basics of rf electronics - cern...a. gallo, basics of rf electronics splitter/combiners can be also...

BASICS OF RF ELECTRONICS( … OR GETTING STARTED WITH LLRF BUILDING BLOCKS)

Alessandro GalloINFNLaboratori Nazionali di FrascatiFrascati (RM), Italy

THE CERN ACCELERATOR SCHOOL

CAS course on “RF for Accelerators”Hotel Ebeltoft Strand, Ebeltoft, Denmark, June 8th to 17th 2010

1

http://cas.web.cern.ch/cas/welcome�

http://www.au.dk/en�

http://www.isa.au.dk/�

SUMMARY• Attenuators• (signal) Amplifiers• RF Transformers• Power Splitters/ /Combiners

• Hybrid junctions/ /Directional Couplers

• Circulators/Isolators• Filters• Modulation Transfer Functions

• Frequency Mixers• Phase Detectors (I&Q)• Bi-phase attenuators/ /I&Q modulators

• Peak Detectors• Step Recovery diodes• PIN diode Switches/ /Attenuators

• Phase shifters• VCOs• PLLs

A. Gallo, Basics of RF Electronics

21 st Lecture 2nd Lecture

Fixed attenuators are widely used in RF electronics to set the proper signal level inthe various circuit branches. Proper level setting is crucial to fully exploit theinstrumentation dynamic range and to avoid circuit overload and damaging.Attenuators are also used as matching pads as they can be designed to connect linesof different impedances. Also, the insertion of attenuators in front of mismatchedloads reduces the VSWR seen at the source side.Fixed attenuators are mainly characterized by the following parameters:• Attenuation ΔdB;

• Max average power rate;

• Max peak power rate;

• Frequency range;

• Attenuation flatness over the specified frequency range;

• VSWR, size and weight, performance over the given temperature range, …

FIXED ATTENUATORS A. Gallo, Basics of RF Electronics

3

Fixed attenuators are passive, 2-ports devices generally made by a network ofresistors with a very broadband frequency response (dc ÷ many GHz, typical). They aredesigned to provide both the required attenuation and matching of the input/outputlines, which might have different characteristic impedances. The attenuation ΔdB,expressed in dB units, and the linear transmission coefficient α are defined as:

( ) ( )2010;log10 dBinoutoutin PPPPdB ∆−==⋅=∆ α

FIXED ATTENUATORS

T-type AttenuatorUnbalanced

Balancedout

outin

outin

in

inout

ZZZ

R

ZZR

ZZZ

R

C

B

A

0200

2

002

0200

2

121

12

121

α

αα

αα

α

αα

−

−+=

⋅−

=

−

−+=

02

0

000

12

11

ZR

ZRR

ZZZif

B

CA

outin

αα

αα

−=

+−

==

==


4

=

00α

αS

Unbalanced

Balancedout

inout

outin

in

outin

ZZZ

R

ZZR

ZZZ

R

C

B

A

000

2

2

00

2

000

2

2

211

21

211

ααα

αα

ααα

−+−

=

⋅−

=

−+−

=

π-type Attenuator

0

2

0

000

21

11

ZR

ZRR

ZZZif

B

CA

outin

αα

αα

−=

−+

==

==

It is important to notice that in order to match unequal input/output line impedancesa minimum attenuation is required, according to (case Z0out ≥ Z0in ):

( )maxmin0000max 1log201 αα ⋅=∆⇒−−= dBZZZZinoutinout

Fixed attenuators are available in a huge variety of packages, power ratings (up to ≈1kW), frequency ranges (to > 18 GHz), any attenuation value and all standard impedancesused in communication electronics.

FIXED ATTENUATORS A. Gallo, Basics of RF Electronics

5

Low-level RF amplifiers are used to increase the signal level whenever it is requiredfor proper treatment and/or manipulation. A very wide subject, can only be mentionedhere. Nowadays almost only solid state technology (silicon or GaAs semiconductors,BJT and FET technology) is used for low/medium power (< 10 W) applications, up to≈ 10GHz.Construction techniques are MIC (Microwave Integrated Circuits) and MMIC(Monolithic Microwave Integrated Circuits). In MIC realization the transistor and itscapacitance and resistors are soldered on microstrip lines laying on a proper substrate;MMIC are completely integrated circuits where all components (the transistors andtheir ancillaries) are fabricated on a common substrate.Concerning small-signal amplifier, the operating class is generally “A” since powerefficiency is not an issue for these kind of applications.

SIGNAL AMPLIFIERS A. Gallo, Basics of RF Electronics

6

• Frequency range;from DC to > 10 GHz, multi-decades covered by a single device.

• Output level;Maximum power at the amp output.

• Gain and gain flatness;Ratio between the output (non-saturated) and input levels, typically expressed in dB.The flatness is defined as half of the gain variation over the entire specified frequency band.

• 1 dB compression point;Output level corresponding to a 1 dB reduced gain because of the incipient device saturation.

• Noise figure;Ratio between the input and output signal-to–noise ratios assuming an input unilateral spectralnoise power density dPin/df=kT, (k=Boltzmann constant, T=290 K). Being G its power gain, theamp generates an extra output spectral noise d(ΔPnout)/df= =G(NF- 1 )kT .

• Dynamic range;Potential excursion of the output level, upper-limited by compression/saturation and lower-limited by the noise power integrated over the application frequency band.

•Two-Tone Third-order intercept point ;Measures the amp linearity. If two-tone fed (two equal-amplitude signals of frequencies f1and f2) the amp generates intermodulation products at m1f1 ±m2f2 frequencies. Theamplitudes of 3rd order products (2f1 - f2, 2f2- f1) grow with the 3rd power of the input signalsso that an input level corresponding to equal fundamental and 3rd order products amplitudescan be extrapolated (laying usually beyond the amp dynamic range).

• Input/Output VSWR or Return Loss;Measures of the input/output matching characteristics of the amplifier.

SIGNAL AMPLIFIERS: BASIC SPECIFICATIONS A. Gallo, Basics of RF Electronics

7

Transformers are widely used in RF electronics. They are veryeffective to:• Match lines of different impedance with negligible insertion loss;• De-couple ground while transmitting RF signals;• Connect balanced and unbalanced circuits (balun)RF transformers are also embedded in a number ofother devices (splitters/combiners, mixers, amplifiers, …)

2nZIVZ

nIIVnV

sppp

ps

ps

−==

−=

=

Ideal Transformer transfer functions

Autotransformer (n ≥ 1)

Center-tap transformer

RF TRANSFORMERS A. Gallo, Basics of RF Electronics

8

Together with transform ratio n and connection topology, real transformers are characterized byoperating bandwidth, insertion loss, max power rating, … The lower cutoff frequency isdetermined by the windings active inductance Lact, while the high frequency cutoff is dominatedby the inter-windings and intra-windings capacitances Cp- p, Cs- s and Cp- s .In-band insertion loss is due to the magnetic core dissipation and to the windings ohmic losses,accounted by the resistances Rloss , Rp and Rs .

Frequency [MHz] Frequency [MHz]

Inse

rtion

Loss

[dB]

Retu

rnLo

ss [d

B]

MCL TMO -1-1+MCL TMO -1-1+

Circuital model of a real RF Transformer

RF TRANSFORMERS A. Gallo, Basics of RF Electronics

9

Power splitters/combiners are usedto divide a signal into N equal copies(N = any number, preferably a powerof 2), or to make a vector sum of Ndifferent signals.Ideally, the power into any outputchannel is Pout = Pin/N.Basic characteristics are:• Number of channels N;• Operating frequency range;

• Max power ratings;• Splitting technique (reactive or

resistive);• Insertion loss (over the nominal

10 Log N );• Isolation between channels;• Phase and amplitude unbalance

among output channels;• …

A 2-way reactive splitter/combinerbased on a double tapped auto-transformer provides impedancematching at all ports and isolationbetween A and B channels.

POWER SPLITTERS/COMBINERS A. Gallo, Basics of RF Electronics

10

Isolation between ports A and B is obtained by a proper choice of the Rint value(Rint =2Z0) in the 2nd auto-transformer, while the 1st transformer is needed to matchthe characteristic input impedance.

( ) 012

2;2

0

2

22

2222

2

0

if

int ZR

ints

sintsB

spsp

RZI

IRVI

IIVV

=

=−=

=−=

==AT2 transformer equations

Kirchhoff current law at node B


Micro-strip design of a 2-way splitter/combiner withsimilar characteristics of atransformer based one.

11

( )22

0

11

2

−=

=⇒

−=⇒

NVV

PPonTransmissi

ZN

NRMatching

in

out

in

out

As the insertion lossgrows linearly with thenumber of ports, practicaluse is restricted to 3-ports devices.


Splitter/combiners can be also resistive, consisting in a “star” connection of equalresistors. The frequency response can be much wider (extending from DC) andflatter in this case, at the expense of a larger insertion loss and no isolation(all ports equally coupled).

12

1 80° Hybrid

90° Hybrid

−−

−−

=

001001

100100

2

1

jj

jj

−

−

−=

0011001111001100

2

j

−−

−−

−−

−−

=

001

001

100

100

2

2

2

2

cjc

ccj

cjc

ccj

S

Directional Coupler

Hybrid junctions and Directionalcouplers are 4-ports passive devicesbased on the same operational principlesbut with different coupling levelsbetween ports. The two class of devicesare used for different purposes.

HYBRID JUNCTIONS/DIRECTIONAL COUPLERS


S

S

13

Distributed coupling between 2 linestravelling close each other is one of thepossible hybrid/coupler configuration.The lines have a characteristicimpedance Z0 when travelling separately,while the 3-conductors system of the 2coupled lines has even and odd excitationimpedances Z0

+ and Z0-, respectively.

The scattering matrix can be worked out by exploiting the 4-fold symmetry of thenetwork. Being β± the propagation constants of the even and odd modes, we get:

( )( )

( )

−−

−−

−−

−−

=⇒

−=

+−

=

−=

=≡=

−+

−+

−+−+

001

001

100

100

2tan1

tan

::

2cos1

2sin

;

2

2

2

2

2x

00

00

22x

x

2000

cc

cc

cc

cc

jj

jj

jj

jj

mac

max

ma

ma

ecjec

ececj

ecjec

ececj

S

dc

ZZZZ

cwith

dc

dcc

ZZZ

φφ

φφ

φφ

φφ

βφ

β

β

βββ

HYBRID JUNCTIONS/DIRECTIONAL COUPLERS- COUPLED LINE


14

If the length of the coupled line is exactly 2d=λ/4=π/2β, the scattering matrix has itssimplest form:

−−

−−

−−

−−

=⇒

=

+−

==

⇓=

−+

−+

001

001

100

100

0

2/2

2xx

x2

x

2xx

x2

x

00

00

mama

mama

mama

mama

c

max

cjc

ccj

cjc

ccj

SZZZZcc

d

φ

πβ

If coupling factors lower than 10 dB (cmax > 0.3) are required,broadside coupled strips can be used. At cmax = .707 (i.e. Z0

+=5.8Z0

-) the device scattering matrix is that of a 90° hybrid junction.Coupled wound coils on ferrite cores are used at low frequencies.

Another possible directional coupler structure isrepresented by 2 parallel lines connected throughcoupling holes. Two equal holes separated by λ/4 aresufficient to produce the proper scattering matrix at agiven frequency. Device bandwidth can be enlarged withmultiple holes with different optimal dimensions.

HYBRID JUNCTIONS/DIRECTIONAL COUPLERS- COUPLED LINE


P1 P3

P4 P2

P1 P3

P2 P4

15

=

−−

−−

−−

−−

=

00

00

00

00

0121

2101

0121

2101

ZjZZZ

ZZZjZ

ZjZZZ

ZZZjZ

S

The branch line coupler is another geometrygiving the desired port-to-port coupling.The scattering matrix can be still worked outby exploiting the 4-fold symmetry of thenetwork. Under the assumptions:I) βd1 = βd2=π/2;II) (1/Z1)

2 -(1/Z2)2= (1/Z0)

2

the scattering matrix has the following form:

90° Hybrid

−=

001

001

100

100

21

j

j

j

j

201

02

ZZZZ

==

HYBRID JUNCTIONS/DIRECTIONAL COUPLERS- BRANCH LINE


16

Directionalcoupler

=

+−

+−

+

+−

+−

+−

+−

+−

+−

++−

+−

=

20

21

20

21

20

21

0120

21

01

20

21

0120

21

20

21

20

21

01

20

21

0120

21

20

21

20

21

01

20

21

0120

21

0120

21

20

21

22

220

22

22

22

220

02

222

22

220

22

22

ZZZZ

ZZZjZ

ZZZjZ

ZZZjZ

ZZZZ

ZZZjZ

ZZZjZ

ZZZZ

ZZZjZ

ZZZjZ

ZZZjZ

ZZZZ

S

The hybrid ring (also called “rat-race”) coupleris a suitable geometry to get 180° hybrids.Under the assumptions:

d1,2=d2,3=d3,4=π/2β=λ/4; d1,4=3π/2β=3λ/4the scattering matrix has the following form:

180° Hybrid

−

−

−=

0101101001011010

2

j01 2 ZZ =

HYBRID JUNCTIONS/DIRECTIONAL COUPLERS- HYBRID RING


17

UnmatchedIsolated

−

+−=

⊗

13

13

3

1

0

0

20

0

VV

VVjV

V

S

( ) 1314

<<+= dwithVdVcV PPP

Directional couplers are used to sampleor to unequal split/sum RF signals .Hybrids are used whenever splitting/combination of RF signals out-of-phase(90°) or counter-phase (180°) arerequired (I&Q modulators/ detectors,differential combination, …).

Basic characteristics are:• Coupling coefficient (3 dB for

hybrids);• Directivity / Isolation between un-

coupled ports;• Operating frequency range;• Max power ratings;• Coupling type (holes, distributed,

rings, …);• Insertion loss (over the nominal

coupling factor);• Phase and amplitude unbalance

among output channels;• Phase and amplitude flatness over

frequency;• …

Coupler Directivity: -20 Log|d|measurement of the imperfect isolation between ideally uncoupled port.

⋅−−

⋅−−

−−⋅

−−⋅

=

01

01

10

10

2

2

2

2

dccjc

dcccj

cjcdc

ccjdc

S

HYBRID JUNCTIONS/DIRECTIONAL COUPLERS


18

CIRCULATORS / ISOLATORS

=

010001100

S

Circulators are non-reciprocal 3-ports (typically) devices whose scatteringmatrix is ideally given by:

Their basic structure is a symmetrical, 120° Y junctionwith a ferrite disk placed at the center biased by an axial magnetic field.Biased ferrites show a tensor magnetic permeability, i.e. an anisotropic behavior. Theincident wave on one port excites 2 unbalanced (because of the anisotropy) waves rotating inthe 2 opposite directions, so that the coupling to the output ports is also unbalanced. Byproper design the junction it is possible to have almost 100% transmission in one port and notransmission in the other in a given frequency band (> 1 octave).

Isolators are circulators with one portinternally terminated. Circulators andisolators are used for a number of tasks, suchus matching sources and loads, protectingsources against backward power, capturingand draining reflected power from a device toanother device.


19

FILTERSFilters are 2-ports devices “tailored” to obtain a specific required frequencyresponse (the s21 of the network). Typically the response is maximized at some bandsof interest, and minimized at other frequency bands that have to be rejected.

Filters are classified on the base of theirnature, topology, dissipation, …• Analog/Digital, depending on the nature

(continuous or sampled & digitized) ofthe input signal;

• Lumped/Distributed, depending on thenature of the internal components (L-C

cells, DR cavities, µ-strip cells, …);• Reflective/Absorbing depending on

the path of the stopbands (reflectedor internally dissipated);

• LowPass/HighPass/BandPass/Notch/Comb, depending on the profile ofthe frequency response.


20

FILTERSFilters with different response around transition between pass and stop bandsare available for different applications. They implements different rationalcomplex polynomials in their transfer functions, the most popular ones being:• Bessel, for a maximally flat group delay;• Butterworth, for a maximally flat frequency response in the pass-band;• Gaussian, providing a gaussian response to a Dirac pulse and no overshoot for

an input step function. The Gaussian filter also minimizes the group delay;• Chebyshev, providing a steep transition with some passband (type I) or

stopband (type II) ripples. They provide the closest possible response w.r.t. anideal rectangular filter;

• Elliptic, providing the steeper possible transition between the pass-band andthe stop-band by equalizing the ripple in both.


Butterworth Chebyshev I Chebyshev II Elliptic 21

FILTERS

• Insertion Loss, defined as thein-band signal attenuation;

• Phase linearity/Group delay: figures of the qualityof the filter phase response across the pass-band,that should present a constant negative slope toavoid distortion of time-profile of in-band pulses.The slope of the response phase is the filter groupdelay, equal to the signal latency while travellingacross the device.

• Input/output impedance and VSWR: characteristicimpedance of the device and reflectivity of signalstransmitted (within pass-band) and rejected (withinstop-band).

A. Gallo, Basics of RF Electronics Design and construction of filters is becoming more and more a specialized activity,so that “home made” devices are seldom used, and mainly for very specific task.Design phase make use of dedicated software packages (such as Touchstone)through various iterative steps. In addition to the typologies already listed, otherimportant characteristics defining filter performances are:

22

DIGITAL FILTERING

∑

∑

=

−−−

=−−−

=+++==

=+++=

N

i

ii

N

iininnnn

zhzhzhhzXzYzH

xhtxhtxhtxhty

0

22

110

022110

...)()()(

....)()()()(

∑

∑

∑∑

=

−

=

−

−−

−−

=−

=−−−

−−

+=

+++

+++==

−=+−−

+++=

M

k

kk

N

i

ii

M

kknk

N

iininn

nnnn

zp

zh

zpzpzhzhh

zXzYzH

ypxhtxptyp

txhtxhtxhty

1

02

21

1

22

110

102211

22110

1...1...

)()()(

....)()(...

....)()()()(

∫∞−

−=t

dtxHty τττ )()()(

Digital filters act on sampled and digitized input signals. There are 2 basic filter architectures:• Finite Impulsive Response (FIR), where the output y is a linear combination of the last N sampled

values of the input x. The coefficients hi of the expansion represent the discretization of the filterGreen’s function.

• Infinite Impulsive Response (IIR), where the output y is a linear combination of the last N and Msampled values of the input x and output y, respectively. IIR filters directly implement a feedbackarchitecture, which may generate sharp frequency responses with a limited number of samples.

FIR

IIR

H(τ) =filter Green’s function


23

DIGITAL FILTERING

Differences between analog and digital filtering are quite evident.Digital filtering is a complex operation requiring many steps suchas down-conversion (necessary in most cases), A-to-D conversion,digital data manipulation, D-to-A conversion and final frequencyup-conversion.However, powerful ICs nowadays available (DSP, FPGA, …) arecapable to perform various tasks in a single chip.On the other hand, digital filtering provide incomparableflexibility and operational adaptivity, since the transfer functioncan be modified and optimized in real time by simply changing theweighting coefficients. Beam feedbacks can greatly benefit thisfeature.

sTezzL zHsH=

= )()( TjezzF zHjH ωω=

= )()(

The z-domain transfer function Hz(z) gives direct information on the filter frequency responsebeing related to the Laplace HL (s) and Fourier HF (jω) transfer functions by the mathematicalexpressions:

kk

z zpzH −−=

11)(421

80−−=

=

kpk

Comb filter transfer function


24

)(ˆ)( sxtxtransformL−

⇒L

r

i

o

i

oQ

withss

ssasasG

2/11

)(ˆ)(ˆ

)(ˆ)(ˆ)( ωσ

σφφ

=+

===

[ ] ( )ttaAtv ciii ωcos)(1)( +=

[ ])(cos)( ttAtv icii φω +=

[ ] ( )ttaAtv cooo ωcos)(1)( +=

[ ])(cos)( ttAtv oocoo φφω +∆+=

cr ωω =

[ ] ( )ttaAtv ciii ωcos)(1)( +=

[ ])(cos)( ttAtv icii φω +=

[ ] ( )[ ]tttaAtv aoocaooo ,, cos)(1)( φφω +∆++=

[ ] ( )[ ]tttaAtv poocpooo ,, cos)(1)( φφω +∆++=

cr ωω ≠

)(ˆ)(ˆ

)(;)(ˆ)(ˆ

)(;)(ˆ)(ˆ

)(;)(ˆ)(ˆ

)( ,,,,

ssa

sGsas

sGss

sGsasa

sGi

popa

i

aoap

i

popp

i

aoaa φ

φφφ

====

LLRF servo-loops and feedback loops often need to apply AM and PMmodulation to the RF drive signal. The response of a resonant cavity to AMand PM excitations depends on its bandwidth and tuning relative to thecarrier:

MODULATION TRANSFER FUNCTIONS A. Gallo, Basics of RF Electronics

25

It may be demonstrated that direct and cross modulation transfer functionsare given by:

( )( )

( )( )

( )( )

( )( )

−−

−+

=−=

−−

++

==c

c

c

cpaap

c

c

c

caapp jA

jsAjA

jsAj

sGsGjAjsA

jAjsAsGsG

ωω

ωω

ωω

ωω

21)()(;

21)()(


zcrr

cav withsssAsA φσωω

ωσσ tan

22)( 220 +≈

++=

with A(s) = transfer function in Laplace domain of the filter applied to themodulated signal. If the signal is filtered by a resonant cavity, one has toconsider A(s)=Acav(s) given by:

where φz is the cavity tuning angle, i.e. the phase of the cavity transferfunction at the carrier frequency ωc. Finally one gets:

( )( ) ( )z

zpaap

z

zaapp

ssssGsG

ssssGsG

φσσφσ

φσσφσσ

222222

22

tan12tan)()(;

tan12tan1)()(

+++−=−=

+++

++==

MODULATION TRANSFER FUNCTION

26

The general form of the modulation transfer functions features 2 poles(possibly a complex conjugate pair) and 1 zero, and degenerates to a singlepole LPF response if the cavity is perfectly tuned (cross modulation termsvanish in this case).


MODULATION TRANSFER FUNCTION

27

In circular accelerators the beam phase depends on the cavity RF phase through thebeam transfer function, while the cavity RF amplitude and phase depend on the beamphase through the beam loading mechanism. The whole generator-cavity-beam linearsystem can be graphically represented in a diagram called Pedersen Model.The modulation transfer functions vary with the stored current and definitely couplethe servo-loops and the beam loops implemented around the system.

Generator Cavity Beam

p

a

p

a

p

a

+

)(sGgaa

)(sGgap

)(sGgpa

)(sGgpp )(sGb

pp

)(sGbpa

)(sGbap

)(sGbaa

)(sB

sφtan1

+

+

MODULATION TRANSFER FUNCTION: THE PEDERSEN MODEL


28


29

Thank you for the moment

and …

See you tomorrow at 1 5. 30

END OF THE 1 ST PART

basics of rf electronics - cern...a. gallo, basics of rf electronics splitter/combiners can be also...

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