-3.3, a)).


4.1-15

Parallel Compensation

a)

b)

B

A

c) Arrow direction → (=) Parallel resistance Rp decrease

Fig. 4.1-3.5 Receiver Tuning with Parallel Compensation a) Evolution of Equivalent Converter Impedance in Parallel with a Resistor. b) Evolution of Equivalent Converter Impedance in Parallel with a Resistor and

Inductance. c) Evolutionary Receiving Sensors Response (or voltage output generated by

sensor): -under a), only sensor and parallel resistance (=) case A; -and under b), sensor, parallel resistance and parallel inductance (=) case B.


4.1-16

When we use an acoustic (piezoelectric) sensor as a receiver, we apply parallel inductive compensation, Fig. 4.1-3.5, since in that case the voltage output and receiving sensitivity of such sensor is maximized (why this is the case, it will be explained). We can notice that in case when we have only a sensor in parallel connection with a variable resistance (Fig. 4.1-3.5, a)), by decreasing resistance we are practically eliminating or attenuating sensor parallel resonance, and only series resonance area remains active (see Fig. 4.1-3.5, c), case A). In case if we have parallel resistance and parallel inductance connected to a sensor (Fig. 4.1-3.5, b)), we shall get similar equivalent impedance as presented on Fig. 4.1-3.2, c and Fig. 4.1-3.3 c), and with resistance decrease, again, only sensor series resonance area will remain active. Now, if we test such sensor (first with parallel resistance = Case A = non-tuned sensor, and then with parallel resistance and parallel inductance = Case B = tuned sensor) as a receiver of variable-frequency, sinusoidal pressure-signal, (Fig. 4.1-3.5, c)), and if we measure the voltage output produced by sensor-receiver (in function of frequency), we shall see that output voltage peaks are always produced in zones where equivalent impedance (sensor impedance + parallel resistor, or sensor + parallel resistor + parallel inductance) is maximal, or in all impedance zones where a kind of parallel resonance is present. All parallel resonance zones (or zones of maximal impedance: in , when sensor is non-tuned, and in f , when sensor is parallel inductance tuned) are low current consuming, because sensor current decreases with equivalent-impedance increase (and voltage on the sensor input terminals will increase), and by analogy VELOCITY-VOLTAGE, we can conclude that sensor in parallel resonance will be a dominant velocity-sensitive sensor (or receiver).

f 2 fPL PH,

Obviously, applying FORCE-CURRENT analogy, we can also conclude that piezoelectric sensors or converters (used as receivers), in zones of low equivalent impedances, or in zones of series resonances ( f f ), are presenting dominant (oscillatory) force-receivers.

fSL SH, ,1

On the Fig. 4.1-3.5, c) we can see that parallel-resistance decrease will produce in case A), when we have only a sensor with parallel resistance, progressive diminishing of output sensor-voltage and frequency shift (of maximal, detected voltage) toward sensor series resonant-frequency f1, and in case B), when we have sensor, parallel resistance and parallel inductance, certain diminishing of output voltage, and sensitivity-curve flattening will be experienced (applicable in cases when we need to create a more-linear receiver response-function: see also Fig. 4.1-3.3, b)). For sensors and converters coupling and inductive matching purposes (presented on Fig. 4.1-3.1 until Fig. 4.1-3.5), we often need to shift characteristic Low and High-frequency resonant frequencies, . If we take as referent points converter’s natural series and parallel resonance/s ( f ), it can be found (experimentally and by software simulation, or calculating using Electric Circuits rules) that, when series

f fL H,f1 , 2


4.1-17

and/or parallel inductance/s ( L ) is/are increasing, then f are coincidently decreasing ( f would asymptotically decrease from certain high frequency point toward , and would decrease toward very low frequencies, below ), see Fig. 4.1-3.6. In the same time, the interval

Ls , p

L

fL H,

H

f 2 f L f1

f fH − would stay almost constant, regardless from L . Ls p, In other words, when making series inductive compensation, regardless of the value of Ls, parallel resonant frequency will always stay the same (constant, or unchanged by L

f 2

s), and only f will be influenced (shifted) by variable LfSH SL& s, but their difference will stay almost constant, f f ConsSH SL t− ≈ . . Also, when making parallel inductive compensation, regardless of the value of Lp, series resonant frequency will always stay the same (constant, or unchanged by L

f1

p), and only f will be influenced (shifted) by variable LfPH PL& p, but their difference will stay almost constant, f f ConstPH PL− ≈ ..

f [Hz]

fSH , fPH , fH

fSL , fPL , fL

Series and/or Parallel Inductance/s Ls , Lp [mH]

Fig. 4.1-3.6 Additional Resonances Generated by Inductive Compensation In Function of Ls and Lp (only qualitatively presented, and applicable to all

situations from: Figs. 4.1-3.1 until 4.1-3.5) Even experienced people working in Ultrasonic engineering are sometimes considering that they are changing series or parallel resonant frequency, , by adjusting the values of parallel or series inductances, , but in realty they are only changing (or shifting) , and often operating converters in some of the side-band frequencies f o , being convinced that converter is operating on its series or parallel resonant frequency (because, depending on the selected value/s of

, it can happen that either f

f f1 2,L Lp S,

f fL & H

H

S

r fL

L Lp , fL ≈ 1 or f fH ≈ 2 ). This situation is very much typical for ultrasonic cleaning converters, when many (of mutually similar) converters


4.1-18

are connected in parallel, and when relatively low value of series inductive compensation is applied, effectively realizing that the best operating frequency would be f f foper SL. = ≈ 1 (and this looks like series resonance operating regime). Also in cases of heavily-attenuated transducers and sensors (like sonar transducers, or NDT transducers…), using Impedance-Phase measurements, we could realize that such structures are dominantly reacting like being capacitive components (with internal losses), and we often find in literature and ultrasonic engineering publications an oversimplified transducers’ modeling, using only two components: a series or parallel connection between a capacitance and resistance (treated as dual, mutually equivalent models). Accepting this non-selective approach, inductive compensation becomes just a matter of neutralizing series or parallel capacitance for certain operating frequency, but comparing such simplified methodology with the methodology described in this paper, we can realize how much of essential matter regarding optimal transducer matching and maximal transducers’ efficiency is simply lost.


4.2-1

4.2 Piezoelectric Converters Operating High Power All models and theoretical conclusions regarding different inductive matching situations, presented on Fig. 4.1-1, until Fig. 4.1-3.6, are equally valid for all piezoelectric converters, sensors, transmitters and receivers. The important difference between sensor operating regimes and high power regimes is mostly in belonging power-supply electronics. In case of sensors, we usually have very low electric-power consumption (in the range of mW or µW), and sensor would never make problems to driving electronics regarding optimal impedance and power matching, or regarding questions like: how much of active power we are delivering to the sensor-transmitter, and how much of reactive power is reflected back to driving electronics (since sensor voltages and currents would be too low, or too far from the safe operating limits of involved electronics). The situation becomes quite different when we should apply hundreds or thousands of watts of output electric power to a single (high mechanical quality factor) converter. In such situations we prefer to deliver almost all electric-power as the Real or Active Power, and to have minimal Reactive Power reflected back to driving electronics (in order to avoid converter overheating and damaging high power electronic components). Practically, we should find a way to compensate a converter (loaded or unloaded) to present purely resistive load (or purely resistive impedance) to its driving electronics (in the operating frequency zone), of course while driving electronic output impedance can also be considered as resistive. There are different impedances and power-matching techniques to realize optimal and maximal power transfer conditions. The most widely known converter resonant operating regimes are when converter operates exactly in its series, or parallel mechanical resonance, which will be analyzed in details. Also, converter can operate high power in certain (relatively large) frequency interval/s if we find a way to make resistive (or zero phase) impedance and power matching between the generator and converter (in such frequency interval/s).


4.2-2

Converter Operating High Power in Series Resonance Applying parallel inductive compensation Lp, we shall enable converter to operate in its series mechanical resonance f1 (or in impedance minimum zone). From Fig. 4.1-3.2, c) and d), we can see that equivalent impedance of Converter & Parallel Inductance has resistive character only when phase function, Fig. 4.1-3.2, d), is crossing zero line in f1. Consequently, applying PLL (for resonant frequency control), in order to keep the input current and voltage in phase, at f1 (measured on the input of the equivalent impedance, Fig. 4.1-3.2, c)), we shall make such (equivalent) load being low resistive impedance. The next important design task, regarding optimal power transfer, is that equivalent load impedance should be matched (or should be equal) to the internal impedance of driving electronics (of ultrasonic power supply, or ultrasonic generator, or power oscillator). This objective we usually realize designing proper primary-to-secondary-ratio of a ferrite-transformer, and adjusting low load impedance to the level of the internal impedance of driving electronics. In order to control output converter power (or output force) under loading, we should also find the best way to extract the motional current im from the output power circuit, and bring this voltage as the feedback to the main supply, or DC supply voltage control unit, effectively keeping im constant (most often by means of PWM regulation), regardless of load variations. Since converter impedance in no-load conditions, and in series resonance (in air) is usually very low, converter can dissipate relatively high energy, and it cannot continuously oscillate long time without applying forced cooling, or without electronic protection circuit that will reduce converter’s current (usually by means of PWM). When converter is mechanically loaded (operating in series resonance), its equivalent impedance is increasing, proportionally with the increase of applied load. Consequently, when converter is operating in series resonance, we should first apply the load, and then start oscillations (or at least apply the load and start oscillating converter in the same time), and we should know that this operating regime is convenient for moderate and heavy-duty, relatively stable acoustical loads. We could also operate converter (with parallel inductive compensation) in two of its side-band parallel-resonance frequencies , but only operating in f can give relatively acceptable results (efficient and stable, high power output) in cases when we adjust parallel inductance to produce

f or fPL PH PH

f foper PH. f= ≈ 2 (see Fig. 4.1-3.6). Also the philosophy of output ultrasonic power regulation, when operating a converter in , would not be the same as operating it in . f or fPL PH f 2

Let us now analyze a parallel-inductance compensated converter, operating on its series resonant frequency. The equivalent circuit of such operating regime, valid only for very close vicinity of converter’s series resonance is presented on Fig. 4.2-1. The meaning of parallel inductive compensation in this case is to eliminate all reactive circuit elements such as capacitances and inductances (of course only for a constant frequency f1, when circuit is operating in resonant regime). This can be realized on the way that Lp should be chosen to create parallel resonance with C*op on f1 (when Lp and C*op would present very high equivalent impedance), and


4.2-3

naturally (in the same time) L1 and C1 would create series resonance, on f1, that has very low (or zero) equivalent impedance.

Generator Compensation Converter Load

Fig. 4.2-1 Converter inductively compensated to operate in series resonant

frequency ( ) /L C L C fp op ≅ ≅1 12

121 4π

This way, we can practically eliminate all reactive elements Lp, C*op, L1 and C1 from the circuit given on Fig. 4.2-1 (by satisfying ), and create a more simplified equivalent circuit (which has only resistive elements), given on Fig. 4.2-2, a). Since R

( ) /L C L C fp op ≅ ≅1 12

121 4π

op usually presents very high resistance (in the range of several MΩ, or higher), we can additionally approximate the same circuit to its final version given on the Fig. 4.2-2, b).


4.2-4

uinp.

uinp.

iLig ig ≈ iL

usip us

uL

uL

⇒ ≈

a) R R R Rop g L>>> ( , ,1 ) b) R R S1 1≅

Fig. 4.2-2 Compensated converter operating on series resonant frequency

Now we can analyze circuits presented on Fig. 4.2-1 and Fig. 4.2-2, and find all characteristic currents and voltages,

gLinp.L1

LLg1inp.

L1

1s

1gigLi

L1g

L1g

L1g

L1opg

L1op

LL1popggginp.

1LLLL1

inp

L1op

inp.Lpg

iRuRR

Ru,iRu

RRR

u

)RR(R,URRRR

URRR

RRU

)R(RRR

)R(RR

)iR(RiRiRUu

)R(R,iRR

u

)R(RR

uiii

≅+

=≅+

=

+=++

=++

+≅

++

+=

=+==−=

=≅+

≅+

=+= −

(4.2-1)

Electrical power losses dissipated on the parallel resistance Rop can be found as,

P R i R RR R R

iuR

RR

R RR R

R

U R RR R R R

U

R RR R R

U R RR R R

U

eo op pL

L opg

inp

op

opg

L

L

op

gL

op g Lg

L

op i Lg

i L

op i Lg

= =+

+ +

LNMM

OQPP

= =

=

++

++L

NMMOQPP

≅++ +

=

=++

≈++

2 1

1

2

22

1

1

2 12

12

2

12

22

2

22

1

1 1

0 5

.

( )

( )( )

( )( )

( . )( )

(4.2-2)

Mechanical, internal converter power-losses, are presented by power dissipated on the resistance R1,


4.2-5

P R i R R RR

i uR

RR R

u

R RR

RR

R RR

U

RR R R

U RR R

U RR R

U

m LL

op

gs

Linp

L g L

op

g

g Lg

i Lg

i

i Lg

1 12

11

2

22

1

1

12

2

11 1

1

22

1

12

2 12

22

2

1

1

1

1 1

0 5

= =+

+

L

N

MMMM

O

Q

PPPP= =

+=

=

+ + ++L

NMMOQPP

≅

≅+ +

=+

≈+

( )

( )

( ) ( ).

( )

.

(4.2-3)

Total power dissipation in a converter is equal to the sum of (electrical and mechanical) loses presented by (4.2-2) and (4.2-3),

P P P R iR

RR R

u

R RR

RU

R R R

RR R R

U RR R

U P RR R

U

d eo m Lop L

inp

L

op

g

g L

g Lg

i Lg m

i

i Lg

= + = = ++

LNMM

OQPP

=

=+

+LNMM

OQPP + +

≅

≅+ +

=+

= ≈+

1 12 1

12

2

12

1

2

12

1

12

2 12

21 2

2

1

0 5

( )

( )( )

( ) ( ).

( )

.

(4.2-4)

Useful acoustical (active) output power is only the power given to a load resistance RL1 = RL ,

P R i uR

RR R

u

R RR

RR

R RR

U

RR R R

U RR R

U

L L LL

L

L

Linp

LL

g

L

L

op

g

L

g Lg

L

i Lg

= = =+

=

=

+ + ++L

NMMOQPP

≅

≅+ +

=+

22

12

2

1 1

2

12

22

2

1

1 1

( )

( )

( ) (

.

)

(4.2-5)

Now we can calculate the internal electromechanical conversion-efficiency of loaded, η emL , and unloaded, η emo , converter as,


4.2-6

η η

η η

emL emm

m eo L

op

emo emL Rm

m eoR

L

op

R

op

PP P R R

R

Lim Lim PP P

Lim R RR

RR

L

L

= =+

=+

+

= =+

=

=+

+=

+≅

→

→

1

1 1

01

10

10

1

1

1

1

1

1

11

,

( ) ( )

( )

L→ (4.2-6)

The internal mechanical-to-acoustical conversion-efficiency of loaded converter, ηma is,

ηmaL

m L

L

L

L

PP P

RR R R

R

=+

=+

=+1 1 1

1

1. (4.2-7)

The total electro-acoustical conversion-efficiency of loaded converter, η ea , is,

η η η

η

ea em mam

m eo

L

m L

L

LL

op

L

L

L

maL

d L

L

eo m L

PP P

PP P

R

R R R RR

RR R

RR

PP P

PP P P

= =+ +

=

=+ +

+≅

+=

=+

= ≅+

=+ +

1

1 1

11 1

1 1

1

1

1

( )( )

.

(4.2-8)

A converter itself, even if it has very high electro-acoustic efficiency, is not sufficient to realize high power output (of acoustic or ultrasonic energy). It is very important to realize proper impedance matching between the generator (as a source of electrical excitation) and the converter, in order to produce high acoustical power output. Let us first find thermal power losses on the internal generator resistance Rg,

P R iR

R R RR R

R

U

R

R R Ru

RR R

u

g g gg

gL

L

op

g

g

op L

inpg

Linp

= =

++

++

L

N

MMMM

O

Q

PPPP

≅

≅+

≅+

2

1

1

22

1

22

12

2

1

( ) ( ). .

(4.2-9)

The total thermal power losses in generator and converter are,


4.2-7

P P P P P P

R

R R Ru

RR

R Ru

R RR R R

U RR R

U PR

R RU

R RR R

u RR R

u RR

P R R R

d tot g d g eo m

g

op L

inpop L

inp

g

g Lg

i

i Lg d

g

i Lg

g

Linp

i

Linp

i

LL i g

− = + = + + =

=+

+ ++

LNMM

OQPP

=

=+

+ +=

+≅ +

+≅

≅+

+=

+≅ =

.

. .

. .

( ) ( )

( ) ( ) ( )

( ) ( ), ( )

1

1

22 1

12

2

1

12

22

22

2

1

12

2

12

21

1

+

(4.2-10)

The total electromechanical conversion-efficiency of loaded and unloaded converter, connected to generator, η em tot− , is,

η

η η

em totm

m g eo

m

m g g i

emo tot em tot Rm

m g eoR

gop

g g i

PP P P

PP P

RR R

RR

Lim Lim PP P P

R

R R RR

RR R R

R

RR

L

−

− − →

=+ +

≅+

=+

=

= =+ +

≅

≅+ +

≅+

=+

=

1

1

1

1

1

1

1

01

10

1

112

1

1

1

11

1

,

( ) ( )L→

(4.2-11)

The total mechanical-acoustical conversion-efficiency of loaded converter connected to generator, ηma tot− , is,

ηma totL

m L

L

Lma

PP P

RR R− =

+=

+=

1 1

,η (4.2-12)

The total electro-acoustical conversion-efficiency of loaded converter connected to a generator, η ea tot− , is,

η η η η

η

ea tot em tot ma tot tot

L

g eo m L

L

d L

L

LL

opg

L

op L

L

g L g

L

i

L

eaL

L

PP P P P

PP PR

R R R RR

R R RR R R

RR R R R R

RRR

RR R

− − −= ⋅ = =

=+ + +

=+

=

=

+ ++

++

+

LNMM

OQPP

≅

≅+ +

=+

+=

+< =

+

1

11 1

1

2

1 1 1

1

1

1

1

1

( )( )( )

( )

(4.2-13)


4.2-8

We also know that mechanical quality factor uniquely qualifies converter power-conversion efficiency and internal losses, and for the converter operating on series resonance, Fig. 4.2-1, can be found as,

Q XR

XR f C R

f LR R

LC

non loaded case

Q XR R

XR R f C R R

f LR R

RR R

QR R

LC

for loaded converter

m oC L

m LC

L

L

L L L

Lm o

L

11

1

1

1 1 1 1

1 1

1 1

1

1

11

1

1

1 1 1 1

1 1

1

1

11

1

1

1

12

2 1

12

2

1

= = = = = −

=+

=+

=+

=+

=

=+

=+

ππ

ππ

( )

( )

( )( ),

,

(4.2-14)

In order to present all (above-found) efficiency, energy and quality parameters of a piezoelectric converter on the very general unique and dimensionless way, in the function of converter’s mechanical load, let us make normalization of above found particular expressions, creating the table T. 4.2-1.

T. 4.2-1 Loaded Converter Quality and Efficiency Parameters

Basic expressions Normalized expressions Equation

P RR R

U RR

PLL

i Lg

L

id tot=

+≅ −( ) 2

2 RU

P R RR

RU

Pi

gL

L i

L

L

gd tot2 21

=+

≅ −

/( / )R i

2 (4.2-5)

P RR R

U

PR

R RU R

RP

d toti

i Lg

dg

i Lg

i

LL

− =+

≅

≅ ++

≅

. ( )

( )

22

22

RU

P RR R

R RRR

PU

i

gd tot

i

i L

L i

i

L

L

g

2

2

2

2

2

2

11

− =+

=

=+

≅

. ( )

( / )

(4.2-10)

η ea totL

L i

L i

L i

RR R

R RR R− ≅

+=

+/

/ 1η ea tot

L

L i

L i

L i

RR R

R RR R− ≅

+=

+/

/ 1 (4.2-13)

P RR

PLL

id tot≅ − .

PP

RR

L

d tot

L

i−

≅.

(4.2-10)

Q RR R

Qm LL

m11

110≅

+ Q

QR

R R R Rm L

m L L

1

10

1

1

11 2

≅+

≈+ / i

(4.2-14)

QR R

RQm L

Lm2

2

220≅ Q

QR R

Rm L

m

L2

20

2

2

≅ (4.2-28)

For purposes of mathematical conveniences, sum of internal generator resistance, Rg, and internal mechanical-circuit converter-resistance, R1, in most of above given expressions is presented as a new resistive (generator and converter specific) parameter R R Ri g= + 1 , since, later we will see that we can easily present mechanical load RL in the function of Ri, or as a number of Ri units, or in a normalized and dimensionless form as x = RL / Ri. This way, the resistance Ri would become the mechanical-load unit. In practice, resistances R1, and Rg are of the same order of magnitude, R Rg ≈ 1 , both in the range (for most of cases) between 5Ω and 50Ω (and when we do not have a better choice, for the purposes of making approximations, we shall consider R R R R Ri g g= + ≈ ≈1 12 2 ). For


4.2-9

instance, if we introduce as normalized mechanical load the ratio x RR

L

i

= , table T.

4.2-1 will become T. 4.2-2. T. 4.2-2 Loaded Converter Quality and Efficiency Parameters

x RR

xL

i

= ∈, ( ,0 ∞)

(=) Normalized Load

Normalized Power and Quality Parameters

Expressions F(x) Equation

RU

P xx

i

gL2 21=

+( ) (4.2-5)

RU

Px

i

gd tot2 2

11− =+. ( )

(4.2-10)

η ea totx

x− ≅+1

(4.2-13)

PP

xL

d tot−

≅.

(4.2-10)

Y = F(x)

QQ x

m L

m

1

10

11 2

≈+

(4.2-14)

Let as now calculate all normalized expressions from T. 4.2-1 and T. 4.2-2 in the function of mechanical load units, when mechanical load is taking values:

, and create new table T. 4.2-3. R RL ∈( , ...)0 10 i

T. 4.2-3 Numerical Values of Loaded Converter Quality and Efficiency Parameters

RL ⇒ 0 0 5. R i R i 1 5. R i 2R i 3R i 10R i ∞

x RR

L

i

= 0 0.5 1 1.5 2 3 10 ∞

RU

Pi

gL2 0 0.22 0.25 0.24 0.22 0.187 0.083 0

RU

Pi

gd tot2 − . 1 0.44 0.25 0.16 0.11 0.0625 0.083 0

η ea tot− 0 0.33 0.5 0.6 0.667 0.75 0.909 1 P

PL

d tot− .

0 0.5 1 1.5 2 3 10 ∞

QQ

m L

m

1

10

1 0.5 0.333 0.25 0.2 0.143 0.0476 0

Now, based on functions and results from T. 4.2-2 and T. 4.2-3, we can present all (normalized) power and quality parameters of a converter on the same multi-functional graph, Fig. 4.2-3 (where x axis is normalized mechanical load, and Y = F(x) presents normalized expressions from T. 4.2-2).


4.2-10

X/(1+X)^2 - Load Power

1/(1+X)^2 - DissipationX/(1+X) - EfficiencyX - Power ratio

1/(1+2*X) - Mechanical Quality

0 2 4 6 80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

10

PP

L

d tot− .

RU

Pi

gL2

η ea tot−QQ

m L

mea tot

2

20

< −η .

RU

Pi

gd tot2 − .

QQ

m L

m

1

10

F(X

)

R

X

Fig. 4.2-3 Normalized Converter Power an

Normalized Mechanic

Before we draw any conclusion based on resultrepeat the same process for a converter operashown that in case of parallel resonance we sfunctions as presented on Fig. 4.2-3, exceptopposite direction (compared to a converter ope In series resonance, using the equivalent conveon Fig. 4.2-1, equivalent mechanical load (hRL=RL1) is starting from zero, for no-load condwith mechanical load increase (theoretically unthe loading character has the opposite directiovery high value, or infinity, and decreases with louse converter model as presented on Fig. 4.2-4mechanical loads, because this is the preferaconversion and to reach high efficiency under mechanical load cannot be presented as a resis(or some other compensation), and well selectshould locally arrive to the conditions when eqwould be presentable as a resistive impedanceits compensation presents an equivalent resistivinterval).

Converters modeling Miodrag Proki

Normalized Mechanical Load: xR

L

i

=

d Quality Pal Load, x

s summarizting in parahall again g that mechrating in ser

rter circuit ere expresitions, and

til infinity), an (starts forad increase). We usuable case fohigh powertive load, bying operatinuivalent me, or to makee impedanc

c L

arameters, F(x), vs.

ed by Fig. 4.2-3, we shall llel resonance. It will be et the same normalized

anical loading will have ies resonance).

on the way as presented sed as load resistance increasing progressively nd in parallel resonance no-load conditions from , under condition that we lly analyze only resistive r making optimal power output. Even if certain making proper inductive g frequency interval, we chanical load impedance that converter including e (in the same frequency

e Locle - Switzerland

4.2-11

Converter Operating High Power in Parallel Resonance

Applying series inductive compensation Ls, we shall enable converter to operate in its parallel mechanical resonance f2 (or in impedance maximum zone). From Fig. 4.1-3.1, c) and d), we can see that equivalent impedance of Converter & Series Inductance has resistive character only when phase function, Fig. 4.1-3.1, d), is crossing zero line in f2. Consequently, applying PLL (for resonant frequency control), in order to keep the input current and voltage in phase, at f2 (measured on the input of the equivalent impedance, Fig. 4.1-3.1 c)), we shall make an equivalent load being high resistive impedance. The next important design task, regarding optimal power transfer, is that equivalent load impedance should be matched (or should be equal) to the internal impedance of driving electronics (of ultrasonic power supply, or ultrasonic generator, or power oscillator). This objective we usually realize designing a proper primary-to-secondary-ratio of a ferrite-transformer, and reducing high load impedance to the level of the internal impedance of driving electronics. In order to control output converter power (or output velocity) under loading, we should also find the best way to extract the motional voltage um from the output power circuit, and bring this voltage as the feedback to the main supply, or DC supply voltage control unit, effectively keeping um constant (most often by means of PWM regulation), regardless of load variations. Since converter impedance in no-load conditions and in parallel resonance (in air) is usually very high, converter is dissipating very low energy, and it can continuously oscillate long time without applying forced cooling. When converter is mechanically loaded (operating in parallel resonance), its equivalent impedance is decreasing, proportionally with the increase of applied load. Consequently, when converter is operating in parallel resonance, we should start converter oscillations just before acoustical load is applied, and we should know that this regime is convenient for low and moderate, largely variable acoustical loads. We could also operate converter (with series inductive compensation) in two of its side-band series-resonance frequencies f o , but only operating in can give good results (efficient and stable, high power output) in cases when we adjust series inductance to produce

r fSL SH fSL

f foper SL. = f≈ 1 (see Fig. 4.1-3.6). Also the philosophy of output ultrasonic power regulation, when operating a converter in , would be not the same as operating it in . The operation in

f or fSL SH

f1 f foper SL. f= ≈ 1 is typical for driving arrays of (mutually similar) ultrasonic cleaning transducers. Let us now analyze regular, series-inductance compensated converter, operating on its parallel resonant frequency. The equivalent circuit of such operating regime, valid only for very close vicinity of converter’s parallel resonance is presented on Fig. 4.2-4. The meaning of series inductive compensation in this case is to eliminate all reactive circuit elements such as capacitances and inductances (of course only for a constant frequency f2, when circuit is operating in resonant regime). This can be realized on the way that Ls should be chosen to create series resonance with Cos on f2 (when Ls and Cos would present very low, or zero equivalent impedance), and naturally (in the same time) L2 and C2 would create parallel resonance, on f2, that has very high equivalent impedance.


4.2-12

Generator Compensation Converter Load

Fig. 4.2-4 Converter inductively compensated to operate in parallel resonant

frequency ( ) /L C L C fs os ≅ ≅2 22

221 4π

This way, we can practically eliminate all reactive elements Ls, Cos, L2 and C2 from the circuit given on Fig. 4.2-4 (by satisfying ), and create a more simplified equivalent circuit (which has only resistive elements), given on Fig. 4.2-5, a). Since R

( ) /L C L C fs os ≅ ≅2 22

221 4π

op usually presents very high resistance (in the range of several MΩ, or higher), we can additionally approximate the same circuit to its final version given on the Fig. 4.2-5, b).

uinp.

usig ≈ i2+ iLig

iLi2ipuL

Rs2 ≈Rosa) b)

Fig. 4.2-5 Com

Converters m

( ) (R R R R Rop L os s>> >>2 2, )

pensated converter operating on parallel

odeling Miodrag Prokic Le

ig ≈ iL

⇒≈

i2

resona

Locle - S

iL

i2 << iL

nt frequency

witzerland

4.2-13

Now we can analyze circuits presented on Fig. 4.2-4 and Fig. 4.2-5, and find all characteristic currents and voltages,

i i iU

R R R RR R

i RR R

i i RR R

i

uR R R

R R R Ru R

R Ru

u u i R i R u i RR R

R R Ru

R RR R R R

u u R R R

g Lg

g os LL L

Lg L

Lg

inpos L

g os Lg

os

LL

L L L inp g osL

os Linp

L

g os Lg inp os s

= + =+ +

=

=+

=+

=+

+ += +

= = = = − =+

=

=+ +

≅ ≅ <<

−22

2

21

2

2

2

2

2 2

2 2 22

2

2

22

1

( ) ( ), ( ),

, ,

( ) ( )(

( ))

( )

( ), (( ) (

.

. .

. 2 R L ))

(4.2-15)

Electrical power-losses, dissipated on the parallel resistance RPeo op (which is in the range of many MΩ) can be neglected, since we made circuit transformation from Fig. 4.2-5, a) to Fig. 4.2-5, b), where all of such loses are transferred to the loses dissipated in series resistance R Ros s≅ 2 ,

P R iR

R R R RU

R

R R Ru

RR R

uR

R Ru

eo os gos

g os L

g

os

os L

inpos

LL

os

Linp

= =+ +

=

=+

= ≅

2

2

22

22

2

22

2

22

2

( ) ( )

( ) ( ) ( ). .

(4.2-16)

Mechanical, internal converter power-losses, are presented by power dissipated on the equivalent internal resistance of mechanical circuit R2,

P P R i uR

R

R R R R R RU

R R

R R R R Ru

R R

R R R Ru

R RR R

iR

R RR R R R

U

m RL L

L g os L

g

L

L os L

inpL

os L

inp

L

Lg

L

os g Lg

2 2 2 22

2

2

2

22

2

22

22

22

22

2 22

2 22

2

22

22

2

2

2

2

2

21

= = = =+ + +

=

=+ +

=+

=

=+

=+ +

LNMM

OQPP

( ) ( )

( ) ( )

( )

( )

( )( )

( )

. . (4.2-17)

Total power dissipation in a converter is equal to the sum of (electrical and mechanical) loses presented by (4.2-16) and (4.2-17),

P P P R R RR R

UR R R Rd eo m os

L

L

g

g os L

= + = ++

LNM

OQP + +2 2

2

22

22( )

( ). (4.2-18)


4.2-14

Useful acoustical (active) output power is only the power given to a load resistance RL2 = RL ,

P R i uR R

R RR R R

u

RR R

R R R R R R RU

RR R

R R R RU R

R RU R

RP

R RR

R R R RR R R

i R R

L L LL

L L

L

os Linp

L

L

os L g os Lg

L

L

g os Lg

L

i Lg

L

id tot

L

L

g os L

os Lg

= = =+

LNM

OQP

=

=+

⋅+ +

≅

≅+ +

LNMM

OQPP

≅+

≅ =

⋅+ +

+

−

22

2

2

2

2

2

2 2

2

2

2

2

22

2

22

2

2

22

1

1 1

1

( )( )

( )( ) ( )

( )( ) ( )

( ) ( )( )

, ((

.

L L i g oR R R R) ,≅ = + s )

(4.2-19)

Now we can calculate the internal electromechanical conversion-efficiency of loaded, η emL , and unloaded, η emo , converter as,

η η

η η

emL emm

m eo

R

R eo

emo emL Rm

m eoR

os

PP P

PP P

Lim LimP

P P RR

L L

= =+

=+

= =+

=+

≅→∞ →∞

2

2

2

2

2

2

2

1

11( ) ( )

(4.2-20)

The internal mechanical-to-acoustical conversion-efficiency of loaded converter, ηma is,

ηmaL

m L L L

PP P

RR R R

R

=+

=+

=+2

2

2

2

1

1. (4.2-21)

The total electro-acoustical conversion-efficiency of loaded converter, η ea , is,

η η η η

η

ea em ma emL

L

d L

L

eo m L

L os L

os L

Lma os L op

RR R

PP P

PP P P

RR

R R

R R R

RR

R R R R

= =+

=+

=+ +

=

=+ +

+

≅+

= <<

2

2 1

2 22

2

21

1

1

1( )

, ( , , ). (4.2-22)

A converter itself, even if it has very high electro-acoustic efficiency, is not sufficient to realize high power output (of acoustic or ultrasonic energy). It is also very important to realize proper impedance matching between the generator (as a source of electrical excitation) and the converter, in order to produce high acoustical power output.


4.2-15

Let us now find thermal power losses on the internal generator resistance Rg,

P R iR

R R R RU

R

R R Ru

RR R

u

g g gg

g os L

g

g

os L

inpg

Linp

= =+ +

=

=+

≅

2

2

22

22

2

22

2

( )

( ) ( ). .

(4.2-23)

The total thermal power losses in generator and converter are, P P P P P P

R RR R

Ru

R R R

R RR R

RU

R R R R

RR R

U RR

PR

R R Ru

R R R R R R

d tot g d g eo m

g osL inp

os L

g osL g

g os L

i

i Lg

i

LL

g

Linp

L L i g os

− = + = + + =

= + +LNM

OQP +

=

= + +LNM

OQP + +

≅

≅+

≅ ≅ +LNM

OQP

≅ = +

.

.

.

( )

( )

( )

( )

( ) ( ),

(( ) , ) .

2

22

2

2

22

22

2

2

2

2

22

22

2

2

2

1

(4.2-24)

The total electromechanical conversion-efficiency of loaded and unloaded converter connected to generator, η em tot− , is,

η

η η

em totm

m g eo

m

m g

emo tot em tot Rm

m g eoR

g os g osg os

PP P P

PP P

Lim Lim PP P P

RR R R R R

R

R R R

L

−

− − →∞

=+ +

≅+

= =+ +

=

=+ +

=+

+≅ >> +

2

2

2

2

1

2

2

2

2

21

11

( ) ( )

, ( ) .

L→∞ (4.2-25)

The total mechanical-acoustical conversion-efficiency of loaded converter connected to generator, ηma tot− , is,

ηma totL

m L L Lma

PP P

RR R R

R

− =+

=+

=+

=2

2

2

2

1

1,η (4.2-26)

The total electro-acoustical conversion-efficiency of loaded converter connected to a generator, η ea tot− , is,


4.2-16

η η η ηea tot em tot ma tot tot

L

g eo m L

L

d LL

g os

L

L L L g os

L

L

L i

L i

L i

PP P P P

PP P

RR RR R R

RR

RR

R R RR R

RR R

R RR R

− − −= ⋅ = =

=+ + +

=+

=

++

+LNM

OQP≅

≅+ + +

+≅

+=

+

2

22

2

2 2 2

1

1 1

1

11

( )

( )( )

//

.

(4.2-27)

We also know that mechanical quality factor uniquely qualifies converter power-conversion efficiency and internal losses, and for the converter operating on parallel resonance, Fig. 4.2-4, can be found as,

Q RX

RX

f C R

Rf L

R CL

for non loaded converter

QR R

XR R

Xf C R R

R RR

Q

R Rf L

R R CL

QRR

R RRR

RR

Q RR

Q R RR R

Q

m oC L

m LL

C

L

LL

Lm o

LL

m o

L

L i

i

L

i

m oL

m oL i

im o

22

2

2

22 2 2

2

2 22

2

2

22

2

2

22 2 2

2

22

2

2 22

2

2

2

2

22

22

22

2

2

2

2 1

= = = =

= = −

= = = =

= = =+

=

=+

≅ =

π

π

π

π

( ),

( ) ( )( )

( )

( )( )

/ //

(for loaded converter).

=

os

(4.2-28)

In order to present all efficiency, energy and quality parameters of a piezoelectric converter on the very general and dimensionless way, in the function of converters mechanical load, let us make normalization of above found particular expressions (valid for converter operating in parallel resonance), creating the table T. 4.2-4. By comparing T. 4.2-1 with T. 4.2-4, we see that all expressions inside are formally identical, or approximately identical, except different expressions for quality factors (between series and parallel resonance), and also mechanical load resistances of series and parallel resonant regimes have mutually opposite directions in relation with load increase (but shear the same load axis). For the purposes of mathematical conveniences, sum of internal generator resistance, Rg, and internal mechanical-circuit converter-resistance, Ros, in most of above given expressions (and in the T. 4.2-4) is presented as a resistive (generator and converter specific) parameter R R Ri g= + , since, this way we can easily treat mechanical load RL as the function of Ri, or as a number of Ri units, or in a normalized and dimensionless form as RL / Ri. This way, the resistance Ri would again become the mechanical-load unit.


4.2-17

T. 4.2-4 Loaded Converter Quality and Efficiency Parameters Basic expressions Normalized expressions Equation

P RR R

U RR

PLL

i Lg

L

id tot≅

+≅ −( ) 2

2 RU

P R RR

RU

Pi

gL

L i

L

L

gd tot2 21

≅+

≅ −

/( / )R i

2 (4.2-19)

PR

R RU

RR

P

d toti

i Lg

i

LL

− ≅+

≅

≅

. ( ) 22

RU

P RR R

R RRR

PU

i

gd tot

i

i L

L i

i

L

L

g

2

2

2

2

2

2

11

− ≅+

=

=+

≅

. ( )

( / )

(4.2-24)

η ea totL

L i

L i

L i

RR R

R RR R− ≅

+=

+/

/ 1η ea tot

L

L i

L i

L i

RR R

R RR R− ≅

+=

+/

/ 1 (4.2-27)

P RR

PLL

id tot≅ − .

PP

RR

L

d tot

L

i−

≅.

(4.2-19)

Q QRR

m Lm o

L

22

21=

+

QQ

R RR R

RR R

RR

RR

RR

R RR R

m L

m

L

L

L i

i

L

i

L L

i

2

20

2

2 2

2 2 2

1

1= =

+=

=+

≅ =/ /

/i

(4.2-28)

Q RR R

Qm LL

m11

110≅

+ Q

QR

R R R Rm L

m L L

1

10

1

1

11 2

≅+

≈+ / i

(4.2-14)

In practice for loaded converter operating in parallel resonance we can use the following circuit elements approximations: ( ) ,R R RL L2 ≈ R R R Ri g os2 > = +( ) ,

. For instance, if we introduce as normalized mechanical load the

ratio

R RL ≤ 2

x RR

L

i

= , table T. 4.2-4 will become T. 4.2-5. Also, when we compare T. 4.2-5

with T. 4.2-2 we will find that all expressions in both of them are either mutually identical or approximately identical, and that only difference between them is related to functions describing mechanical quality factors ratios. T. 4.2-5 Loaded Converter Quality and Efficiency Parameters

x RR

xL

i

= ∈, ( ,0 ∞)

(=) Normalize Load

Normalized Power and Quality Parameters Expressions F(x) Equation

RU

P xx

i

gL2 21≅

+( ) (4.2-19)

RU

Px

i

gd tot2 2

11− ≅+. ( )

(4.2-24)

Y = F(x) η ea tot

xx− ≅

+1 (4.2-27)


4.2-18

PP

xL

d tot−

≅.

(4.2-19)

QQ x

m L

m

1

10

11 2

≈+

(4.2-14)

QQ

xRR

x

xx

m L

m

i

ea tot2

20 2 11=

+<

+= ≤−η

(4.2-28)

Let as now calculate all normalized expressions from T. 4.2-4 and T. 4.2-5 in the function of mechanical load units, when mechanical load is taking values:

, and create new table T. 4.2-6, which is almost identical to T. 4.2-3, R RL ∈( , ...)0 10 i

T. 4.2-6 Numerical Values of Loaded Converter Quality and Efficiency Parameters

RL ⇒ 0 0 5. R i R i 1 5. R i 2R i 3R i 10R i ∞

x RR

L

i

= 0 0.5 1 1.5 2 3 10 ∞

RU

Pi

gL2 0 0.22 0.25 0.24 0.22 0.187 0.083 0

RU

Pi

gd tot2 − . 1 0.44 0.25 0.16 0.11 0.0625 0.083 0

η ea tot− 0 0.33 0.5 0.6 0.667 0.75 0.909 1 P

PL

d tot− .

0 0.5 1 1.5 2 3 10 ∞

QQ

m L

m

1

10

1 0.5 0.333 0.25 0.2 0.143 0.0476 0

QQ

m L

m

2

20

0 < 0.33 < 0.5 < 0.6 < 0.667 < 0.75 < 0.909 1

Now, based on functions and results from T. 4.2-5 and T. 4.2-6, we can again present all (normalized) power and quality parameters of a converter operating in parallel resonance on the same multi-functional graph, Fig. 4.2-3, which is already made for a converter operating in series resonance (where x axis is normalized mechanical load, and Y = F(x) presents normalized expressions from T. 4.2-5). Of course, conclusions based on Fig. 4.2-3 should be drawn differently, regarding mechanical loading, for a case when converter is operating in series or in parallel resonance. In series resonance with an increase of mechanical load, RL is also increasing, and in parallel resonance when mechanical load is increasing, RL is decreasing (here we should make a difference between RL and mechanical load resistance: RL only represents a quantitative equivalent, or analogical replacement, for real mechanical load). Of course, this explanation regarding character of mechanical loading is strictly related to the use of dual converter models (inductively compensated in series or parallel) presented on Fig. 4.2-1 and Fig. 4.2-4.


4.2-19

The next conclusion from Fig. 4.2-3 is that if converter is operating in series resonance, with its load increase, the total electro-acoustical efficiency is also progressively increasing, total power dissipation decreasing, and until certain maximum (x = 1) the output load power is increasing and then slowly decreasing. Consequently, we can conclude that series resonance regime is very good for moderate and heavy loads, since when load is zero or very small (x < 0.5), total power dissipation is maximal, efficiency minimal, and output power minimal. Practically, it is preferable that converter operating in series resonance is always reasonably or highly (mechanically) loaded (for instance in applications like ultrasonic metal welding). If converter is operating in parallel resonance, with its load increase, the total electro-acoustical efficiency is also progressively, but slowly decreasing, total power dissipation is slowly increasing, and until certain maximum (x = 1) the output load power is increasing, and then slowly decreasing. Consequently, we can conclude that parallel resonance regime is very good for low and moderate loads, since when load is zero or very small (x > 10), total power dissipation is minimal, efficiency maximal, and output power still reasonably high. Practically, converter in parallel resonance is perfectly operating in no-load or moderate loading conditions (like in ultrasonic atomizers, majority of applications in ultrasonic plastic welding etc.). It is also very important to notice that useful loading interval (∆x), where converter efficiency is relatively high, power losses minimal, load power almost constant… is significantly larger in cases of parallel resonance regime than in cases of series resonance regime. We should also pay attention on loading Ri = Rg + Rs units, which are quantitavelly different for series and parallel resonance regimes (regardless that normalized diagrams presented on Fig. 4.2-3 are looking the same for series and parallel resonance). In cases of series resonance, Ri units are relatively small (typically in the range of several ohms), and in cases of parallel resonance Ri units can be 10 to 100 times higher (typically in the range between 100 to 1000 ohms). This is related to realizing maximal real-power transfer, when generator internal resistance should be equal (or closely matched) to the load resistance, RL = Rg. Since RL in series resonance takes values between 0 Ω (for no-load situation) and few hundreds of Ω, for heavy loaded situation, and in parallel resonance RL varies between infinity (for no-load) and a few hundreds of Ω, for heavy loaded situation, it is clear that loading Ri units of series and parallel resonance circuits would be mutually different (at least for the order of magnitude). After understanding the nature of mechanical loading of piezoelectric converters (regarding the order of magnitude/s of Ri units) we are in a position to understand why parallel resonance regime has much wider loading interval (delivering reasonably high power, under reasonably low losses), than series resonance operating regime. Of course, there are some other, exceptional (mutually exclusive) advantages (and disadvantages) of both regimes, sometimes making only one of them particularly more useful in certain of critical applications. If we would like to create converter models where in series and parallel resonance, both load resistances RL would proportionally grow with mechanical load increase, we should make circuit modification/s in the converter model given on Fig. 4.2-4 (see Fig. 4.2-6).


4.2-20

For instance, operating only with dual converter models proposed in the first chapter, and again presented on Fig. 4.2-6, a), will give us the chance to treat mechanical (or acoustical) load uniquely, as a mobility, or velocity divided by force, measured on the element where output power is dissipated, and in both cases (of series and parallel resonances), with an increase of mechanical loading, RL (= RLs1 or RLs2) would proportionally increase. Also, when using only dual converter models given on Fig. 4.2-6, b), will give us another chance to treat mechanical (or acoustical) load uniquely, too, as a mechanical impedance, or force divided by velocity, measured on the element where output power is dissipated, and in both cases (of series and parallel resonances), with an increase of mechanical loading, when we are measuring input, equivalent, AC impedance of a loaded converter, operating in series resonance, we find that it starts with low impedance values for no-load conditions and it increases with mechanical loading increase, meaning that Fig. 4.2-1 is a natural modeling for such situations. (= RLp1 or RLp2) would proportionally decrease.

RLS1 RLp1

RLS2 RLp2

L*1sC1

C*op

C1

L*1pC*opRop Rop

L*2s

L*2p

C*os

C2 C2

C*os

Rop Rop

⇔

⇔

⇓⇑ ⇓⇑ a) b)

Electrical Input

Series Mechanical

Load/s

M

Fig. 4.2-6 Single-Mode, Series and Parallel, Resisand mutually equivalent, Piezoelectric Converter

a) RLs1 and RLs2 are increasing with load increb) RLp1 and RLp2 are decreasing with load incr

Elements marked with asterisk (*) are presenting real cinternal resistive losses (as already explained in the firs

Converters modeling Miodrag Prokic

ElectricalInput

tive Load-Impedanc Models: ase ease

apacitances and inductt chapter).

Le Locle - Swit

Parallel echanicalLoad/s

e, Dual

ances with

zerland

4.2-21

Of course, the most logical and most natural situation would be to use only mobility based analogies and models presented on the Fig. 4.2-6, a), (left side), and to treat mechanical loading-increase as an increase of RL (= RLs1 or RLs2). In existing literature, models on Fig. 4.2-1 and Fig. 4.2-4 are more often found (just because of historical reasons of their development, and because of certain intellectual rigidity against radical changes in already accepted methodology and logic regarding converters modeling). The other, more tangible reason for using models on Fig. 4.2-1 and Fig. 4.2-4 is in the fact that when we are measuring input, equivalent AC impedance of a loaded converter, operating in series resonance, we will find that it starts with low impedance values for no-load conditions, and it increases with mechanical loading increase, meaning that Fig. 4.2-1 presents the natural modeling for such situations, since RL = RL1 is also increasing with loading. Also when we are measuring input, equivalent AC impedance of a loaded converter, operating in parallel resonance, we will find that it starts with very high impedance values for no-load conditions, and it decreases with mechanical loading increase, meaning that Fig. 4.2-4 presents the natural modeling for such situations, since RL = RL2 is also decreasing with loading. The same problematic becomes more complicated if generator and load impedances are not resistive (for instance, instead of Rg we could have Zg = Rg + jXg and instead of RL, ZL= RL +jXL). In such situations, ZL is contributing to the input complex impedance of a loaded-converter, making that free no-load converter impedance would become a new complex impedance Zc= RcL +jXcL. Now, maximal power transfer (from generator to the converter) would be realized when we find a way to make RcL = Rg and XcL = -Xg. Matching condition XcL = -Xg we usually name as reactive impedance matching, or reactive elements compensation (in fact elimination of all reactive circuit elements), which can be realized only for certain discrete (resonant) frequencies, or sometimes for limited frequency intervals. After realizing XcL = -Xg we enable generator to send only the active power to its load (regardless if the power transfer is maximal or not). A matching condition RcL = Rg we can qualify as an active generator-to-load impedance and power matching, when we shall obtain maximal active power transfer from generator to its load.


4.2-22

Converter Operating High Power Between Series and Parallel Resonance Applying convenient, mixed series and parallel inductive compensation Lp & Ls (and some more of electric Circuit and Filter Theory combinations) we can enable converter (on a number of different ways) to operate in a stable and efficient, high power regime, between its series and parallel mechanical resonances f1 and f2 (or between impedance minimum and maximum zone), or even to extend this frequency interval towards lower and higher frequencies (outside of interval f1 - f2). Basically, by making mixed (series and parallel) inductive compensation we should realize that equivalent converter’s electrical impedance, including all compensating elements, has fully resistive character in a desired frequency band (for instance between f1 and f2, or in a bit wider interval). Of course, similar design strategy can be applied if piezoelectric converter is used as a sensor (to become an efficient detector of mechanical vibrations in a certain frequency interval). Such Mixed-mode resonant regimes (conveniently compensated converters to become resistive loads, and operational high power, over one frequency interval) can unite advantages of single resonant-frequency operating regimes, with an excellent electromechanical efficiency (practically without using PLL for resonant frequency control). In order to control output converter power in a mixed-mode regime under loading conditions, we should also find the best way to extract representative motional current im and motional voltage vm from the output power circuit (then, in some cases, multiply them to produce active power signal), and bring them as the feedback to the main supply or to transistors driving circuit (usually realized by means of PWM regulation). The most promising future of piezoelectric converters (and sensors) applications will be experienced in mixed-mode resonant regimes. Mixed-mode resonant operating regimes can also be found (unintentionally and by game of chance) in traditional ultrasonic cleaning systems when many of similar ultrasonic converters are connected (electrically and mechanically) in parallel, or in other familiar situations regarding sensors composed of number of transducers. Since in a group of converters (in parallel connection) each of them is usually slightly different than any other, after coupling them mechanically to an ultrasonic cleaning bath, small initial differences (in electric and mechanical characteristics) are becoming even much higher (because of number of geometrical and acoustical coupling factors). After connecting all single converters in parallel electrical connection, it can happen that particular Impedance-Phase characteristics of every single converter are mutually (and randomly) shifted in frequency scale, making that inductive areas of certain converters are partially overlapping capacitive areas of others. Naturally, overall equivalent phase characteristic of the converter group can be equal (or close) to zero in certain frequency intervals, making that (equivalent) converters’ group impedance becomes resistive in the same (zero-phase) areas, producing high power output when driven inside of such intervals. The design strategy in ultrasonic cleaning, regarding transducers can be:

a) To take similar, but not well selected converters (slightly different by frequency and impedance), what minimizes the production cost and slightly reduces overall efficiency of electro-acoustic conversion, making simple and easy high power, frequency sweeping operation, and


4.2-23

b) To take perfectly frequency and impedance selected converter group (of almost mutually identical converters, what will increase production cost), make proper external inductive compensation, and produce high electro-acoustic efficiency on a single resonant frequency, but the frequency sweeping becomes less power-efficient than in the first case, and driving electronics should be more sophisticated, compared to the case under a).

c) In cases, a) and b), single resonant frequency (series or parallel) and mixed-mode resonant regimes (operating between average series and parallel resonant frequency or in wider frequency interval) can be produced by proper external inductive (as well capacitive) compensation (basically applying methods of electric filters theory).

If converters group is composed of highly mutually-dissimilar-converters (case a)), equivalent impedance of such group looks similar to certain series or parallel connection between a capacitive and resistive (frequency dependant) components, and if all converters are almost identical (case b)), equivalent impedance of such group looks similar to a typical piezoelectric impedance of a single transducer (with clear presence of series and parallel resonances).


4.3-1

4.3 Acoustically Sensitive Zones of Piezoelectric Converters In order to experimentally support conclusions regarding piezoelectric (acoustic) FORCE and VELOCITY-sensitive zones, let us take the same piezoelectric converter (see Fig. 4.3-1), as already presented on T. 4.1-1, Figs. 4.1-2, 4.1-3.1 and 4.1-3.2. When we measure Impedance-Phase curves using Network Impedance Analyzer and relatively high sinusoidal frequency-sweeping signal usweep. ≥ 1 V-rms, we shall get the typical piezoelectric converter curves such as presented on Fig. 4.3-1. If we now make the same Impedance-Phase measurements reducing the amplitude of the sinusoidal frequency-sweeping signal to usweep. = 0.1 V-rms, we shall get similarly looking curves, presented on Fig. 4.3-2. We can already notice (on Fig. 4.3-2) that Impedance and Phase curves in the zone of high impedance are no more as smooth and perfectly continuous lines, as curves on Fig. 4.3-1. The cause of getting such result is still not well recognizable, but as we will see later, by reducing amplitude of the frequency-sweeping signal, we are approaching closer to the environmental acoustic-noise excitation of the laboratory where we make measurements (in this case measurements were performed in Branson Ultrasonic Corp. in USA). In other words, if the same piezoceramic were used as a receiver, it would detect certain laboratory (background, building vibrations) acoustic-noise, and produce voltage output that becomes (in its peak values) comparable to 0.1 V-rms. If we now again make the same Impedance-Phase measurements, once more reducing the amplitude of the sinusoidal frequency-sweeping signal to usweep. = 0.01 V-rms, we shall get new Impedance-Phase curves, presented on Fig. 4.3-3. We can now notice (on Fig. 4.3-3) that Impedance and Phase curves in the zone of high impedance are very much influenced by background, laboratory acoustic-noise. In other words, the piezoceramic converter in question is now detecting the laboratory (background) acoustic-noise, which can easily be compared to sinusoidal frequency sweeping signal usweep. = 0.01 V-rms. After reducing the sinusoidal frequency-sweeping signal to usweep. = 0.001 V-rms, we shall get new Impedance-Phase curves, presented on Fig. 4.3-4. We can now notice (on Fig. 4.3-4) that the background, laboratory acoustic-noise, has no influence on Impedance and Phase curves in the zone of low impedance, but in all other frequency zones, we would not be able to recognize the shape of typical piezoelectric Impedance-Phase curves. In other words, now, the piezoceramic converter in question is non-selectively detecting the laboratory (background) acoustic-noise in a very wide frequency band (completely loosing phase information), and basically we cannot consider such operating sensor regime as engineering-wise useful (and obviously, in this latest case, the background-noise is behaving stochastically, and it is producing much higher voltage than usweep. = 0.001 V-rms). As the conclusion based on measurements presented on Fig. 4.3-1 until Fig. 4.3-4, we can say that converter parallel-resonance (high impedance) area is very much VELOCITY-VOLTAGE sensitive. We also see, based on the same measurements that converter series resonance (low impedance) area is totally insensitive on VELOCITY-VOLTAGE background-excitation, but we could also find that series resonance (low impedance) area is very much FORCE-


4.3-2

CURRENT sensitive. Of course, here we are talking about oscillating-velocity and force signal forms.

Impedance

Phase

a)

Impedance

Phase

b)

Fig. 4.3-1 Piezoelectric Impedance-Phase Curves (usweep. ≥ 1 V-rms) a) Logarithmic Impedance-amplitude (in dB) b) Linear Impedance-amplitude of the same converter


4.3-3

Impedance

Phase

a)

Impedance

Phase

b)

Fig. 4.3-2 Piezoelectric Impedance-Phase Curves (usweep. = 0.1 V-rms) a) Logarithmic Impedance-amplitude (in dB) b) Linear Impedance-amplitude of the same converter


4.3-4

Impedance

Phase

a)

Impedance

Phase

b)



4.3-5

Impedance

Phase

a)

Impedance

Phase

b)



4.3-6

Let us again take the same piezoelectric converter, as already presented on T. 4.1-1, Figs. 4.1-2, 4.1-3.1, 4.1-3.2 and 4.3-1, inductively compensated in series and/or in parallel, and make similar sensitivity detection as before (measuring equivalent Impedance-Phase curves, and reducing sinusoidal frequency-sweeping signal, starting from usweep. ≥ 1 V-rms, until usweep. = 0.01 V-rms; -see Fig. 4.3-5 until Fig. 4.3-8). Again, we can consistently conclude that all parallel-resonance (and high impedance) zones are VELOCITY-VOLTAGE sensitive, regardless if this is the natural, parallel resonance of a single piezoceramic, or equivalent high-impedance, where piezoceramic is in certain connection with some other passive electrical components (see Fig. 4.1-3.5). We also see, again based on the same measurements (Figs. 4.3-5 – 4.3-8), that all equivalent series resonances (low impedance zones) are totally insensitive on VELOCITY-VOLTAGE background-noise excitation, but we can also conclude that series resonance/s (low impedance/s) zone/s are very much FORCE-CURRENT sensitive (regarding alternative velocity and force signal forms), since all series resonance zones are producing high-pressure acoustic fields; - see Fig. 4.1-3.4 (and, of course, we can also base our conclusions on electro-mechanical analogies, CURRENT-FORCE & VOLTAGE-VELOCITY, and on dual-circuits’, equivalent models). In order to increase piezoelectric-sensor receiving or transmitting-sensitivity (to operate in the band between two of its impedance peaks, or between fL and fH) we can simply play with connecting parallel and/or series resistances to it, as presented on Figs. 4.1-3.4 and 4.1-3.5. There are some other possibilities regarding how to increase sensor sensitivity in the frequency areas outside of the resonant frequency couple ( 0 1 2≤ < ≤f f f f& ), when using capacitance connected in series to a piezoelectric sensor or converter, Fig. 4.3-9 (meaning that such sensor will have increased sensitivity starting from very low frequency (close to zero Hz) until series resonance f1, and again starting from parallel resonance f2 toward higher frequencies, and it will not be significantly sensitive in the frequency interval between series and parallel resonant frequency). It is interesting to notice that only if series capacitance, Cs, is smaller than static (low frequency) capacitance of a piezoelectric sensor, Co, the resulting sensitivity will increase (testing sensitivity on the same way as before, by reducing sinusoidal frequency-sweeping signal of Network Impedance Analyzer, starting from usweep. ≥ 1 V-rms, until usweep. = 0.01 V-rms). For instance, the highest increase of sensitivity (of the same piezoelectric converter presented on T.1, Figs. 4.1-2, 4.1-3.1, 4.1-3.2 and 4.3-1) will be measured if series capacitance is approximately equal to ½ of the static sensor capacitance, Cs ≅ 0.5 Co, see Fig. 4.3-10. Surprisingly, when Cs ≅ Co, this would make sensor-sensitivity non-selective, behaving randomly, and almost useless (phase curve would become unrecognizable, at least in the example of piezoceramics tested here; - given in T. 4.1-1). And again, when Cs >> Co, such sensor would behave as sensor without series capacitance. Generalizing the conclusions about converters activity and/or sensitivity, once more, we can see that everything we are talking about, here, is related to generator-to-load matching (where internal generator or receiver impedance should be close to the


4.3-7

load impedance), and where complex impedance of the load should be reactively compensated in order to become closer to resistive impedance.

Equivalent Impedance

Equivalent Phase

a)


Equivalent Phase

b)

Fig. 4.3-5 Converter with Series Inductive Compensation Ls = 7.69 mH (usweep. ≥ 1 V-rms)

a) Logarithmic Impedance-amplitude (in dB) b) Linear Impedance-amplitude of the same converter


4.3-8


Equivalent Phase

a)


Equivalent Phase

b)

Fig. 4.3-6 Converter with Series Inductive Compensation Ls = 7.69 mH (usweep. = 0.01 V-rms)



4.3-9


Equivalent Phase

a)


Equivalent Phase

b)

Fig. 4.3-7 Converter with Parallel Inductive Compensation Lp = 7.69 mH (usweep. ≥ 1 V-rms)



4.3-10


Equivalent Phase

a)


e

Fig.

a) Logarithmb) Linear Im

Converters

Equivalent Phas

b)

4.3-8 Converter with Parallel Inductive Compensation Lp = 7.69 mH (usweep.= 0.01 V-rms)

ic Impedance-amplitude (in dB) pedance-amplitude of the same converter

modeling Miodrag Prokic Le Locle - Switzerland

4.3-11

Impedance

Phase

a)

Impedance

Phase

b)

Fig. 4.3-9 Converter (from T. 4.1-1& Fig. 4.1-2) with Series Capacitance, Cs = 1.368 nF, Cos = 3.0221 nF (usweep. ≥ 1 V-rms)



4.3-12

Impedance

Phase

a)

Impedance

Fig. 4.3-10 Conv

a) Logarithmic Impeb) Linear Impedanc

Converters modeling

Phase

b)

erter with Series Capacitance (usweep.= 0.01 V-rms) Cs = 1.368 nF, Cos = 3.0221 nF

dance-amplitude (in dB) e-amplitude of the same converter

Miodrag Prokic Le Locle - Switzerland

4.4-1

4.4 Practical and simplified procedure for inductive impedance compensation, optimal impedance and optimal power matching of ultrasonic transducers

(The first part of this document is only valid for ultrasonic cleaning transducers operating on a series resonant frequency. The second part explains operation of welding transducers in parallel and series resonance.) 1. Find the input capacitance of a single transducer measured on 1 kHz, Cp(1kHz) = Cp = ___(nF),

and multiply it by number of transducers, n = 1, 2, 3…, that will be assembled in parallel connection (on the same ultrasonic tank). Cpn(1kHz) = Cpn = nCp(1kHz) = ___(nF). Use LRC-meter.

2. Find the total input capacitance of the transducer group (n transducers in parallel electrical

connection, glued to ultrasonic tank) measured on 1 kHz. Cpn(1kHz) = Cpn= ___(nF). This value should be very similar to the value of Cpn(1kHz), calculated in point 1. Use LRC-meter.

3. Find the resonant frequency and resonant impedance of a free and single transducer (not loaded,

measured in air) fr = fr = ___kHz (at minimum impedance Zr = Zr = ___Ω). This should be the value declared by the producer of a transducer. Check if your measurement is in agreement with declared value. If you cannot make your own frequency and impedance measurements, take data from producer’s catalogue. For measurements it is necessary to have recommended IEC circuit arrangement, or HP 4194A Network Impedance Analyzer. The simple improvisation for measurements is possible to be organized using sinus function generator, oscilloscope and IEC resistive network circuit.

4. Find the resonant frequency and resonant impedance of all transducers (as a group) when they

are in parallel connection (n = number of transducers), glued to ultrasonic tank or to immersible transducer box (and when tank, or immersible transducer box are fully water loaded): frn = frn = ___kHz (at minimum impedance Zrn = Zrn = ___Ω). This could also be the value declared by the producer of transducers, but if it is not declared, it would be found in the vicinity of about 10% lover frequency than resonant frequency measured under 3. Check if your measurement is in agreement with declared value/s. If you cannot make your own measurements (missing instruments), take data from producer catalogue. If producer did not give any data, take this frequency as an approximate number that will be equal 0.9 multiplied by frequency fr found under 3. ⇒ frn = 0.9fr ___kHz. If possible, find resonant impedance of the transducer group (when fully water loaded) Zrn = Zrn = ___Ω. Zrn should be in the range of value measured under point 3., or more precisely in the range 10Zr/n ≤ Zrn ≤ 100Zr/n (for efficient transducer group, and for good gluing-bonding technology). For above-mentioned measurements it is necessary to have recommended IEC circuit arrangement, or HP 4194A Network Impedance Analyzer. The improvisation is possible to be organized using sinus function generator, oscilloscope and IEC resistive network circuit.

5. Calculate the approximate (maximal) value of necessary inductive compensating coil using the

formula: Ls ≤ 1/(4π2 frn2Cpn) = ___(µH or mH) (in order to cancel the static capacitance of the

transducer group). Resonant frequency frn is taken from 4., and parallel capacitance Cpn is taken from 2. Make, adjust and fix compensating inductance Ls, changing the air gap in the ferrite core. In real operation, the best ultrasonic power will be found adjusting a bit lower inductance Ls than previously calculated (but sometimes, the real inductive compensation has until 2 times smaller value than here calculated Ls, because of non-linear and self-compensating, interference-coupling effects between transducers in high power operation, especially when transducers have well operating higher harmonics, and when large frequency sweeping is applied; -since in certain limited frequency band/s every particular piezoelectric transducer reacts like inductive element, while other transducers are dominantly capacitive, and not all of them from the same group are mutually 100% identical, enabling mutual inductive compensation). Use LRC-meter for inductance measurements (usually on 1 kHz), or HP 4194A, or HP 4195A.

6. Optimal ultrasonic power (on ultrasonic transducers, operating on lowest basic series resonant

frequency) will be achieved when: a) High frequency current and voltage measured on the primary input side of the ferrite (output power) transformer are mutually in phase, and b) When generator

Miodrag Prokic, Marais 36, CH-2400, Le Locle, e-mail: [email protected]

4.4-2

current reaches its maximum (measured on the primary side of the ferrite transformer, or (approximately speaking) on the main supply input of generator). c) In the same time when conditions a) and b) would be satisfied, the input power factor (cosines theta) on the main supply, low frequency power input (115/230 VAC, 50/60 Hz) should be very close to 1 (to one), since ultrasonic generator would consume only active power. High frequency transducers’ current measured on the primary or secondary side of the ferrite transformer has sinusoidal shape. Voltage shape measured on the primary side of ferrite transformer has rectangular shape (usually 1:1, or 50% Duty Cycle, except when PWM signal is used for direct transducers driving). For highly efficient transducers operation (and for good impedance matching) it is extremely important to oversize the primary windings cross-section area of the Litz wire (on the output, high power ferrite transformer). For instance, select the Litz wire with the total cross section surface, that corresponds to the RMS primary current of: 2.5 until 1.5 A/mm2. To avoid any heating in output ferrite transformer, the best would be to use criteria 1.5 A/mm2. Similar logic is valid for inductive compensating choke, and for secondary windings of the ferrite transformer (less rms. current per mm2 is always a better choice). For inductive compensating inductance we can apply the criteria 2.5 until 2 A/mm2 and for secondary transformer windings, 2 until 1.5 A/mm2 (always based on RMS primary or secondary current). The most important of all above-mentioned conditions is to increase the cross section area of primary transformer windings, in order to minimize heating in output transformer.

7. The natural or forced aging of ultrasonic transducers is extremely important process (in ultrasonic

cleaning or welding applications, when operating in any of resonant or multifrequency regimes). For instance, after assembling a group of cleaning transducers (after gluing/bonding to a cleaning tank), it is necessary to make minimum 24 hours of continuous-operating aging: Start with about 30% of ultrasonic power-loading (of an ultrasonic tank filled with water), and after several hours increase the tank power until 100% of ultrasonic generator power (and keep this way during long time: in total minimum 24 hours). Material aging and vibrating-surfaces micro-friction would stabilize transducers parameters, as well as stabilize adhesive bonding layers. Some famous ultrasonic companies are often performing 7 days (24 hours a day) forced aging of their cleaning tanks (and immersible transducer boxes). During the process of forced aging the water level in the tank should be kept on the same level (adding water when necessary). After certain time of such operation, it appears significantly noticeable that operating regime of ultrasonic transducers becomes quieter, smoother, more stable, and cleaning efficiency increases. When initial forced aging is finished, and transducers reach room temperature (for instance after 24 hours), if we perform measurements of transducers’ parameters, we shall notice parameters-changes between 5% and 10% compared to measurements before forced aging. Forced aging should be performed using certain (maximally acceptable) level of frequency-modulation (sweeping), and applying sufficiently high amplitude modulation of ultrasonic power. Practically, all above described measurements and tuning steps (from 1. to 6.) should be repeated in order to make new and final frequency and impedance tuning and matching. This way we avoid the situation that transducers aging would be naturally finalized when ultrasonic cleaner is sold to a customer (because usually a customer is not able to make retuning and new impedance matching). The aging is finalized successfully if after 24 hours (when transducers again reach room temperature) we get the same full-power operating-conditions as during the final tuning. A similar aging procedure should be applied to single plastic-welding transducers operating on parallel or series resonance (only the conditions for treatment, acoustic loading, and time intervals should be optimally adjusted for every specific situation). Once (after long empirical data collection) when we know the final and optimal operating regimes for certain transducer configurations, we can apply direct settings for best frequency and impedance matching, and immediately start with optimal artificial aging.

8. In the case when transducer group is operating on some higher-harmonic frequency, we could

easily have the case that transducers’ current has 2 to 4 times higher frequency than the frequency of the voltage applied on primary side of ferrite transformer. Such (ringing and high frequency) resonant regimes are not described here. The inductive compensation Ls for such situations can be calculated similar as for basic frequency (see point 5) and then experimentally adjusted for the best ultrasonic operation on the high frequency harmonic, increasing the gap of the ferrite core of Ls, what is equivalent to decreasing the Ls. For the operation on high frequency harmonics it is


4.4-3

necessary to apply relatively wide band frequency sweeping voltage input into output ferrite transformer (and to produce well balanced ON-OFF signal group modulation in order to give sufficient OFF time to transducer group to produce high frequency, self-generated ringing current response). Ultrasonic transducers operating on higher harmonics usually have strong acoustic coupling between number of higher harmonics and basic (lowest frequency) resonant mode. If we are driving transducer on its basic resonant mode, it will also be possible to excite all other, coupled, higher frequency modes (just giving them sufficient OFF time to produce self-generated, ringing current signals). Knowing this, we also know that inductive compensation should be (first) well made for basic resonant mode, and later on, manually tuned (reduced to lower value) to enable transducers operating on higher frequency modes.

9. Optimal high frequency RMS-voltage (ultrasonic frequency RMS-voltage) measured directly on

input side of ultrasonic transducers (on piezoceramics) should never be higher than 50 Volts (RMS) per each millimeter of single piezoceramic ring thickness (measured using high frequency and wide band, AC, RMS voltmeter, in any possible operating regime). For instance, if one transducer has two piezoceramic rings (in parallel electrical connection, as usual), each of them t = 5mm thick, than RMS-voltage measured directly on ceramics (PZT4 and PZT8) should be ut ≤ 50 x t = 50x5 = 250 V (RMS). If we cannot measure RMS voltage on ultrasonic frequency (missing RMS voltmeter), then using oscilloscope we can (approximately) find maximum (half period peak) transducer voltage as utpeak ≤ 1.41 x 50 x t ≅ 70 x t, or we can also find peak-to-peak transducer voltage as utpeak-to-peak ≤ 2 x 1.41 x 50 x t ≅ 2 x 70 x t ≅ 140 x t. For the same example when thickness of one piezoceramic ring is 5mm, it will be: utpeak ≤ 1.41 x 50 x t ≅ 70 x 5 = 350 V, utpeak-to-

peak ≤ 2 x 1.41 x 50 x t ≅ 2 x 70 x 5 ≅ 140 x 5 = 700 V. In any other case, if we try to make different frequency and impedance adjustments, it would become destructive for ultrasonic transducers to increase voltage on piezoceramics for more than: ut ≤ 100 x t , or utpeak ≤ 1.41 x 100 x t , or utpeak-to-

peak ≤ 2 x 1.41 x 50 x t. Voltage and current shapes measured directly on transducers are (approximately) sinusoidal and mutually shifted between 60° and 90° angle degrees (usually shifted for almost 90° in cases of high quality transducers and good impedance matching). Transducer voltage should be adjusted by changing output (secondary side) of the ferrite transformer and slightly adjusting inductive compensation coil (changing its air gap).

10. Continuous and safe, RMS power of one single ultrasonic, cleaning transducer (when glued to

ultrasonic tank and water loaded) should be in the range of 3 Watts per cubic centimeter of the total volume of piezoceramic rings for PZT4, and 4 to 5 Watts per cubic centimeter for PZT8 piezoceramics. For instance, if one single ultrasonic (cleaning) transducer has 2 of PZT4 piezoceramics, where total ceramics volume (of both ceramics) is approximately 13 cubic centimeters, its continuous and safe operating power would be 13 x 3 ≅ 40 Watts. If the same transducer has PZT8 piezoceramic rings, then it can operate producing RMS power of 13 x 4 = 52 Watts, or not higher than 13 x 5 = 65 Watts. In a short-time loading conditions, cleaning transducers can produce up to 5 times higher ultrasonic power, providing that gluing or bonding technology is guarantying sufficiently strong mechanical coupling between transducers and ultrasonic tank (to resist such stress). Here we are talking only about real, active or RMS power that can be measured using high frequency wide band power meters (connected directly to transducers group or to the primary side of output ferrite transformer). Ultrasonic generator should be adjusted (by inductive compensation and selection of output voltage on secondary side of the ferrite transformer) not to produce higher output power than safe and continuous operating power of transducer group is (given by transducers supplier, or found empirically and tested very long time). The same transducers, when operating as plastic welding transducers could produce between 30W and 50 Watts per cubic centimeter of total piezoceramics volume installed in the sandwich transducer (valid for PZT 4 and PZT 8). The second and extremely important criteria for efficient transducer/s operation is to achieve the maximal input power factor (cosine theta) of 1, or 0.98, or always higher than 0.9 (meaning that ultrasonic generator is consuming only the real and active power (when operating full power on its power maximum), and that input electrical energy conversion to ultrasonic energy is maximal). This will be achieved by choosing the proper inductive compensation, proper output transformer ratio, and proper filtering capacitor in the DC power diode-bridge section (input main supply AC side). When using modern PFC and PWM controller circuits for input DC power control, we can achieve the best power conversion.


4.4-4

11. Every single cleaning transducer can accept the input RMS high frequency current, It, in the range given by the following relation:

(3 until 5) V

100 tI (Amps,RMS) (3 until 5) V

50 t(for PZT4, use 3, and for PZT8 use 5),

V total piezoceramics volume for one single transducer ( ) cm

t thickness of the single piezoceramic ring ( ) mm

t

3

⋅⋅

≤ ≤⋅

⋅

= =

= =

The total transducers current for the group of transducers in parallel connection (operating on series resonance) is equal to the above calculated current multiplied by the number of transducers. For welding transducers operating on parallel resonant frequency, the similar input transducer current criteria can be expressed by the equation:

I (Amps,RMS) (30 until 50) V200 t

(for PZT4 Sonox P - 4, use , and for PZT8 use ),



t

3

≤⋅

⋅≈ ≈

= =

= =

30 8 50SonoxP −

If welding transducer is operating on its series resonant frequency, use the criteria:

(30 until ) V100 t

I (Amps,RMS) (30 until ) V50 t

(for PZT4 Sonox P - 4, use , and for PZT8 use ),



t

3

50 50

30 8 50

⋅⋅

≤ ≤⋅

⋅≈ ≈

= =

= =

SonoxP −

12. Continuous and safe operating temperature of modern (PZT) piezoceramics (in ultrasonic

cleaning) should be lower than 100°C (sometimes up to 120°C, for high temperature resistant piezoceramics), depending of water temperature in ultrasonic tank, but always lower than 150°C (for the best piezoceramics, found on the market at this time). It is recommendable to operate ultrasonic transducers on a lowest possible temperature. For welding transducers (when operating continuously in air) the piezoceramic temperature should not be higher than 50°C to 60°C (measured after very long time). The continuous operating temperature of piezoceramics can be measured using remote non-contact temperature infrared (laser beam) thermometers (directing the laser beam to the ceramics area).


4.4-5

ELECTRICAL SAFE OPERATING LIMITS OF PIEZOCERAMICS (PZT-8 GRADE)

(All values are in RMS and SI units)

Definitions of Piezoceramic parameters (used in T.1), which are important for transducer power calculation (valid for piezoceramic rings):

Geometry parameters: t – thickness of the ring (=) [m] , R – outer radius of the ring (=) [m] , r – inner radius of the ring ID (=) [m] , S1 – surface of one, flat side, of the ring = π(R2 – r2) (=) [m2] , V1 – volume of one piezoceramic ring = S1t = π(R2 – r2)t (=) [m3] , n –Total number of PZT elements (rings) in assembled transducer (n= 2, 4, 6…). Applied voltage and current (directly on a piezoceramic): ut ≤ 0.5 ud – acceptable operating (rms.) voltage on a piezoceramic (=) [v], ud – depolarization voltage (=) [v]

[ ] )()(

)()(2

/,)('9.0*

,)(,)('

21

1

rmscurrentInputtransducerTotalAI

desPZTelectroaondensitycurrentSurfaceAveragemA

nSII

icpiezoceramaonappliedfieldoperatingloadingContinuousmVuu

icpiezoceramoneonfieldelectricalappliedAcceptablemV

tuu

t

ts

tt

tt

−=

−⎥⎦⎤

⎢⎣⎡=

⋅=

−⎥⎦⎤

⎢⎣⎡=⋅≤

−⎥⎦⎤

⎢⎣⎡==

Specific ultrasonic power density:

PZTaofsideflatoneondensitypowersurfaceAveragemW

nSP

P

densitypowervolumetricAveragemWIu

nVP

P

crS

stcr

v

−⎥⎦⎤

⎢⎣⎡=

⋅=

−⎥⎦⎤

⎢⎣⎡=⋅==

−

−

21

max)/(1

3111

max)/(

)(2

,)(21

Average resonant frequency (applicable for power calculation): (For the purpose of calculation and modeling, fres is taken between two characteristic resonant frequencies) fres = (f1 + f2)/2 = (fp + fs)/2 (=) [Hz]

ELECTRICAL SAFE OPERATING LIMITS OF BOLTED LANGEVIN TRANSDUCERS The following table, T.1, is applicable on single and free (in air) ultrasonic (sandwich) transducers that are not bonded/glued to some other solid-state mass (data established based on measurements for the operating frequency fres. ≅ 20 kHz ≅ (f1 + f2) / 2, and operating peak-to-peak, sinusoidal amplitude app = 20


4.4-6

µm). Later on, we shall extrapolate this situation to other operating frequencies and amplitudes (as well as for cleaning transducers that are glued to a cleaning tank). T.1

BLT is operating in series resonance: fs = f1Current driven transducers, Z = Zmin.

BLT is operating in parallel resonance: fp = f2Voltage driven transducers, Z = Zmax.

Non-Loaded

(in air) Fully-Loaded

(in water) Non-Loaded


(in water) n.a. Pv = 60 [W/cm3]

for Pr-max.

n.a. Pv = 60 [W/cm3] for Pr-max.

n.a. Pv = 30 [W/cm3] for Pc-max.


n.a. Ps1 = 30 [W/cm2] for Pr-max.


n.a. Ps1 = 15 [W/cm2] for Pc-max.


Pvdiss. = 0.23 – 0.60 [W/cm3]

Pvdiss. = 0.92 – 2.77 [W/cm3] Pvdiss. = 0.11 – 0.35 [W/cm3]

Pvdiss. = 0.39 – 0.46 [W/cm3]

u’t = 1.3 - 10 [V/mm] u’t = 10 - 100 [V/mm] u*t = 50 - 60 [V/mm]

u’t = 50 - 220 [V/mm] u*t = 200 [V/mm]

u’t = 50 - 220 [V/mm] u*t = 200 [V/mm]

Is1max. = 80 [A/m2] Is1 = 80 - 1200 [A/m2] Is1max. = 20 [A/m2] Is1 = 20 - 300 [A/m2] n.a. max.(u’t Is1) ≤ Pv n.a. max.(u’t Is1)≤ Pv

(Zmin.S1/t) ≤ (0.65 to 5) [Ωm]

n.a. (Zmax.S1/t) ≥ (33 to 135) [KΩm]

n.a.

Pr-max. [Repetitively driven, Fully-Loaded (in water)] (=) Ton : Toff = 1 : 1, pulse-repetitive train, Pr-max. = 2 Pc-max.Pc-max. [Continuously driven, Fully-Loaded (in water)] (=) Ton = continuous, Toff = 0 Pdiss. (=) Thermal-dissipation power (losses) u’t – For variable voltage regulation, u*t – Constant (stable) voltage for continuous operating regime n.a. = Not applicable In the case if transducer/s is/are bonded/glued or differently fixed to some other solid-state mass, above given (safe-operating) maximal power limits are 5 to 10 times lower (as in the case of ultrasonic cleaning applications). The current and voltage limits (operating intervals and safe operating limits) for piezoceramics will stay the same in any situation, providing that maximal power limits are respected. If transducer/s is/are operating in some other, frequency sweeping regime, passing over many series and parallel resonant points, all safe operating limits listed in T. 1 should be respected in the same time (max. Applied voltage, max.-Input-current, max.-Power-density). The corresponding electrical limitations, ultrasonic generator regulations and protections should be implemented. The most important criteria in such situations are to minimize reactive power of the transducer and to maximize its active power. Operating temperature of piezoceramic elements (in any operating regime) should be lover as possible (we can say between 20°C and 60°C is preferable situation, but in any case, for continuous and very long operation, no more than 90°C). If piezoceramics in operation reach temperatures between 150°C and 180°, efficiency of transducer will drop significantly. Generally valid (semi-empirical) formula for calculating the output power of one sandwich, ultrasonic transducer (Bolted Langevin transducer = BLT) is:

P a f P P Const Q MM

d u f

a Transducer s front emitting surface peak to peak amplitudemeasured when transducer isoscillating harmonically

output pp res r c em effPiezo

convertert res

pp

= ⋅ ⋅ = = ⋅ = ⋅

= − −

− − −η 2 233

2 22. max. max. .. ( )

( ),' ,

( )

factor 2 is taken as the security margin

sin. oscilating function− ,


4.4-7

The most important quality parameters of power ultrasonic converters Static & Low signal parameters; Low power parameters (converter is operating in air = no-

load) C at kHz High enough nFinp. ( )1 → Total input capacitance

E max. N / m 2→ Piezoceramics Young modulus

Q of assembled, non- loaded converterQ of single, non- loaded piezoceramics max.mo1,2

mo1,2

((

≥

≥

))→

Relations between Mechanical Quality Factors of assembled, non-loaded converter and Quality factor

of piezoceramics (that is the part of the same converter)

Q* Q Q max.,( 1000 , for instance)

mo mo1 mo2= ⋅ →

≥ Effective mechanical quality factor

(here invented parameter)

Q * Q * Q max.,( 1000 , for instance)

emo mo e= ⋅ →

≥ Effective electromechanical quality factor (here

invented parameter)

tan minimumδ ( )at kHzQe

1 1= → Dielectric loss factor

Qe = Dielectric quality factor

d High enough, 10 mV

CN33

12→ ⋅LNMOQP

− = Piezoelectric charge constant

Z1000 Z

R1000 R

max.

( 100, for

max.

min.

2

⋅≅

⋅→

≥1

,

example) Maximum to minimum impedance ratio

T High enough (max.T > 300) , Cc c→ ° Curie temperature

∆∆

f f ff

f

= − →

⋅ ≥

( )

. %

2 1

100 10

max.

for ex : Frequency gap between series and parallel resonant frequency

Dynamic parameters in resonance

Power parameters (converter is operating in air = no-load) f f series resonanceres. ,= 1 f f parallel resonanceres. ,= 2

D Output ForceMotional Current

Fi

f ai

f ai

f aP

u

f aP

u

Im

m

m inp inpinp

dissinp

= =

= ≅ =

≅ →

4 4 4

4

21

21

21

21

21

21

21

21

π π π

π

. ..

.. maximum.

=

D Output VelocityMotional Voltage

Vu

f au

f au

f aP

i

f aP

i

Um

m

m inp inpinp

dissinp

= =

= ≅ =

≅ →

2 2 2

2

2 2 2 2 2 2

2 2

π π π

π. .

.

.. maximum

=

a output amplitudei i motional currentiuP P

m inp

inp

inp

inp diss

1 =≅ =

=

=

≅ =

.

.

.

. .

converter's input currentconverter's input voltage

converter's dissipation in air

a output amplitudeu u motional voltageiuP P

m inp

inp

inp

inp diss

2 =≅ =

=

=

≅ =

.

.

.

. .


converter's dissipation in air


4.4-8

d Output AmplitudeMotional Current

ai

ai

Df

Im inp

I

= =

= →

1 1

21

24

.

πmaximum.

≅

dOutput AmplitudeMotional Voltage

au

au

Df

Um in

U

= =

= →

2 2

22

.

πmaximum

p

≅

f f series resonanceres. ,= 1

u Vmmt ≈LNMOQP50

f f parallel resonanceres. ,= 2

u Vmmt ≈LNMOQP200

P a f

d i fD i

fconst D i

f

Const Q MM

d u f

const

output pp

I m

I m I m

em effPiezo

convertert

= ⋅ ⋅ =

= ⋅ ⋅ ⋅ =

= ⋅ = ⋅ =

= ⋅

=

−

η

η

ηπ

η

2 21

21

2

2

41

21

2

332

12

16

.

( )( )

( )

. ( ) ,

( .)

P a f

d u fD u const D u

Const Q MM

d u f

const

output pp

U m

u mU m

em effPiezo

convertert

= ⋅ ⋅ =

= ⋅ ⋅ ⋅ =

= ⋅ = ⋅ =

= ⋅

=

−

η

η

ηπ

η

2 22

22

2

22

332

22

4

.

( )( )

( )

. ( ) ,

( .)

The most important is to respect the Safe Operating Limits of piezoceramics, as the part of BLT structure, given in T. 1.


5-1

5. Power Performances of Ultrasonic Converters

Generally valid (semi-empirical) formula for calculating the average output power of one sandwich ultrasonic converter, when radiating into resistive mechanical load (for instance radiating in water), could be formulated as: P a f P P

a d u nMM

Q f

output pp res r c

pp tpiezo

transdem res

= ⋅ ⋅ = = =

= ⋅ ⋅ ⋅

− −η

η

2 2

233

2

2. max. max.

..( , , , * ) ,

(1.14)

where η is the converter (geometry and design related) form factor, η η= = ⋅ ⋅ =( * , , ) ( ) * , ( ). . .Q f f C f Q f C femo res res emo res∆ ∆ Const. Pr−max. is the maximal acceptable pulse-repetitive (short time ON) converter power, Pc−max. is the maximal acceptable continuous converter power (long time ON). The factor 2 as the relation between P and , is taken as the (overload) security margin, and can be verified empirically, meaning that high power converter can safely deliver about 2 times higher output power ( ) in a short time intervals, but if we would like to operate the same converter continuously ( P ), we better reduce its operating power to a half of P . For instance, we know that BRANSON 502/932R converter is able to deliver 3000 Watts (=P ) in a short time intervals, but if we would like to operate it safely and continuously, very long time, we better limit its power to 1500 Watts (or maximum 2000 Watts =P , with proper air cooling).

r−max. Pc−max.

Pr−max.

c−max.

r−max.

r−max.

c−max.

- a a d u nMM

Q n app pp tpiezo

transdem pp= ⋅( , , , * )

.33 0 1= ⋅

n

f or f f

-is converter’s front-emitting-surface,

peak-to-peak amplitude measured when converter is oscillating harmonically. - -is converter’s average, front-emitting-surface, peak-to-peak amplitude calculated as the contribution of only one piezoceramic ring assembled in the same converter ( is the total number of piezoceramic rings assembled in a converter = 2,4,6…).

a app pp1 = /

n

f f

f f f f fres s p

p s

. ( )= = ≅ ≅

= − = − =

resonant frequency of assembledconverter(measuredin air converter is not under load)

1 2

2 1∆

Q Q Q

Qtg

Q Electricalquality factor of assembled converter in air no load at kHz

Q Q Q Effective electromechanical quality factor

mo mo mo

e eo

emo mo e

*

( ,

* *

= = =

= = = = −

= =

1 2

1 1

Mechanical quality factor of assembledconverter (in air non - loaded)

δ)

M n M Total massof allpiezo elements in one assembled converterM V Mass of onlyone piezoceramic element

piezo = ⋅ =

= ==

1

1 1ρV Volume of one piezo element1

5-2

n Total number of piezoelements (rings) in assembled converterTotal mass of assembled converter

== + =

=

M M MM Total mass of metal parts elements of assembled converter

transd piezo metal

metal

.

/

Example: BRANSON 502/932R, welding converter operating in parallel resonance BRANSON converter model 502/932R (see Fig. 17) will serve only to help us to calculate (or estimate) empirical constants of approximate formulae for power calculation, (1.14), since for this converter all relevant power and model/s parameters are already well known.

Fig. 17 BRANSON converter model 502/932R

T 1.5 BRANSON 502/932R, Typical Model Parameters Variations

In Series Resonance In Parallel Resonance Model Parameters

for Non-Loaded Converter

(Measured on the random, standard-production-quality sample > 100 pcs.

of converters, taken after

assembling)

C nF

C nF

L mH

R

f Hz

Q Q

p

m m

0

1

1

1

1

01 01

15 3 18 1 3

3 92 4 05

17 53 18 7

175 4 6 20

18435 18905 0 5

20

∈ − ±

∈ −

∈ −

∈ − ±

∈ − ±

∈ ±

. . , %

. . ,

. .

. . , %

, . %

%

Ω

C nF

C nF

L H

R K

f H

Q Q

s

m m

0

2

2

2

2

02 02

18 7 22 05 3

79 10153

570 50 747

94 250 20

20635 20912 0 5

20

∈ − ±

∈ −

∈ −

∈ − ±

∈ − ±

= ±

. . , %

. ,

. ,

, %

, . %

%

µ

Ω

z

For instance, one of well-selected (excellent quality) BRANSON 502/932R converter, Fig. 17, has the following characteristics (see T 1.6, T 1.7 and T 1.8).

5-3

T 1.6 Parameters and Properties of the Well-Selected 502/932R Converter Ton : Toff = 1:1 (Pulse-Repetitive, ON-OFF regime in parallel resonance, ) 02res. ff ≅app = 20 µm = 6 x 3.33* µm = 6 x a1pp a1pp = (20/6) µm = 3.33* 10-6 m = 0.66* t 10-9 [m], (f1+f2)/2 ≅ 20 kHz , ∆f = f2- f1 ≅ 2 kHz, (ut)rms. 1500 V, E≤ t = ut /5≤ 300 V/mm ( = voltage & field on the converter input) (um)rms. = 930 V, Em = um /5 ≤ 186 V/mm ( = converter’s motional voltage & field) Poutput = 3000 W = Pr-max. = Maximal pulse repetitive output power under load Pd02 = 5 W = Power dissipation in air (non-loaded converter; more often Pd02 ≅ 10 W)

[ ]W)(fa18750faηP2PP

msW)(18750

)102010(203000

faP

faP

res.2

pp2

res.2

pp2

max.cmax.routput

2

2

236res.

2pp

2max.r

res.2

pp2

output

=⋅⋅=⋅⋅=⋅==

⎥⎦

⎤⎢⎣

⎡ ⋅==

⋅⋅⋅===

−−

−−η

(see eq. (1.14))

n = 6 piezoceramic rings: ≅ φ50 x φ20 x 5 mm (PZT-8, Vernitron – Morgan-Matroc) t = 5 mm = 0.005 m, S1 = 16.5 cm2 = 16.5 10-4 m2, V1 = S1 x t = 8.25 10-6 m3, V = 6V1 = 49.5 10-6 m3 = total volume of piezoceramics d33 = 245 x 10-12 [v/m] ± 10% , k33 = 0.640, Cinp.(1 kHz) = 19.55 nF, tg δ (1 kHz) = 0.000279 Rs(1 kHz) = 56 Ω, Rp(1 kHz) = 52 MΩ C nFC nFL mHRZf HzQ

p

m

0

1

1

1

01

1

01

17 324013 904418 3335187

19818815

1139

=

===

=

==

. ,. ,.

. ,. ,

,

Ω

Ω

C nC nL HR KZ Kf HzQ

s

m

0

2

2

2

02

2

02

21 22897 08423595 6216191

208282485

====

=

==

. ,. ,

. ,,

,

µΩ

Ω

FF

, Z

ZR

R02

01

2

11000 100096 46 115 51≅ ≅ ( . , . )

Q*mo = 1682, Qe = 3584 = 1/tg δ, Q*emo = 2455

kEnergystored in mechanical form

Total input energy


(The ideal converter design would be when (k .

c2

c

= =+

≅ ⋅ ≅ ≅

≅ ≅

≅

CC C

CC

ff

k

kk

k

op

osc

c

1

1 2

12

22

33

33

0 1811 0 4256

0 42560 640

0 665

1

. , .

..

. ,

/ ) )

=

,

5-4

If we simply apply DC (or low frequency AC) voltage to the electrical input of BRANSON 502/932R converter, in the acceptable upper voltage limits u t ∈ 930 1500, V , we shall produce converter’s total elongation proportional to:

∆

∆

∆

x d u u u d ud u or

d u

u V u x u m mu V u x u m m

t t t converter tconverter m

converter m

t m m

t m m

≅ ⋅ ⋅ = ⋅ ⋅ ⋅ = ⋅ ⋅ = ⋅ =⋅

⋅ ⋅

RST= = ⇒ ≅ ⋅ ⋅ = ⋅ =

= = ⋅ ⇒ ≅ ⋅ ⋅ ≅ ⋅ =

− −−

−

−

− −

− −

6 6 225 10 1350 1016129

930 1350 10 1 2555 10 1 225551500 16129 2177 419 10 2 025 10 2 025

3312 12

3333

33

12 6

12 6

,

.

. .. . . . .

µ

µ The same situation, when converter is driven with the same high voltage in parallel resonance ( u V um t= ∈930 930 1500, , V ) will result in converter peak-to-peak amplitude equal to:

a x u u d ud ud u

u V x m d

pp m m resonant mresonant t

resonant t

m resonant

= = ⋅ = ⋅ ⋅ = ⋅ =⋅⋅ ⋅

RST= ⇒ = ⋅ ⋅ = =

− −−

−

−

−−

∆

∆

20930

10 21505 376 100 62

930 21505 376 930 10 20 21505 376

6 1233

33

33

1233

.,

.

. , ( .µ

or

) .

Obviously, the voltage-amplitude conversion factor for BRANSON 502/932R converter in parallel resonance (21505.376) is for the order of magnitude greater, comparing to the voltage-amplitude conversion factor (2177.419, or 1350) when converter is driven the same voltage and far from resonance area (let us say below 1 kHz): 21505.376/2177.419 = 9.8765; 21505.376/1350 = 15.93. It looks that d33 constant that is valid for non-resonant (low or DC) frequency area of piezoceramic (here for PZT8, d33-converter = d33 x 6 = 225 x 6 = 1350) should be multiplied by the factor close to 10 (or a bit higher than 10), in order to get d33-resonant valid for parallel resonance: d33-

resonant ≥ 10 d33-converter . This numerical example is given only for the purpose of creating an approximate picture of the order of magnitudes of converter amplitudes in DC and low frequency operating conditions, and operations in parallel resonance. Similar calculation (with similar conclusions) should be applicable for converter operating in series resonance, but this time important amplitude conversion factors will be linked to converter motional current (as well as to converter input current).

5-5

T 1.7 Dynamic Power Parameters In Resonance (operating in air = no-load)

f fres. ,= 1 u u Vmmt inp= ≤ LNMOQP. 50 f fres. ,= 2

u u u Vmm

u Vmm

m t inp

m

≤ = ≤ LNMOQP

≤ LNMOQP

. 300

186

a output amplitudea a peak to peaki iiuP P

F

pp

m inp

inp

inp

inp diss

m

1

1 12== = − −

≅ =

=

=

≅ =

==

.

.

.

. .

motional currentconverter's input currentconverter's input voltage

converter's dissipation in airV converter's motional velocity

converter's motional forcem

a output amplitudea a peak to peaku u motional voltageiuP P

F

pp

m inp

inp

inp

inp diss

m

2

2 22== = − −

≅ =

=

=

≅ =

==

.

.

.

. .


converter's dissipation in airV converter's motional velocity

converter's motional forcem

5-6

Total Output Power Under Load P a f

d i fuD

Const QM

Md u f

const

output pp

I minp

IF

emoPiezo

convertert

= ⋅ ⋅ =

= ⋅ ⋅ ⋅ = ⋅ =

= ⋅

= ≅⋅L

NMOQP

η

η ηπ

η

12 2

1

12

12

12

332

12

1

2( ) ( )

. ( * ) ,

( .

)

.

18750 W sm

for BRANSON 502 / 932R

2

2

Total Output Power Under Load P a f

d u fD u

Const QM

Md u f

const

output pp

U mUV m

emoPiezo

convertert

= ⋅ ⋅ =

= ⋅ ⋅ ⋅ = ⋅ =

= ⋅

= ≅⋅L

NMOQP

η

η ηπ

η

22 2

2

22

22

22

332

22

2

2( ) ( )

. ( * ) ,

( . ,

)

18750 W sm

for BRANSON 502 / 932R

2

2

Force transformation constant

DOutput Force

Motional CurrentFi

uf d i

uf a

PV i

Pf a i

s

N for

IFm

m

inp

I m

inp

pp

diss

m m

diss

pp m

= = =

≅ ≅ ≅⋅

≅

≅⋅

→

≅ ⋅ ⋅ ≅⋅

=

2 2 2

2

2 216 18815 20 36 54

36 54

1 1 1

1 1

. . .

.

. .

. ,

π π

π

π

maximum

( /(V)

m

(A) 502 / 932R)

rms

rms

Velocity transformation constant

DOutput Velocity

Motional VoltageVu

f au

f au

f d

sfor

UVm

m

m

pp

mU

= = =

= = = →

≅ ⋅ ⋅ ≅ ⋅⋅

−

2

20828 1407 2 10

2 2 2 22

6

π ππ

π

max.

( 20 / 930 m(V)

BRANSON 502 / 932R)rms

.

Amplitude transformation constant

dOutput AmplitudeMotional Current

ai

ai

uf D i

Ipp

m

pp

inp

inp

IF m

= = ≅

≅ →

≅ ≅

1 1

1

2.

.

(, )

πµ

maximum.

( 20 / 2.98 6.71m)

(A)for 502 / 932Rpp

rms

Amplitude transformation constant

dOutput AmplitudeMotional Voltage

au

au

Df

Upp

m

pp

inp

UV

= = ≅ =

= →

≅ ≅

2 2

22

02

.

.(

, )

πµ

maximum

( 20 / 930 0m)

(V)for 502 / 932Rpp

rms

5-7

T 1.8 Additional Power Parameters In Resonance (converter is non-loaded)

f fres. ,= 1 u u Vmmt inp= ≤ LNMOQP. 50 f fres. ,= 2

u u u Vmm

u Vmm

m t inp

m

≤ = ≤ LNMOQP

≤ LNMOQP

. 300

186

Velocity transformation constant

DOutput Velocity

Motional CurrentVi

PD i

PD i

ms

P W for

IVm

m

diss

IF m

diss

IF inp

diss

= = =

≅⋅

≅⋅

→

≅ ⋅ ≅⋅

≅

. .

.

.

. .

. ,

2 2

264 368 36.54 2 98 0 1984

64 368

maximum

( / .(A)

502 / 932R)rms

Force transformation constant

DOutput Force

Motional VoltageFu

Pu D

Pu D

mN

VP W for 502

UFm

m

diss

m UV

diss

inp UV

rms

diss

= = =

≅⋅

≅⋅

→

≅ ⋅ ⋅ ≅

≅ ⋅⋅

= ⋅

≅

−

− −

. .

.

.

/ .

. .( )

,

2 2

2 6

3 3

10 930 1407 2 10

8 82 10 8 82 10

10

maximum

((A) s

/ 932R)

rms

All voltages and currents in this table should be treated as RMS values. Tables T 1.7 and T 1.8 are only establishing the definitions of important dynamic converter parameters (related to low signal and low power converters modeling), and it is obvious that such parameters should be conveniently re-defined in the function of converter’s loading (starting from no-load situation until a fully loaded converter).

5-8

ELECTRICAL SAFE OPERATING LIMITS OF BOLTED LANGEVIN TRANSDUCERS (Piezoceramics PZT-8 GRADE, All values are in RMS, and SI units)

The following table, T.1, is applicable on single and free (in air) ultrasonic (sandwich) transducers that are not bonded/glued to some other solid-state mass (data established based on measurements for the operating frequency fres. ≅ 20 kHz ≅ (f1 + f2) / 2, and operating peak-to-peak, sinusoidal amplitude app = 20 µm). Later on, we shall extrapolate this situation to other operating frequencies and amplitudes (as well as for cleaning transducers that are glued to a cleaning tank). T.1

BLT is operating in series resonance: fs = f1Current driven transducers, Z = Zmin.

BLT is operating in parallel resonance: fp = f2Voltage driven transducers, Z = Zmax.

Non-Loaded


(in water) Non-Loaded


(in water) n.a. Pv = 60 [W/cm3]

for Pr-max.

n.a. Pv = 60 [W/cm3] for Pr-max.







Pvdiss. = 0.23 – 0.60 [W/cm3]

Pvdiss. = 0.92 – 2.77 [W/cm3]

Pvdiss. = 0.11 – 0.35 [W/cm3]

Pvdiss. = 0.39 – 0.46 [W/cm3]

u’t = 1.3 - 10 [V/mm] u’t = 10 - 100 [V/mm] u*t = 50 - 60 [V/mm]

u’t = 50 - 220 [V/mm] u*t = 200 [V/mm]

u’t = 50 - 220 [V/mm] u*t = 200 [V/mm]

Is1max. = 80 [A/m2] Is1 = 80 - 1200 [A/m2] Is1max. = 20 [A/m2] Is1 = 20 - 300 [A/m2] n.a. max.(u’t Is1) ≤ Pv n.a. max.(u’t Is1)≤ Pv

(Zmin.S1/t) ≤ (0.65 to 5) [Ωm]

n.a. (Zmax.S1/t) ≥ (33 to 135) [KΩm]

n.a.

Pr-max. [Repetitively driven, Fully-Loaded (in water)] (=) Ton : Toff = 1 : 1, pulse-repetitive train, Pr-max. = 2 Pc-max.Pc-max. [Continuously driven, Fully-Loaded (in water)] (=) Ton = continuous, Toff = 0 Pdiss. (=) Thermal-dissipation power (losses) u’t – For variable voltage regulation, u*t – Constant (stable) voltage for continuous operating regime n.a. = Not applicable In the case if transducer/s is/are bonded/glued or differently fixed to some other solid-state mass, above given (safe-operating) maximal power limits are 5 to 10 times lower (as in the case of ultrasonic cleaning applications). The current and voltage limits (operating intervals and safe operating limits) for piezoceramics will stay the same in any situation, providing that maximal power limits are respected. If transducer/s is/are operating in some other, frequency sweeping regime, passing over many series and parallel resonant points, all safe operating limits listed in T. 1 should be respected in the same time (max. Applied voltage, max. Input current, max. Power density). The corresponding electrical limitations, generator regulations and protections should be implemented. The most important criteria in such situations are to minimize reactive power of transducer and to maximize its active power. Operating temperature of piezoceramic elements (in any operating regime) should be lover as possible (we can say between 20°C and 60°C is preferable situation, but in any case, for continuous and very long operation, no more than 90°C). If piezoceramics in operation reach temperatures between 120°C and 180°, efficiency of transducer will drop significantly.

5-9

Piezoceramic parameters (used in T.1), which are important for transducer power calculation (valid for piezoceramic rings):

t – thickness of the ring (=) [m] , R – outer diameter of the ring (=) [m] , r – inner diameter of the ring (=) [m] , S1 – surface of one, flat side, of the ring = π(R2 – r2) (=) [m2] , V1 – volume of one piezoceramic ring = S1t = π(R2 – r2)t (=) [m3] , ut – rms. voltage on one piezoceramic (=) [v], fres. – resonant frequency (=) [Hz]

densitycurrentsurface,sideflatonemA)(

SII

,icpiezoceram)one(onfieldelectricalmV)(

tuu

,densitypowersurface,sideflatonemW)(

SP

P

,densitypowervolumetricmW)(

VP

P

21

1s

1t

21

max)c/r(1S

31

max)c/r(v

−⎥⎦⎤

⎢⎣⎡==

−⎥⎦⎤

⎢⎣⎡==

−⎥⎦⎤

⎢⎣⎡==

−⎥⎦⎤

⎢⎣⎡==

−

− Applied voltage and current (directly on a piezoceramic): ut ≤ 0.5 ud – acceptable operating (rms.) voltage on a piezoceramic (=) [V], ud – depolarization voltage (=) [V]

[ ] )()(

)()(2

/,)('9.0*

,)(,)('

21

1

rmscurrentInputtransducerTotalAI

desPZTelectroaondensitycurrentSurfaceAveragemA

nSII

icpiezoceramaonappliedfieldoperatingloadingContinuousmVuu

icpiezoceramoneonfieldelectricalappliedAcceptablemV

tuu

t

ts

tt

tt

−=

−⎥⎦⎤

⎢⎣⎡=

⋅=

−⎥⎦⎤

⎢⎣⎡=⋅≤

−⎥⎦⎤

⎢⎣⎡==

For rough and fast estimation of the quality and type of piezoceramics (applicable for

high power BLT), we can use the following empirical criteria: PZT-8 , SP-8 TYPE III

DOD-STD-1376A (Vernitron USA)

Excellent

Very good

Good

Acceptable

ZZ

max.

min.1000≥

For single piezoceramics, non-loaded, in air

80

36

18

7

ZZ

max.

min.1000≥

For assembled converter, non-loaded, in air

100

75

50

20

5-10

Piezoceramic materials that have higher dielectric constants and higher (low frequency) static capacitance are generally preferable for designing high power ultrasonic transducers. Also important parameters for high power transducers are: very high Young modulus, high mechanical quality factors, very fine ceramic granulation, very low porosity of sintered piezoceramics, high d33, low dielectric losses and high Curie temperature.

6-1

6. IMPORTANT COMMENTS REGARDING HIGH POWER BLT ASSEMBLING, PIEZOCERAMICS QUALITY AND TOLERANCES (BLT = Bolted Langevin Transducer)

For producing high power ultrasonic transducers it is necessary to use “hard” Piezoceramic materials (such as PZT-8 and SP8). Hard piezoceramic materials have mechanical quality factor higher than 1000 (often between 1000 and 2000). There is also a big difference among hard piezoceramics (from different sources) regarding implemented internal polarization. It is preferable to ask suppliers of piezoceramics to use maximal as possible, high DC voltage for implementing very strong internal electrical polarization-field in piezoceramics. Piezoceramics in process of implementing internal polarization are permanently connected to a very high voltage DC source, immersed in silicone oil (or some other electrically isolating liquid), and kept on elevated temperature during the poling process. After certain time, internal dipole domains of piezoceramics are sufficiently well oriented in the direction of externally applied electric field, and in order to permanently stabilize and freeze such internal polarization, oil heating is disconnected and piezoceramics and surrounding oil start cooling down, slowly, naturally, until reaching room temperature, or lower, without disconnecting the high DC polling voltage, during more than 24 hours. If piezoceramics producers were trying to accelerate this process and to reduce the costs of pooling, this would result in weaker internal polarization (consequently reducing the maximal power of assembled ultrasonic converters). A good, high-DC polarization-voltage should be close to the piezoceramic material breakdown-voltage. The next difference between hard piezoceramics from different sources is in values of materials Young Modulus, which should be sufficiently high (comparable to Vernitron-USA, PZT-8 material, and to CeramTec SP8). Higher Young Modulus will produce higher power BLTs, if all other BLT design elements are well selected. Piezoceramic materials that have higher dielectric constants and higher (low frequency) static capacitance are generally preferable for designing high power ultrasonic transducers. Also important parameters for high power transducers (in addition to very high Young modulus) are: high mechanical quality factors, as the consequence of very fine ceramic granulation and very low porosity of sintered piezoceramics (what basically makes high strength of piezoceramics), high d-33, low dielectric losses, low aging rate and high Curie temperature. In order to accelerate natural aging of “green” piezoceramics, many suppliers are applying additional thermal aging (or thermal cycling). The best artificial aging can be realized when piezoceramic is externally in electrical short circuit, during thermal cycling, and in the same time mechanically agitated (placed on some variable-frequency, vibrating platform). Such aging arrangement should be discussed and arranged with piezoceramic supplier. There are some more innovative and excellent methods for piezoceramic artificial aging and stabilization, but not as easy applicable, as above described methods. Piezoceramic, if not well homogenized and properly sintered, could have structural defects, voids, certain porosity, non-uniform internal-stress distribution etc. The best option to check piezoceramic quality from that aspect (structural stability) is, to perform (suitable) variable-frequency mechanical agitation (on a convenient vibrating platform), when piezoceramics containing internal structural defects will brake (this way being naturally eliminated). Practically vibrations are internally accelerating the process of structural space stabilization of polarized domains, and accelerating the migration and mobility of internal defects and structural imperfections (until all of them reach final and stable conditions). Development of modern ultrasonic generators (or power supplies) gives us the chance to use hard piezoceramic materials, not only for ultrasonic welding BLTs, but also for ultrasonic cleaning BLTs. This is possible because modern power regulation applies PWM concept for power regulation, operating frequency is automatically controlled using fast PLL, overload protection is reacting very fast, active power is maximized, reactive power minimized, etc.


6-2

The next important aspect of making high power BLTs is to know the how to determine the optimal prestress on piezoceramics (realized by central bolt) in order to have the highest operational margin for sinusoidal oscillations. Usually, on “soft” piezoceramic materials we apply static pressure on piezoceramics in the range of 25 MPa (valid for PZT-4, SP4), and for hard piezoceramics around 50 MPa (valid for Vernitron PZT-8 and for CeramTec Sonox P8, P88 materials). For any other piezoceramics, optimal prestress should be found in collaboration with piezoceramics supplier, or experimentally. We should also have in mind that every piezoceramics is much more resistant on positive, compressive-pressure than on negative, elongating-pressure (about 5 times). Applying compressive pressure (to a piezoceramic stack of a BLT) in the range of 60 MPa and higher is not recommendable. Most of piezoceramics for high power BLTs should be fastened until no more than 50 MPa. Braking or cracking point of CeramTec SP8 is +125 MPa on compressive pressure, and –25 MPa on elongation. The total length (or available pressure gap) of this interval (for SP8) is 125 + 25 = 150 MPa (where 125 MPa is below zero line and 25 MPa is above zero line). The middle pressure point of 150 MPa is 75MPa, and since, negative pressure of 25 MPa (25 MPa is the limit of piezoceramics maximal elongation or extension) should be reduced from 75 MPa, we can calculate the best optimal (static, and positive) pressure on SP8 as 75 – 25 = 50 MPa. In reality, any pressure between 40 MPa and 50 MPa, applied on SP8, and/or PZT-8 is very good, because BLT should also be able to oscillate linearly in the rest of allowed pressure interval (without damaging piezoceramics). The highest prestress pressure-limit of 50 MPa is also producing higher power BLTs. Branson is applying 51 MPa for the highest power converters. Also: 1°, as the prestress increases, the Curie temperature decreases; 2°. The prestress must be large enough to handle not only the ultrasonic stress in the ceramics due to normal axial vibration but also any stress that is caused by bending (e.g., due to an asymmetric horn). In order to determine the optimal static pressure on certain BLT we should implement the following process: First, we shall produce a test-BLT and gradually start increasing the pressure on its piezoceramics (making impedance measurements whenever we change the static prestress). During fastening process, step-by-step, we are measuring different resonant frequencies and different belonging impedances of a test-BLT, depending on applied pressure. We should make the table of measured points and produce corresponding curves using the most suitable curve fitting method (in the function of applied pressure). When all (measured) characteristic impedances and frequencies reach their relatively stable, almost constant and saturated-like values, and when mechanical quality factor of such test-BLT reaches its maximum, we can be sure that this is the optimal static pressure for BLT under testing. The pressure found this way takes into account all design-specific parameters, quality of used piezoceramics, quality of metal masses and quality of central bolt. For such measurements we need HP-4194A Network Impedance Analyzer (or an equivalent instrument). If we continue with fastening (after optimal point is reached), applying stronger pressure, characteristic impedances, frequencies and belonging mechanical quality factor will stay relatively stable and constant only during certain (relatively short) pressure interval, and later on, with further pressure increase, mentioned BLT parameters will start changing (non-linearly) and degrading rapidly, until to the fatal ceramic or central bolt damage/s, or the bolt pulls the threads out of the front driver. Optimal static pressure should be chosen to be in the linear stability-interval of BLT parameters (in the first 50% of that interval, but not just on its starting point, because after certain time BLT will naturally loose about 10% of its initial static pressure (because of spontaneous static-stress release and natural aging of the converter). If we change something (significantly) in BLT design, above described measurement procedure should be repeated and new (optimal) static pressure should be found. Optimum bolt diameter (considering desired preload and strength of front driver threads); undercutting bolt diameter next to ceramics to increase bolt compliance. Note: if bolt is


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overly stiff, the coupling coefficient k is reduced. Optimum bolt engagement length in front driver (how many threads) don't want fatigue of Al front driver. Best bolt mat: steel vs. Ti? Optimum bolt-ceramic clearance (for arc protection). Undercutting bolt to give additional clearance while still retaining large threads required for static preload. Front driver material: Aluminum (as opposed to Ti) gives higher amplitude but has lower fatigue strength and front threads (where booster attaches) wear out more quickly. Some designs have the stud permanently installed in the converter, rather than the booster, to prevent wear to front converter threads. Design of BLT’s metal masses, surface finishing, position of piezoceramics and BLT nodal plane, size and quality of central bolt could have significant influence on static and dynamic stress distribution in the structure of BLT. In the case if piezoceramic contact surfaces are non-uniformly loaded (because of bad surface finishing, voids, surface defect…), this will produce non-uniform stress and oscillating-amplitude distribution and partial piezoceramic depolarization, followed by increased thermal dissipation. Such piezoceramic (non-uniformly loaded) will easily loose a part of its piezoelectric properties. In order to uniformly equalize mechanical loading, stress and oscillating-amplitude distribution on metal and ceramic contact surfaces, some ultrasonic companies are implementing relatively simple design tricks on all surfaces of BLT end-metal masses (that are in contact with piezoceramics). They counterbore the metal mass in the central zone, around the central bolt axis (often from both sides on both metal masses in contact with piezoceramics). This looks like reducing the total front surfaces of BLT masses, introducing one empty space zone in the middle, but as a result, stress and vibration-amplitude distribution on such (reduced surface/s) will be much more uniform than if surface has only one flat level. The same reasoning is applicable on the front emitting aluminum mass that is in contact with a booster or sonotrode (reduced contact area produces uniform amplitude and stress distribution on the remaining contact surface, exciting dominantly longitudinal vibration modes of a booster). The next good reason for introducing two axial-levels in contact surfaces (removing metal in the middle zone of contact area) is in minimizing the effects of coupling between longitudinal and radial oscillating modes, and to stimulate only axial-longitudinal oscillations, since radial, or planar surface coupling is minimized if the contact surface is minimized. To be sure in previously described design modifications (how and where to apply them), we should use the Finite Elements Analysis method for BLT design optimization. For electrode materials it is very important to use 99% nickel foils, or nickel-beryllium alloys, or some other nickel-rich alloys (which are relatively easy solderable, or where brazing can easily be applied, for creating good electrical contacts). Surface hardness of applied electrodes (empirically found as the best) should not be higher than 140 HV (Vickers) and not softer than 100 HV. Electrode-foil bands should be produced with cold drawing, without applying thermal treatment (resulting in relatively softer and more elastic alloys, with long fatigue-life). Electrode surfaces should be mirror and brilliant shiny, and extremely flat. Electrode thickness should be around 0.25mm. It is preferable not to apply any mechanical treatment on electrode surfaces (like polishing), because there is a danger to implement some hard particles in electrode surfaces (in any case, proper electrode cleaning should be applied in cases when any surface treatment is used). The best would be to buy an excellent foil material and to produce proper electrodes for BLT, without making any surface treatment. All solid surfaces that are in contact with piezoceramics (BLT metal masses, foil-electrodes, piezoceramic metal plating) should secure/maintain: -Maximum as possible contact surface/s, -Excellent acoustic and mechanical coupling, -Excellent heat transfer and very low thermal resistance,


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-Very low surface-electrical-resistance, or high electrical conductivity, -High flexibility and very low friction in planar-radial contact-zone, because different materials anyhow have different elasticity and different thermal expansion properties, -Very long fatigue-life under vibrations, -As well as high resistance against surface corrosion. Nickel is an excellent material for reaching the most of above listed performances. After assembling a BLT, when BLT is selected (after measurements) as a good one, we should apply a convenient surface coating using some protecting, liquid mass (rubber, adhesive, lacquer) that will not be able to make capillary penetration in the capillary zones of contact between piezoceramics and metal masses. For this purpose, some high quality (and relatively high density) transformer or magnet-wires lacquers could be used (usually part of the specific know-how). It is extremely important to protect external (lateral) surfaces of piezoceramics against moisture (since piezoceramic materials are known as slightly hygroscopic), without giving the chance to a coating mass (lacquer) to penetrate into internal (capillary) contact areas. Some companies apply silicone seal to electrode tabs to damp the vibration and reduce fatigue failures. Electrode wire material and braid type: The best are highly flexible and highly electrically conductive, Nickel-rich alloys, litz-type, multiple wires. Many other, tinned or silver-plated copper-alloys wires are also in use. Wire size (to handle amps): ≥ 2 A, rms., per mm2. Attachment method of wire to electrode tabs: Brazing, crimping, and soldering, eventually silicone coated.


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SUMMARY OF THE MOST IMPORTANT ELEMENTS RELATED TO PIEZOCERAMICS ASSEMBLING PROCEDURE FOR MAKING HIGH POWER CONVERTERS

During high-power transducer assembling, piezoceramic could be damaged, especially when we apply (very) high static pressure, comparable to 40-50 MPa. In the following text, only very critical and very important aspects of piezoceramic assembling process will be summarized. It is clear that all other, standard and traditionally known technological conditions for transducer assembling should be respected (such as: tolerances of metal parts, very high quality of surface finishing, plan-parallel contact surfaces, necessary metal plating, quality of fastening bolts...). The following procedure (given in a form of description including necessary comments) is securing the highest quality of assembled high power, BLT transducers: 1. Some producers of ultrasonic equipment think that when they buy piezoceramics that has

good mechanical tolerances and good electro-mechanical properties they can immediately assemble such piezoceramics, without any special preparation. This is not correct because metal (or silver) plated surfaces of piezoceramics usually have certain non-uniform roughness, oxides and attached solid particles. Just before transducer assembling, it is extremely recommendable to make a kind of (additional) piezoceramic fine polishing and cleaning, to eliminate any possible roughness, particles, remains of non-uniform silver plating, oxidized surface elements… For this operation, experienced ultrasonic companies are using a kind of vibrating polishing-platform that is producing only planar vibrations (X-Y vibrations). This could be a very flat metal surface driven by two electromagnets (only in x-y plane), or driven by one fast rotating electro-motor, transferring rotation to X-Y motion (via one eccentric coupling). On the surface of such vibration platform, a kind of very fine polishing paper should be placed (like silky fabric, impregnated with some very fine abrasive: quality 600 or finer). The operator, who is assembling transducers, should take every single piezoceramic ring, place it on such vibrating surface, slightly pressing on it, during 5 to 10 seconds, then change the side of piezoceramics, then apply pressurized air-jet for cleaning of remaining dust (on the ceramics surfaces), and eventually piezoceramic is ready to be assembled in the transducer. Of course, silver plating, which was mat-white, nice, uniform and clean (before such vibration polishing process), becomes much more shiny, partially covered with slightly transparent spots, and we can (almost) see certain fogy zones with the piezoceramic original color in the spots where some hard particles are removed by fine polishing (of course, silver plating should not be removed by applied polishing). This way, the remaining roughness on piezoceramic rings would be eliminated, and such piezoceramic is ready to be assembled in high power transducers (there will be no more cracking and braking of piezoceramics caused by imperfect electrode surfaces).

2. The same procedure (as specified in 1.) could be applied on metal masses and foil

electrodes that will be directly in contact/s with piezoceramics (just to realize slight polishing and micro grinding to remove possible particles, roughness, sharp edges, and similar surface imperfections). Modern industrial production of ultrasonic transducers is applying high standards for mechanical surface rectification and surface finishing of metal masses (to make them plan-parallel and perpendicular to the central axis). Then, a liquid-abrasive-polishing is applied (lapping and honing), and in the end, we apply ultrasonic cleaning to remove remains of abrasive particles. It looks sufficient, but in reality, a lot of abrasive particles are mechanically implemented in such surfaces and not easy visible. This way, mechanical friction and electrical resistance of such contact surfaces is remaining higher than acceptable for assembling high power BLT. Consequently, we should once more apply dry polishing and surface cleaning. Above described procedure is important because when piezoceramics is oscillating in the structure of sandwich transducers, it is performing significant planar/radial friction with surrounding/contacting


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metal masses (and foil electrodes), and this friction (and elevated electrical resistance) should be minimized. Also, because of very high pressure (40-50 MPa), applied on piezoceramics, if contact surfaces are not very flat, plan-parallel and shiny polished, piezoceramics could be damaged (because surface friction and elevated electrical resistance are producing intensive local heating on certain spots of piezoceramics). Basically, planar/radial friction between piezoceramic rings and surrounding metal masses should be minimized, and contact surface between ceramics and metal masses should be maximized. Later on, when ultrasonic generator drives assembled transducer, an additional surface self-polishing and reduction of surface electrical resistance (caused by immanent radial/planar friction between vibrating piezoceramics and metal masses) will (again) contribute to BLT structure stabilization. The first several minutes of transducer’s operation are very critical, because radial surface friction and electrical resistance (between piezoceramics and metal masses) are much higher in the beginning, and getting lower (and stable) after certain time (and it would be better to start operating a converter with reduced power or reduced amplitude).

3. Using convenient metal plating and convenient foil-electrodes (in the case of very high

power BLT) should also minimize radial and planar surface friction. The best metal plating (that could be applied to reach highest quality of high power ultrasonic transducers) is pure nickel (realized by galvanic plating, which gives hard nickel layer, and makes mirror shiny surfaces: 5 to 10µm thick). Some ultrasonic companies are applying shiny, hard-nickel plating, after polishing, even on aluminum masses, in order to minimize surface friction and surface contact-electrical-resistance. Nickel is an excellent material because of its mechanical properties, very low surface friction, high stability in the large temperature interval, and anticorrosion stability. In fact, any relative movement of ultrasonic components in direct contact is undesirable (i.e., all components should be essentially "welded" together, so friction should not play a role), but in all practical design situations this undesirable effects can only be minimized, but not eliminated, since different materials have different relative elongations, different elasticity, and different temperature expansion-coefficients. Most probably that some other metal-alloys could have similar surface-friction and electrical and thermal conductivity properties as Nickel.

4. The best materials for foil-electrodes of BLT are approx. 0.25mm thick foils, such as pure

nickel (99%), or nickel, cooper, beryllium alloys (where nickel content is dominant). Material for production of foil electrodes should not be hardened (has to be very flat, very bright, to have almost mirror shiny surfaces, to be sufficiently soft and flexible, or half-hard, like in the case of elastic springs production: surface hardness between 100 HV and 140 HV is preferable). Foil electrodes could be produced using photo-chemical-etching technology to avoid formations of sharp edges. It is a common opinion that beryllium alloys are the best materials for contact electrodes of BLTs, but this is not correct, except if additional brilliant nickel plating, over such electrodes is applied (because beryllium itself is not sufficiently corrosion resistant, it has relatively high surface-friction coefficient, and under strong planar-radial vibrations starts degrading, separating cooper from beryllium).

5. Threaded body and head of a central bolt, that is fastening piezoceramics and metal

masses, should be slightly lubricated with (high temperature stable) Molybdenum Disulphide grease, before transducer fastening, in order to minimize unnecessary friction during fastening. Sometimes, 30% to 60% of applied fastening torque/pressure is lost in metal-to-metal friction, if central bolt is not properly lubricated. We should not forget that central bolt, beside its function as a mechanical spring element, is also very important as electrical contact, connecting two sides of piezoceramic electrodes, or two end metal masses of a BLT (creating ground-mass electrode). The best method of applying consistent preload is: using torque reading versus using d-c voltage across shunt capacitor (note: we will still need to record torque to check later if preload has relaxed).


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6. After transducer is assembled/fastened, in order to be sure in the stable quality of transducers, it is also very recommendable to place, just finished/assembled, transducers into an oven on 80°C to 120°C, during several hours, or longer. Such elevated temperature is performing thermal treatment/aging (on metal parts and ceramics), consequently all BLT parts are moving (or elongating) a little bit, and finding better (more stable) micro-positions in the structure of the transducer. This is in the same time a kind of residual stress deliberation in complete transducer structure. After thermal treatment, transducers should be on a room temperature (20°C, without humidity) during several hours, or 24 hours, to cool down naturally. This way, mechanical structure of transducers is again getting more stabilized and finds its optimal position. During thermal treatment (and later, during natural cooling), electrical input terminals of transducers should be permanently in a short circuit connection, in order to avoid effects of internal electrical depolarization of piezoceramics. Thermal treatment is contributing significantly to the quality and stability of assembled transducers. In order to test more severely if certain converter design is optimal, we can make gradual converter heating in an oven, on a several converters only, during one hour, until reaching 160°C, then to keep converter on this temperature during 2 hours, and then again gradually reduce the temperature until room temperature is reached (20°C). If after such severe test, piezoceramics or central bolt are not broken or differently damaged, and if static pressure on the BLT is still in the initially given range (not lower than for -10%, compared to initially realized pressure; see 9 below), we can say that such BLT is well designed (and that its components are well selected).

7. Later, when transducers are again cold, we can put them in real operation, during 10 to 20

minutes (each), on some specially made testing place, driven with real ultrasonic generator, but on low power (without mechanical loading, hanging in air). This way, transducers are performing additional structural stabilization, contact surface self-polishing, aging and more of residual stress elimination.

8. In the next step, such transducers should be tested, driven full power (or in over-

power/overload), during 10 to 30 seconds. This way, vibrating transducers are forcing piezoceramic and metal parts to create additional strong friction and mutual self-polishing on their contact surfaces, making transducer structure more stable. During this test, we should control carefully if some electrical discharge, corona or micro arcing appears in the contact areas between piezoceramics and metal masses (followed by ozone creation). This is a very dangerous phenomenon that will produce local over-heating and cause crack of piezoceramics. After certain time of such self-polishing between metal masses and piezoceramics, mentioned corona and micro arcing can disappear, or if not, we should not continue forcing this situation, before we protect the lateral surfaces of piezoceramics using special coating.

9. After all above-mentioned operations, if transducers are still visibly compact (without

cracks and defects on piezoceramics) an additional refastening of central bolt could be applied. This is just to check/control if previous fastening is still correct (for instance, we can again use the same torque-key (locked in the same position as before) and check if during previous thermal treatment transducer did not loose the initially realized pressure on piezoceramics (of course, without unscrewing the central bolt of transducer). Refastening should not be applied systematically, if on several randomly chosen transducer samples we find that fastening torque/pressure is still the same as before (or lower not more than 5% to 10%, compared to initial fastening). Contrary, if we find that fastening pressure (realized by a central bolt) drops more than 10%, after initial transducer assembling and its thermal treatment, this usually tell us that we have a big problem regarding used metals, components and transducer design. Fastening and refastening should be always performed on the same room temperature (when metal parts are relatively cold: 20°C).


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10.Electrical and mechanical measurements should be applied after the last assembling step

(as a final quality control). Transducer fastening and all measurements should be applied only in a controlled temperature (and very low humidity) environment, when all solid, transducer parts are relatively cold (say, always on 20°C to 25°C). Converters insulation resistance should be much higher than 10 MΩ, measured on 1000 V DC (it can go until 1000 MΩ on 1000 V DC). Also DC breakdown voltage of a piezoceramic stack should be measured on the assembled converter, by applying 1000 V-DC/mm test field, and converter should resist in such test conditions during 60 seconds, without producing any electrical discharge or arcing effects.

Eventually, we should protect transducer surface (against humidity and moisture, and against corona and micro arcing phenomena on electrode surfaces) using convenient (high temperature and high breakdown-voltage resistant) coating: Silicone spray, epoxy, magnet-wires lacquers... (such as Conap, etc.) This should be very fine protective layer (0.1 to 0.3 mm thick). Transducer surface coating should be applied as the last step, after connecting its electrodes to output electric terminals (and after thermal treatment). The most important is to protect externally visible area of piezoceramics and all places where moisture and humidity could penetrate into internal area of a transducer structure (because piezoceramics are slightly hygroscopic materials. A protective coating could also be done before the DC breakdown test, since one purpose of the insulation is to prevent arc-over? The bolt area passing the internal zone of piezoceramic rings should be protected by certain PTFE (teflon) cylinder, in order to prevent arc-over. For some coatings, the converter loss may increase after the coating has first been applied. Subsequently, the converter loss may decrease for several days as the coating cures. Therefore, converter loss tests should not be conducted during this time in order to assure that the loss data is valid.


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MECHANICAL TOLERANCES OF PIEZOCERAMIC RINGS

For making very high power and high efficiency ultrasonic converters (with minimal internal losses), it is extremely important to use mechanically well-finished piezoceramic rings, produced under the strongest tolerance limits. For instance, PZT8 or SP8 piezoceramic rings used for assembling very high power ultrasonic welding converters usually have the following mechanical and surface-finishing tolerances:

T 1.6 Tolerance limits applicable on piezoceramic rings TYPE III

DOD-STD-1376A Tolerance limits

specified by BRANSON

CeramTec Catalogue (SP8)

Morgan Electro Ceramics Ruabon (PZT8)

TAMURA SA

Material

Parallelism top and bottom surface

0.001”= 0.0254mm until 0.0127mm

from ≤ 0.02mm to ≤ 0.002mm

Standard Within 0.012mm ---------------------- Lapped Silver Within 0.007mm

0.01mm max.

Surface Flatness top and bottom surface

0.001”= 0.0254mm until 0.0127mm 0.004mm

Standard Within 0.012mm ---------------------- Lapped Silver Within 0.002mm to 5 light bands

0.01mm max.

Thickness ±0.005”= ±0.127mm ±0.05mm, without electrodes

Standard +/- 0.05mm Without electrodes

Standard type : -0/+0.05, High cost type : -0/+0.005, Without electrodes

Concentricity ? max. 0.1mm 0.2mm T.I.R. ?

Outer Diameter ±0.381mm from ±0.1mm to ±0.3mm

Standard +/- 0.15mm ± 0.4

Inner Diameter ±0.381mm from ±0.1mm to ±0.3mm

Standard +/- 0.15mm ± 0.4

Squareness (Edge to Face) ? ? Within 0.5o ?

Surface Finish 35 micro-inches = 0.000889mm (?!)

Standard, Polished K-Schliff

Standard Within 0.003mm --------------------- Lapped Silver Within 0.0005mm

Standard type: polished (#800) High cost type: polished (#2,000) Before silver electrodes are printed.

Roughness ? ≤1,6 µm (Ra – value) ? ?


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Morgan Electroceramics best spec. for PZT8:

Polished ElectrodesFor applications where surface flatness and surface finish to very high standards are required e.g. ultrasonic welding transducers, parts can be supplied with ("Flat Lapped Silver") the silver electrodes polished to the following: Surface flatness within 5 light bands Surface finish within 0.3 microns. Evaporated Electrodes Metal electrodes such as copper, aluminum, gold on nickel, silver on nickel applied by the evaporation process can also be supplied as an alternative to "fired-on" silver.

CeramTec spec. for piezoceramic rings (SP8):

1. For transducer rings for Ultrasonic Cleaning and Welding applications we don’t specify Squareness values because they don’t influence the transducer performance and would create additional costs to measure them.

2. Surface finish of the piezoceramic is not relevant because in any case a metallization is added to the piezoceramic. Only flatness and parallelism are important.

3. Standard means: Screen-printed ductile silver surface with no further treatment. 4. Polished and K- Schliff are more or less the same; there we grind the parts after the

metallization process a second time to create a more flat surface over the silver. Polished is mirror - like. K-Schliff is a technical grinding which creates only a flat surface with no consideration of the visual appearance.

5. Roughness measured over metallization is ≤ 1,6 µm (Ra - value). 6. We don’t specify or measure a Friction value. 7. Talking about piezoelectric parameters like capacitance, coupling coefficient, charge

constant etc. if nothing special is specified we have a tolerance of +- 20 %. Basis is the data in our brochure.

8. Depending on the part and the price the tolerances for capacitance and resonance frequency can be narrowed to different levels: +- 15 %: Often agreed with customers for selected parameters. +- 10 % +- 5 %: Normally our tightest tolerance level +- 2 %: Extreme, seldom for very special parts.

9. All these data are based on high volume production (>10000 parts per year) and include "in - batch - variation" and "batch to batch - variation".


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ACCEPTABLE STATISTICAL QUALITY LEVEL OF ELECTRICAL AND PIEZOELECTRIC PARAMETERS OF PIEZOCERAMICS USED FOR BLT ASSEMBLING (FOR MASS PRODUCTION OF POWER CONVERTERS) When specifying the acceptance quality level of certain piezoceramics, we should also specify its particular electrical characteristics and piezoelectric-constants, including acceptable tolerances, using HP 4194A, (an impedance-phase network analyzer), applied on a statistical sample of ≥ 1000 pcs., for verification. All measurements should be taken on a room temperature, 20°C, and very low humidity, in an air-conditioned laboratory. For instance, PZT-8 or SP8 piezoceramic rings (OD x ID x t = 2” x 0.8” x 0.2” ≅ 50 x 20 x 5 mm), should be specified (for input factory control, on the reception) as (see Table T 2.6):

Table 2.6 Input control, acceptance limits for piezoceramic rings 2” x 0.8” x 0.2” PZT-8 , TYPE III DOD-STD-1376A

Average Values

Maximal Tolerances Conditions

Cinp. (1 kHz), [nF] 3.005 ± 5% 90 days after poling f1 = fs, [Hz] 32880 ± 1% 90 days after poling f2 = fp, [Hz] 34880 ± 1% 90 days after poling

∆f : fs = (f2 - f1) : fs 0.063 ± 3% 90 days after poling Zmin = Zs, [Ω], at f1 (≤) 13.7 ± 10% At fs

Zmax = Zp, [kΩ], at f2 (≥) 245 ± 20% At fpQm10 (≥) 1000 ± 15% At fs

tan δ (≤) 0.002 ± 20% At 1 kHz d33 , [E-12 C/N] 245 - 260 ± 5% Reference only

Y (Young Modulus) [N/m2] Short circuit: 8.1496 E+10 Open circuit: 8.9632 E+10

Poisson R = 0.31, Density: 7600 kg/m^3

Calculated using Finite Elements Analysis, based on

data for f1 and f2 Silver electrodes on both surfaces d33-meter: Berlincourt, Piezo d33 meter, Channel Products Inc.

Table 2.6-1 Input control, acceptance limits for piezoceramic rings 1.6” x 0.8” x 0.2”

PZT-8 , TYPE III DOD-STD-1376A

Average Values


Cinp. (1 kHz), [nF] 1.86 ± 5% 90 days after poling f1 = fs, [Hz] ± 1% 90 days after poling f2 = fp, [Hz] ± 1% 90 days after poling

∆f : fs = (f2 - f1) : fs 0.043 ± 3% 90 days after poling Zmin = Zs, [Ω], at f1 ± 10% At fs

Zmax = Zp, [kΩ], at f2 ± 20% At fpQm10 ≥ 1000 ± 15% At fs

tan δ ≤ 0.002 ± 20% At 1 kHz d33 , [E-12 C/N] 245 - 260 ± 5% Reference only

Table 2.6-2 Input control, acceptance limits for piezoceramic rings 0.75” x 0.25” x 0.1”

PZT-8 , TYPE III DOD-STD-1376A

Average Values



∆f : fs = (f2 - f1) : fs 0.071 ± 3% 90 days after poling Zmin = Zs, [Ω], at f1 ± 10% At fs

Zmax = Zp, [kΩ], at f2 ± 20% At fpQm10 ≥ 1000 ± 15% At fs



6-12

In cases of all other piezoceramic rings (other sizes) used for high power BLT, we can apply the following selection criteria; see Table 3.6 (again using for measurements HP 4194A, on a statistical sample ≥ 1000 pieces). Table 3.6 Input control, acceptance limits for any-size piezoceramic rings

PZT-8 , SP-8 TYPE III DOD-STD-1376A Excellent Very good Good Acceptable

Cinp. (1 kHz), [nF] nominal ± 1%

nominal ± 2.33%

nominal ± 3.67%

nominal ± 5%

f1 = fs, f2 = fp[Hz ]

nominal ± 0.1%

nominal ± 0.4%

nominal ± 0.7%

nominal ± 1%

∆f : fs = (f2 - f1) : fsnominal ± 1%

nominal ± 2.33%

nominal ± 3.67%

nominal ± 5%

Zmin = Zs, [Ω], at f1

nominal ± 5%

nominal ± 10%

nominal ± 15%

nominal ± 20%

Zmax = Zp, [kΩ], at f2

nominal ± 10%

nominal ± 16.67%

nominal ± 23.33%

nominal ± 30%

Qm10nominal ± 5%

nominal ± 10%

nominal ± 15%

nominal ± 20%

d33nominal ± 3%

nominal ± 5.33%

nominal ± 7.67%

nominal ± 10%

nominal = average or mean value, f1 = fs, f2 = fp first resonant frequency couple of free piezoceramics (radial-planar mode), d33-meter: Berlincourt, Piezo d33 meter, Channel Products Inc. Measurements should be taken on a room temperature (20°C, very low humidity, air-conditioned).

.,1000,102 1.103

sresm fffatQtg ==≥⋅≤δ − Silver electrodes applied on both surfaces.

All above given tolerances expressed in percentage-intervals (around average values), are calculated based on the Gauss, Standard, Statistical Distribution curve, where 99% of all measured cases are captured by the interval of 6 (six) standard deviations, and this criteria is in the perfect agreement with all practical situations, when measured sample is relatively large; -minimum 1000 pieces of piezoceramics. The distinction between "Excellent", "Very good", "Good" and "Acceptable" is made by splitting the interval between “Excellent” and “Acceptable” on four, almost equal intervals (while “excellent” and “Acceptable” limits should be empirically known, in advance, after enough testing and experimental results).


6-13

T 4.6 Young modulus of different piezoceramics [N/m2] Source FEM calculated Declared in the Catalogue

Morgan: PZT8

Short circuit: 8.1496 E+10 [N/m2] Open circuit: 8.9632 E+10 [N/m2] Poisson R = 0.31 Density: 7600 kg/m^3

Short circuit: Y11= 8.7 E+10 [N/m2] Y33= 7.4 E+10 [N/m2] Open circuit: Y11= 9.9 E+10 [N/m2] Y33= 11.8 E+10 [N/m2]

CeramTec: SP8



Philips: PXE43



TAMURA SA Material

Y11E

Y33E

Y11D

Y33D

Poisson’s ratio Density

8.54 * 1010 N/m2

7.33 * 1010 N/m2

9.71 * 1010 N/m2

14.1 * 1010 N/m2

0.31 7910 kg/m3

Data in this table (red color) probably should be corrected!!! Piezoceramic rings used in Branson production of ultrasonic welding converters (PZT-8, SP8):

2” x 0.8” x 0.2” = 50.80 x 20.32 x 5.08 mm ≅ 50 x 20 x 5 mm 1.6” x 0.8” x 0.2” = 40.64 x 20.32 x 5.08 mm ≅ 40 x 20 x 5 mm 0.75” x 0.25” x 0.1” = 19.05 x 6.35 x 2.54 mm ≅ 19 x 6 x 2.54 mm

Piezoceramic ring and disc used in Branson production of ultrasonic cleaning converters (PZT-4, SP4):

Ring (for 40 kHz cleaning transducer): 1.5” x 0.594” x 0.2” = 38.10 x 15.0876 x 5.08 mm ≅ 38 x 15 x 5 mm Disk (7.48 nF, for 50 kHz plate transducers): 2” x 0.1” = 50.80 x 2.54 mm ≅ 50 x 2.5 mm

Now, we can illustrate the complete impedance measurements and modeling results, realized using HP-4194A, an Impedance-Phase Network Analyzer, on the well selected (an excellent quality sample of) PZT-8, piezoceramic rings produced by Vernitron (Morgan Matroc, USA): OD x ID x t = 2” x 0.8” x 0.2” ≅ 50 x 20 x 5 mm (see T 4.7).


6-14

T 4.6 Piezoceramics measurements and Models

Piezoceramic Ring:Vernitron, PZT8, 45 / 89, First Radial Resonance Parameters

∅ ∅2 0 8 0 2" . " .x x "

f Hzf HzZZ kQQ

m

m

1

2

10

20

3288034880

12 8724256 861

11291385

==

=

=

==

min. .

max. .

Ω

Ω

A

ZA

C pFHL m

R

1

1

1

336 81269 556512 6210

===

... Ω

A

ZA

C pL mR M

p

p

1

1

1

336 81269 556516 8762

=

==

... Ω

FH

B

⇔

ZB

C nL HR k

2

2

2

24 2667857 904260 045

===

...

µΩ

F

B

C nL HR m

s

s

2

2

2

24 2667857 903142 611

===

...

µΩ

F

C

⇔ ZB

C nFC pFL mR

op =

===

2 68554336 55269 616412 741

1

1

1

..

.

. ΩH

C

C C CnF

C nL HR k

os op= +

F

=

====

1

2

2

2

3 022124 26857 9260 045

....µ

Ω

⇔Z Z

Converters modeling Mio

drag Prokic Le Locle - Switzerland

6-15

The electric characteristics and tolerances of TAMURA Rings

Example: Ring dimensions (SA material): 38.0 x 15.5 x 6.2 mm

Items Average value Standard with tolerance fr 41.43 kHz 41.5 kHz ± 10%

Zmin 7.76 Ω - fa 45.08 kHz -

Zmax 649.25 kΩ - ∆f 3.66 kHz - kr 44.51 % 39.7 % min. Qm 2,295.87 - C 1,818 pF 1,800 pF ± 20%

tan δ 0.18 % - Cop 1.55369 nF - C1 286.77 pF - L1 51.49 mH - R1 5.99 Ω -

Equivalent circuit

L1R1C1

Cop

TAMURA PIEZOCERAMICS TECHNOLOGY (comments) 1. Density Tamura applies normal, conventional sintering method to produce high-density piezoceramic. Granulated ceramic powder, which contains some kind of binder, is die-pressed to form compacts, whose green density is in the range from 4.7 to 4.8 grams per volume centimeter. Tamura powders are also fired traditionally, in the muffle furnace, after the binders were removed, to obtain the density of about 7.9 grams per cubic centimeter. 2. Specific ultrasonic power The value of specific ultrasonic power of Tamura SA material (according to laboratory testing) must not be much different from CeramTec's (30 Watts per Centimeter Square). 3. Important parameters The most important parameter for both of SS and SA Tamura materials is high Qm value, because they are used specially for high power converters, and its heat losses are very low. For SA material relative permitivity is the most important parameter. The value is almost the same as the value of Japanese other big ceramic makers. SA material's relative permitivity do not change significantly, before and after ring ceramic is assembled into BLT. This is also the reason why SA materials are suitable for ring-shaped piezoelectric transducers.


6-16

"Best" Piezoceramics

The following is from a conversation on 6/10/99 with a ceramic engineer who was previously employed by Morgan-Matroc.

His ranking of ceramic manufacturers:

a. First tier

i) Morgan Matroc

ii) Piezokinetics Inc (PKI) – buys from EDO. 4/3/00: Has special high ampl piezos designed for medical applications. Suresh Viswanathan seems very knowledgeable.

iii) ?Valpey – buys from EDO

iv) ?Stavley Sensors – buys from EDO

b. Second tier

i) EDO

ii) Sensor Tech

iii) Transducer Products of South Hampton

iv) Hoescht – material is between PZT4 and PZT8; lower loss material similar to Morgan Matroc 4S and 8M, but "odd" formulation so may not be able to obtain from other companies as secondary suppliers

v) Unilator

c. Third tier

i) APC – uses Czech ceramics; Craig had some problems

ii) Channel

Ceramic tolerances (disks, fully machined):

Company Parameter Tolerance Notes

Channel Industries (p. 13) Diameter ±.003 Ø.125 Ø1.500

Thickness ±.003 .080 .200

Parallelism .001"/" Up to Ø2.000

Flatness .001"/" Up to Ø2.000

Silver electrode thickness (p. 11)

.0006 -- .001 Dimension, not tolerance

Stavley Sensors Diameter +.000/-.003

Thickness ±.0003 Up to Ø3.000

Finish 35 (fired electrodes) 32 (electroless Ni) 8 (electroless Cu or Au)

Valpey Fisher (p. 4) Diameter


6-17

Thickness

Parallelism .00005" max.

Flatness .0001" max.

EDO (p. 27) Diameter ±.003 Ø.125 Ø1.500

Thickness ±.003 .080 .200

Parallelism

Flatness Within thickness tolerance if dia/thickness < 10:1

Vernitron (PD-9247, p. 5) Diameter ±.015

Thickness ±.010 .080 .500

Parallelism .007"

Flatness

Within thickness tolerance up to Ø1.0 and up to .080 thick; .005" above .080 thick



Nickel electrode thickness (p. 26)


Piezokinetics Silver electrode thickness


Nickel electrode thickness

.0003 Dimension, not tolerance

Stavley Sensors Diameter ±.002

Thickness ±.003

Parallelism

Flatness



Nickel electrode thickness (p. 26)


Electroless copper or gold

< 100 microns

Channel (p. 11) – Electroless nickel is a low temperature process that gives thinner, higher resistance electrodes than silver. Used mainly for shear ceramics. Insulation for ceramics

Loctite Lite-tak 375: brush on, ultraviolet curing in several seconds, very low mechanical loss (per Tim Meyers at Vernitron, 6/9/92).

Conap


6-18

Parylene: Vapor deposition, therefore excellent coverage (even on sharp corners and cavities) with no pinhole leaks. Masking required preventing adherence to unwanted surfaces. Thickness typically <= 1 mil. Dielectric strength 5500 – 7000 volts/mil. Used by Morgan Matroc on their ceramics (per Tim Meyers, Morgan Matroc, 4/9/96)

Others: Hysol, Master Bond (per American Piezoceramics, 3/28/93); Eccocoat 831 (Emerson & Cumming)


6.1-1

6.1 MECHANICAL TOLERANCES OF PIEZOCERAMIC RINGS

For making very high power and high efficiency ultrasonic converters (with minimal internal losses), it is extremely important to use mechanically well-finished piezoceramic rings, produced under the strongest tolerance limits. For instance, PZT-8 or SP8 piezoceramic rings (OD x ID x t = 50 x 20 x 5 mm = 2” x 0.8” x 0.2”), often used for assembling very high power ultrasonic welding converters usually have the following mechanical and surface-finishing tolerances:

T 1.6 Tolerance limits applicable on piezoceramic rings: 50 x 20 x 5 mm

PZT Rings BRANSON tolerance limits CeramTec Vernitron

Parallelism top and bottom surface

0.001”= 0.0254mm until 0.0127mm

from ≤ 0.02mm to ≤ 0.002mm 0.012mm

Surface Flatness top and bottom surface

0.001”= 0.0254mm until 0.0127mm

0.004mm from 0.012mm to 5 light bands

Thickness ±0.005”= ±0.127mm

±0.05mm, without electrodes

±0.05mm, without electrodes

Concentricity ? max. 0.1mm 0.2mm T.I.R.

Outer Diameter ±0.381mm from ±0.1mm to ±0.3mm ±0.15mm

Inner Diameter ±0.381mm from ±0.1mm to ±0.3mm ±0.15mm

Squareness (Edge to Face) ? ?

Within 0.5° Max. deviation less than 1 micrometer, both on ID and OD

Surface Finish 35 micro-inches = 0.000889mm (?!)

Standard, Polished K-Schliff

from 0.003mm until 0.0003mm

Roughness Surface Friction


6.1-2

ACCEPTABLE STATISTICAL QUALITY LEVEL OF ELECTRICAL AND PIEZOELECTRIC PARAMETERS OF PIEZOCERAMICS USED FOR BLT ASSEMBLING (FOR MASS

PRODUCTION OF POWER CONVERTERS) When specifying the acceptance quality level of certain piezoceramics, we should also specify its particular electrical characteristics and piezoelectric-constants, including acceptable tolerances, using HP 4194A, applied on a statistical sample of ≥ 1000 pcs., for verification. All measurements should be taken on a room temperature, 20°C, and low humidity, in an air-conditioned laboratory. For instance, PZT-8 or SP8 piezoceramic rings (OD x ID x t = 50 x 20 x 5 mm = 2” x 0.8” x 0.2”), should be specified (for the purpose of input control on the reception) as:

T 2.6 Input control, acceptance limits for piezoceramic rings 50 x 20 x 5 mm PZT-8 , SP-8 TYPE III

DOD-STD-1376A Average Values


Cinp. (1 kHz), [nF] 3.005 ± 5% 90 days after poling f1 = fs, [Hz] 32880 ± 1% 90 days after poling f2 = fp, [Hz] 34880 ± 1% 90 days after poling

∆f : fs = (f2 - f1) : fs 0.063 ± 3% 90 days after poling Zmin = Zs, [Ω] ≤ 13.7 ± 10% At fs

Zmax = Zp, [kΩ] ≥ 245 ± 20% At fpQm10 ≥ 1285 ± 15% At fs


Silver electrodes on both surfaces d33-meter: Berlincourt, Piezo d33 meter, Channel Products Inc.

T 2.6-1 Input control, acceptance limits for piezoceramic rings 1.6” x 0.8” x 0.2” PZT-8 , SP-8 TYPE III

DOD-STD-1376A Average Values



∆f : fs = (f2 - f1) : fs 0.043 ± 3% 90 days after poling Zmin = Zs, [Ω] ± 10% At fs

Zmax = Zp, [kΩ] ± 20% At fpQm10 ≥ 1000 ± 15% At fs


T 2.6-2 Input control, acceptance limits for piezoceramic rings 0.75” x 0.25” x 0.1”

PZT-8 , SP-8 TYPE III DOD-STD-1376A

Average Values



∆f : fs = (f2 - f1) : fs 0.071 ± 3% 90 days after poling Zmin = Zs, [Ω] ± 10% At fs

Zmax = Zp, [kΩ] ± 20% At fpQm10 ≥ 1000 ± 15% At fs



6.1-3

In cases of all other piezoceramic rings (other sizes) used for high power BLT, we should apply the following selection criteria (again using HP 4194A, on a statistical sample ≥ 1000 pieces): T 3.6 Input control, acceptance limits for any-size piezoceramic rings

PZT-8 , SP-8 TYPE III

DOD-STD-1376A

Excellent

Very good

Good

Acceptable

Cinp. (1 kHz), [nF] nominal ± 1%

nominal ± 2.33%

nominal ± 3.67%

nominal ± 5%

f1 = fs, f2 = fp[Hz, kHz]

nominal ± 0.1%

nominal ± 0.4%

nominal ± 0.7%

nominal ± 1%

∆f : fs= (f2 - f1) : fs

nominal ± 1%

nominal ± 2.33%

nominal ± 3.67%

nominal ± 5%

Zmin = Zs, [Ω] nominal ± 5%

nominal ± 10%

nominal ± 15%

nominal ± 20%

Zmax = Zp, [kΩ]

nominal ± 10%

nominal ± 16.67%

nominal ± 23.33%

nominal ± 30%

Qm10nominal ± 5%

nominal ± 10%

nominal ± 15%

nominal ± 20%

d33nominal ± 3%

nominal ± 5.33%

nominal ± 7.67%

nominal ± 10%

nominal = average or mean value, f1 = fs, f2 = fp first resonant frequency couple of free piezoceramics (radial-planar mode), d33-meter: Berlincourt, Piezo d33 meter, Channel Products Inc. Measurements should be taken on a room temperature (20°C, very low humidity, air-conditioned).

.,1000,102 1.103

sresm fffatQtg ==≥⋅≤δ − Silver electrodes applied on both surfaces.

All above given tolerances expressed in percentage-intervals (around average values), are calculated based on the Gauss, Standard, Statistical Distribution curve, where 99% of all measured cases are captured by the interval of 6 (six) standard deviations, and this criteria is in the perfect agreement with all practical situations, when measured sample is relatively large; -minimum 1000 pieces of piezoceramics. Piezoceramic rings used in Branson production of ultrasonic welding converters (PZT-8, SP8):

2” x 0.8” x 0.2” = 50.80 x 20.32 x 5.08 mm ≅ 50 x 20 x 5 mm 1.6” x 0.8” x 0.2” = 40.64 x 20.32 x 5.08 mm ≅ 40 x 20 x 5 mm 0.75” x 0.25” x 0.1” = 19.05 x 6.35 x 2.54 mm ≅ 19 x 6 x 2.54 mm

Piezoceramic ring and disc used in Branson production of ultrasonic cleaning converters (PZT-4, SP4):

Ring (for 40 kHz cleaning transducer): 1.5” x 0.594” x 0.2” = 38.10 x 15.0876 x 5.08 mm ≅ 38 x 15 x 5 mm Disk (7.48 nF, for 50 kHz plate transducers): 2” x 0.1” = 50.80 x 2.54 mm ≅ 50 x 2.5 mm

Now, we can illustrate the complete impedance measurements and modeling results, realized using HP-4194A, an Impedance-Phase Network Analyzer, on the PZT-8, piezoceramic rings produced by Vernitron (Morgan Matroc, USA): OD x ID x t = 50 x 20 x 5 mm ≅ 2” x 0.8” x 0.2”.


6.1-4

T 4.6 Piezoceramics measurements and Models

Piezoceramic Ring:Vernitron, PZT8, 45 / 89, First Radial Resonance Parameters

∅ ∅2 0 8 0 2" . " .x x "

f Hzf HzZZ kQQ

m

m

1

2

10

20

3288034880

12 8724256 861

11291385

==

=

=

==

min. .

max. .

Ω

Ω

A

ZA

C pFHL m

R

1

1

1

336 81269 556512 6210

===

... Ω

A

ZA

C pL mR M

p

p

1

1

1

336 81269 556516 8762

=

==

... Ω

FH

B

⇔

ZB

C nL HR k

2

2

2

24 2667857 904260 045

===

...

µΩ

F

B

C nL HR m

s

s

2

2

2

24 2667857 903142 611

===

...

µΩ

F

C

⇔ ZB

C nFC pFL mR

op =

===

2 68554336 55269 616412 741

1

1

1

..

.

. ΩH

C

C C CnF

C nL HR k

os op= +

F

=

====

1

2

2

2

3 022124 26857 9260 045

....µ

Ω

⇔Z Z

Converters modeling Miod

rag Prokic Le Locle - Switzerland

THE ULTRASONIC HAMMER TRANSDUCER

Miodrag Prokic* *MP Interconsulting, Marais 36, CH-2400 LeLocle, Switzerland

Jon Tapson ϒ ϒ Department of Electrical Engineering, University of Cape Town, Rondebosch 7701, South Africa

Bruce J.P. Mortimer •• Centre for Instrumentation Research, Cape Technikon, PO Box 652, Cape Town 8000, South Africa

Abstract – A new high power ultrasonic transducer is described. This transducer consists of a front mass and rear mass and two piezoelectric drivers on either side of a center mass which is also the dominating resonance mass. The characteristics of the transducer resemble a force transducer combined with the small signal characteristics of a sandwich transducer. Several variations can be achieved by manipulating the mechanical coupling as well as the electrical connections. The transducer is suitable for driving high mass loads such as the contents of pipes.

I. INTRODUCTION Conventional Langevin or sandwich transducers [1] (Figure 1) are well known and are widely used in high power piezoelectric sonar and ultrasonic applications. These transducers are usually characterized by a relatively narrow resonant frequency bandwidth (or a relatively high Q) and a resonant characteristic that is coupled to the load i.e. its resonance is affected by changes in the medium. Most conventional ultrasonic systems account for this variation by employing various frequency and motional locking schemes to insure that the load is optimally driven within the limits of the transducer and the energy supply [2]. In spite of the popularity of the Langevin transducer, these transducers cannot effectively drive certain loads such as those contained within thick-walled pipes. This is due the load mass being directly connected to the front oscillating mass and thereby introducing an additional order of complexity into the system as well as an increase in lumped element mass. This problem has become more prevalent as many emerging applications such as sonochemistry or

high pressure cleaning [3], require ultrasonic energy to be transmitted through thick walled vessels. A separate problem with conventional ultrasonic cleaning and liquid processing systems is that one generally requires a uniform coverage over the treatment vessel. Since the transducers used have of relatively narrow resonant characteristics, standing waves usually form. Many systems make use of some frequency sweeping, amplitude modulation or a combination of different transducers to reduce this effect. However, this usually results in some compromise on the transducer efficiency. Increasing the transducer system frequency response would therefore have the potential to couple energy into the load without the formation of standing waves, but this is not practical with conventional sandwich transducer systems.

Load

Front Mass

Back Mass

Piezoelectricelements

Figure 1 : Conventional Langevin transducer. The front mass is coupled to the back mass via a tensional bolt and the piezoelectric ring elements in the middle of the transducer. The load is coupled to the front mass and therefore influences the resonance response.

This paper presents a novel configuration that we have termed the hammer transducer [4]. The transducer has a resonant mass element that is independent of the acoustic load and, in certain circumstances, this configuration has a multi-resonant characteristic that is well suited to multi-frequency driving (over a significant bandwidth).

II. THE HAMMER TRANSDUCER This transducer uses a minimum of two series driven piezoelectric elements to drive a fixed resonator mass element that is clamped between the piezoelectric rings. The piezoelectric elements are in turn sandwiched between a front and back mass and held together by a central bolt (which is also used to pretension the piezoelectric elements) as shown in Figure 2.

H

Type A Type B

1.

2.

3.

4.

h

+

(Load) (Load)

+++-

-

-

-

++

+ +- - - -

Figure 2 : Schematic of the Hammer transducer showing 1) tail mass, 2) four piezoelectric rings, 3) hammer mass and 4) front mass that is usually coupled to the load. Two configurations are possible: Type A - Center mass connected directly to bolt and Type B - Center mass moves independently of the bolt.

Although the construction of the transducer is similar to a conventional sandwich transducer, the operation is different as the central mass is usually designed to be the predominant resonant mechanism. This mass is

in turn forced in one direction and then the other as a consequence of the piezoelectric elements being driven in series – as one disk contracts the other expands (in thickness mode). We have termed this type of transducer a “hammer transducer” as the central mass “hammers” the front and back masses [4]. Two different constructions are possible; firstly Type A the central bolt mechanically connects all masses and participates as an important (spring and mass) element of the total mechanical oscillating circuit (all three masses are oscillating) and secondly, Type B where the middle mass is not mechanically (directly) connected to the central bolt and central bolt can be approximately considered as the rigid connection between two end metal masses (only the middle mass is oscillating).

III. TRANSDUCER OPERATION An approximate mechanical-equivalent oscillating circuit demonstrating the different hammer transducers configurations is shown in Figure 3. Mechanical excitation is symbolically represented as a mechanical generator/s, em(F,V), where F and V are the force and the velocity respectively.

M1

M2

M3

(Load)

em

+

+

(F, V)

M1

M2

M3

(Load)

Type A : Type B :

(F, V)

em

em+

+

(F, V)

em

Figure 3 : Simplified equivalent oscillation system for the Hammer transducer options.

In the Type A resonator, we create a double resonant, 3 degree of freedom system. This can be rather complex to analyze (especially if the load interacts

with M3). This configuration is similar to a hybrid magnetorestrictive / piezoelectric combinational transducer that has been reported to be capable of achieving a significantly greater bandwidth than conventional tonpilz type transducers [5]. In Type B we have a hammer-mode where we can assume that only the middle metal mass is moving (in piston-mode with the end masses confined by the central bolt). With suitable parameter selection, Type A resonators can be designed with a dominant central mass, also approximating hammer-mode resonance. In this case, the end-masses are also oscillating, but significantly less (power-wise) than the middle-mass. Hammer-mode resonance : The total length of the active transducer stack h (the length of the central mass and two piezo-ceramic layers, without counting the end metal) remains approximately constant and fixed by the bolt. This is because the two piezo-ceramic layers are mechanically identical and are in opposite electrical polarization (and equally driven). When the first piezo-ceramic layer is contracting, the second one is extending by the same displacement (and vice versa) mutually compensating the total displacement of the active middle-stack length. Of course, the total transducer length, including both end-masses will not stay absolutely constant, except in case/s of fully symmetrical hammer transducer structure where upper and lower end-masses, and piezo-ceramic layers are identical. An unusual property of the hammer transducer is that the dynamic center of mass, or center of inertia also oscillates. This is in contrast with conventional transducers where the center of mass will remain in a stable position. The overall hammer-mode vibration is as a force transducer as opposed to a displacement transducer. This property means that the transducer will require a coupling to a load (such as a cavity resonator), but the characteristics of the load will not necessarily affect the resonance of the hammer.

Electrical driving options : There are two basic configurations in which the hammer transducers electrical terminals can be connected; piezoelectric elements series driven and piezoelectric parallel driven. This is demonstrated by drawing the system in the format of a two-port network [6]. This is shown for the case of a hammer Type B with piezoelectric elements driven in series in Figure 4 and for parallel driving in Figure 5.

M1

M2

M3

Piezo-ceramic

Piezo-ceramic

(Load)

UIN

U1

U2

I2

I1

+

V11F11

F1V1

F21

F32

F33

F3

V21

V32

V33

V3

Figure 4 : Two-port system representation [6] showing piezoelectric elements driven in series. In this case, F1≈F3, V1≈V3, F21=F32, V21=V32, I1=I2 and U1=U2=0.5Uin

M1

M2

M3

Piezo-ceramic

Piezo-ceramic

(Load)

UIN

U1

U2

I2

I1

+

V11F11

F1V1

F21

F32

F33

F3

V21

V32

V33

V3

Figure 5 : Two-port system representation of a Type B hammer transducer [6] showing piezoelectric elements driven in parallel. In this case, F1≈F3, V1≈V3, F21=F32, V21=V32 and U1=U2=Uin

The two electrical configurations shown in Figure 4 and 5 allow for various driving options (including motional current and motional voltage). A complete analysis however, must take into account the transmission line characteristics of the components (including the load).

IV. EXPERIMENTAL RESULTS Hammer transducers have been tested in various metal stamping, forging and high pressure cleaning [3] applications. Coupling from the hammer transducer to the load was via an aluminum or steel bar and in the case of high pressure cleaning, an additional acoustical coupling ring (round the exterior of the liquid filled pipe).

Variations in the load have very little effect on the operation of this transducer. Further the drive frequency can be modulated in a much wider frequency interval than in the case of traditional two-mass transducers (sometimes up to ±30% around central operating frequency), without dropping mechanical quality factor.

V. CONCLUSIONS Several configurations of a force transducer, termed the hammer transducer have been presented. A central mass is the dominant resonant mechanism and the transducer has several advantages; namely a certain degree of isolation from the load, a transducer capable of generating large forces in an attached work-piece (even of high mass) and the potential of driving over a wide power bandwidth. Two electrical configurations are possible, each resulting in a complex system filter characteristic that cannot be accurately represented as a lumped equivalent circuit.

VI. REFERENCES [1] A.P. Hulst, “On a family of high-power

transducers”, Ultrasonics International Conference Proceedings, 1973.

[2] B. Mortimer, T. du Bruin, J. Davies, J. Tapson, “High power resonant tracking amplifier using admittance locking”, Ultrasonics, vol. 39, pp. 257-261, 2001.

[3] M. Prokic, “Multifrequency Ultrasonic Actuators with Special Application to Ultrasonic Cleaning in Liquid and Supercritical CO2”, UIA Conference, Atlanta, 10-12 October 2001.

[4] European Patent Application: EP 1 060 798 A1, Miodrag Prokic, 2400 Le Locle Switzerland, Date of publication: 20.12.2000, Bulletin 2000/51, (patent pending in several countries).

[5] S. Butler, F. Tito, “A broadband hybrid magnetorestrictive piezoelectric transducer array”, MAGSOFT UPDATE, vol. 7, No. 1 1-7 2001.

[6] R. S. Woollett, Scientific and Engineering Studies : Section II The Longitudinal Vibrator, Naval Underwater Systems Centre, Newport, Rhode Island, pp 66-79.

ANNEX: MMM Technology

Analogies MMM-basics Analytic Signal and Power presentation

PRESENTATION OF BASIC ELECTROMECHANICAL ANALOGIES Let us first establish or summarize the basic set of electromechanical analogies that will be used, later on, to formulate some of the most important ideas and messages in this paper. Comparing different configurations of electric circuits in Fig.1.1a),b),c),d), consisting of electrical elements (resistance (R), inductance (L), and capacitance (C)), to different configurations of mechanical circuits, consisting of mechanical elements (mechanical resistance or damper (Rm), spring (s), and mass (m)), it is possible to determine six levels of different electro-mechanical analogies (see [1], pages 9-18, [2] and [3]). Here, we shall not present or analyze the way of getting these analogies (since this can be found in literature), but, as a general conclusion, we shall state that all of the known analogies have been created after noticing similarity of the forms of corresponding differential equations (between dual or similar electric and mechanical networks), which are describing currents, voltages, forces and velocities in those electric or mechanical networks. It is clear that we are thus creating mathematical analogies (which present the first criteria in creating analogies). The mathematical conclusion is that all the possible six analogy situations ([1] and [2]) are mutually equal, or equally useful. We shall now introduce the second analogy criteria (which is not mathematical) to give more weight and power to certain analogies (from the existing six). Let us say that beside mathematical analogies we would like, for equivalent electric and mechanical circuits, to have the same circuit configuration (topography) of their elements (if we look at how they are mutually connected in corresponding circuits or where they are placed). By introducing this second analogy criterion, the previous set of six analogies (which is systematically analyzed in [1], [2] and [3]) is reduced to the set of only two analogy situations (Fig.1.1a), b), c), d)). The latest mathematical and topography analogy can be represented by velocity-to-voltage and force-to-current analogy (known in literature as the Mobility type of analogy). Fig. 1.1c

C R L

u

i i icR L

vm R S

f ffm s

m

fL

Rmr

mr

Fig.1.1a Fig. 1.1b Fig. 1.1d

Fig.1.1 Equivalent electric and mechanical circuits

Miodrag Prokic E-mail: [email protected] Jul-04 Electric and Mechanical Analogies...

2

The content that results from this double-level analogy-platform is shown in T.1.1. In this paper it will be shown that the analogies from T.1.1 are presenting the most important, most predictive, most practical and most natural (electro-mechanical) analogy platform in physics (and later, the same Mobility type analogy platform from T.1.1 will be widely extended; -see T.1.2 until T.1.7). T.1.1

Electric parameter / [unit] Mechanical parameter / [unit] Voltage (=) u (=) [V = volt] Velocity (=) v (=) [m/s]

Current (=) i = dq/dt (=) [A = ampere] Force (=) f = dp/dt (=) [N = Kg m/s2 = newton] Resistance (=) R (=) [Ω = ohm] Mech. Resistance (=) Rm (=) [m /N s = s/Kg]

Inductance (=) L (=) [H = Henry] Spring Stiffness (=) S (=) [m/N = s2/Kg] Capacitance (=) C (=) [F = farad] Mass (=) m (=) [Kg]

Charge (=) q = Cu (=) [C = coulomb] Momentum (=) p = mv (=) [Kg m/s] Magn. Flux Φ (=) [Wb = V s = Weber] Displacement (=) x (=) [m]

In order to establish perfect (1:1) analogy and symmetry between elements of electrical and mechanical circuits, as given in T.1.1 (L ⇔ S and R ⇔ Rm), it was necessary to redefine the unit of spring stiffness and unit of mechanical resistance or damping. In literature we find for spring stiffness S (=) [N/m], and here we use S (=) [m/N], and for mechanical resistance in today’s literature is very usual to find Rm (=) [Ns/m], and here we use Rm (=) [m/Ns]. The consequences of these parameter redefinitions are that usual (traditional) meaning of Mechanical Impedance = Force/Velocity should also be redefined into “New Mechanical Impedance” = Velocity/Force (presently known in Mechanics as Mobility), in order to fully support the VOLTAGE-VELOCITY and CURRENT-FORCE analogies. Similar redefinitions will be also applied to mechanical-resistance (or friction constant) and spring-stiffness valid for rotation (see all equations from (1.1) until (1.9)). Needless to say, all the other analogies are equally useful (but only at the mathematical, formal level of analogies), and if we consider certain (additional) rules, correct use of any of these analogies will lead to the same result (but often in a more complicated, more difficult and not so natural way). Let us come back to the equivalent models given in Fig.1.1 and introduce rotational motion, as yet another level of study for additional creation of analogies. Let us imagine two more (mechanical), analog, oscillating circuit-models added to this situation, presenting the case of rotation (in series and parallel configuration of oscillatory-rotating mass having a moment of inertia J, angular velocity ω, angular momentum L = J ω, torque M = dL/dt, rotational friction RR, and spring stiffness for rotational oscillations SR). In this situation (analog to cases shown in Fig.1.1a),b),c),d)) we can establish all the relations between the parameters from T.1.1, adding to them similar relations concerning the rotational oscillatory motion (here not presented with any additional figure, but easy to visualize). Generalized Ohm’s Laws and Generalized Kirchoff’s Laws are directly applicable to the circuit situation in Fig.1.1 (see [2], Vol. 2).


3

Kirchoff’s Voltage Law states: “The sum of all the voltages in a loop is equal to zero.” Based on the situation in Fig.1.1 a), we shall have:

,)(

,1,0 2

2

dtdqiiii

Cq

dtdqR

dtqdLidt

CRi

dtdiLuuuuu

CRL

CRLi

====

++=++=++== ∫∑ (1.1)

For mechanical circuits, the analogy to Kirchoff’s Voltage Law will state: “The sum of all the velocities in a loop is equal to zero.” Based on the situation in Fig.1.1b), we shall have:

.)(

,1,0 2

2

dtdpffff

mp

dtdpR

dtpdsfdt

mfR

dtdfsvvvvv

mRmS

mmmRmSi

====

++=++=++== ∫∑ (1.2)

For a series rotational-elements circuit (similar to Fig.1.1 b)), the analogy to Kirchoff’s Voltage Law will state: "The sum of all the angular velocities in a loop is equal to zero",

.)(,

1,0

2

2

dtdMMMM

dtdR

dtds

MdtMRdt

dMs

JRRSRRR

RRJRRSRi

LJLLL

J

====++=

=++=ω+ω+ω=ω=ω ∫∑ (1.3)

Kirchoff’s Current Law states: "The sum of all the currents into a junction is equal to the sum of the currents out of the junction". Given the situation in Fig.1.1 c), we shall have:

=++=++== ∫∑ ∑ udtLR

udtduCiiiiii LRCoutpinp

1,..

.)(,12

2

dtduuuu

Ldtd

RdtdC LRC

Φ====

Φ+

Φ+

Φ= (1.4)

For mechanical circuits, analogy to Kirchoff’s Current Law will state: "The sum of all the input forces is equal to the sum of the output forces". Looking at the situation in Fig.1.1 d), we shall have:

.)(,1

1,

2

2

..

dtdxvvvv

sx

dtdx

Rdtxdm

vdtsR

vdtdvmffffff

SRmmm

mSRmmoutpinp

====++=

=++=++== ∫∑ ∑ (1.5)


4

For a parallel rotational-elements circuit (similar to Fig.1.1 d)), the analogy to Kirchoff’s Current Law will state: “The sum of all the input torque is equal to the sum of the output torque”,

.)(,1

1,

2

2

..

dtd

sdtd

Rdtd

dtsRdt

dMMMMMM

SRRRJRR

RRSRRRJoutpinp

α=ω=ω=ω=ω

α+

α+

α=

=ω+ω

+ω

=++== ∫∑ ∑

J

J

(1.6)

To avoid possible confusion in understanding the previous system of analogies, we must make a difference between the electric potential and voltage, which is a difference of potentials between the two points (u = ϕ2 - ϕ1 = ∆ϕ). The same is valid for velocities concerning a certain reference level and for the velocity « on » a certain mechanical element, which is equal to the difference of two referential velocities. A next important difference between the electric and mechanical analogue elements is in the fact that all electric elements from T.1.1 are scalars, and some of mechanical elements are vectors (v, p, f). On the other hand, if we now start from the mechanical conservation law of momentum, we shall have:

....... ∑∑ ∑∑ ∑∑ =⇔=⇒= outpinpoutpinpoutpinp ffpdtdp

dtdpp (1.7)

Clearly, momentum conservation (1.7) directly corresponds (or leads) to "Kirchoff's Force/Current Laws" (1.5). Following the pattern of conclusion in (1.7), we can return to the electric system and develop the total electric charge conservation law:

....... ∑∑ ∑∑ ∑∑ =⇔=⇒= outpinpoutpinpoutpinp qqidtdi

dtdii (1.8)

If we now take the mechanical conservation law of angular momentum, we shall have:

....... ∑∑ ∑∑ ∑∑ =⇔=⇒= outpinpoutpinpoutpinp MMdtd

dtd

LLLL (1.9)

Obviously, angular momentum conservation (1.9) directly corresponds (or leads) to "Kirchoff's Force/Current Laws" for rotational motion (1.6), or, in other words, this is just a torque conservation law. In the previous way, step-by-step, we introduced the third level of electro-mechanical analogies, connecting them to the well-known and proven conservation laws in physics ((1.1) to (1.9)). This task has to be completed by taking into account all other, known conservation laws in physics in such a way as to obtain parallelism between electric and mechanical formulas, as before. By using this method, we


5

could (probably) establish some new (but up to now not explicitly expressed) conservation laws. Now we can summarize all the additionally presented analogies (comparing the equations from (1.1) to (1.9)), and extend the table of analogies T.1.1 to T.1.2 (by adding the elements originating from “rotational” circuit situations, (see [3])). T.1.2

Electric parameter / [unit] Mechanical parameter / [unit] Velocity (=) v (=) [m/s] Voltage (=) u (=) [V = volt] Angular Velocity (=) ω (=) [rad/s] Force (=) f = dp/dt (=) [N = Kg m/s2 = newton]

Current (=) i = dq/dt (=) [A = ampere] Torque (=) M = dL/dt (=) [Kg m2/s2] Mech. Resistance (=) Rm (=) [m /N s = s/Kg] Resistance (=) R (=) [Ω = ohm] Mech. tors. Resistance (=) RR (=) [s/Kg m2] Spring Stiffness (=) S (=) [m/N = s2/Kg] Inductance (=) L (=) [H = henry] Torsion. Spring Stiffness (=) SR (=) [s2/Kgm2] Mass (=) m (=) [Kg]

Capacitance (=) C (=) [F = farad] Moment of Inertia (=) J (=) [Kg m2] Momentum (=) p = mv (=) [Kg m/s] Charge (=) q = Cu (=) [C = coulomb] Angular Momentum (=) L = J ω (=) [Kg m2/s] Displacement (=) x (=) [m]

Magn. Flux Φ (=) [Wb = V s = weber] Angle (=) α (=) [rad] (=) [1]

Along with T.1.2, let us dimensionally compare relevant expressions for rectilinear motion in a gravitational field, the situation in electromagnetic fields, and the situation related to the rotation of masses, T.1.3, T.1.4 and T.1.5. For the common and generic names of the corresponding columns (in the following tables), we shall take the names and symbols of relevant electro-magnetic parameters (see [2], Vol. 1).

T.1.3

[W] = [ENERGIES] [Q] = [CHARGES] [C] =[CAPACITANCES]

Electro-Magnetic Field [Cu2] (=) [qu1] [Li2] (=) [Φi1]

[Cu1] (=) [qu0] (=) [q] [Cu0] (=) [qu-1] (=) [C]

Gravitation [mv2] (=) [pv1] [mv1] (=) [pv0] (=) [p] [mv0] (=) [pv-1] (=) [m] Rotation [J ω2] (=) [L ω1] [Jω1] (=) [Lω0] (=) [ L ] [Jω0] (=) [Lω-1] (=) [J]

T.1.4

[U] = [VOLTAGES] [I] = [CURRENTS] [Z] = [IMPEDANCES] (=[mobility] in mechanics)

Electro-Magnetic Field [u] (=) [dΦ/dt] [i] (=) [dq/dt] [Ze] (=) [u] / [i] Gravitation [v] (=) [dx/dt] [f] (=) [dp/dt] [Zm] (=) [v] / [f] Rotation [ω] (=) [dα/dt] [M] (=) [dL/dt] [ZR] (=) [ω] / [M]

T.1.5

[L] = [INDUCTANCES] [R] = [RESISTANCES] [Φ] =[DISPLACEMENTS]

Electro-Magnetic Field [L] [R] [Φ] (=) [Li] Gravitation [S] [Rm] [x] (=) [Sf] Rotation [SR] [RR] [α] (=) [SRM]


6

[♣ COMMENTS & FREE-THINKING CORNER: Let us make one digression towards the Special Relativity Theory. In T.1.2, we find that mass m is analogue to electric capacitance C (m0 - rest mass), meaning that this analogy can possibly be extended in the following way:

2

20

2

20

11cv

CC

cv

mm−

=⇒

−

= (1.10)

The analogy (1.10) can be strongly supported by the following example. Let us imagine the plane electrodes capacitance, where the surface of a single electrode is S, and the distance between the electrodes is d. In that case, electric capacitance by definition is:

.0 dS

uqC ε== (1.11)

Let us suppose that the electric capacitance is moving by velocity v in the direction of its electrode distance (perpendicular to the electrode surface). In this case, the distance between the electrodes is described by the relativistic contraction formula:

.1 2

2

cvdd o −= (1.12)

Introducing (1.12) in (1.11), we get the form for capacitance that is equal to (1.10):

.111 2

20

0

2

20

2

20

0

cv

uq

cv

C

cv

dS

uqC

−

=

−

=

−

ε== (1.13)

From (1.13) we can get the relativistic formula related to the electric charge (on the capacitance electrodes):

.1 2

20

0

cv

uu

qq−

= (1.14)

Because we know that electrical charge is not dependent on its velocity, (1.14) will be transformed in:

.1 2

2

0 cvuuqq o −=⇒= (1.15)

In (1. 15), voltage u is equal to the potential difference between the capacitance electrodes,

.1)(1 2

2

2

2

12 cv

cvuu oo −ϕ∆=−=ϕ∆=ϕ−ϕ= (1.16)

Also, velocities «on» mechanical elements are equal to the difference of their corresponding referential velocities (if we want to treat the previous analogies correctly). Another interesting aspect of analogies can be developed if we compare the relativistic formulas for additions of the referential velocities with electrical voltages and potentials (but this might be somewhat premature and complicated at this stage).


7

Following the same analogy and the same supporting description as in (1.10) - (1.16), it is possible to conclude that the moment of inertia (for rotating particle in rectilinear motion), most probably, could be presented as (see T.1.2):

.111 2

20

2

20

2

20

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

−

ω=⇒

−

=ω

=⇒

−

=

cv

cv

cv

mm JL

JLJ (1.17)

The analogical prediction of the formula (1.17) is almost obvious as a correct result, since by definition of the moment of inertia, this is (dimensionally) the product of a certain mass and certain surface. Of course, it is clear that a rotating mass, which has the moment of inertia J (described by formula (1.17)) should be moving along its axis of rotation, having a linear (axial) velocity v. In the same case, combined, non-relativistic, motional (kinetic) energy can be expressed as (v << c):

,21

21 22 ω+= JmvEk (1.18)

which is usually applicable for observations in a center of gravity system. In relativistic situations (using (1.17)), we can generalize (1.18) into (1.19), hypothetically, assuming the applicability of used analogies:

.))cv(1)

cv()(,)((

)()(

pv)()(

1/22

222 −−=⎥

⎦

⎤⎢⎣

⎡−+=+⇒

−++

−+=

=+−=

c

2k

22

2c

20k

cv11Epv

cv11

cv11

cmmE

ωω

ω

ω

ω

L

L

J-J 0

(1.19)

From (1.19) we could try to go back to analogous electromagnetic equivalents, using again (1.13) and T.1.2 - T.1.5, (with an objective to create corresponding and meaningful expressions, in the frame of the system of analogies presented here, if shows possible). It would be interesting to make a comparison between electric charge q, and momentum p (its analogue match in Mechanics of rectilinear motion). As we already know, electric charge is invariant (always stays the same, constant amount), regardless of its velocity, or independent from velocity. By analogy, the same is valid for Linear Momentum (or quantity of motion), p = mv. If we (regardless this is correct or not) want to make them fully analogous, than it would be:

q Cu inv. p mv* inv. , m m / 1 v / c ,

C C / 1 v / c u u 1 v / c v* v * 1 v / c

p mv* = (m / 1 v / c ) (v * 1 v / c ) m v * inv..

02 2

02 2

02 2

02 2

02 2

02 2

0 0

= = ⇒ = = = −

= − = − ⇒ = −

RS|T|

UV|W|⇒

⇒ = − ⋅ − = =

l q l q , (1.20)

A similar concept (based on analogies) should be extendible to Angular Momentum L = Jω (for rotational motion). At present, nothing confirms the possibility that momentum p and L could be independent (invariant) regarding velocity, meaning that we still miss some important elements (regarding analogy between electric and mechanical charges) to make (1.20) possible. The only possibility “to save” (1.20) is to imagine somehow that we are taking about internal, intrinsic, stable and constant particle momentum p and L.


8

Since many contributions and discussions regarding boundary areas and basic assumptions of Einstein Relativity Theory are currently on the rise, it would be better to leave aside (for certain time) all premature and analogies-based conclusions (as for instance (1.10)-(1.20)) that are in direct connection with the Relativity Theory. ♣] A careful analysis of the previously established analogies, T.1.1 - T.1.5, shows that all the electric and mechanical parameters can be very conveniently classified into two groups according to their definition or nature: Spatial/Geometry parameters, and Action parameters, T.1.6. T.1.6 Spatial/Geometry

parametersLinear Motion

Gravitation Electromagnetism Rotation Spatial/Geometry parameters

Velocities and voltages v = dx/dt u = dΦ/dt ω = dα/dt Velocities and voltages

Displacements x = sf Φ = Li α = sR M Displacements Reactances (+) s L sR Reactances (+)

Resistances and Impedances

(*) Rm = (v / f)Real

Zm = (v / f)Complex

(*) R = (u / i)Real Z = (u / i)Complex

(*) RR = (ω / M)Real ZR = (ω/M)Complex


Reactances (-) m C J Reactances (-) Charges p = mv q = Cu L = Jω Charges

Forces and Currents f = dp/dt i = dq/dt M = dL/dt Forces and Currents

Action Parameters

Linear Motion Gravitation Electromagnetism Rotation Action Parameters

(*) R = (u / i)Real ⇔ u(t) and i(t) in phase (=) Real, Active Impedance or Resistance, Z = (u / i)Complex ⇔ u(t) and i(t) out of phase (=) Complex Impedance (=) Resistance + Reactance.

As we can see, in the table T.1.6 we do not have an explicit separation regarding parameters relative only to electric and only to magnetic fields. In the next chapters of this paper it will be shown that such parameters-separation (by analogy criteria) can be realized on the following way (see table T.1.7):

T.1.7 Spatial/Geometry

parametersElectric field Magnetic field Spatial/Geometry

parametersVelocities and voltages u = dΦ/dt i = dq/dt Velocities and voltages

Displacements Φ = Li q = Cu Displacements Reactances (+) L C Reactances (+)


(*) Rel.= (u / i)Real Zel. = (u / i)Complex

(*) Rmag.= (i / u)Real Zmag.= (i / u)Complex


Reactances (-) C L Reactances (-) Charges q = Cu Φ = Li Charges

Forces and Currents i = dq/dt u = dΦ/dt Forces and Currents Action Parameters Electric field Magnetic field Action Parameters

The classification given in T.1.6 and T.1.7 represents an almost complete and basic picture of electromechanical analogies, structure and order of physics parameters known at present. Based on the above-systematized analogies, several hypothetical statements and predictions will be formulated later, regarding new aspects of Gravitation, Faraday-Maxwell Electromagnetic Theory and Quantum Mechanics (see (2.1)-(3.5), (4.1)-(4.4), (5.15) and (5.16)). At the same time, a specific unification platform will be proposed (in later chapters of the same paper), based on generic


9

Symmetries, T.1.8, that will underline possible and most probable unification areas for Gravitation, Electromagnetism and Quantum Mechanics. The concept of basic Symmetries is additionally integrating and connecting all analogies given in T.1.3 – T.1.7. In 1905, a mathematician named Amalie Nether proved the following theorem (regarding universal laws of Symmetries): -For every continuous-symmetry of the laws of physics, there must exist a conservation law. -For every conservation law, there should exist a continuous symmetry. Let us only summarize already-known conservation laws and basic symmetries in Physics, by creating the table T.1.7 (without entering into a more-profound argumentation, since in later chapters of this paper we will again discuss basic symmetries, introducing much wider background). T.1.7 Symmetries of the Laws of Physics (mutually-conjugate variables)

Original Domains ↔(spatial, geometry or static parameters)

↔ Spectral Domains (motional, dynamic parameters)

Time = t Energy = E (or frequency = , or mass = m) fTime Translational Symmetry Law of Energy Conservation Displacement = Force)(,SpSx === FF& Momentum = mvxmp == & Space Translational Symmetry Law of Conservation of Momentum Angle = R RS Sα = =&L M Angular momentum = = α = ω&L J JRotational Symmetry Law of Conservation of Angular Momentum Electric Charge = el. el . mag. mag.q Cq Ci= Φ = =& Magn. Charge = mag. mag. el . el .q Lq Li= Φ = =& Law of Total Electric Charge Conservation The Electric Charge-reversal Symmetry The Magnetic Charge-reversal Symmetry “Total Magnetic Charge” Conservation

(Magnetic-monopole-charge is not a free and self-standing entity)

In the T1.7, mutually-conjugate variables are formally presenting Fourier integral transformations (in both directions) between all values from one side of the table to the other side of the table, showing deeper connections and symmetry between conditionally spatial or static parameters of our universe (left side of the table T.1.7), and their motional or dynamic conjugated-couples (right side of the table T.1.7). Obviously (we can see from T.1.7) that couples of conjugate, Original-Spectral domains, can be extended on both sides, by creating T.1.8, using simple analogies already present in T.1.3 – T.1.7. In fact, the proper merging between Analogies and basic or generic Symmetries (related to most important conservation laws, generally applicable: as given in T.1.8) would be the strongest predictive and unifying platform of future Physics (very much applied allover this paper).


T.1.8 Generic Symmetries and Analogies of the Laws of Physics Original Domains ↔ (spatial, static parameters)

↔ Spectral Domains (motional, dynamic parameters)

Time = t Energy = E Time Translational Symmetry Law of Energy Conservation

Velocity Voltage

analogies

Force Current

analogies

Displacement = x Momentum = p

xdtdxv

&=

= Space Translational Symmetry Law of Conservation of

Momentum vmpdtdp

&& ==

=F

Angle = α Angular momentum = L

αdtdαω

&=

=

Rotational Symmetry Law of Conservation of Angular Momentum ω

dtd

&& JL

LM

==

=

Electric Charge =

el. el . mag. mag.q Cq Ci= Φ = =& Magnetic Charge =

mag. mag. el . el .q Lq Li= Φ = =&

Law of Total Electric Charge Conservation

The Electric Charge-reversal Symmetry

mag.

mag.

Φdt

dΦu

&=

=

The Magnetic Charge-reversal Symmetry

“Total Magnetic Charge” Conservation

el.

el.

Φdt

dΦi

&=

=

(magnetic monopoles do not exist) Of course, the concept of Symmetries could be extended much more than given in T.1.8, but in this paper we will consider having the strongest predictive platform when using unified analogies and symmetries, as given in T.1.8. [♣ COMMENTS & FREE-THINKING CORNER: Developing specific (or new) theory in physics (separately from other fields) is usually not an easy job, and can sometimes be an arbitrary and meaningless process, if we do not have a general picture of the natural place and (most probable) basic structure of the new theory in question. The author’s position, and the main idea of this paper, is to show that multi-level analogies and symmetries, if correctly established, create a very common, simple and very much acceptable (conceptual and theoretical) platform for further generalizations in physics, and present an easy guide to new scientific discoveries (while respecting all conservation laws). Also, in the process of creating analogies we could be in a position to notice possible irregularities, missing links, or weak points in already known and well accepted theories, or to transfer positive achievements known (only) in one field to its analogous domain/s. Often analogies are helping to initiate and motivate a creative and intuitive thinking, supporting valuable brainstorming ideas. Another feeling of the author of this paper is that some of contemporary Physics’ theories and models are originally established and promoted by their founders, which were very strong and authoritative personalities, and later being dogmatized by the army of their faithful (but often non sufficiently critical) followers, and professionally-dependant assistants, or in some aspects also existentially dependent coworkers and students, becoming successors and defenders of the theories-founders, and natural barriers for penetration of new and different theories. In order to painlessly introduce new ideas and concepts into such environment of old and “well established theories”, the simplest and commonly acceptable strategy, proposed by the author of this paper, is to test them creating here explained multilevel analogies-symmetries platform (and to see which elements of the larger Physics-picture are missing or contradicting to other elements).


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Modern science realized its biggest achievements (in 20th century) based on the concept that nature prefer symmetry in a physics theory and that symmetry is the key to constructing physics laws without disastrous anomalies and divergences (see [10] and [11]). The objective of this paper is to show that any symmetry concept should be supported and united with wide (and multilevel) background analogy-platform in order to become the strongest framework in describing and mastering natural phenomena.

One can argue that analogy platform itself is a very week theoretical and practical platform (for making relevant conclusions and predictions) in comparison with modern Topology, but this is absolutely not correct since we are obviously living in an already united universe, where all forces and fields are coincidentally and harmonically present in every sequence of our existence (internally and externally). Most probably that our particular field theories are sometimes too primitive, too simple or incorrectly formulated, that we cannot see (or describe) the areas of their unification (from the point of view of modern Topology). If we just conclude that (today’s) Maxwell Theory and Gravitation cannot be easily united because of problems with some second-rank and curvature tensors, this is maybe temporarily (and conditionally) correct, but it also indicates that most probably some important topology components are missing both in Maxwell and Gravitation Theory, and that we should continue our search for better mathematical models and for missing field components (as for instance, to find them by including rotational, or torsion field components and motions in Maxwell Theory, Relativity Theory and Gravitation). Really, to realize something like that, we do not have better starting points than using well-established and multilevel analogies, and later to test them experimentally and theoretically using modern Topology Theory, Symmetries, Group Theory... ♣]

Works Cited: [1] Hueter, F., and Richard H. Bolt. SONICS, Techniques for the Use of Sound and Ultrasound in Engineering and Science, New York: John Willey & Sons, Inc., 1955. Pages: 9-18. Library of Congress Catalog Card Number: 55-6388, USA [2] Cyril M. Harris and Charles E. Crede. Shock and Vibration Handbook (Vol. 1 & 2): McGraw-Hill Book Company, 1961. (-Volume 1, Chapter 10, Pages: 10-1 to 10-59; -Volume 2, Chapter 29, Pages: 29-1 to 29-47). Library of Congress Catalog Card Number: 60-16636, USA [3] Prokic Miodrag. “Energetic Parallelism” , Diploma Work (B.S. Thesis), November 1975. University of Nis, Electronics University, YU. [4] L’ubomir Vlcek. New Trends in Physics: Slovak Academic Press, Slovakia, 1996. ISBN 80-85665-64-6. [5] José-Philippe Pérez, Nicole Saint-Cricq-Chéry. Relativite et quantification: MASSON SA, Paris Milan Barcelone 1995. ISBN: 2-225-80799-X, ISSN: 0992-5538. Pages: 175-179 [6] R. Boskovic. Theoria Philosophiae naturalis; -theoria redacta ad unicam legem virium in natura existentium; Viennae 1758 (and the same book in Venetian edition from 1763). [7] Shie Qian and Dapang Chen. Joint Time-Frequency Analysis; PRENTICE HALL PTR, 1996, Upper Saddle River, NJ 07458. ISBN: 0-13-254384-2. Pages: 29-37 [8] Papoulis Athanasios. Signal Analysis; McGraw-Hill, USA, 1977. ISBN: 0-07-048460-0.


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[9] Robert Eisberg and Robert Resnick. Quantum Physics, Second Edition; John Wiley & Sons, USA, 1985. ISBN: 0-471-87373-X. [10] Michio Kaku & Jennifer Thompson. Beyond Einstein: Anchor Books Edition, USA, 1995. ISBN 0-385-47781-3. Pages: 100-110. [11] Michio Kaku. Hyperspace: First Anchor Books Edition, USA, March 1995. ISBN 0-385-47705-8. Copyright by Oxford University Press. Printed in USA. [12] Lawrence C. Lynnworth. Ultrasonic Measurements for Process Control: Panametrics, Inc., Waltham, Massachusetts. ACADEMIC PRESS, INC., 1989. ISBN 0-12-460585-0. Printed in USA. Pages: 325-335. [13] Nick Herbert. Quantum Reality, Beyond the New Physics: Anchor Books, DOUBLEDAY, 1985. ISBN 0-385-23569-0. Printed in USA. [14] Heinz R. Pagels. The Cosmic Code: A Bantam New Age Book, Bantam DOUBLEDAY Dell Publishing, 1983-1990. ISBN 0-553-24625-9. Printed in USA and Canada. [15] R. M. Kiehn. An Interpretation of the Wave Function as a Cohomological Measure of Quantum Vorticity, Physics Department, University of Houston. (Presented in Florida 1989) Houston TX 77004, USA. See also number of papers written by the same author regarding Torsion Fields and Topology.


MPI Providing challenging ultrasonic solutions

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Basic Elements of MMM Systems & How MMM Systems Operate

Ultrasonic systems based on our unique MMM (Multi-frequency, Multimode, Modulated) technology may be used as structural actuators capable of delivering high power sonic and ultrasonic energy to a large or small loads. MMM uses proprietary techniques to initiate ringing and relaxing multimode (wideband high and low frequency) mechanical oscillations in a mechanical body to produce pulse-repetitive, frequency, phase and amplitude modulated bulk-wave-excitation on that body. MMM (Modulated, Multimode, Multifrequency) ultrasonic generators utilize a new and proprietary technology capable of stimulating wideband sonic and ultrasonic energy, ranging in frequency from infrasonic up to the MHz domain, that propagates through arbitrary shaped solid structures. Such industrial structures may include heavy and thick walled metal containers, pressurized reservoirs, very thick metal walled autoclaves, extruder heads, extruder chambers, mold tools, casting tools, large mixing probes, various solid mechanical structures, contained liquids, and ultrasonic cleaning systems.

Every elastic mechanical system, body, or resonator that can oscillate has many vibrating modes as well as frequency harmonics and sub harmonics in the low and ultrasonic frequency domains. Many of these vibrating modes can be acoustically and/or mechanically coupled while others would stay relatively independent. The MMM technology can utilize these coupled modes by applying advanced Digital Signal Processing to create driving wave forms that synchronously excite many vibrating modes (harmonics and sub harmonics) of an acoustic load. This technique produces uniform and homogenous distribution of high-intensity acoustical activity to make the entire available vibrating domain acoustically active while eliminating the creation of potentially harmful and problematic stationary and standing waves structures. This is not the case for traditional ultrasonic systems operating at a stable frequency where creation of standing waves structures is the norm.

The MMM or multimode excitation techniques are very beneficial to many applications including liquid processing, fluid atomization, powders production, artificial aging of

MPI, www.mpi-ultrasonics.com, [email protected]


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solids and liquids, accelerated stress relief, advanced ultrasonic cleaning, liquid metal treatment, surface coating, accelerated electrolysis, mixing and homogenizing of any fluid, waste water treatment, water sterilization, materials extrusion, wire drawing, improved molding and casting, and surface friction reduction to name a few.

Modulated, Multimode, Multifrequency sonic & ultrasonic vibrations can be excited in most any heavy-duty system by producing pulse-repetitive, phase, frequency and amplitude-modulated bulk-wave-excitation covering and sweeping an extremely wide frequency band. Every elastic mechanical system has many vibration modes, plus harmonics and sub harmonics, both in low and ultrasonic frequency domains. Many of these vibrating modes are acoustically and/or mechanically coupled, others are relatively independent. The MMM multimode sonic and ultrasonic excitation has the potential to synchronously excite many vibrating modes through the coupled harmonics and sub harmonics in solids and liquid containers to produce high intensity vibrations that are uniform and repeatable. Such sonic and ultrasonic driving creates uniform and homogenous distribution of acoustical activity on a surface and inside of the vibrating system, while avoiding the creation of stationary and standing waves, so that the whole vibrating system is fully agitated.

Every MMM system consists of (see Fig. 1, below):

A) A Sweeping-Frequency, Adaptively Modulated Wave Form generated by an MMM Ultrasonic Power Supply (including all regulations, controls and protections);

B) High Power Ultrasonic Converter(s);

C) Acoustical Wave-Guide (metal bar, aluminum, titanium), which connects the ultrasonic transducer with an acoustic load, oscillating body, or resonator;

D) Acoustical Load (mechanical resonating body, sonoreactor, radiating ultrasonic tool, sonotrode, test specimen, vibrating tube, vibrating sphere, a mold, solid or fluid media, etc.);

E) Sensors of acoustical activity fixed on, in, or at the Acoustical Load (accelerometers, ultrasonic flux meters, cavitation detectors, laser vibrometer(s), etc.), which are creating regulation feedback between the Acoustical Load and Ultrasonic Power Supply. In most of cases the piezoelectric converter can function as the feedback element, avoiding installation of other vibrations sensors.



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A strong mechanical coupling between the high power Ultrasonic Converter (B) to the Acoustical Load (D) is realized using a metal bar as an Acoustic Wave-Guide (C).

The Ultrasonic Converter (B) is electrically connected to the Ultrasonic Multimode Generator Power Supply (A).

The Acoustic Activity Sensor (E) relays physical feedback (for the purpose of automatic process control) between the Acoustical Load (D) and Ultrasonic Power Supply (A).

Fig. 1

MMM Generator Technology:

A new approach to Ultrasonic power supplies and systems

As depicted in Figure 1 above the Ultrasonic Converter (B), driven by Power Supply (A), is producing a sufficiently strong pulse-repetitive multifrequency train of mechanical oscillations or pulses. The Acoustical Load (D), driven by incoming frequency and amplitude modulated pulse-train starts producing its own vibration and transient response, oscillating in one or more of its natural vibration modes or harmonics. As the excitation changes, following the programmed pattern of the pulse train, the amplitude in these modes will undergo exponential decay while other modes are excited.

A simplified analogy is a single pulsed excitation of a metal bell that will continue oscillating (ringing) on several resonant frequencies for a long period following the initial pulse. How long each resonant mode will continue to oscillate after a pulse depends on the mechanical quality factor for that mode.

Every mechanical system (in this case the components B, C and D) has many resonant modes (axial, radial, bending, and torsional) and all of them have higher frequency harmonics. Some of the resonant modes are well separated and mutually isolated, some of them are separated on a frequency scale but acoustically coupled, and some



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will overlap each other over a frequency range and these will tend to couple particularly well.

Since the acoustical load (D) is connected to an ultrasonic converter (B) by an acoustical wave-guide (C), acoustical relaxing and ringing oscillations are traveling back and forth between the load (D) and ultrasonic converter (B), interfering mutually along a path of propagation. The best operating frequency of the ultrasonic converter (B) is normally found when the maximum traveling-wave amplitude is reached and when a relatively stable oscillating regime is found. The acoustical load (D) and ultrasonic converter (B) are creating a “Ping-Pong Acoustical-Echo System”, like two acoustical mirrors generating and reflecting waves between them. For easier conceptual visualization of this process we can also imagine multiple reflection of a laser beam between two optical mirrors. We should not forget that the ultrasonic converter (B) is initially creating a relatively low pulse frequency mechanical excitation, and that the back-and-forth traveling waves can have a much higher frequency.

In order to achieve optimal and automatic process control, it is necessary to install an amplitude sensor (E) of any convenient type (e.g. accelerometer, ultrasonic flux sensor) on the Acoustical Load (D). The sensor is connected by a feedback line to the control system of Ultrasonic Power Supply (A).

There is another important effect related to the ringing resonant system described above. Both the ultrasonic source (B) and its load (D) are presenting active (vibrating) acoustic elements, when the complete system starts resonating. The back-forth traveling-waves are being perpetually reflected between two oscillating acoustical mirrors, (B) and (D). An immanent (self-generated) multifrequency Doppler Effect (additional frequency shift, or frequency and phase modulation of traveling waves) is created, since acoustical mirrors, (B) and (D), cannot be considered as stable infinite-mass solid-plates. This self-generated and multifrequency Doppler Effect is able to initiate different acoustic effects in the load (D), for instance to excite several vibrating modes at the same time or successively, producing uniform amplitude distributions of acoustic waves in the acoustic load (D). For the same reasons, we also have permanent phase modulation of ultrasonic traveling waves since opposite-ends of the acoustic mirrors are also vibrating. We should strongly underline that the oscillating system described here is very different from the typical and traditional half-wave, ultrasonic resonating system, where the total axial length of the ultrasonic system consists of integer number of half-wavelengths. In the case of MMM systems we, generally speaking, do not care much about the specific ultrasonic system geometry and its axial (or any other) dimensions. Electronic multimode excitation continuously (and automatically) searches for the most convenient signal shapes in order to excite many vibration modes at the same time, and to make any mechanical system vibrate and resonate uniformly.

In addition to the effects described above, the ultrasonic power supply (A) is also able to produce variable frequency-sweeping oscillations around its central operating frequency (with a high sweep rate), and has an amplitude-modulated output signal (where the frequency of amplitude modulation follows sub harmonic low frequency vibrating modes). This way, the ultrasonic power supply (A) is also contributing to the multi-mode ringing response (and self-generated multifrequency Doppler effect) of an acoustical load (D). The ultrasonic system described here can drive an acoustic load (D) of almost any irregular shape and size. In operation, when the system oscillates we cannot find stable nodal zones, because they are permanently moving as a result of the specific signal modulations coming from the MMM Ultrasonic Power Supply (A)).



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It is important to note that by exciting an acoustical load (D) we could produce relatively stable and stationary oscillations and resonant effects at certain frequency intervals, but also a dangerous and self-destructive system response could be generated at other frequencies. The choice of the central operating frequency, sweeping-frequency interval and ultrasonic signal amplitudes from the ultrasonic power supply (A) are critical elements to be carefully selected. Because of the complex mechanical nature of different acoustic loads (D), we must test carefully and find the best operating regimes of the ultrasonic system (B, C, D), starting with very low driving signals (i.e. with very low ultrasonic power). Therefore an initial test phase is required to select the best operating conditions, using a resistive attenuating dummy load in serial connection with the ultrasonic converter (A). This minimizes the acoustic power produced by the ultrasonic converter and can also dissipate accidental resonant power. When the best driving regime is found, we disconnect the dummy load and introduce full electrical power into ultrasonic converter.

The best operating ultrasonic regimes are those that produce very strong mechanical oscillations, or high and stable vibrating mechanical amplitudes, with moderate electric output power from the ultrasonic power supply. The second criterion is that thermal power dissipation on the total mechanical system continuously operating in air, with no additional system loading, is minimal. In other words, low thermal dissipation on the mechanical system (B, C, D) means that the ultrasonic power supply (A) is driving the ultrasonic converter (B) with limited current and sufficiently high voltage, delivering only the active or real power to a load. The multifrequency ultrasonic concept described here is a kind of “Maximum Active Power Tracking System”, which combines several PLL and PWM loops. The actual size and geometry of acoustical load are not directly and linearly proportional to delivered ultrasonic driving-power. Its possible that with very low input-ultrasonic-power a bulky mechanical system (B, C, D) can be very strongly driven (in air, so there is no additional load), if the proper oscillating regime is found.

Traditionally, in high power electronics, when driving complex impedance loads (like ultrasonic transducers) in resonance, a PLL (Phase Locked Loop) is related to a power control where load voltage and current have the same frequency. In order to maximize the Active Load Power we make zero phase difference between current and voltage signals controlling the driving voltage frequency. In modern Power Electronics we use Switch-Mode operating regimes for driving Half or Full Bridge, or some other output transistors configuration(s). The voltage shape on the output of the Power Bridge is square shaped (50% Duty Cycle), and current in the case of R/L/C resonant circuits as electrical loads always has a sinusoidal shape. Here we are dealing with a time domain current and voltage signals.

We can summarize the traditional PLL concept as:

Input values, Source CAUSE ⇒

(driving voltage)

Produced Response CONSEQUENCE/s

(output current)

Regulation method for maximal Active Output Power

Square (or sine) shaped driving-voltage on the output Power Bridge

Sinusoidal output current

Control the driving-voltage frequency for minimal phase difference between output Load Voltage and Current signals.

Relatively Stable driving frequency (or resonant frequency)

Load Voltage and Current have the same frequency

Control the current and/or voltage amplitude/s for necessary Active Power Output (and to realize correct impedance matching)



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The new MMM concept can be summarized as:

Input values, Source CAUSE ⇒

(driving voltage)

Produced Response CONSEQUENCE/s

(output current)

Regulation method for maximal Active Output Power:

The average phase differences between the output HF current

and voltage and the sub-harmonics, on the output ferrite

transformer, should be minimal in average.

Square shaped voltage on the output Power Bridge: PWM + Band Limited, Frequency Modulation (+ limited phase modulation in some applications)

Multi-mode or single sinusoidal output current (or ringing decay current) with Variable operating frequency + Harmonics

First PLL at resonant frequency: • To control the central

operating frequency of a driving-voltage signal to produce Active Load Power much higher than its Reactive Power.

• To realize the maximal input (LF) power factor (PF = cos(theta) = 1).

Stable central operating, driving frequency + band limited frequency modulation (+ limited phase modulation in some applications)

Stable mean operating (Load) frequency coupled with the driving-voltage central operating frequency, as well as with harmonics

• To make that complete power inverter/converter looks like a resistive load to the principal Main Supply AC power input.

• To realize the maximal input (LF) power factor (PF = cos(theta) = 1).

Output transformer is “receiving” reflected harmonics (current and voltage components) from its load.

Particular frequency spectrum/s of a Load Voltage and Current could sometimes cover different frequency ranges.

Second PLL at modulating (sub-harmonic) frequency: • To control the modulating

frequency in order to produce limited RMS output current, and maximal Active Power (on the load).

• To realize maximal input (LF) power factor (PF = cos(theta) = 1).

All over-power, over-voltage, over-current and over-temperature regulations, limitations and protections (pulse-by-pulse and in average) should be implemented. Safe operating components margins should be chosen sufficiently higher than in the cases of traditional, single frequency PLL systems.

The MMM concept is the most general case of Maximum Active Power Tracking and it covers the Traditional PLL concept. A number of variations of MMM concept are imaginable depending on resonant-load applications (like suppressing or stimulating certain operating frequencies or harmonics, implementing frequency sweeping, or randomized frequency and phase modulation/s etc.). Traditionally the PLL concept is applied to immediate load current and voltage signals, and in MMM we apply the similar concept to the immediate active load-power signal. In any case the principal objectives are to realize optimal and maximal active power transfer to the load, and that complete power system (in-average, time-vise) looks like resistive load to the main supply input, and this is exactly how MMM systems operate.



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In conventional Ultrasonics technology the transducers and connected elements are designed to satisfy precise resonant conditions. To achieve maximum efficiency, all oscillating elements must be tuned to operate at the same resonant frequency. In contrast the patented MMM technology was developed to breakaway from this restrictive “tuned mode” by using advanced Digital Signal Processing (DSP) techniques to implement an intelligent feedback loop that allows adaptation to most any un-tuned, changing, or evolving mechanical system. Instead of optimizing acoustic elements to accept a specific resonant frequency operation, MMM systems use the intelligent DSP to adapt to the un-tuned load. The system continuously analyzes system feedback and optimizes a complex shaped electrical driving signal customized to each specific oscillating structure.

To remain compatible with standard transducers the MMM generators use an adjustable primary resonant frequency as a central carrier frequency that efficiently drives standard transducers in a modulated mode. The MMM driving oscillations are not fixed or random, rather they follow a consistent and evolving pulse-repetitive pattern, where frequency, phase and amplitude are simultaneously modulated by the control system. The optimized modulations provide a highly efficient transfer of electrical to mechanical energy and prevent the creation of problematic stationary or standing waves as typically produced by traditional ultrasonic systems operating at a single frequency.

MMM systems offer a high level of control through regulation and programming of all vibration, frequency, and power parameters using either a handheld control panel or a Windows PC software interface. The system's fine control extends excellent repeatability and produces highly efficient active power that may range from below 100 W up to many kW. MMM technology can drive, with high efficiency, complex mechanical system up to a mass of several tons and consisting of arbitrary resonating elements.

Due to the flexible nature of the MMM technology, a wide range of new or improved applications are possible. For example applications requiring high temperatures represent a problem to conventional transducers that are extremely sensitive to heat. Since MMM systems are not restricted to specific tuned elements it is now possible to address high temperature applications through the use of extended acoustic wave-guides (e.g. 1 to 3 meters in length). An extended wave-guide puts the necessary physical distance between the heat sensitive transducer and the high temperature load. A long wave-guide also provides a convenient mounting point for cooling jackets that will draw away excessive heat and protect the transducer. Other fields of possible MMM Technology application are: Advanced Ultrasonic Cleaning, Material Processing, Sonochemistry, Liquid Metals and Plastics treatment, Casting, Molding, Injection, Ultrasonically assisted sintering, Liquids Atomization, Liquids Mixing and Homogenization, Materials Testing, Accelerated Aging, and Stress Release.

Please make contact with us to discuss any new or challenging application (www.mpi-ultrasonics.com)



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Example Ultrasonic Applications That Can Benefit From MMM Systems

1. Ultrasonic liquid processing

a. mixing and homogenization

b. atomization, fine spray production

c. surface spray coating

d. metal powders production and surface coating with powders

2. Sonochemical reactors

3. Water sterilization

4. Heavy duty ultrasonic cleaning

5. Pulped paper activation (paper production technology)

6. Liquid degassing, or liquid gasifying (depending of how sonotrode is introduced in liquid)

7. De-polymerization (recycling in a very high intensity ultrasound)

8. Accelerated polymerization or solidification (adhesives, plastics…)

9. High intensity atomizers (cold spay and vapor sources). Metal atomizers.

10. Profound surface hardening, impregnation and coating

a. surface hardening (implementation of hard particles)

b. capillary surface sealing

c. impregnation of aluminum oxide after aluminum anodizing

d. surface transformation, activation, protection

11. Material aging and stress release on cold

a. Shock testing. 3-D random excitation

12. Complex vibration testing (NDT, Structural defects detection, Acoustic noise…)

a. accelerated 3-dimensional vibration test in liquids

b. leakage and sealing test

c. structural stability testing of Solids



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d. unscrewing bolts testing

13. Post-thermal treatment of hardened steels (cold ultrasonic treatment)

a. elimination of oxides and ceramic composites from a surface

b. profound surface cleaning

c. residual stress release, artificial aging, mechanical stabilization

14. Ultrasonic replacement for thermal treatment. Accelerated thermal treatment of metal and ceramic parts in extremely high intensity ultrasonic field in liquids.

15. Surface etching

a. abrasive and liquid treatment

b. active liquids (slightly aggressive)

c. combination of active liquids and abrasives

16. Surface transformation and polishing

a. combination of abrasives and active liquid solutions

b. electro-polishing and ultrasonic treatment

17. Extrusion (of plastics and metals) assisted by ultrasonic vibrations

a. special ultrasonic transducers in a direct contact with extruder

18. Founding and casting (of metals and plastics) assisted by ultrasound

a. vacuum casting, homogenization, degassing

b. micro-crystallization, alloying, mixing of different liquid masses

19. Adhesive testing

a. aging test

b. accelerated mechanical resistance testing

c. accelerated moisture and humidity testing

20. Corrosion testing

a. in different liquids

b. in corrosive liquid, vapor phase



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MMM GENERATOR SELECTION GUIDE

Products, Applications, & Understanding Acoustical Loading

We offer a complete line of ultrasonic generators to address a wide range of traditional and new ultrasonic applications. Due to the complex nature of the applications we address our ultrasonic generators are available with a variety of standard and custom features. This guide will help you understand our unique features and help you define the best feature set for your application.

Active Ultrasonics’ generators are based on our unique and proprietary MMM technology (MMM = Multi-frequency, Multimode, Modulated). This technology employs advanced Digital Signal Processing (DSP) for input signal analysis and output signal conditioning techniques to produce wideband sonic and ultrasonic Vibrations in acoustic loads. (See the detailed explanation of MMM technology below)

Open Frame MMM Modules for OEM, ODM, & System Integrator Clients

MMM open frame generator modules (OF and OW models) are available to clients producing equipment or systems requiring integration of ultrasonic generators into their own cabinets or housings. Our open frame generators are available without the manufacturers brand mark meaning that clients can use them as a component part of a system marked with the client’s private label. Although these generator modules offer a full set of programmable functions and features the housing and interface options are simplified to make them cost competitive. The simplified design uses terminal block connectors and requires our clients to have sufficient technical background and experience to safely make all electrical connections. Open Frame generator clients are often in the business of producing ultrasonic related equipment for various applications (Cleaning, Water Processing, Sonochemistry, etc…), and are experienced in systems integration including installation of transducer elements, source power, additional sensors, PLCs, automatic controls.

The greatest advantage of Open Frame MMM generators is that they are easily adjusted to drive almost any kind of piezoelectric transducers, including single transducers or arrays of transducers operating in parallel mode. As such our generators have the flexibility to drive a client’s existing transducers. In most cases clients can make adjustments and settings by carefully following the instructions given in our equipment operation manuals. In some cases the necessary impedance adjustments to adapt to a clients transducer may be outside the range of our standard equipment and will require a simple factory modification.

Our systems also provide a programmable frequency adjustment for adapting to shifting system resonant frequency due to changes in the acoustic load. Open Frame generators allow user frequency tuning within a 5 kHz window (e.g. 17.5 kHz – 28.5 kHz), or much more. This frequency window may be customized in our factory to address your application needs.

Fully Featured MMM Generator IX Models for R&D or Application Development

For laboratory and scientific research, extraordinary, challenging, unusual, or complex high-tech applications we are delivering fully featured version of the MMM generator. Standard systems labeled as IX models are housed in a finished stainless housing with standard connectors for power, transducer, and controller electrical connections. IX



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models have front panel controls for basic power ON/OFF and ultrasonic power output adjustment.

Advanced features and parameter settings including frequency controls, power adjustments, modulations, modulation timing, digital and smooth frequency settings are controlled via a hand-held control panel or user friendly Windows PC interface. The PC adapter may also be used by clients for development of custom software control using external PC, PLC, or mechanical relay controls. This complete feature set plus advanced overload protection circuits allow clients to quickly become fully-operational, fully-protected and fully-independent in the initial system development phase.

We can also offer customized versions to address unusual applications. In such cases we recommend our clients consult with us to define the requirements and the desired ultrasonic goal.

MMM Generator Control

RS485 Interface: All MMM generators may be ordered with an optional RS485 interface used for connection to our hand-held control panels or PC interface adaptor for Windows PC software control or a custom developed PLC or PC software control. With these tools clients may perform external manual control, remote control, PLC control, or utilize our software graphic-interface control on a Windows PC.

RS458 Network Adapter: This optional adapter is connected to generator RS485 connector and allows connecting up to 16 MMM generators in a single network. We also have an adapter for a network of up to 64 generators.

MMM Generator Set-Up

Our MMM generators are able to drive heavy and arbitrary-shaped solid masses, however due to physics and acoustics certain limits must be respected. In unusual applications it is highly recommended that clients review all available MMM information and consult with Active Ultrasonics. There is no universal answer, solution, or recommendation for driving arbitrary and complex mechanical systems that usually have number of resonances and harmonics since in every specific case the “acoustic-reality” is different. Based on our experience and knowledge we can advise clients on the best possible hardware options and system set-up.

Resonant Frequency: If you would operate a single piezoelectric converter in resonance, and your converter by its design/geometry has a well-defined and sharp resonance, 20 kHz for example, you would need an MMM generator that is factory set with a frequency window covering 20 kHz (e.g. 18 kHz to 22 kHz). Having a wider interval of carrier frequency range will not offer additional benefit for your application, and you could only destroy, overload, or over-heat your mechanical system by trying to drive it against its “acoustical-nature”. In such cases it is always better to consult with Active Ultrasonics to make a factory adjustment of frequency interval range to match your system. If your converter would have 30 kHz resonant frequency, then we will limit carrier-frequency settings to a window covering 30 kHz (e.g. 28 kHz to 32 kHz). This allows for safe operation while giving some flexibility to adjust to shifts in the resonant carrier frequency.

As a special order we can provide a custom MMM generator that has a wide window of frequency adjustment up to 24 kHz (e.g. 18 kHz to 42 kHz). Although such custom systems offer increased research and development flexibility they also offer a much greater risk of damage to the generator and acoustic system. Driving a transducer/converter outside of its primary resonant frequency will likely damage your equipment if operation is forcing unnatural, acoustically-unacceptable operating



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regimes. To recognize and avoid such regimes you would need a great deal of specific ultrasonic system experience and we do not recommend that you take such actions without consulting Active Ultrasonics.

Generator Adjustment to Various Transducers: All MMM generators offer some flexibility to adapt to transducer/converter changes. Any change of transducers/converters within the range of the generators capabilities will require performing important inductive compensation adjustments. The adjustments are detailed in the generator Operations Manual and require some level of electronics knowledge for safe operation. The range of possible adjustment is a factory setting based on an installed inductive component. When ordering the MMM generator you will need to specify the primary type of transducer(s) used in your system.

If you would like to drive different ultrasonic transducers, each of them having different resonant frequencies and very different input capacitances, one single MMM generator will most likely not have the necessary range of adjustment for proper operation. Operating transducers/converters without optimum impedance matching may cause inefficient power consumption and damage to the transducer. Electronically we could try to drive a resonant system in a frequency band larger than its “acoustic-nature” is dictating but the mechanical system itself would not accept to be driven too far from its resonance, regardless how many of signal-modulating tricks we would apply. In unusual or demanding applications we strongly recommend that clients consult with Active Ultrasonics and provide details of the transducers/Converters to be used and the systems physical environment including boosters, sonotrodes, tools, and mechanical devices (“Acoustic Load”) to be driven. Such consulting is necessary to avoid non-realistic and out of “acoustics-reality” expectations and situations.

Systems Set-Up: To reduce new application development time and create safer operating systems we are often providing initial consulting services to clients. In many cases we are making complete mechanical systems or critical component parts and making initial MMM generator testing to realize the best adjustments and tuning in our laboratories. Clients without previous experience in ultrasonics or with our new MMM technology normally experience problems and make mistakes that give less that desirable results or damage equipment. In other cases we cooperate with clients who produce mechanical parts or devices to our specification and they send these parts to us for final fitting of an appropriate converter, making the best inductive compensation, optimizing the generator operating regime, and making the best tuning.

Although the generator parameters are well defined in the Operations Manual we find that clients are often accidentally, naively, and in some cases against our specific instruction trying some exceptional operating situations that are damaging to the system. Due to the wide range of options available to the MMM generator we strongly recommend that clients first consult with Active Ultrasonics about the application and operational environment before exploring risky options. We have implemented many levels of internal overload protections to protect our equipment from mishandlings and possible mistakes, however, over zealous clients can find a way to damage the MMM generator.

MMM Technology

While the MMM technology (MMM = Multifrequency, Multimode, Modulated Sonic & Ultrasonic Vibrations) is the industries most flexible sonic & ultrasonic solution it is very important to understand the underlying concept to avoid unrealistic expectations. Physics and acoustics dictate the applications that may be addressed by our MMM technology. Following are some important technical points to be considered and understood:



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a) Carrier Frequency: Every MMM generator has a predefined factory set carrier frequency interval window (e.g. 18 kHz to 42 kHz, or 17.5 kHz to 22.5 kHz). The generator may be set to operate at a specific constant frequency inside of the factory set window. It is not possible to operate the MMM generator outside of the factory predefined carrier frequency range.

b) Frequency Modulations: In addition to the pre-defined carrier frequency setting, there are number of user-selectable frequency-modulating options such as mathematically-predefined modulation and MMM-dynamic (time-evolving and load-dependant) carrier-frequency modulating options.

a. For optimum system set-up the user should first test and select the best central operating (carrier) frequency for every particular application.

i. While performing such initial frequency settings the generator power setting should be limited to between 10% and 30% of its maximal output power to avoid sudden overloads and system damage.

ii. All modulating parameters should be initially disabled or giving a zero values.

b. Once the converter is producing measurable, constant-frequency amplitudes, start gradually implementing different MMM modulating parameters.

i. When optimal acoustic regime is reached and well tested, user can gradually increase the output power. If in process of power-increasing we notice that system is not optimally tuned, carrier frequency and frequency-modulating parameters should be slightly readjusted.

c) Multifrequency Effects: The MMM generator itself is not operating in a large frequency band, but the acoustical load attached to the transducer, especially arbitrary shaped elements, can effectively-oscillate in a large frequency band when complex frequency modulations are applied (modulating the carrier or central operating frequency = MMM modulations). This means that a constant carrier frequency generator with MMM modulation can excite a wide range of resonant modes and harmonics in the acoustic load. The multifrequency effect is coming form the excited acoustic spectrum inside of an acoustic load as the consequence of applying MMM modulation. The width of the acoustic loads frequency spectrum is depend on acoustic and mechanical properties of the load, on its geometry, and on the MMM parameters-settings. For example, we can use a 20 kHz piezoelectric transducer, connect it to a certain mechanical load, drive it with 20 kHz carrier signal, then start performing different MMM-modulations, and conclude by applying spectral measurements (microphones, hydrophones, accelerometers, laser vibrations meters, etc.) that our mechanical load starts vibrating uniformly, without creating standing waves if MMM modulating parameters are optimally selected. We can also see that the system is generating a large band of many resonant frequencies. In other words, MMM-modulated acoustic loads are becoming dynamically-controlled multi-resonant systems producing complex acoustic spectrum, starting from very low frequencies until very high frequencies (often from several Hz to MHz). If a mechanical system in its static (non-vibrating) state can be characterized by lumped-circuit models and constant electromechanical parameters, after applying MMM modulation, the same system is manifesting interval-type parameters definition, producing extraordinary wide-frequency band effects, eliminating standing waves and



14

giving an impression that the complete acoustic load is vibrating uniformly. In some applications, such as Liquids-processing, Sonochemistry, and Cleaning, applying a constant carrier frequency anywhere in the range between 20 and 40 kHz plus MMM modulation we can measure acoustic spectral components from infrasonic vibrations until MHz range, being produced as secondary effects of MMM modulation, and being mixed with other acoustics-related effects that are naturally producing large frequency band emission (e.g. cavitation).

d) Acoustic Loads and Multifrequency Effects: Acoustic loads or solid objects that already have many different resonant frequencies and harmonics are preferable loads for MMM technology. Spectral (mechanical) complexity of the acoustical system is very important if we would like to realize very large-band and uniformly distributed sonic and ultrasonic oscillations (without creating standing waves). In other words, well tuned mechanical systems that have strong and single resonant frequency (like constant frequency plastic welding sonotrodes) would not be the best choice as acoustic loads for MMM systems. In contrary, non-tuned and complex geometry objects have a strong chance to be well driven by MMM Power Supplies. MMM signal processing is able to initiate strong oscillations in many resonant frequencies and their harmonics at the same time, giving the impression that the complete object under such vibrations is oscillating uniformly (externally and internally). Hollowed objects with circular and cylindrical holes and slits are normally showing excellent performances regarding MMM, multifrequency operating regimes.

e) MMM Operating Regime: Every well-selected MMM operating regime can be recognized by smooth, uniform and easy-going load-oscillations. If in any part of a mechanical system (converter-wave-guide-acoustic-load) excessive heat is being generated, this is usually the sign that operating parameters are not well selected. If the mechanical system is producing randomized, cracking, braking or impulsive, low-frequency noise, this is also the sign that operating regime is not well selected and that MMM parameters should be changed. Monitoring of such regimes and detecting zones of high stress and overheating could be realized using real-time infrared cameras. A hot spot having a significantly different temperature compared to its surrounding area is a sign of a poorly selected operating regime or a sign of structural/mechanical defects.

f) MMM generators can also operate as ordinary constant-frequency ultrasonic generators if we disable or set to zero all modulating parameters (in such cases operating on given carrier frequency).

g) MMM technology is highly recommendable primarily for unique, extraordinary and new applications, where the user is expecting results that could not be reached by using standard, traditional ultrasonic technologies. MMM technology is also providing superior performance in most liquid processing and cleaning applications.

h) Meaning of Loading of piezoelectric transducer/s in traditional ultrasonic technologies (constant operating frequency systems) is a very complex subject and can be explained as follows:

In applications such as Ultrasonic Welding, single operating, well-defined, resonant frequency transducers are usually used (operating often on 20, 40 and sometimes around 100 kHz and higher). In recent time, some new transducer designs can be driven on sweeping frequency intervals (applied to a single transducer).



15

In Sonochemistry and Ultrasonic Cleaning we use single or multiple ultrasonic transducers (operating in parallel), with single resonant frequency, two operating frequencies, multi-frequency regime, and all of the previously mentioned options combined with frequency sweeping. Frequency sweeping is related to the vicinity of the best operating (central) resonant frequency of transducer group. Frequency sweeping can also be applied in a low frequency (PWM, ON-OFF) group modulation (producing pulse-repetitive ultrasonic train, sometimes-called digital modulation).

Also, multi-frequency concept is used in Sonochemistry and Ultrasonic Cleaning when we can drive a single transducer on its ground (basic, natural) frequency and on several higher frequency harmonics (jumping from one frequency to another, without changing transducer/s).

Real time and fast automatic resonant (or optimal operating frequency) control/tuning of ultrasonic transducers is one of the most important tasks in producing (useful) ultrasonic energy for different technological applications, because in every application we should realize/find/control:

• The best operating frequency regime in order to stimulate only desirable vibrating modes.

• To deliver a maximum of real or active power to the load (in a given/found operating frequency domain/s).

• To keep ultrasonic transducers in a pulse-by-pulse, real time, safe operating area regarding all critical overload/overpower situations, or to protect them against: overvoltage, overcurrent, overheating, etc.

All of the previously mentioned (control and protecting) aspects are so interconnected, that none of them can be realized independently, without the other two. All of them also have two levels of control and internal structure:

a) Up to a certain (first) level, with the design and hardware, we try to insure/incorporate the most important controls and protecting, (automatic) functions.

b) At the second level we include certain logic and decision-making algorithm (software) which takes care of real-time and dynamic changes and interconnections between them.

It is necessary to have in mind that in certain applications (such as ultrasonic welding), operating and loading regime of ultrasonic transducer changes drastically in relatively short time intervals, starting from a very regular and no-load situation (which is easy to control), going to a full-load situation, which changes all parameters of ultrasonic system (impedance parameters, resonant frequencies…). In a no-load and/or low power operation, ultrasonic system behaves as a typically linear system; however, in high power operation the system becomes more and more non-linear (depending on the applied mechanical load). The presence of dynamic and fast changing, transient situations is creating the absolute need to have one frequency auto tuning control block, which will always keep ultrasonic drive (generator) in its best operating regime (tracking the best operating frequency).



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The meaning of mechanical loading of ultrasonic transducers

Mechanical loading of the transducer means realizing contact/coupling of the transducer with a fluid, solid or some other media (in order to transfer ultrasonic vibrations into loading media). All mechanical parameters/properties (of the load media) regarding such contact area (during energy transfer) are important, such as: contact surface, pressure, sound velocity, temperature, density, mechanical impedance. Mechanical load (similar to electrical load) can have resistive or frictional character (as an active load), can be reactive/imaginary impedance (such as masses and springs are), or it can be presented as a complex mechanical impedance (any combination of masses, springs and frictional elements). In fact, direct mechanical analogue to electric impedance is the value that is called Mobility in mechanics, but this will not influence further explanation. Instead of measuring complex mechanical impedance (or mobility) of an ultrasonic transducer, we can easily find its complex electrical impedance (and later on, make important conclusions regarding mechanical impedance).

Mechanical loading of a piezoceramic transducer is transforming its starting impedance characteristic (in a no-load situation in air) into similar new impedance that has lower mechanical quality factor in characteristic resonant area/s. There are many electrical impedance meters and network impedance analyzers to determine/measure full (electrical) impedance-phase-frequency characteristic/s of certain ultrasonic transducers on a low sinus-sweeping signal (up to 5 V rms.). However, the basic problem is in the fact that impedance-phase-frequency characteristics of the same transducer are not the same when transducer is driven on higher voltages (say 200 Volts/mm on piezoceramics). Also, impedance-phase-frequency characteristics of one transducer are dependent on transducer’s (body) operating temperature, as well as on its mechanical loading. It is necessary to mention that measuring electrical Impedance-Phase-Frequency characteristic of one ultrasonic transducer immediately gives almost full qualitative picture about its mechanical Impedance-Phase-Frequency characteristic (by applying a certain system of electromechanical analogies).

We should not forget that ultrasonic, piezoelectric transducer is almost equally good as a source/emitter of ultrasonic vibrations and as a receiver of such externally present vibrations. While it is emitting vibrations, the transducer is receiving its own reflected (and other) waves/vibrations and different mechanical excitation from its loading environment. It is not easy to organize such impedance measurements (when transducer is driven full power) due to high voltages and high currents during high power driving under variable mechanical loading. Since we know that the transducer driven full power (high voltages) will not considerably change its resonant points (not more than ±5% from previous value), we rely on low signal impedance measurements (because we do not have any better and quicker option).

Also, power measurements of input electrical power into transducer, measured directly on its input electrical terminals (in a high-power loading situation) are not a simple task, because we should measure RMS active and reactive power in a very wide frequency band in order to be sure what is really happening. During those measurements we should not forget that we have principal power delivered on a natural resonant frequency (or band) of one transducer, as well as power components on many of its higher and lower frequency harmonics. There are only a few available electrical power meters able to perform such selective and complex measurements (say on voltages up to 5000 Volts, currents up to 100 Amps, and frequencies up to 1 MHz, just for measuring transducers that are operating below 100 kHz).



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Optimal driving of ultrasonic transducers

For optimal transducer efficiency, the best situation is if/when transducer is driven in one of its mechanical resonant frequencies, delivering high active power (and very low reactive power) to the loading media. Since usually resonant frequency of loaded transducer is not stable (because of dynamical change of many mechanical, electrical and temperature parameters), a PLL resonant frequency (in real-time) tracking system has to be applied. When we drive transducer on its resonant frequency, we are sure that the transducer presents dominantly resistive load. That means that maximum power is delivered from ultrasonic power supply (or ultrasonic generator) to the transducer and later on to its mechanical load.

If we have a reactive power on the transducer, this can present a problem for transducer and ultrasonic generator and cause overheating, or the ultrasonic energy may not be transferred (efficiently) to its mechanical load. Usually, the presence of reactive power means that this part of power is going back to its source. The next condition that is necessary to satisfy (for optimal power transfer) is the impedance matching between ultrasonic generator and ultrasonic transducer, as well as between ultrasonic transducer and loading media.

If optimal resonant frequency control is realized, but impedance/s matching is/are not optimal, this will again cause transducer and generator overheating, or ultrasonic energy won’t be transferred (efficiently) to its mechanical load. Impedance matching is an extremely important objective for realizing a maximum efficiency of an ultrasonic transducer (for good impedance matching it is necessary to adjust ferrite transformer ratio and inductive compensation of piezoelectric transducer, operating on a properly controlled resonant frequency). Output (vibration) amplitude adjustments, using boosters or amplitude amplifiers (or attenuators) usually adjust mechanical impedance matching conditions. Recently, some ultrasonic companies (Herman, for instance) used only electrical adjustments of output mechanical amplitude (for mechanical load matching), avoiding any use of static mechanical amplitude transformers such as boosters (this way, ultrasonic configuration becomes much shorter and much more load-adaptable/flexible, but its electric control becomes more complex). By the way, we can say that previously given conditions for optimal power transfer are equally valid for any situation/system where we have energy/power source and its load (To understand this problem easily, the best will be to apply some of the convenient systems of electromechanical analogies).

It is important to know that Impedance-Phase-Frequency characteristics of one transducer (measured on a low sinus-sweeping signal) are giving indicative and important information for basic quality parameters of one transducer, but not sufficient information for high power loaded conditions of the same transducer. Every new loading situation should be rigorously tested, measured and optimized to produce optimal ultrasonic effects in a certain mechanical load.

It is also very important to know that safe operating limits of heavy-loaded ultrasonic transducers have to be controlled/guaranteed/maintained by hardware and software of ultrasonic generator. The usual limits are maximal operating temperature, maximal-operating voltage, maximal operating current, maximal operating power, operating frequency band, and maximum acceptable stress. All of the previously mentioned parameters should be controlled by means of convenient sensors, and protected/limited in real time by means of special protecting components and special software/logical instructions in the control circuits of ultrasonic generator. A mechanism



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of very fast overpower/overload protection should be intrinsically incorporated/included in every ultrasonic generator for technologically complex tasks. Operating/resonant frequency regulation should work in parallel with overpower/overload protection. Also, power regulation and control (within safe operating limits) is an additional system, which should be synchronized with operating frequency control in order to isolate and select only desirable resonances that are producing desirable mechanical output.

Electronically, we can organize extremely fast signal processing and controls (several orders of magnitude faster than the mechanical system, such as ultrasonic transducer, is able to handle/accept). The problem appears when we drive ultrasonic configuration that has high mechanical quality factor and therefore long response time, which is when mechanical inertia of ultrasonic configuration becomes a limiting factor. Also, complex mechanical shapes of the elements of ultrasonic configuration are creating many frequency harmonics, and low frequency (amplitude) modulation of ultrasonic system influencing system instability that should be permanently monitored and controlled. We cannot go against physic and mechanical limitations of a complex mechanical system (such as ultrasonic transducer and its surrounding elements are), but in order to keep ultrasonic transducer in a stable (and most preferable) regime we should have absolute control over all transducer loading factors and its vital functions (current, voltage, frequency…). This is very important in case of applications like ultrasonic welding, where ultrasonic system is permanently commuting between no-load and full load situation. In a traditional concept of ultrasonic welding control we can often find that no-load situation is followed by the absence of frequency and power control (because system is not operational), and when start (switch-on) signal is produced, ultrasonic generator initiates all frequency and power controls. Some more modern ultrasonic generators memorize the last (and the best found) operating frequency (from the previous operating stage), and if control system is unable to find the proper operating frequency, the previously memorized frequency is taken as the new operating frequency. Usually this is sufficiently good for periodically repetitive technological operations of ultrasonic welding, but this situation is still far from the optimal power and frequency control. In fact, the best operating regime tuning/tracking/control should mean a 100% system control during the totality of ON and OFF regime, or during full-load and no-load conditions. Previously described situation can be guaranteed when Power-Off (=) no-load situation is programmed to be (also) one transducer-operating regime which consumes very low power compared to Power-ON (=) full-load situation. This way, transducer is always operational and we can always have the necessary information for controlling all transducer parameters. Response time of permanently controlled/driven ultrasonic transducers can be significantly faster than in the case when we start tracking and control from the beginning of new Power-ON period.

When transducers are driven full power, it happens in the process of harmonic oscillation, so input electrical energy is permanently transformed to mechanical oscillations. What happens when we stop or break the electrical input to the transducer? - The generator no longer drives the transducer, and/or they effectively separate. The transducer still continues to oscillate certain time, because of its elastomechanical properties, relatively high electro-mechanical Q-factor, and residual potential (mechanical) energy. Of course, the simplest analogy for an ultrasonic transducer is a certain combination of Spring-Mass oscillating system. Any piezoelectric or magnetostrictive transducer is a very good energy transformer. It means that if the input is electrical, the transducer will react by giving mechanical output; but, if the active, electrical input is absent (generator is not giving any driving



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signal to the transducer) and the transducer is still mechanically oscillating (for a certain time), residual electrical back-output will be (simultaneously) generated. It will go back to the ultrasonic generator through the transducer’s electrical terminals (which are permanently connected to the US generator output). Usually, this residual transducer response is a kind of reactive electrical power, sometimes dangerous to ultrasonic generator and to the power and frequency control. It will not be synchronized with the next generator driving train, or it could damage generator’s output switching components.

Most existing ultrasonic generator designs do not take into account this residual (accumulated) and reversed power. In practice, we find different protection circuits (on the output transistors) to suppress self-generated transients. Obviously, this is not a satisfactory solution. The best would be never to leave the transducers in free-running oscillations (without the input electrical drive, or with “open” input-electrical terminals on the primary transformer side). Also, it is necessary to give certain time to the transducers for the electrical discharging of their accumulated elastomechanical energy.

Resonant frequency control under load

Frequency control of high power ultrasonic converters (piezoelectric transducers) under mechanical loading conditions is a very complex situation. The problem is in the following: when the transducer is operating in air, its resonant frequency control is easily realizable because the transducer has equivalent circuit (in the vicinity of this frequency) which is similar to some (resonant) configuration of oscillating R-L-C circuits. When the transducer is under heavy mechanical load (in contact with some other mass, liquid, plastic under welding…), its equivalent electrical circuit loses (the previous) typical oscillating configuration of R-L-C circuit and becomes much closer to some (parallel or series) combination of R and C. Using the impedance-phase-network analyzer (for transducer characterization), we can still recognize the typical impedance phase characteristic of piezo transducer. However, it is considerably modified, degraded, deformed, shifted to a lower frequency range, and its phase characteristic goes below zero-phase line (meaning the transducer becomes dominantly capacitive under very heavy mechanical loading). If we do not have the transducer phase characteristic that is crossing zero line (between negative and positive values, or from capacitive to inductive character of impedance) we cannot find its resonant frequency (there is no resonance), because electrically we do not see which one is the best mechanical resonant frequency.

Active and Reactive Power and Optimal Operating Frequency

The most important thing is to understand that ultrasonic transducers that are used for ultrasonic equipment (piezoelectric or sometimes magnetostrictive) have complex electrical impedance and strong coupling between their electrical inputs and relevant mechanical structure (to understand this we have to discuss all relevant electromechanical, equivalent models of transducers, but not at this time). This is the reason why parallel or serial (inductive for piezoelectric, or capacitive for magnetostrictive transducers) compensation has to be applied on the transducer, to make the transducer closer to resistive (active-real) electrical impedance in the operating frequency range. The reactive compensation is often combined with electrical filtering of the output, transducers driving signals. Universal reactive compensation of transducers is not possible, meaning that the transducers can be tuned as resistive impedance only within certain frequencies (or at maximum in band-limited frequency



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intervals). Most designers think that this is enough (good electrical compensation of the transducers), but, in fact, this is only the necessary first step.

This time we are coming to the necessity of making the difference between electrical resonant frequency and mechanical resonant frequency of an ultrasonic converter. In air (non-loaded) conditions, both electrical and mechanical resonant frequencies of one transducer are in the same frequency point/s and are well and precisely defined. However, under mechanical loading this is not always correct (sometimes it is approximately correct, or it can be the question of appearance of some different frequencies, or of something else like very complicated impedance characteristic). From the mechanical point of view, there is still (under heavy mechanical load) one optimal mechanical resonant frequency, but somehow it is covered (screened, shielded, mixed) by other dominant electrical parameters, and by surrounding electrical impedances belonging to ultrasonic generator. To better understand this phenomenon, we can imagine that we start driving one ultrasonic transducer (under heavy loading conditions), using forced (variable frequency), high power sinus generator, without taking into account any PLL, or automatic resonant frequency tuning. Manually (and visually) we can find an operating frequency producing high power ultrasonic (mechanical) vibrations on the transducer. As we know, heavy loaded transducer presents kind of dominantly capacitive electrical impedance (R-C), but it is still able to produce visible ultrasonic/mechanical output (and we know that we cannot find any electrical pure resonant frequency in it, because there is no such frequency). In fact, what we see, and what we can measure is how much of active and reactive power circulates from ultrasonic generator to piezoelectric transducer (and back from transducer to generator). When we say that we can see/detect a kind of strong ultrasonic activity, it means most probably that we are transferring significant amount of active/real electrical power to the transducer, and that much smaller amount of reactive/imaginary power is present, but we cannot be absolutely sure that such loaded transducer has proper resonant frequency (it could still be dominantly capacitive type of impedance, or some other complex impedance). In fact, in any situation, the best we can achieve is to maximize active/real power transfer, and to minimize reactive/imaginary power circulation (between ultrasonic generator and piezoelectric ultrasonic transducer). If/when our (manually controlled) sinus generator produces/supplies low electrical power, the efficiency of loaded ultrasonic conversion is also very low, because there is a lot of reactive power circulating inside of loaded transducer (and back to the generator).

Here is the most interesting part of this situation: if we intentionally increase the electrical power that drives the loaded transducer (keeping manually its best operating frequency, or maximizing real/active power transfer), the transducer becomes more and more electro-acoustically efficient, producing more and more mechanical output, and less and less reactive power. Also, thermal dissipation (on the transducer) percentage-wise (compared to the total input energy) becomes lower. What is really happening: under heavy mechanical loading and high power electrical driving (on the manually/visually found, best operating frequency, when real power reaches its maximum) the transducer is again recreating/regaining (or reconstructing) its typical piezoelectric impedance-phase characteristic which, now, has new phase characteristic passing zero line, again (like in real, oscillatory R-L-C circuits). Somehow, high mechanical strain and elastomechanical properties of total mechanical system (under high power driving) are accumulating enough (electrical and mechanical) potential energy, and the system is again coming back, mechanically decoupling itself from its load (for instance from liquid) and/or starting to present typical R-L-C structure that is



21

easy for any PLL resonant frequency control (having, again, real/recognizable resonant frequency).

Of course, loaded ultrasonic transducer (optimally) driven by high power will have some other resonant frequency, different than the frequency when it was driven by low power, and also different than its resonant frequency (or frequencies) in non-loaded conditions (in air), because resonant frequency is moving/changing according to time-dependant loading situation (in the range of ±5% around previously found resonant frequency).

To better understand the importance of active power maximization, we know that when we have optimal power transfer (from the energy source to its load), the current and the voltage time-dependant shapes/functions (on the load) have to be in phase. This means that in this situation electrical load is behaving as pure resistive or active load. (Electrically reactive loads are capacitive and inductive impedances). The next condition (for optimal power transfer) is that load impedance has to be equal to the internal impedance of its energy source (meaning the generator). In mechanical systems, this situation is analogous or equivalent to the previously explained electrical situation, but this time force and velocity time-dependant shapes/functions (on the mechanical load) have to be in phase, which means that in such situations mechanical load is behaving as pure (mechanically) resistive, or active load. Active mechanical loads are basically frictional loads (and mechanically reactive loads/impedances are masses and springs in any combination). We usually do not know/see exactly (and clearly) if we are producing active mechanical power, but by following/monitoring/controlling electrical power, we know that when we succeed in producing/transferring certain amount of active electrical power to one ultrasonic transducer, that corresponds, at the same time, to one directly proportional amount of active mechanical power (dissipated in mechanical load). Delivering active power to some load usually means producing heat on active/resistive elements of this load. We also know that productivity, efficiency and quality of ultrasonic action (in Sonochemistry, plastic welding, ultrasonic cleaning…) strongly and directly depend on how much active mechanical power we are able to transfer to a certain mechanical load (say to a liquid or plastic, or something else). When we have visually strong ultrasonic activity, but without transferring significant amount of active power to the load, we can only be confused in thinking (feeling) that our ultrasonic system is operating well, but in reality, we do not have big efficiency of such system. Users and engineers working in/with ultrasonic cleaning know this situation well. Sometimes, we can see very strong ultrasonic waving in one ultrasonic cleaner (on its liquid surface), but there is no ultrasonic activity and cleaning effects are missing.

In conclusion, it is correct to say that: active electrical power & active mechanical power, for an electromechanical system where we transfer electrical energy to the mechanical load. Another conclusion is that we also need to install convenient mechanical/acoustical/ultrasonic sensors which are able to detect, follow, monitor and/or measure resulting ultrasonic/acoustical/mechanical activity (in real-time) on the mechanical load, in order to be 100% sure that we are transferring active mechanical power to certain mechanical load, and to be able to have a closed feedback loop for automatic (mechanical, ultrasonic) power regulation. For instance, in liquids (Sonochemistry and ultrasonic cleaning applications), the appearance of cavitation is the principal sign of producing active ultrasonic power. To control this we need sensors of ultrasonic cavitation. Also, we know that the last step in any energy chain (during electromechanical energy transfer) is heat energy. By supplying electrical resistive load with electrical energy we produce heat. The same is valid for supplying mechanical



22

resistive/frictional load with ultrasonic energy, when the last step in this process is again heat energy (but, again, force and velocity wave shapes of delivered ultrasonic waves have to be in phase, measured on its load). From the previous commentary we can conclude that the best sensors for measuring active/resistive ultrasonic energy transfer in liquids are real-time, very fast responding temperature sensors (or some extremely sensitive thermocouples, and/or thermopiles).

There can be a practical problem (for resonant frequency tracking) if we start driving certain transducer full power, under load, if we are not sure that we know its best operating mechanical resonant frequency (because we can destroy the transducer and output transistors if we start with a wrong frequency). In real life, every well designed PLL starts with a kind of low power sweep frequency test (say giving 10% of total power to the transducer), around its known best operating frequency taking/accepting one frequency interval that is given in advance. When the best operating resonant frequency is confirmed/found, PLL system tracks this frequency, and at the same time the power regulation (PWM) increases output power (of ultrasonic generator) to the desired maximum. Of course, when the transducer is in air (mechanically non-loaded), previous explanation is readily applicable because we can easily find its best resonant frequency, and later on we can start gradually increasing the power on the transducer. If starting and operating situation is with already heavy loaded transducer (which can be represented by dominantly R-C impedance), the problem is much more serious, because we should know how to recognize (automatically) the optimal mechanical resonant frequency (without the possibility of using phase characteristic that is crossing zero line). There are some tricks, which may help us realize such control. Of course, before driving one transducer in automatic PLL regime, we should know its impedance-phase vs. frequency properties (and limits) in non-loaded and fully loaded situations. In order to master previous complexity of driving ultrasonic transducers (and to explain this situation) we should know all possible and necessary equivalent (electrical) circuits of non loaded and loaded ultrasonic transducers, where we can see/discuss/adjust different methods of possible PLL control/s. Since ultrasonic transducer is always driven by using ultrasonic generator which has output ferrite transformer, inductive compensation and other filtering elements, it is necessary to know the relevant (and equivalent) impedance-phase characteristics in all of such situations in order to take the most convenient and proper current and voltage signals for PLL.

All previous comments are relevant when driving (input) signal is either sinusoidal or square shaped, but always with a (symmetrical, internal) duty cycle of 1:1 (Ton: Toff = 1:1), meaning being a regular sinusoidal or square shaped wave train. There is a special interest in finding a way/method/circuit capable of driving ultrasonic transducers directly using high power (and high ultrasonic frequency), PWM electrical (input) signals, because of the enormous advantages of PWM regulating philosophy. Applying special filtering networks in front of an ultrasonic transducer can be very useful when we want to drive ultrasonic transducer with PWM signals.

Influence of External Mechanical Excitation

One of the biggest problems for PLL frequency tracking is when ultrasonic (piezoelectric) transducer under mechanical load, driven by ultrasonic generator, produces mechanical oscillations, but also receives mechanical response from its environment (receiving reflected waves). Sometimes, received mechanical signals are so strong, irregular and strangely shaped that equivalent impedance characteristic of loaded transducer becomes very variable, losing any controllable (typical impedance) shape. It looks like all the parameters of equivalent electrical circuit of loaded



23

transducer are becoming non-linear, variable and like transient signals. There is no PLL good enough to track the resonant frequency of such transducer, but luckily, we can introduce certain filtering configuration in the (electrical) front of transducer and make this situation much more convenient and controllable (meaning that external mechanical influences can be attenuated/minimized).

Sometimes loaded ultrasonic transducer (in high power operation) behaves as multi-resonant electrical and mechanical impedance, with its entire equivalent-model parameters variable and irregular. Optimal driving of such transducer, either on constant or sweeping frequency, becomes uncontrollable without applying a kind of filtering and attenuation of external vibrations and signals received by the same transducer. In fact, the transducer produces/emits vibrations and at the same time receives its own vibrations, reflected from the load. There is a relatively simple protection against such situation by adding a parallel capacitance to the output piezoelectric transducer. Added capacitance should be of the same order as input capacitance of the transducer. This way, ultrasonic generator (frequency control circuit) will be able to continue controlling such transducer, because parallel added capacitance cannot be changed by transducer parameters variation. In case of large band frequency sweeping, we can also add to the input transducer terminals certain serial resistive impedance (or some additional L-R-C filtering network). This way we avoid overloading the transducer by smoothly passing trough its critical impedance-frequency points (present along the sweeping interval).

In any situation we can combine some successful, useful and convenient PLL procedures with a real/active power maximizing procedure incorporated in an automatic, closed feedback loop regulation (of course, trying in the same time to minimize the reactive/imaginary power).

Objectives and new R&D tasks

Traditional ultrasonic equipment exploits mainly single resonant frequency sources, but it becomes increasingly important to introduce/use different levels of frequency and amplitude modulating signals, as well as low frequency (ON – OFF) group PWM digital-modulation in low and high frequency domains. Several modulation levels and techniques could be applied to maximize the power and frequency range delivered to heavily-loaded ultrasonic transducers (and, this way, many of the above-mentioned loading problems could be avoided or handled in a more efficient way). Discussing such situations can be a subject of a special chapter.



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MMM, Wideband (Multi-Frequency) Ultrasonic Power Supplies

Our Wideband (Multi-Frequency) Generator employs a completely new approach to frequency generation, frequency sweeping, frequency tracking, applied power, and most importantly a new way of adapting to large or varying loads. It's unique feature set makes it well suited for unusual application where ultrasonics must be applied to a large mass, a thick walled chamber, or dynamic loads that present an un-tuned mechanical system that normal generators cannot drive efficiently. Key Features

• Wideband Multi-Frequency Effect (300 Hz to > MHz)

o Adaptive and rapidly shifting frequency and phase modulation creates a highly efficient power shifting distribution across a broadband of harmonic and sub-harmonic frequencies.

• Adaptive to large and heavy loads

o The Problem: Other conventional ultrasonic systems must work in a tuned mechanical



25

environment. Careful selection of sonotrodes and reaction chamber designs are critical. Changes to the load or large loads create conditions that are destructive to the transducers and generators of other traditional systems.

o The Solution: Our new technology eliminates these problems by monitoring and adapting to the natural harmonics of the mechanical system. The adaptive modulation techniques mimic feedback of harmonic signals from the mechanical systems and use these signal to drive the system in a highly efficient and synergistic manner. More about the MMM technology....

o The Result: We can apply effective ultrasonic power to un-tuned mechanical systems such as large heavy industrial tanks, thick walled chambers, complex forms, and large metal masses like extruders and injection mold tools.

• Power Options from 100 watts to 120,000 watts

• Advanced option programming via handheld control panel or PC Windows’s software.

• Complete System Protection Circuits for Over Voltage, Over Current, and Driver fault.

• Modular options for industrial applications.

• Reactor Options:

o This Multi-Frequency Generator greatly increases our options for reaction chamber design.

o Now we can make most any size or shape and provide the necessary power to meet your complex needs.

• Compatible with all transducers options

Windows PC Software

Control Option


http://www.activeultrasonics.com/Products/Generators/MMM.htm

http://www.activeultrasonics.com/Products/Generators/MMM.htm


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MMM, Universal and Wideband Multifrequency Power Supplies

MMM Transducers & SystemsMMM Transducers & Systems

MuMMswsunu

ltiple modulations operation: MMM technology M, Universal Ultrasonic Power Supplies are replacing all other types of constant or

eeping frequency power supplies for driving all kind of piezoelectric transducers, bmersible transducers, bench top cleaners, Sonochemical reactors… bringing mber of advantages and new options.

(OF) (OW) (IX)



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OF, MMM Power Supplies

Technical characteristics

MSG.300.OF MSG.600.OF MSG.1200.OF

Main Supply Voltage

220/230 V; 50/60 Hz 220/230 V; 50/60 Hz 220/230 V; 50/60 Hz

Max. Input Power 400 W 700 W 1300 W Non-modulated, carrier frequency range

19.020kHz ÷ 46.728 kHz

19.020kHz ÷ 46.728 kHz

19.020kHz ÷ 46.728 kHz

Modulated acoustic frequency range

Wideband, from Hz to MHz



Average Continuous Output Power

300 W 600 W 1200 W

Peak Output (max. pulsed power)

1500 W 3000 W 6000 W

Output HF Voltage

~ 500 V-rms ~ 500 V-rms ~ 500 V-rms

Dimensions (h x w x d)

170x150x150mm 250x150x150mm 230 x 160 x 370

Weight 2 kg 3.6 kg 4 kg MasterSonic open frame generator modules (OF series) are designed for internal mounting in the control cabinets of Ultrasonic Systems. Such cabinets should be very well ventilated, protecting the generator module from excessive dust, moisture, and harmful chemical agents. The installation and electrical connections of the generator should be performed by a qualified specialist in electronics who is experienced in Power Ultrasonics. MSG.X00.OF is designed as a component part for integration into Ultrasonic systems. Therefore it is not equipped with a Power Supply ON/OFF switch. Make sure the Ultrasonic System you are assembling is provided with such switch. Please read manuals for more information.



28

OW, MMM Power Supplies

Technical characteristics

MSG.300.OW MSG.600.OW MSG.1200.OW

Main Supply Voltage

220/230 V; 50/60 Hz 220/230 V; 50/60 Hz 220/230 V; 50/60 Hz

Max. Input Power 400 W 700 W 1300 W Non-modulated, carrier frequency range

21.435kHz ÷ 40.560 kHz

21.435kHz ÷ 40.560 kHz

21.435kHz ÷ 40.560 kHz

Modulated acoustic frequency range




Average Continuous Output Power

300 W 600 W 1200 W

Peak Output (max. pulsed power)

1500 W 3000 W 6000 W

Output HF Voltage

~ 500 V-rms ~ 500 V-rms ~ 500 V-rms

Dimensions (h x w x d)

170x150x150mm 250x150x150mm 230 x 160 x 370

Weight 2 kg 3.6 kg 4 kg All MSG modular ultrasonic generators, MSG X00.OW, utilize the MMM Technology and are constructed with an open frame design intended for integration into Ultrasonic Systems providing appropriate housing and protection. OW series generators have much higher frequency resolution than OF series generators, making them convenient when precise frequency settings are important. The MSG.X00.OW generators are intended mainly for application in ultrasonic cleaning tanks and systems. MasterSonic generator modules (OW series) are designed for internal mounting in the control cabinets of Ultrasonic Systems. Such cabinets should be very well ventilated, protecting the generator module from excessive dust, moisture, and harmful chemical agents. The installation and electrical connections of the generator should be performed by a qualified specialist in electronics who is experienced in Power Ultrasonics. MSG.X00.OW is designed as a component part for integration into Ultrasonic systems. Therefore it is not equipped with a Power Supply ON/OFF switch. Make sure the Ultrasonic System you are assembling is provided with such switch. Please read manuals for more information.



29

IX, MMM Power Supplies

Technical characteristics MSG.1200.IX

Main Supply Voltage 220/230 V; 50/60 Hz

Max. Input Power 1300 W

Non-modulated, carrier frequency range 19.020kHz ÷ 46.728 kHz

Modulated acoustic frequency range Wideband, from Hz to MHz

Average Continuous Output Power 1200 W

Peak Output (max. pulsed power) 6000 W

Output HF Voltage ~ 500 V-rms

Dimensions (h x w x d) 250mm x 150mm x 450mm

Weight 10 kg

MSG modular ultrasonic generators (MSG XXX.IX) utilize the MMM Technology and are constructed with a separate housing as an independent power supply of piezoelectric acoustic loads. IX series generators have maximum of available options of MMM technology (practically all of the best options of OF and OW series generators, including many of new options), and can be operated by people without background in High Power Ultrasonics. IX series power supplies are also very convenient for challenging R&D projects, laboratory applications and other scientific projects. IX generators are fully protected against overloading and load short-circuits. Please read manuals for more information.



30

ACCESSORIES, INTERFACES, REMOTE, PLC AND PC CONTROLL TOOLS FOR ALL MMM GENERATORS

Handheld Control Unit For manual control and settings

All Mastersonic, MMM generators can be controlled, being connected by RS485 link to a PC, using the software interface for enabling easy visual and multi-parameter control and settings.

Home page: http://www.mpi-ultrasonics.comServer: http://www.mastersonic.com E-mail: [email protected]@bluewin.ch

MMM-Link-2339 Adapter RS485 / RS232C+software MMM-Link-2339_16 Option RS485 Link extender16 generator MMM-Link-2339_64 Option RS485 Link extender16 generator Interface cable



http://www.mastersonic.com/




31

Also available single Power Supply units until 100 kW

PS Cabinet

PS Programming Interface and Display

15 kW acoustic load (Extractor: H = 3.6 m, OD = 1 m)



32

MMMMMM TTeecchhnnoollooggyy:: MMuullttiiffrreeqquueennccyy,, MMuullttiimmooddee,, MMoodduullaatteedd

SSoonniicc && UUllttrraassoonniicc TTeecchhnnoollooggyy No other manufacturer has yet achieved and matched MMM exciting standards in precision cleaning. MMM is not only more efficient and effective than any other ultrasonic cleaning technology, it is UNIQUE. Seeing is the believing! Try the aluminum foil test for yourself! Place the

foil sample into our ultrasonic bath and hold the foil for approx. 5 -10 seconds and you'll discover why there's simply no comparison with any other conventional ultrasonic cleaning machine.

Left: Perfectly, uniformly perforated aluminum foil, after 5 to 10 seconds of exposure to MMM ultrasonic vibrations in an ultrasonic cleaner. Frequency Range: From Hz to MHz; From Infrasonic to Supersonic. Right: Load current and voltage shapes (modulated and carrier).

Superior and deep penetration, independent of water levels. Reliability with extra power spread throughout the bath. Even distribution of ultrasonic energy throughout the liquid gives uniform

and thorough cleaning of the surface without the risk of damage to fine parts and sensitive instrument.

Extremely efficient electronics and transducer coupling to ultrasonic bath (overall approx. 95% efficiency) eliminates or reduces the additional need for heating.

Spatial distribution of ultrasonic activity inside of a cleaning liquid is homogenous (no dead zones, no standing waves, fast and large frequency sweeping, broadband spectrum, complex modulation).



33

Cleaning solvents, detergents and additives can be significantly reduced, or even eliminated because of the very high cleaning activity of the acoustic broadband spectrum.

Cleaning time can be several times shorter comparing to traditional ultrasonic cleaning technology. Fast liquid conditioning and de gassing because of very large regulating

between acoustic spectrums. zone between maximal and average ultrasonic power and because of the ability to switch instantaneously

Smooth Ultrasonic, PWM-power regulation from 1% to 100%. Ultrasonic energy can be easily adjusted in order to clean very fine and sensitive parts



34

Important Background regarding Settings of the MMM-technology Ultrasonic Power Supply based

systems Modulated Multimode Multifrequency

The technology was originally developed for application of ultrasonics to large and arbitrary shaped mechanical systems. It turns out that water tanks and submersible box transducer are essentially arbitrary shaped mechanical systems where all of the MMM technological advantages are exhibited. We find that in liquid baths or liquid processing chambers our systems greatly improve basic cleaning and more complex functions like sonochemical reactions. Following is a brief comparison between our MMM system and conventional ultrasonic systems. For more information please have a look at our web site www.mpi-ultrasonics.com and for details on or MMM technologies go directly to: http://mastersonic.com/documents/mmm_basics_presentation.pdf Conventional Ultrasonics Cleaning Systems: As you may know, conventional ultrasonic systems are based on the relatively-fixed resonant frequency of the transducers used. The generator drives the transducers at the fixed frequency without regard to the attached mechanical system including the steel tank surface and walls, the water, the parts loaded in the tank, temperature changes, etc. Each of these factors can significantly shift the resonant frequency of the transducers and conventional ultrasonic generators are not equipped to adapt to the change. The problem is compounded because industrial systems are continuously acoustically-evolving causing all of the load parameters to change. The result is inefficient transducer driving and reduced cavitation capabilities. Conventional fixed frequency systems are also creating standing waves with areas of high acoustic activity and areas of low acoustic activity. When cleaning parts or making sonochemical treatment these problematic standing waves can over treat or damage some areas and leave other areas untreated. Some systems try to solve this problem with a small amount of generator frequency sweeping around the fixed center frequency. This method helps but does not normally correct the problem. MMM Ultrasonic Liquid Treatment Systems: Unlike conventional systems the flexibility of our generators starts with an adjustable primary frequency option. This feature allow us to consider shifts to the system resonance (e.g. 28 kHz shifting to 28.7 kHz) caused by the entire load factors mentioned above. Such adjustment and fine-tuning to the primary driving signal will greatly improve efficiency and improve the system response. The MMM generator uses the adjusted resonant frequency (e.g. 28.7 kHz) as an improved carrier frequency that is further modified by special system feedback techniques to create an optimized and complex driving signal. We have developed advanced Digital Signal Processing techniques to monitor transducer(s) response to the load, and to follow the evolving changes of the load. Using such a real-time feedback loop the system creates an evolving and complex Modulated Multimode driving signal to stimulate coupled harmonics in the mechanical load (bath or chamber) to produce an effective wideband Multi-frequency acoustic field from infrasonic up to the megahertz range. As a result MMM systems using conventional transducers are capable of producing a very wide range of cavitation bubble sizes



http://mastersonic.com/documents/mmm_basics_presentation.pdf


35

and a greater density of cavitation bubbles. This provides faster and better cleaning, faster sonochemical reactions, faster physical reactions, and faster liquid degassing. Furthermore our unique Modulation methods use wide frequency sweeping and signal phase shifting techniques to eliminate the standing waves typically seen in fixed frequency systems. This advantage allows our systems to clean or process materials evenly with reduced risk of parts damage. Our systems can provide homogeneous energy throughout the sonication chamber resulting in faster and more uniform reactions.

CRITICAL SETTINGS EXPLANATIONS

PWM is Pulse Width Modulation.

• This feature allows Low Frequency ON / OFF Pulsing of the ultrasonic power. It works very well to make strong shocking of the mechanical system.

• During the ON period the system is delivering modulated high frequency ultrasonic power. During the OFF period no ultrasonic power is given.

• The total maximum PWM Period is 1 second (1,000 ms).

• Using the PC Software control window the minimum PWM Period Steps adjustment is 10 ms. (use keyboard arrow key or mouse scroll wheel to make 10 ms fine tune adjustment)

• The PWM Ratio is the relative ON/OFF time. A 50% setting gives 50% ON Time and 50% OFF Time.

What does “sweeping” and “fast sweeping” mean?

Answer: Sweeping and Fast Sweeping are additional modulation techniques added to the fixed-frequency carrier signal. In other words, we start with a fixed-frequency signal adjusted to the system resonance (example: 21.3 kHz) to work as a primary signal carrier (or central operating frequency). Then by using one of the following methods we make complex modulations to cause shifting and sweeping of the primary fixed-frequency signal. This gives many benefits including reduce or eliminate standing waves, better ultrasonic stimulation of large and arbitrary shaped mechanical systems.



36

• “Sweeping” is a forced or static pre-programmed method of sweeping and modulating the carrier signal (central operating frequency) within an interval of up to 1,000 Hz. The sweeping is symmetrically placed, equally wide on both sides of the carrier frequency, and may be selected to cover frequency intervals from 0 Hz to 1,000 Hz. The mathematical function describing this sweeping is pre-programmed to produce large frequency-band acoustic-response effects in the mechanically-oscillating system including many harmonics and sub-harmonics.

o When set to “0” (zero) there is no sweeping applied (system oscillates only on constant carrier frequency).

o When set to the maximum the sweep range is 1,000 Hz.

OF generator model sweeping steps are 0 to 7.

OW generator model sweeping steps are 0 to 255.

IX generator model sweeping steps are 0 to 1.000 kHz.

• “Fast Sweeping” is a load-dependent Dynamic method of sweeping the carrier signal. This method is monitoring signal response from the mechanical load and using special algorithms to extract several resonant modes of the load. This information is transformed into a convenient time-evolving voltage signal that is used to make dynamic changes and modulated driving of the carrier frequency signal.

o When set to “0” (zero) there is no dynamic sweeping applied.



37

o When set to highest value the maximum dynamic sweeping effect is applied.

OF and OW generator model sweeping steps are 0 to 255 (0 to 1,000 Hz).

IX generator models sweeping steps are 0 to 1,000 Hz.

• Sweeping + Fast Sweeping: It is also possible to make a modulated signal that is a combination of forced pre-programmed “sweeping” plus dynamic “fast sweeping”. When both are turned on, the system makes a complex hybrid signal modulation using both techniques at the same time.

What does “Tracking Range” mean?

Answer: MMM generators use system feedback signals (phase between current and voltage on the transducer) to keep the central operating frequency close to the resonant frequency of the transducer (operating in parallel resonance). The “Tracking Range” correction is minimized when set to 0 units. The “Tracking Range” correction is maximized when set to 30 units. One unit is approximately 100Hz. So when the “Tracking Range” is set, for example to 12, the central operating frequency will be corrected (if it is necessary, regarding the phase between current and voltage) in the range of +- 1200 Hz. The “Tracking Range” is a very important feature of MMM generators, because it makes them self adaptive to the mechanical changes of the load, for example when you put parts in a cleaning tank.

• When set to “0” (zero) no tracking is implemented.

• When set to 30 (1,000 Hz on IX models) the tracking range is set to the maximum correction mode. The internal oscillator, which is generating carrier frequency-signal, is trying to follow wide-band oscillations of the mechanical system. This maximum setting may be inefficient for many applications. Consequently, the optimal tracking range should be experimentally adjusted.



38

• In general systems using one or few transducers (e.g. Pipe-Clamp or Sieving systems) will work well if the Tracking Range is set to a narrow small range.

• Systems using many transducers (e.g. Cleaning Bath or Submersible Transducer Arrays) often benefit from wider Tracking Range settings.

Please visit our website for more details, or contact us directly with any

inquiries: www.mpi-ultrasonics.com & www.mastersonics.com



http://www.mastersonics.com/


39


1

Importance of waves and signal analysis based on Analytic Signal Modeling

Here it will be made an attempt to significantly rectify and generalize concepts dealing with wave-motions. The first and very important step would be to establish (or accept) the most applicable, most common, and sufficiently rich and universal mathematical model that would be able to cower all kind of (physics-related) wave and oscillatory motions (what presently could look as the very ambitious task, but it will be shown as realistic if we would use Analytic Signals modeling).

Let us start from an arbitrary waveform Ψ , transformed using the Analytic Signal model, in a form of the product of an amplitude and phase function (1.1), since this is the most-closest, ready-made model where we have amplitude and phase function separated (like in cases of signals modulating techniques), and where we will make the association (and later prove it) that the amplitude function would propagate with a group velocity, and the phase function would have phase velocity. It is also important to underline that all kind of wave functions (here marked as ) describing real energy-finite motions, known in Physics (and Nature), are presentable using the Analytic Signal modeling (first time introduced by D. Gabor), and this is one reason more to establish this analysis using the Analytic Signal concepts. Of course, later we will present more-profoundly and with more arguments why Analytic Signal modeling is taken as the very much representative natural waveforms modeling.

(t)

Ψ(t)

Most of contemporary (and old) analyses elaborating this problematic (and wave functions analyses in any other aspect) are still not using the Analytic Signal concepts for wave functions modeling, and this will be shown as a really big disadvantage in the house of modern mathematical physics (regarding analyses of wave motions), since different interpretation-languages are in use, making that important unifications and generalizations are presenting difficult tasks. The formal differences when using the Analytic Signal or Fourier Transform for presenting waveforms are not so much evident and immediately clear, that science-community is in most of cases continuing (by intellectual inertia) using old and well-established, but not far-reaching, methods based on Fourier analysis (and almost repeating in their publications what somebody else made long time ago). It is really necessary to make an effort and enter into a new universe of Analytic Signals and Hilbert transform-based waveforms-modeling in order to catch all finesse and advantages of such modeling, and to distinguish it from the ordinary Fourier-transform-based methods. We could say that the differences between Fourier analysis and Analytic Signal concepts are comparable to differences between the operations with Real Numbers and Complex Numbers Analysis (where Real Numbers are just a small area of the Complex Numbers Analysis). Briefly summarizing, Analytic Signal modeling is giving the chance to extract the immediate (time-evolving) amplitude and phase signal-functions, and immediate signal-frequency, both in time-space and frequency domains (including all mutually-analogical amplitude and phase spectral functions), from any arbitrary waveform which could be of certain relevance in Physics. It is also worth to mention that starting from a wave function presented as a Complex Analytic Signal, development of Schrödinger, Klein-Gordon and many other well-known differential wave equations (known in Physics) is a relatively easy task.

Miodrag Prokic E-mail: [email protected] Jul-04 Wave Function Concepts 1/

2

Going directly to the most useful analytic-signal forms for this analysis, we will consider that is our time-domain wave-function, wave-packet (or wave group), presented as a Real Analytic Signal function,

Ψ(t)

[ ] [ ]

[ ][ ] [ ]

[ ] ,transformHilbert)(H(t)H(t)a(t)sin

dωΦ(ω))ωt )sin( A(ωπ1dω t ω )cos (ωUωt )sin (ωU

π1(t)Ψ

,(t)-H(t)a(t)cos

dωΦ(ω))ωt )cos( A(ωπ1dω t ω )sin (ωUωt )cos (ωU

π1Ψ(t)

00sc

00sc

=Ψ==

=+=+=

Ψ==

=+=+=

∫∫

∫∫

∞∞

∞∞

,

ˆ

ˆ

ϕ

ϕ (1.1)

where is the immediate signal-amplitude or signal-envelope, a(t) (t)cosϕ is the signal-carrier function, (t)ϕ t) ( f 2π(t)/dtdω(t) == ϕ is the signal-phase function, and is the immediate signal-frequency. Analogical functions in the signal frequency-domain are as the signal amplitude, and Φ as the signal phase function. ) A(ω (ω)

The same, time-domain Real Analytic Signal function, transformed into the Complex Analytic Signal has the following form,

.a(t)e

dω )e A(ωπ1dω )e U(ω

π1t) ( Ψ jH)(1(t)ΨjΨ(t)(t)Ψ

(t)j0

Φ(ω))t j(ω

0

t jω

ϕ=

==+=+= ∫∫∞

+∞

ˆ (1.2)

The frequency-domain wave functions of an Analytic Signal are,

[ ].) (ω jΦ

-

t jω-sc

cs) (ω -jΦ

)e A(ωdtΨ(t)e) (ωjU) (ωU) (ωU

1j,) (ω)/U (ωUarctan) Φ(ω,)e U(ω) A(ω

==−=

−=−==

∫∞+

∞

(1.3)

In this analysis of waveform-velocities we will use all (here mentioned) real and complex time-domain and frequency-domain Analytic Signal wave-functions, (1.1)-(1.3).

For more-complete understanding regarding creating Analytic Signal functions and regarding Analytic Signal advantages in comparison to all other waveform presentations, it would be useful to go to Analytic Signal and Hilbert Transform literature (in order to focus analysis in this article only to wave velocities and closely-related topics).

Signal Energy Content

Let us first find the energy carried by the analytic signal waveform (t)a(t)cosΨ(t) ϕ= . Parseval’s theorem is connecting time and frequency domains of signal’s wave functions, and such expression can have the meaning of signal’s (or wave) energy in cases when /dtEdP(t)(t)Ψ 2 ~== is modeled to present an immediate signal-power, as follows,


3

[ ]

.)(0,ω,π

) A(ωdω

Edω) P(),,(-t,2

a(t))((t)ΨdtEdP(t)

,P(t)dt

dωπ

) A(ωdω) A(ωπ1dω

2π) (ωUdω) (ωU

2π1

dt2

a(t)(t)dta21dt(t)Ψ

21(t)dtΨ(t)dtΨE

222

-

2

0

2

0

22

22222

+∞∈⎥⎦

⎤⎢⎣

⎡==+∞∞∈⎥⎦

⎤⎢⎣

⎡⇔==

=

=⎥⎦

⎤⎢⎣

⎡====

=⎥⎦

⎤⎢⎣

⎡=====

∫

∫∫∫∫

∫∫∫∫∫

∞+

∞

∞∞∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

∞+

∞−

~~

ˆ~

(1.4)

From the expressions for signal energy (1.4), it is obvious that a total signal energy-content is captured only by the signal amplitude (or envelope) function or , both in time and frequency domain,

a(t) ) A(ω

dωπ

) A(ωdt2

a(t)E2

0

2

∫∫∞∞+

∞− ⎥⎦

⎤⎢⎣

⎡=⎥⎦

⎤⎢⎣

⎡=~ , (1.5)

and it is also obvious that signal phase functions, (t)ϕ and do not directly participate in a total-signal energy-content. This is very important fact to notice in order to understand the meaning of signal velocities, or saying the same differently, the wave velocity of the signal-amplitude (or signal-envelope) is the velocity of the total signal-energy propagation, and this velocity is usually named as a group velocity. The speed of the signal-carrier phase-function is usually named as a phase velocity, and it should present the signal-carrier velocity, or velocity of signal-elements (which are creating a wave-packet). Here, the signal-carrier functions in time-domain are

) Φ(ω

(t)cosϕ , or , and in a frequency-domain or , and we can use them equally and analogically, depending on our preferences for operating with trigonometric or complex functions.

(t)jϕ ) (ω jΦe ) (ω Φ cos e

In the end of this paper we will again take much closer look to physics-related wave-functions structure and properties.

Since there is a lot of available literature regarding Hilbert transform and Analytic Signal modeling, let us only summarize important properties and expressions valid for presenting arbitrary waveforms as Analytic Signals, T.1.1.


4

T.1.1 Analytic Signal Parallelism between

Time and Frequency Domains Time Domain Frequency Domain

Complex Signal

j (t )

-j t

(0, )

-j( t ( ))

(0, )

(t) a(t)eˆ(t) j (t)

= U( )e d

= A( )e d

ϕ

ω

+∞

ω +Φ ω

+∞

Ψ =

= Ψ + Ψ

ω ω

ω ω

∫

∫

[ ]

[ ]

-j ( )

c s

j t

t

j( t + (t))

t

U( ) A( )eU ( ) - j U ( )

= (t) e dt

a(t)e dt

Φ ω

ω

ω ϕ

ω = ω= ω ω

Ψ

=

∫

∫

Real and imaginary signal components

[ ]

[ ]

(t) a(t)cos (t)ˆH (t) ,

ˆ (t) a(t)sin (t)H (t)

ψ = ϕ

= − ψ

ψ = ϕ

= ψ

[ ]

[ ],

,

) (ωUHΦ(ω) )sin A(ω ) (ωU

) (ωUHΦ(ω) s )co A(ω ) (ωU

c

s

s

c

===

−===

Signal Amplitude 2 2ˆa(t) (t) (t)= ψ + ψ 2 2

c sA( ) U ( ) U ( )ω = ω + ω

Instant Phase ˆ (t)(t) arctg

(t)ψ

ϕ =ψ

s

c

U ( )( ) arctan

U ( )ω

Φ ω =ω

Instant Frequency (t)(t)t

∂ϕω =

∂

( )( ) ∂Φ ωτ ω =

∂ω

Signal Energy

∫

∫∫∫

∞+

∞−

∞+

∞−

∞+

∞−

+∞

∞−

⎥⎦

⎤⎢⎣

⎡=

==

==

==

dt2

a(t)

(t)dtΨ

(t)dtΨ

dt(t)ΨE

2

2

2

2

ˆ

~

0(t)dtΨΨ(t)-

=⋅∫+∞

∞ˆ

dωπ

) A(ω

dω) (ωU2π1

dω) (ωU2π1

dω) (ωU2π1E

2

0

2s

2c

∫

∫

∫

∫

∞

∞+

∞−

∞+

∞−

∞+

∞−

⎥⎦

⎤⎢⎣

⎡=

==

==

==2~

0)dω (ωU) (ωU- sc =⋅∫+∞

∞

Central Frequency [ ]

[ ]

c

t

2t

2

c f 2π(t)dta

(t)dtaω(t)ω =

⋅

=∫

∫

[ ][ ]

[ ][ ]

c

ω

2ω

2

c f 2πdω) A(ω

dω) A(ωωω =

⋅

=∫

∫

“Central Time Point” [ ]

[ ]∫

∫ ⋅

=

t

2t

2

c (t)dta

(t)dtatt

[ ][ ]

[ ][ ]∫

∫ ⋅

=

ω

2ω

2

c dω) A(ω

dω) A(ωτ(ω)t

Standard Deviation [ ]

dtωω(t)∆t1σ

2

tc

2ω ∫ −=

[ ]dωtτ(ω)

∆ω1σ

2

ωc

2t ∫ −=


5

δωt δπ

δftδ2πTΩπδft δ2πδωt δ0

?)(!δωt δ

πδft δ2

πTΩtωτ(ω)ω(t)σσπ

2

2

cctω

⋅=

⋅⋅≤⋅<<⋅⋅=⋅<

⋅=

⋅⋅≤⋅≅⋅≅⋅<⋅≤

Uncertainty Relations

.δftδ4

1TF21δft δ0

πTΩ,πδωt δ,21δft δ

,21TF,

2T1δf,

2F1t δ

1/2,FTδftδ2δftδ

⋅⋅≤⋅<<⋅<

>⋅<⋅<⋅

>⋅≤≤

≤⋅⋅⋅⋅≤⋅

Shannon-Kotelnikov optimal signal-sampling relations in time and frequency domains: = maximal time sampling interval, = maximal frequency sampling interval.

and T are the total, or absolute signal-lengths in its frequency and time domains; (

t δδf

F 2πΩ =δf 2πδω, f 2πω == ).

Other forms of Uncertainty Relations (see later in this chapter for more complete development of such relations):

21,

2

,

≅⋅≅⋅⋅⋅

≤⋅<⋅≤<⋅<

≅⋅≅⋅⋅

=⋅⋅

≤⋅<⋅≤<⋅⋅=⋅<

cctf

cc

2

tω

tfτ(ω)f(t)δft δ4

1TFσσ1δft δ0

tωτ(ω)ω(t)δωt δ

πδft δ2

πTΩσσπδft δ2πδωt δ0 π

.δfx δ4

hδft δ4

hLPTEσσσσ2π2hδpx δEδt δ0

,δfx δ4

1δft δ4

1LFTFσσσσ21δfx δδft δ0

xpxEt

xxx-fxftx

⋅⋅=

⋅⋅≤⋅=⋅<⋅=⋅≤<⋅=⋅<

⋅⋅=

⋅⋅≤⋅=⋅<⋅=⋅≤<⋅=⋅<

~~~

RESUME OF DIFFERENT ANALITIC SIGNAL REPRESENTATIONS

Analytic Signal can be presented in number of ways, giving the chance to reveal the internal structure of waveforms from different points of view. Here are summarized several of such possibilities.


6

[ ]

,

ˆ 0

∑ ∑Ψ==∑

=

=ϕϕ=ΨΨΨ

ϕϕ

ϕ

(k) (k)k

(t)ik

(t)i

0

000(t) I

0

(t)(t)ea(t)ea

(t)sin I+(t)cos(t)a(t)ea=(t)I+(t)=(t)

kk(k)kk

[ ] I (t )

2

ˆH (t) (t) (t) I (t) a(t) e

H 1 I H , I 1

⋅ϕΨ = Ψ = Ψ + ⋅Ψ = ⋅

= + ⋅ = −

AS SUPERPOSITION

k k(k ) (k ) (k )

ˆ(t) (t) (t) I (t)Ψ = Ψ = Ψ + Ψ∑ ∑ ∑ k

AS MULTIPLICATION

n 1 n 1

n i n(i 0) (i 0)

(t) (t) cos (t) I (t) sin (t)− −

= =

Ψ = Ψ ϕ + Ψ ϕ∏ ∏ i

RELATIONS BETWEEN ADDITIVE AND MULTIPLICATIVE ELEMENTS

[ ]

k k

n 1

k n i 0 0(k ) (i 0)

n 1

k n i 0 0(k ) (i 0)

i (t )k k k k k k k

ˆ(t) (t) (t) cos (t) a (t)cos (t) a(t)cos (t) H (t)

ˆ ˆ(t) (t) (t) sin (t) a (t)sin (t) a(t)sin (t) H (t)

ˆ ˆ(t) (t) i (t) a (t)e , (t) H (t) ,

−

=

−

=

ϕ

⎡ ⎤Ψ = Ψ = Ψ ϕ = ϕ = ϕ = − Ψ⎣ ⎦

Ψ = Ψ = Ψ ϕ = ϕ = ϕ = Ψ

⎡ ⎤Ψ = Ψ + Ψ = Ψ = − Ψ⎣ ⎦

∑ ∏

∑ ∏

[ ]

k k k k

k k

n nI (t ) I (t ) I (t ) I (t )n

n+1 nk 0 k 0

ˆ (t) H (t)

a (t) 1ˆ(t)= (t)+I (t)= (e e ) (e e )2 (i)

ϕ − ϕ ϕ − ϕ

= =

Ψ = Ψ

⎧ ⎫Ψ Ψ Ψ + + −⎨ ⎬

⎩ ⎭∏ ∏

Signal Amplitude or Envelope

n n2 2

0 n i n 1(i 1) (i 1)

n2 2

k k 1 k k k k 1 k 1 n i(i k 1)

2 2n 1 n n 1 n 1 n 1 n n

â(t) a (t) (t) (t) (t) a (t) cos (t) (t) cos (t)

â (t) (t) (t) (t) a (t)cos (t) a (t) cos (t) , k n

â (t) (t) (t) (t) (t) a (t)cos (

+= =

+ + += +

− − − −

= = Ψ = Ψ + Ψ = ϕ = Ψ ϕ

= Ψ = Ψ = Ψ + Ψ = ϕ = ϕ <

= Ψ = Ψ = Ψ + Ψ = ϕ

∏ ∏

∏

i

0n 1

i(i 0)

(t)t)

cos (t)−

=

Ψ=

ϕ∏


7

Signal Phase

k k

2 k0 k k

(k ) k

2 2 2 21 2 n j k

n

0 1 1 2 2 n n k kk 1

i (t )k k k

ˆˆ (t)(t)(t) (t) arctg (t), (t) arctg(t) (t)

I i i ... i 1 , i i 0 , j k (hypercomplex imaginary units)

I (t) I (t) i (t) i (t) ... i (t) i (t)

e cos (t) i sin (t

=

ϕ

ΨΨϕ = ϕ = = ϕ ϕ =

Ψ Ψ

= = = = = − = ∀ ≠

ϕ = ϕ = ϕ + ϕ + + ϕ = ϕ

= ϕ + ϕ

∑

∑n

2 20 k

k 1

) , (t) (t)=

ϕ = ϕ∑

Signal instant frequency

i

i i(t)

(t) 2 f (t) , i 0,1, 2, ...k, ...nt

∂ϕω = π = =

∂

Some more of interesting relations

Ψ Ψ Ψ

ΨΨ

Ψ Ψ Ψ

Ψ Ψ Ψ

k ki t)

k k k

kk

k0

2k

2

(k)

k

k2

k-12

k-12

k+12

k2

k2

02 2 2

k

(t) = a (t)e (t) + i (t) ,

cos = 12

( sin = 12i

(

(t) = arctg(t)(t)

(t) = (t)(t)

a (t) = a (t) + a (t) = (t) = (t) + (t) ,

a (t) = (t) (t) + (t) = a

kϕ

ϕ ϕ ϕ ϕϕ ϕ

ϕ ϕ ϕ ωϕ

π

k

k k k kk

I t I tk

I t I t

k k

e e e e

tt

f t

(

( ) ( ) ( ) ( )

$

), ),

$, , ( ) ( ),

$ $

$

=

+ −

=∂∂

=

=

− −

∑ 2

2 2

(k)i j

(i j)

nn+1

nn+1

nn+1 n

(t) (t) (t) i, j,k 1,n

(t) =a2

(t) =a(2i)

(t) = (t) + I (t) =a2

1(i)

∑ ∑

∏ ∏

∏ ∏

+ ∀

+ −

+ + −RST

UVW

≠

−

=

−

=

−

=

−

=

2

0 0

0 0

Ψ Ψ

Ψ Ψ

Ψ Ψ Ψ

, .

( )( ), $

( )( ) ,

$ ( )( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

te e

te e

te e e e

I t I t

k

nI t I t

k

n

I t I t

k

nI t I t

k

n

k k k k

k k k k

ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

∈

.

The Analytic Signal modeling of the wave function can easily be installed in the framework of the Fourier Integral Transform, which exist on the basis of simple-harmonic functions cos ωt,

[ ]

[ ] [ ] ,f2,(t)F=dtftcos2H(t)=dt(t)e = )e2

A(=)2

U(

, )2

(UF=dfftcos2H)U(=dfe)2

U(=(t) cos a(t)=(t)

-

*

-

ftj2-)(j

1-

--

ftj2

πωΨπΨΨπω

πω

πω

ππω

πω

ϕΨ

ππω

Φ

π

=

⎥⎦⎤

⎢⎣⎡

∫∫

∫∫

∞

∞

∞

∞

∞

∞

∞

∞

2

2

where the meaning of symbols is:


8

F (=) Direct Fourier transform,

F (=) Inverse Fourier transform, -1

jH+1=H (=) Complex Hilbert transform, j2 = -1,

jH-1=H* (=) Conjugate complex Hilbert transform.

[ ] [ ][ ] [ ]

[ ][ ] [ ][ ] [ ] (t).ˆj-(t)=(t)jH - (t)=(t)(t)H

, (t)ˆj+(t)=(t)jH + (t)=(t)(t)H

,tcosjH)(et,costsinH,etcosH

t,sintcosH,etcosH

**

tj

tj-*

tj

ΨΨΨΨΨΨ

ΨΨΨΨΨΨ

ω

ωωω

ωωω

ω

ω

ω

=

=

±=

−==

==

± 1

The further generalization of the Fourier integral transformation can be realized by convenient replacement of its simple harmonic functions basis cos (ωt) by some other (compatible) signal basis α(ω, t). Now, the general wave function (in the generalized framework of Fourier transform) can be represented as,

Ψ

Ψ Ψ

(t) = (2

H ( , t) df = (2

(2

= (t) H ( , t) dt = (t) .

-

*

-

U F

U F

-1ωπ

α ω Uωπ

ωπ

α ω

) )

)

m r

m r

∞

∞

∞

∞

z

z

,LNMOQP


Miodrag Prokic E-mail: [email protected] Jul-04 Wave Function Concepts

9

9/

Extension of the Electrical Power Definition to Arbitrary (Non-Sinusoidal) Voltage and Current Signals and Consequences Regarding Novel Understanding of Quantum-Mechanical Wave Function List of Symbols:

Z R jX Z Z R X

R

tFrequency

signalFrequency signal

theCurrent signalFrequency Current signal

jHPF

t Wave functiona t Amplitude of a Wave function

t Phase of a Wave function

p

p

u

i

= − = = = +

=

=

=

====

= − =

=

= ====

Complex Impedance ,

Resistance, X = ReactanceT =Time interval

Phase function of the Instantaneous Powerof the Instantaneous Power

Phase function of the Voltageof the Voltage

Phase function ofof the

Imaginary unitHilbert Transform

cos Power Factor

u

i

2 2

1

ϕ

ω

ϕωϕω

θ

ϕ

( )

( )

( )( )( )

Ψ

p(t) Instantaneous PowerHilbert Transform of Instantaneous Power

p(t) Instantaneous Power in the form of Complex Analytic Signalp(t) Absolute Value of Instantaneous PowerP(t) Amplitude function of Instantaneous Power

Amplitude function of Instantaneous PowerQ(t) = Amplitude function of Instantaneous Reactive Power

===

=

==

$ ( )

( )( )

p t

real or activeS t Apparent

u(t) = Instantaneous Voltage in the form of complex Analytic Signalu(t) = Instantaneous Voltageu(t) = Hilbert Transform of Instantaneous Voltagei(t) = Instantaneous in the form of Complex Analytic Signali(t) Instantaneous Current

i(t) = Hilbert Transform of InstantaneousU(t) = Amplitude function of Instantaneous VoltageI(t) = Amplitude

$

$

,,

Current

Current

U Effective RMS VoltageI Effective RMS Current

rms

rms

=

==

function of Instantaneous Current

Analytic Signal and Electrical Power Characterization (comparative tables) Instantaneous Power

(Apparent Power & Analytic Signal) Averaged Complex Power

(Averaged Apparent Power & Phasor notation) Generalized Complex Power

(Generalized Phasor Notation) p t p t j p t p t e

t arctg p tp t

t t

t t t t

t t tt

tj

j t

p u i

u i

i u pp

p( ) ( ) $ ( ) ( ) ,

( )$ ( )( )

( ) ( )

( ) ( ) ( ) ( ) ,

( ) ( ) ( ) ,( )

, ( )

( )= + ⋅ = ⋅

= = + =

= + = −

= − =∂

∂= −

ϕ

ϕ ϕ ϕ

θ ϕ ϕ θ

θ ϕ ϕ ωϕ

2 2

12

S U I P jQ Se U I e

u(t), i(t) (t) (t)(t)t

(t)t

(t) = t + (t) = t +

* -j rms rms

-j

i u i u

u i

u u i i

= ⋅ = − = =

= ∠ = − = − =

= − = = =∂∂

=∂∂

12

θ θ

θ ϕ ϕ ϕ ϕ

θ θ θ ωϕ ϕ

ϕ ω θ ϕ ω θ

( )

,

( , )

t

arctg QPi u

S u i P jQ

S(t)e U(t)I(t)e

(t) (t) = arctg Q(t)P(t)

*

-j -j

i u

( ) ( ) ( ) ( ) ( )

( )

( ) ( )

t t t t t

t

t t

= ⋅ = − =

= =

= −

12

12

θ θ

θ ϕ ϕ

p(t) u(t) i(t) cos (t) == U(t) I(t) cos (t) cos (t)

p

u i

= ⋅ = ⋅ ⋅ +

⋅ ⋅ ⋅

= + = =

= = ⋅

p t p t t t

p t p t p t U t I tt t

t

p tt

S tt t

t

u i

u i

p

p

u i

p

( ) ( ) cos( ( ) ( ))

( ) ( ) $ ( ) ( ) ( )cos ( ) cos ( )

cos ( )

( )cos ( )

( )cos ( ) cos ( )

cos ( )

ϕ ϕ ϕ

ϕ ϕϕ ϕ

ϕ

ϕϕ ϕ

ϕ

2 2

2

$ ( ) ( ) ( ) ( ) ( ) ,

( )

( ) ( ) cos ( ), ( ) ( ) $ ( ) ( ) ,

( ) ( ) $ ( ) , ( )$ ( )( )

, ,

( ) ( ) cos ( ), ( ) ( ) $( ) ( )

( )

p t H p t p t t t

H

u t U t t u t u t j u t U t e

U t u t u t t arctg u tu t t

i t I t t i t i t j i t I t e

u i

uj t

u uu

ij

u

i

= = ⋅ ⋅

=

= = + ⋅ = ⋅

= + = =∂∂

= = + ⋅ = ⋅

sin (t) = p(t) tan( + )

Hilbert transformation ,pϕ ϕ ϕ

ϕ

ϕ ωϕ

ϕ

ϕ

ϕ

2 2

( ) ,

( ) ( ) $( ) , ( )$( )( )

, .

t

i iiI t i t i t t arctg i t

i t t= + = =

∂∂

2 2 ϕ ωϕ

ui +ui = U(t)I(t)cos (t), ui - ui = U(t)I(t)sin (t)ui = U(t)I(t)cos (t)cos (t),

ui = U(t)I(t)sin (t)sin (t)u i

u i

$ $ $ $

$ $

θ θϕ ϕ

ϕ ϕ

U = 2U I = 2I I

I IP U I cos Active Power ( ) W

Q U I sin Reactive Power ( ) VAR

S P Q U I ) VA

PF PS

Power Factor

rms rms rms

rms rms

rms rms

rms rms

2 2rms rms

e e e

I e e

UT

u t dt IT

i t dt

j j j

j j

rmsT

rmsT

u i u

i u

ϕ ϕ ϕ θ

ϕ ϕ θ

θ

θ

θ

, ,

,

(

cos ,

( ) , ( ) .

( )

* ( )

=

= =

= = =

= = =

= + = =

= = =

= =

+

− − +

z z

2

2 2

1 12 2

Z Z eUI

e R jX

R X e

j rms

rms

j

j

= = = − =

= + ⋅

− −

−

θ θ

θ2 2

P t S t t

Q t S t t

S t P t Q t U t I t

p tt

t tp tt t

P tt

PF t P tS t

tU t I t

U t u t u t

p

u i

u i

( ) ( ) ( ) = 12

(ui +ui)

( ) ( ) ( ) = 12

(ui - ui)

( ) ( ) ( ) =

=( )

( ) ( )( )

( ) ( ) ( )

( ) ( )( )

( ) =ui +ui( ) ( )

( ) ( ) ( )

=

=

= + =

=

= =

= =

= +

cos $ $

sin $ $

( ) ( )

( )cos

cos cos

cos cos( )

cos,

cos$ $

$

θ

θ

ϕϕ ϕ

ϕ ϕ θ

θ

2 2

2 2

12

,

( ) ( ) ( )

( ) ( ) ,

( ) ( ) ( )

( )

( ) ( )+ (t)

I t i t i t

u t U t e

i t I t e I t e

j t

j t j t

u

i u

= +

=

= =

2 2$

ϕ

ϕ ϕ θ

Z t u ti t

U tI t

e

R t jX t R t X t e

j t

j t

( ) ( )( )

( )( )

( ) ( ) ( ) ( )

( )

( )

= = =

− = + ⋅

−

−

θ

θ2 2

Extension of the Electrical Power definition, M.Prokic, [email protected], http://www.mpi-ultrasonics.com/

Evolution of the RMS concept Based on traditional Phasor (complex) presentation of sinusoidal currents and voltages we have:

p(t) Average Instantaneous Power

p(t)

Average Power

= = =

= = = = =

= = = = =

= = = ⇒

⇒ = =

z zz z z

z z zz zz

1 1

1 1 1

1 1 1

1

1 1

2

2 2

2

2 2

Tp t dt

Tu t i t dt

UR T

p t dtT

u t i t dtT

u t u tR

dt

RTu t dt RI

Tp t dt

Tu t i t dt

TRi t i t dt R

Ti t dt Active

UT

u t dt IT

i t

T T

rms

T T T

Trms

T T

T T

rmsT

rms

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) , ( )dt

Average

Average

Average Apparent Power

U U e I I e I I e S U I

T

rmsj

rmsj

rmsju i i

z= = =

= = =

= + = = =

= = =

= = = = ⋅−

P U I cos Active Power ( ) W

Q U I sin Reactive Power ( ) VAR

S P Q U I ) VA

PF PS

Power Factor

rms rms

rms rms

2 2rms rms

θ

θ

θ

θ θ θ

(

cos ,

, , , .* *2 2 2 12

It is important to underline that only the Active Power P is the power delivered to the load, and that the Reactive Power Q is the power reflected from the load and sent back to its energy source. We already know all of that from the basic electro-technique regarding alternative currents and voltages (usually only related to sinusoidal and constant operating frequency signals in electric energy distribution systems). Here we shall extend and generalize the same concept of Active, Reactive and Apparent Power to the propagation of any arbitrary shaped and multifrequency, or large frequency-band signals. The generalization platform for new Active, Reactive and Apparent power definition will be related to Analytic, complex signal (and Hilbert Transform) that gives the possibility to present an arbitrary shaped time domain signal/s into corresponding Complex and Sinusoidal-like signals. After replacing the Instantaneous Power with its Analytic Signal form we will find that the traditional concept of RMS currents and voltages remains basically unchanged:


12

p(t)

p(t) p(t) Average Instantaneous Power

p(t)

= = =

= + ⋅ =

= + = = + =

= + = + =

= ⋅ + ⋅ = + = +

z z

z z zz z

z z

− −

− −

1 1

1 1 1

1 1 1

1 1

2 2

22

2 2 2 2

Tp t dt

Tp t e dt

j Complex

UR

jU

R Tp t dt

Tp t dt j

Tp t dt

RTu t dt j

Tp t dt

UR

j

R I jR I RT

i t dt jT

p t dt RI

T

j t

T

A rms R rms

T T T

T T

rms

A rms R rmsT T

rms

p( ) ( )

$ ( )

( ) ( ) $ ( )

( ) $ ( ) ( )

( ) $ ( ) (

( )ϕ

j

UT

u t dt RT

p t dt U RT

p t dt U

IT

i t dtRT

p t dt IRT

p t dt I

p t p t j p t p t j p t j

A rmsT T

R rmsT

rms

A rmsT T

R rmsT

rms

)

( ) ( ) $ ( ) ,

( ) ( ) $ ( ) ,

( ) ( ) $ ( ) ( ) ( ) $ ( ) ( ) .

⇒

⇒ = = = = =

= = = = =

= + ⋅ = + = +

− −

− −

z z zz z z

1

1 1 1

1 1

2

2

Based on complex Analytic Signal forms of voltage and current functions we can now generalize the Phasor notation concept defining Instantaneous, Complex, Apparent, Active and Reactive Power (to be applicable to arbitrary signal shapes), as follows:


13

S u i P jQ S(t)e U(t)I(t)e

(t) (t) = arctg Q(t)P(t)

arctg i(t)i(t)

arctg u(t)u(t)

( ) ( ) ( ) = 12

(ui +ui) = Power delivered to a load ,

( ) ( ) ( ) = 12

(ui - ui) = Power reflected from a load ,

( ) ( ) ( ) =( )

* -j -j

i u

( ) ( ) ( ) ( ) ( ) ,

( )$ $

,

cos $ $

sin $ $

( ) ( ) ( )cos

cos

( ) ( )t t t t t

t

P t S t t

Q t S t t

S t P t Q t U t I t p tt

t t

p

u

= ⋅ = − = =

= − = −

=

=

= + =

θ θ

θ ϕ ϕ

θ

θ

ϕϕ

2 2 12 ( ) ( )

( )( ) ( ) ( ) ( )

( ) ( )( )

( ) =ui +ui( ) ( )

( ) = 1 ui = ui)

( ) ( ) ( ) ( ) = 1 ,

( ) ( ) (

t tp tt t

P tt

Q tt

PF t P tS t

tU t I t

t

U t u t u t u t ii

u t uu

for t

I t i t i

i

u i

cos

cos cos( )

cos( )

sin,

cos$ $

, cos ( $ $

$ ( ) ($) ( ) (

$) , cos

$

ϕ

ϕ ϕ θ θ

θ θ

θ

=

= = =

= = ⇔

= + = + = +RS|T|

UV|W|

= +

n s

2 2 2 2

2

1 1

t i t ii

i t uu

for t

u t U t e i t I t e I t e

i t I t e I t e

Z t u ti t

U tI t

e R t jX t R t X

j t j t j t

j t j t

j t

u i u

i u

) (

( ) ( ) , ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )+ (t)

( ) ( )+ (t)

2 2 2

2 2

1 1= + = +RS|T|

UV|W|

= = =

= =

= = = − = +

− −

−

( ) ($) ( ) (

$) , cos ,

,

*

( ) ( )( )

( )( )

( ) ( ) ( ) (( )

θ

ϕ ϕ ϕ θ

ϕ ϕ θ

θ

) = 1

t e

u ti t

e for t

j t

j t

)

( )( )

, cos .

( )

( )

⋅

=RST

UVW

−

−

θ

θ θ( ) = 1

In majority of energy-transfer systems we do not care too much about immediate, transient, time-domain signals, since mathematically it could be complicated to use such functions (especially for arbitrary shaped signals). What count much more in the practice of power and energy conversion and distribution systems are effective, rms, mean, averaged and other numerically-expressed values (applicable to sufficiently representative frequency and/or time signal intervals). In order to avoid time dependence of above expressions, we shall determine all corresponding average values, as follows:


14

p(t)

Average Power

= = =

= =

= = = =

= =

= =

=

z z =

⇒

zzz

zz zz

1 1 1

1

1

1 1

1

22 2 2

2 2

2 2 2

2 2

Tp t dt

Tu t i t dt

Tp t t dt

TU t I t t t dt

UR RT

U t t dt RI

RT

I t t dt Active

UT

U t t dtT

u t dt

IT

I t t t dt

T Tp

T

u iT

rmsu

Trms

i

T

rms u

T T

rms i

T

( ) ( ) ( ) ( ) cos ( )

( ) ( ) cos ( ) cos ( )

( ) cos ( )

( ) cos ( ) ( )

( ) cos ( ) ( ) ,

( ) ( ) cos ( )

ϕ

ϕ ϕ

ϕ

ϕ

ϕ

ϕ =

= =

= =

=

= − = =

= =

= + = =

= + =

zz

zz

z

z

−

1

1

1

1

1

1

2

2 2

Ti t dt

tT

P t dt

TS t t dt Average

tT

Q t dt

TS t t dt Average

S P jQ Se Complex Apparent Power

arctg QP

arctgtt

Tdt

P Q Average Apparent Power

T

T

T

T

T

j

T

( )

( ) ( )

( ) cos ( )

( ) ( )

( ) sin ( )

( )( )

P U I cos =

= = Active Power

Q U I sin =

= Reactive Power

sincos

S P(t) Q(t) U I

rms rms

rms rms

2 2rms rms

θ

θ

θ

θ

θθθ

θ

Now it will be possible to extend the meaning of average electrical impedance for the general case of arbitrary voltage and current signals, as for instance:


15

S P jQ Se ZI e UZ

e R jX I

Z R jX Ze

Z ZUI

R X

u t dt

i t dt

U t t dt

I t t dt

tt

tt

R PI

jrms

j rms jrms

j

rms

rms

T

T

u

T

i

T

rms

= − = = = = −

= − = =

= = = + = =

= = = =LNM

OQP

=LNM

OQP

= = ⋅

− − −

−

zz

zz

θ θ θ

θ

ϕ

ϕ

θ θθθ

θ θθ

θ

22

2

2 2

2

2

2 2

2 2

2

( )

( )

( )

( ) cos ( )

( ) cos ( ),

( )( )

, ( )( )

(

Average Complex Impedance

arctan XR

arctan QP

arctansincos

arctan sincos

Z cos t

X QI

t

QP

tt

Average Quality Factor tt

t t t Different Power Factors

rms

)

( )

( )( )

, ( )( )

( ) , ( ) , ( ) ,

=

= = ⋅ =

= = = =

=

Average Active (Real) Impedance

Z sin Average Reactive (Imaginary) Impedance, or Reactance

XR

sincos

= tan tan sincos

cos cos cos cos

2 θ

θθ

θ θ θθ

θ θ θ θn s

Practically, in power management systems (after introducing Analytic Signal methodology) we shall be able to apply innovative concepts based on averaged and rms signal values, easily measurable using existing technology, and valid for arbitrary-shaped signals (without need to have explicit time and frequency expressions). This way, many of traditionally known concepts of power and frequency regulation will be generalized and could be significantly optimized. Also Quantum-mechanical energy-exchanges, quantum states and wave functions can be explained using “active and reactive wave functions” (similar to active and reactive power components). Stable atoms could be described as reactive resonant (multidimensional) circuit structures, where Quality Factor = tanθ , approaches to

infinity (θ π→

2), or where all internal, stationary atom waves and other movements

are behaving (analogically) like currents and voltages in pure loss-less capacitive-inductive oscillating circuits, without resistive components.

Here it will be made an attempt to connect arbitrary Power Function (product between current and voltage, or product between force and speed, or product between any other relevant, mutually conjugated functions creating power) to a Wave function as it is known in Quantum Mechanics. Energy-wise analyzed, any wave-propagation in time and frequency domain can be mutually (time-frequency) correlated using Parseval’s identity. Consequently, the immediate (time-domain) Power signal can be presented as the square of the wave-function , and analysis of the optimal power transfer can be extended to any wave propagation field (and to arbitrary shaped signals), and profit enormously (in booth, directions) after unifying traditional concepts of Active, Reactive and Apparent power with the wave-function mathematics, based on Analytic Signal methodology.

Ψ (t)2

2Ψ (t)


16

In Quantum Mechanics the wave function is conveniently modeled as a probability function, but in many aspects effectively behaves like normalized and dimensionless Power function, and here it will be closely related to Active Power, or power delivered to a load expressed in Watts as its units:

Ψ 2 (t)

Let us start from the immediate electrical-power found as the product between corresponding voltage and current signals, where both of them are arbitrary-shaped functions. We can show that such active-power function (which is transferring the power from its source to its load) can be presented as,

Ψ 2 ( ) cos $ $t P t S t t t= = ⋅( ) ( ) ( ) = 12

(ui +ui) = Q(t) cotan ( ) (=) Wθ θ .

The power reflected from a load, or Reactive Power, can be given as:

Q t S t t t t P t t VAR( ) ( ) ( ) = 12

(ui - ui) = 2= ⋅ = ⋅sin $ $ ( ) tan ( ) ( ) tan ( ) ( )θ θ θΨ =

Electric Power and Energy transfer-analysis (especially for arbitrary voltage and current signal forms) can be related to a wave-function analysis if we establish the wave function (or more precisely, the square of the wave function) on the following way:

P t a t t t T

t a t t t a t t

t t j t t jH t a t e

dt

j t

( ) ( ) cos ( ) , ,

( ) ( ) cos ( ) , $ ( ) ( ) sin ( ) ,

( ) ( ) $ ( ) ( ) ( ) ( )( )

,

( )

( )

( , )

( ,

= = ∈

= =

= + = + = =

= =

= =

−∞

+∞

+∞

+∞

−∞

zz z

Ψ

Ψ Ψ

Ψ Ψ Ψ Ψ Ψ

Ψ

2

-j t

-j t -j t

c sj t

(t) = Wave function

1 U( )e d

1 U( )e d 1 A( )e d

U( ) U ( ) - j U ( ) t e

ϕ

ϕ ϕ

πω ω

πω ω

πω ω

ω ω ω

ϕ ω

ω ω

ω

2

2

20 0

2=

+∞

∞

+∞

∞

+∞

∞

+∞

−∞

+∞

∞

+∞

∞

z

z z zzz

= + =

⋅ = = = = =

= = ⋅ = =

)

( ) ( ) $ ( ) , ( )$

,

( ) $

$ $ ( ) $

= A( )e ,

U ( ) = A( )cos ( ), U ( ) = A( )sin ( ),

(t)(t)

,

A ( ) = U ( ) + U ( ), ( ) = arctgU ( )U ( )

P(t)dt (t)dt (t)dt 12

(t) dt

12

(t) + j (t) dt P(t)dt

-j ( )

c s

2 2c

2s

s

c

-

2 2

-

2

-

2

- -

ω

ω ω ω ω ω ω

ϕ

ω ω ω ωωω

ωΦ

Φ Φ

Ψ ΨΨΨ

Φ

Ψ Ψ Ψ

Ψ Ψ

a t t t t arctg

T P t

T P t

2 2

+∞

∞

+∞

∞

+∞ ∞

zz z z= = =

1 a (t)dt 12

U( ) d 1 A( ) d2

-

2

-

2

02 πω ω

πω ω ,


17

As we can see, every single wave function has at least two wave components (since to create power function it is essential to make the product of two relevant, mutually conjugated signals, like current and voltage, velocity and force, or some other equally important couple of conjugated signals:

Ψ Ψ Ψ2 2 2( ) cos $ $ ( ) ( )t P t S t t t t= = +( ) ( ) ( ) = 12

(ui +ui) = 1 2θ , 2iuΨ

2uiΨ 2

221

ˆˆ, ==

t

t

t

,

and it shouldn’t be too big success to causally explain quantum-mechanical diffraction, superposition and interference effects, when a “single-wave-object” and/or a single particle (like an electron, or photon) is passing the plate with (at least) two small, diffraction holes, because in reality there isn’t a single object (there are always minimum 2 of mutually conjugated wave elements and their mixed products, somehow force-energy-coupled with their environment, extending the number of interaction participants). What maybe could look as a bit unusual quantum interaction, or interference of a single wave or particle object with itself, in fact presents an interaction of at least 2 wave entities with some other, third object ( ). Somehow the Nature is always creating complementary and conjugated-couples of important elements (signals, particles, energy states…) belonging to all kind of matter-motions, or we can also say that every object (or energy state) in our universe has its non-separable and conjugated image (defined by Analytic Signal concepts). Consequently, the quantum-mechanical wave function and wave energy should represent only a motional energy (or power) composed of minimum two mutually coupled wave functions (Ψ Ψ ).

Ψ Ψ Ψ2 2 2( ) ( ) ( )t t= +1 2

Ψ2 2 2( ) ( ) ( )t t= +1 2

Here applied mathematics, regarding wave functions , after making appropriate normalization/s and generalizations, would start looking as applying Probability Theory laws, like in the contemporary Quantum Theory. Consequently, modern Quantum Theory could also be treated as the generalized mathematical modeling of micro world phenomenology, by conveniently unifying all conservation laws of physics in a joint, dimensionless, mutually well-correlated theoretical platform, creating new mathematical theory that is different by appearance, but in reality isomorphic to remaining Physics structure.

Ψ Ψ Ψ2 2 2( ) ( ) ( )t t= +1 2

Also, a kind of generalized analogy with Norton and Thevenin’s theorems (known in Electric Circuit Theory) should also exist (conveniently formulated) in all other fields of Physics and Quantum Theory related to wave motions, since the cause or source of certain action is producing certain effect, and vice versa, and such events are always mutually coupled.


18

[♣ COMMENTS & FREE-THINKING CORNER: Standard Deviation in relation to Load Impedance

Since the standard deviation is often used to explain Uncertainty relations, let us analyze the meaning of the signal Standard Deviation regarding electric circuits where it is possible to measure voltage, u, and current, i, on a certain electric load. Electric load can have fully resistive (or active) impedance, R, or in general case it can present the complex impedance, Z (as the combination of R, L and C elements).

Since the Standard Deviation presents the power of the average signal deviation from the mean power, and since we are able to measure coincidently (by sampling) the load current and voltage (both of them mutually dependant), we are in the position to formulate the following generalized expression for Standard Deviation, by multiplying voltage and current deviations,

22 2g i i i i

(i ) (i )

i(i ) (i )

1 1(u u)(i i ) (u u) (i i )N N

1 1N(=) number of samples, u u , i iN N

u, i (=) runnung mean values (time evolving)

σ = − − = − −

= =

∑ ∑

∑ ∑

2

i (1.1)

Only in case/s when a load is resistive, above-formulated Standard Deviation, (1.1) can be transformed into the following expressions (fully equivalent to the contemporary definition of the Standard Deviation),

2 2g

222 2 2i i i i

(i ) (i ) (i ) (i )

2 2 2 2i ii i i

( i ) (i ) (i ) (i )i i

2 2 2i i

(i ) (i )

u Ri, u R i

1 1 1 1 1 1u u R i i (u u) R (i i )N R N N R N

i u1 i 1 u 1 1(u u) (i i ) (u u) (i i )N u N i N u N i1 (u u) (i i )N

= = ⇒ σ → σ ⇒

σ = − = − = − = −

σ = − = − = − = −

σ = − ⋅ −

∑ ∑ ∑ ∑

∑ ∑ ∑ ∑

∑ ∑

2i

(1.2) If after sampling of a load current and voltage we find that expressions (1.1) and (1.2) would produce significantly different results, this should mean that the load impedance has non-resistive, complex character (presenting a combination of R, L and C elements). The factor of “Impedance Complexity” can be formulated as,

2 2i i2

(i ) (i )22 2 2g i i

(i )

(u u) (i i )= 1 Resistive Load

1 Complex Load(u u) (i i )

− ⋅ −→⎧ ⎫σ

γ = = ⎨ ⎬≠ →σ − ⋅ − ⎩ ⎭

∑ ∑∑

(1.3)

In other words, when the ratio (1.3) is different than 1, contemporary formulation of Standard Deviation (found in all Statistics books) is not the best (and not the most general) representation of the natural signal deviation. If we generalize the same situation on arbitrary signals, we could say that the complexity of any signal, which can be formulated as a product of two mutually-conjugated signals, can be tested and additionally classified by (1.1), (1.2) and (1.3), meaning that often used Standard Deviation in Quantum Theory and Physics in number of cases could be very much limited (or wrong), since not all loads are resistive (electrically, or mechanically, or in some other analogical meaning). Many very important laws of Statistics, Thermodynamics, Quantum Theory… such as Normal, Gauss distribution, Black body radiation law, Uncertainty Relations etc. are formulated using the definition of the Standard Deviation, which is intrinsically limited to active or resistive loads (where conjugated functions, defining the signal power, are in phase), influencing that some of conclusions based on such laws could be wrong (the fact never noticed until present, because of the present incomplete definition of the Standard Deviation). ♣]


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