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2005 16: 809Journal of Intelligent Material Systems and StructuresShad Roundy

On the Effectiveness of Vibration-based Energy Harvesting

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On the Effectiveness of Vibration-based Energy Harvesting

SHAD ROUNDY*

LV Sensors, 1480 64th St., Suite 175, Emeryville, CA 94608, USA

ABSTRACT: There has been a significant increase in the research on vibration-based energyharvesting in recent years. Most research is focused on a particular technology, and it is oftendifficult to compare widely differing designs and approaches to vibration-based energyharvesting. The aim of this study is to provide a general theory that can be used to comparedifferent approaches and designs for vibration-based generators. Estimates of maximumtheoretical power density based on a range of commonly occurring vibrations, measured bythe author, are presented. Estimates range from 0.5 to 100mW/cm3 for vibrations in the rangeof 1–10m/s2 at 50–350Hz. The theory indicates that, in addition to the parameters of the inputvibrations, power output depends on the system coupling coefficient, the quality factor of thedevice, the mass density of the generator, and the degree to which the electrical load maximizespower transmission. An expression for effectiveness that incorporates all of these factorsis developed. The general theory is applied to electromagnetic, piezoelectric, magnetostrictive,and electrostatic transducer technologies. Finally, predictions from the general theory arecompared to experimental results from two piezoelectric vibration generator designs.

Key Words: vibrations, energy scavenging, energy harvesting, efficiency, effectiveness.

INTRODUCTION

THE rapidly decreasing size, cost, and power con-sumption of sensors and electronics has opened up

the relatively new research field of energy harvesting.The goal, of course, is to harvest enough ambient energyto power a standalone sensor and/or actuator system.A lot of research has been done in recent years on usingambient vibrations as a power source. Most of thisresearch has been focused on technology-specific solu-tions. For example, Amirtharajah and Chandrakasan(1998), El-hami et al. (2001), and Ching et al. (2002)have developed electromagnetic generators. Meningeret al. (2001), Miyazaki et al. (2003), Roundy et al.(2002), and Sterken et al. (2003) have worked onelectrostatic generators. Ottman et al. (2003), Glynne-Jones et al. (2001), and Roundy et al. (2004) havedeveloped piezoelectric vibration-based generators.All three of these basic methods of generating powerfrom vibrations have been successfully demonstrated.Additionally, some of these generators have successfullypowered wireless transceivers (Ottman et al., 2003;Roundy et al., 2003).While there have been many publications that

document successfully developed generators, a solidbasis for comparison between basic technologies has not

been published, to the author’s knowledge. It is,therefore, appropriate to step back and ask what is themaximum amount of power available from real-worldvibration sources, and what is a good basis to comparethe effectiveness of different vibration-to-electricityconversion methods? The aim of this study, then, is topresent such a basis for comparison and a basis for veryquickly determining how much power could be gener-ated from a given vibration source.

A basic, technology independent, theory is presentedfirst. The theory is then applied to electromagnetic,piezoelectric, magnetostrictive, and electrostatic trans-ducer technologies. The results of measurements on awide range of commonly occurring vibrations are thenpresented, and the basic theory is applied to thosevibration sources. Finally, the general theory is com-pared to the experimental results for two piezoelectricgenerators.

BASIC THEORY

The efficiency of a given design is often requested,and sometimes supplied, while discussing a particularvibration-based generator. However, in the contextof vibration-based energy harvesting, the concept ofefficiency is usually not very well defined. Sometimesefficiency is defined by summing up the losses dueto factors like resistance of the coil, internal resistance

*E-mail: [email protected]

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 16—October 2005 809

1045-389X/05/10 0809–15 $10.00/0 DOI: 10.1177/1045389X05054042� 2005 Sage Publications



of the piezoelectric material, etc. It can also bedefined by summing the losses in the power conversioncircuitry. However, neither of these definitions revealshow much of the potential power supplied by thevibrations actually gets converted to electricity. Thestandard definition of efficiency is �¼Pout/Pin, wherePout is the power delivered to the electrical load, and Pin

is the power supplied by the vibrations. An equivalentdefinition of efficiency is �¼Uout/Uin, where U is energyper cycle instead of power. Efficiency in this contextis mathematically identical to what is sometimes referredto as the transmission coefficient (�) of a transducer(Wang et al., 1999). The definition of the commonlyused coupling coefficient (k) is sometimes worded indifferent ways depending on the specific applicationunder consideration (Wang et al., 1999; Bright, 2001;Yaralioglu et al., 2003). However, the concept is alwaysthe same, both conceptually and mathematically. Thecoupling coefficient relates the total energy put intoa system to the amount of energy that is converted bythe transducer. If there is no load on the output ofthe transducer (such as a resistor or externally imposedmechanical force), then the square of the couplingcoefficient is simply the energy stored at the output portdivided by the total energy put into the system, which isequal to the total energy stored in the system if there isno external load. Mathematically, the coupling coeffi-cient can be written as k2¼Ust/Uin, where Ust is theenergy stored at the output port when there is noexternal load and Uin is the total input energy, and isthe same value used in the definition of transmissioncoefficient. While the coupling coefficient is related toefficiency and the transmission coefficient, they are notone and the same.Using general equations for a linear transducer, the

coupling coefficient (k), transmission coefficient (�), andmaximum output power (Pmax) are derived. Assume astandard two-port transducer as shown in Figure 1.Assuming that the transducer is linear, the constitu-

tive equations are given in Equation (1).

A1

A2

� �¼

q11 q12q21 q22

� �T1

T2

� �ð1Þ

where A1 and A2 are the across variables, T1 and T2 arethe through variables, and qij are proportionalityconstants. The names ‘across’ and ‘through’ for thevariables indicate the states acting ‘across’ and ‘through’

an element. For example, the across variable for anelectrical element is voltage (or flux linkage) and thethrough variable is current (or charge). The acrossvariable for a spring is velocity (the velocity mustbe measured on either side, thus velocity is acting‘across’ the spring) and the through variable is force.Multiplying the across and through variables at aninstant in time results in the power (or energy) of theelement. In contrast to simple elements, a transducer hastwo sets of across and through variables representingthe states in each of its two domains. For example, theacross and through variables at the mechanical port ofan electromagnetic transducer are velocity and force,respectively, and the across and through variables at theelectrical port are voltage across the coil and currentthrough the coil.

Thecouplingcoefficient (k) isdefinedas inEquation (2).As this term is well defined (Bright, 2001), it neednot be derived here. However, note that it is onlydependent on the proportionality constants and not onany load condition. This is because it is defined usingthe energy stored at the output port rather than energytransferred to a load.

k2 ¼q212

q11q22ð2Þ

To derive the transmission coefficient, the input andoutput energy terms need to be defined. We assume thatenergy is put into the system by changing the throughvariable at the input port while keeping the acrossvariable constant. Then, the input energy is defined byEquation (3).

Uin ¼ A1

Z T1

0

dT1 ð3Þ

Substituting A1¼ q11T1þ q12T2 into Equation (3) yieldsthe expression in Equation (4).

Uin ¼1

2q11T

21 þ q12T1T2 ð4Þ

In a similar manner, the output energy, or energytransferred to an external load, can be defined as shownin Equation (5).

Uout ¼1

2q22T

22 þ q21T1T2 ð5Þ

The transmission coefficient is defined as �¼Uout/Uin

and therefore is given by Equation (6).

� ¼T2 q21T1 þ ð1=2Þq22T2ð Þ

T1 ð1=2Þq11T1 þ q12T2ð Þð6Þ

If we substitute T1 and T2 with the ratio a¼T2/T1, theresulting expression for the transmission coefficient is:

� ¼ð1=2Þq22a

2 þ q21a

ð1=2Þq11 þ q12að7Þ

A1 A2

T2T1

Transducerin out

Figure 1. Standard two-port model of a transducer. A1 and T1 arethe input across and through variables, respectively. A2 and T2 arethe output variables (Bright, 2001).

810 S. ROUNDY



Clearly, the transmission coefficient, unlike thecoupling coefficient, is dependent on the load condition(expressed here as the ratio a). If the output isconstrained such that the through variable (T2) iszero, then �¼ 0. Likewise, if the output is constrainedsuch that the across variable (A2) is zero, thenT2¼�q21/q22T1, a¼�q21/q22, and �¼ 0 again. Itshould also be noted that the transmission coefficientis not necessarily constant in time. The load conditionsmay change. In particular, power circuits incorporatingnonlinear electrical loading can improve power trans-mission (Ottman et al., 2003). However, regardless ofwhether the external load is linear or not, Equation (7)accurately describes the transmission coefficient aslong as the underlying behavior of the transduceritself is linear.We are interested in the transmission coefficient at

which the output energy is maximum (�max). Assuminglinear transducer relationships, the potential values ofthe output variables (A2 and T2) are shown by the line inFigure 2. A2o is the value of the across variable whenT2 is constrained to be zero, and T2o is the value of thethrough variable when A2 is constrained to be zero.Note that the procedure used here is similar to thatused by Wang et al. (1999). The maximum outputenergy state (Umax) occurs when A2¼A2o/2 andT2¼T2o/2, as shown in Figure 3. Thus the maximumoutput energy is given by:

Umax ¼1

4A2oT2o ð8Þ

From Equation (1), if A2¼ 0, then T2o is:

T2o ¼ �q21q22

T1 ð9Þ

If T2¼T2o/2, then A2¼A2o/2 (see Figure 2). Thus,referring to Equation (1):

1

2A2o ¼ q21T1 þ

1

2q22T2o ð10Þ

Substituting Equations (9) and (10) into Equation (8)yields the following expression for Umax:

Umax ¼ �1

4

q221q22

T21 ð11Þ

The maximum output energy appears as a negativenumber because it represents energy leaving the systemas opposed to that entering the system. In practice, weare only concerned about the magnitude of the outputenergy term, and so it will be treated as a positive valuein deriving the maximum transmission coefficient.

AssumingthatT2¼T2o/2,andsubstitutingEquation(9)into Equation (4) results in the following expression forinput energy:

Uin ¼ q11 �q12q212q22

� �T21 ð12Þ

The maximum transmission coefficient is then �max¼

Umax/Uin and is given by Equation (13).

�max ¼q221

4q11q22 � 2q12q21ð13Þ

We now assume that q21¼ q12, which is generally thecase for linear transducers. Substituting Equation (2)into Equation (13) and simplifying yields the followingexpression for �max:

�max ¼k2

4� 2k2ð14Þ

Thus, the maximum transmission coefficient dependsonly on the coupling coefficient, but is not equivalentto it. Note also that the �max is independent of loadingconditions. This is because it represents the theoreticalmaximum output to input ratio. Figure 3 shows thecurve relating �max to the coupling coefficient (k).As noted earlier, the actual transmission coefficient (�)is dependent on the load condition and can varyanywhere from zero to �max. Thus, the system needs tobe designed such that the apparent load maximizes theaverage actual transmission coefficient as nearly aspossible.

T2

A2

A2o

Output energy

T2o

Figure 2. Relationship of the output variables T2 and A2, showing themaximum energy state at A2¼A2o/2 and T2¼ T2o/2.

0

0.1

0.2

0.3

0.4

0.5

0 0.2 0.4 0.6 0.8 1Coupling coefficent (k)

Max

tra

nsm

issi

on

co

effi

cien

t (

λmax

)

Figure 3. Maximum transmission coefficient (�max) vs couplingcoefficient (k).

Effectiveness of Vibration-based Energy Harvesting 811



Finally, it is useful to develop a generic expression formaximum output power in terms of the parameters ofthe input vibrations and the system coupling coefficient.Naturally, the maximum output power is Pmax¼

dUmax/dt. Assuming that the vibrational excitation,and thus the AC power signal is sinusoidal in nature,and recalling that Umax¼ �maxUin, the output power canbe expressed as:

Pmax ¼ �max!Uin ð15Þ

where ! is the circular frequency of driving vibrations,Uin is given by Equation (12), and �max is given byEquation (14).

BASIC THEORY APPLIED TO

SPECIFIC TECHNOLOGIES

The basic theory already presented is applied toelectromagnetic, piezoelectric, magnetostrictive, andelectrostatic transducer technologies. In the applicationto specific technologies, a more specific general equationfor power density will emerge.

Electromagnetic

The constitutive equations for a simple oscillatingelectromagnetic transducer are given by

F�0

� �¼

ksp BlBl L

� �zi

� �ð16Þ

where F is force, �0 is the flux linkage, ksp is the stiffnessof a restoring spring, B is the magnetic field, l is thetotal length of the conductor (usually a coil), L is theinductance of the conductor, z is the relative displace-ment of the conductor and magnetic field, and i is thecurrent in the conductor.By direct substitution into Equation (2), the coupling

coefficient is given by the following expression

k2 ¼Blð Þ2

kspLð17Þ

Direct substitution into Equation (11) yields

Umax ¼Blð Þ2

4Lz2 ð18Þ

Noting the similarity between Equations (17) and (18),Umax can be written as:

Umax ¼kspk

2

4z2 ð19Þ

Likewise, direct substitution into Equation (12) yields

Uin ¼ ksp �ðBl Þ2

2L

� �z2 ð20Þ

Again noting the similarity between Equations (20) and(17), Uin can be written as:

Uin ¼ 1�k2

2

� �kspz

2 ð21Þ

The maximum transmission coefficient (�max) canbe obtained by substituting Equation (17) intoEquation (14). These substitutions will yield anexpression for maximum power output in terms ofthe relative displacement (z). However, a more usefulexpression would give the maximum output power interms of the parameters of the input vibrations. Thus, ageneral relationship between the displacement (z) andthe parameters of the input vibrations needs to befound.

Most inertial vibration-based generator consists of anoscillating spring and mass system as shown in Figure 4(Mitcheson et al., 2004). The input vibrations arerepresented as y(t). The relative motion betweenmagnetic field and conductor is represented by therelative motion between the housing and proof mass(or the spring deflection in Figure 4). Assuming that thisis a resonant system, the spring deflection (z(t)) can beexpressed as z(t)¼Qy(t) where Q is the quality factor,given as Q¼ 1/(2�). � is the dimensionless damping ratioand is related to the damping constant (b) shown inFigure 4 by b¼ 2m�!n, where !n is the system naturalfrequency.

The AC magnitude of the spring deflection is then|Z|¼Q|Y|. However, it is more common to refer tovibrations in terms of acceleration magnitude ratherthan displacement magnitude. Recalling that |A|¼!2|Y|where |A| is the acceleration magnitude of the inputvibrations (referred to as just A hereafter), we can writethe magnitude of the spring deflection as

Zj j ¼QA

!2ð22Þ

m

y(t)

z(t)ksp

b Fe(t)

Figure 4. Schematic of basic, non-technology specific, vibration-based generator. Fe(t) is the force generated by the mechanical–electrical coupling, ksp is the spring constant, b is the dampingcoefficient, y(t) is the displacement of the input vibrations, and z(t) isthe spring deflection.

812 S. ROUNDY



Substituting Equation (22) into Equation (21) yields

Uin ¼ 1�k2

2

� �ksp QAð Þ

2

!4ð23Þ

Note that ksp¼m!2n where !n is the natural frequency

of the system. As mentioned before, we are assuming aresonant system. Therefore, !n¼!. Substituting intoEquation (23) yields the final expression for Uin.

Uin ¼ 1�k2

2

� �m QAð Þ

2

!2ð24Þ

Substituting Equations (24) and (14) into Equation (15)and simplifying results in

Pmax ¼k2m QAð Þ

2

4!ð25Þ

Normalizing for size, the expression for powerdensity is

pmax ¼k2� QAð Þ

2

4!ð26Þ

where � is the density of the proof mass material, andpmax is power density in W/m3 or the equivalent mw/cm3.Note that the output power is dependent on the level

of mechanical damping, which determines Q. However,the maximum transmission coefficient (�) does notdepend on damping. The reason is that the inputenergy (or power) used in the derivation of � onlyconsiders the energy at the input port of the transducer,which is the energy imparted by the oscillating proofmass, not the energy input by the vibrations. However,when Uin is defined as in Equation (24) the relationshipbetween the displacement of the proof mass and theinput accelerations is accounted for. The maximumtransmission coefficient depends on the suitability of thetechnology or material chosen as a transducer while Uin

depends primarily on the design of the overall oscillatingsystem and the parameters of the input vibrations. As aresult, effects of all three (material, design, and inputvibrations) appear in the power expression.

Piezoelectric Application

The constitutive equations for a piezoelectric materialare given by

SD

� �¼

s dd "

� �TE

� �ð27Þ

where S is the strain, D is the electrical displacement, s isthe compliance, d is the piezoelectric strain coefficient," is the dielectric constant, T is the stress, and E is theelectric field.

By direct substitution into Equation (2), the couplingcoefficient is given by the following familiar expression:

k2 ¼d 2

s"¼

d 2Y

"ð28Þ

where Y is the elastic constant or Young’s modulus.Again by direct substitution into Equations (11) and

(12), the maximum output energy (umax) and the inputenergy (uin) are given by:

umax ¼ �1

4

d 2

"T 2 ¼ �

1

4

k2

YT 2 ð29Þ

uin ¼1

Y�d 2

2"

� �T 2 ¼ 1�

k2

2

� �T 2

Yð30Þ

Note that the energy terms here are lower case u instead ofupper case U. That is because the piezoelectric variablesyield energy density rather than energy. The couplingcoefficient and the transmission coefficient can just aseasily be defined with energy density as with energy.

The maximum transmission coefficient (�max) andmaximum output power can be obtained by substitutingEquations (28) and (30) into Equations (14) and (15).These substitutions will yield an expression for max-imum power output in terms of the stress in thepiezoelectric material (T ). We prefer an expression interms of the input vibrations. Therefore, we will proceedto relate the stress to the input vibrations.

Referring back to Figure 4, we note that for mostdesigns the piezoelectric material will constitute therestoring spring (ksp). The mass will exert some force (F )on the piezoelectric material, given by F(t)¼m €zz(t).Therefore, the AC magnitude of this force can be writtenas |F|¼mQA. Assuming a simple stack structure,the stress in the piezoelectric element is just T¼ |F|/apwhere ap is the cross-sectional area of the piezoelectricstack. Then

uin ¼ 1�k2

2

� �ðmQAÞ2

Ya2pð31Þ

The stiffness of that spring is ksp¼Yap/h where h isthe height of the piezoelectric element. As before, weassume a resonant system such that ksp¼m!2. Then wecan write the following relationship:

!2 ¼Yaphm

ð32Þ

Substituting Equation (32) into Equation (31) yields

uin ¼ 1�k2

2

� �ðQAÞ2m

!2hapð33Þ

The term hap is the volume of the piezoelectric material.Thus Uin¼ hapuin and

Uin ¼ 1�k2

2

� �ðQAÞ2m

!2ð34Þ




Finally, substituting Equations (34) and (14) intoEquation (15) yields

Pmax ¼k2mðQAÞ2

4!ð25Þ

And, normalizing for the size of the proof mass resultsin the following expression for power density:


2

4!ð26Þ

As the equations for power and power density derivedfor the piezoelectric case are identical to those derivedfor the electromagnetic case, they have been numberedas Equations (25) and (26), referring back to theelectromagnetic power equations. These two equationsturn out to characterize the power and power density forany linear transducer technology.The assumption of a simple piezoelectric stack at

the geometric structure is somewhat unrealistic as thenatural frequency would be far too high. A more likelyscenario would be to use a bending element or a similarstructure. However, in the case of other geometricstructures, the input force can usually be related to theaverage stress in the piezoelectric material by somegeometric constant (ap), the effective spring stiffness (ksp)can still be related to the elasticity of the material (Y ) bysome other geometric constant (ap/h). Regardless of thestructure, the units of h are length and the units of ap arelength squared. Therefore, the volume of the materialis just then k0aph where k0 is some constant. ThenEquations (25) and (26) still apply for a piezoelectricmaterial with a constant multiplicative factor (1/k0).

Magnetostrictive Application

Assuming small signal linear behavior, the constitutiveequations for a magnetostrictive material are given by

SB

� �¼

s dd �

� �TH

� �ð35Þ

where S is the strain, B is the magnetic flux density,s is the compliance, d is the strain coefficient, � isthe permeability constant, T is the stress, and H is themagnetic field strength.This set of equations is identical to a piezoelectric

material except that D and E are replaced by B and H,respectively, and " is replaced by �. Therefore, thederivation of the coupling coefficient and maximumoutput power are identical to the piezoelectric case, andonly the results are given here.

k2 ¼d 2

s�¼

d 2Y

�ð36Þ


2

4!ð26Þ

Note again, that the equation for power density isthe same as for electromagnetic and piezoelectrictransducers, and so has been numbered 26 as donepreviously.

While these equations do assume small signalbehavior and ignore hysteresis, they do provide a goodstarting point for a broad comparison of technologiesthat could be used for inertial energy scavenging.It should be stated that, to the author’s knowledge,no magnetostrictive energy scavengers have yet beendesigned.

Electrostatic Application

The constitutive equations for electrostatic trans-ducers depend heavily on the geometry and operatingconditions (e.g., constant voltage or constant charge).Furthermore, they are usually not linear relationships.It is not possible to present the complete range ofpossibilities within this article. Rather, one commonrepresentative example will be covered.

Assume a parallel plate capacitive transducer asshown in Figure 5. The top plate of the capacitoroscillates between a minimum capacitance position(maximum gap between plates) and a maximumcapacitance position (minimum gap). Assume thatthe capacitor is charged and discharged very rapidlycompared to the period of mechanical oscillation.Therefore, the capacitive structure is operating underconstant charge virtually all the time. See Meningeret al. (2001) and Roundy et al. (2002) for a furtherexplanation of the operation of capacitive vibration-based generators.

Assuming constant charge operation and ignoringany parasitic capacitance, the coupled constitutiveequations are:

F ¼q2

2ae"0� kspz ð37Þ

q ¼ CðzÞV ð38Þ

CðzÞ ¼ae"0d� z

ð39Þ

z(t)

ksp

d

Top electrode

Bottom electrode

Figure 5. Standard parallel plate capacitive transducer.

814 S. ROUNDY



where F is the force acting on the top plate, q is thecharge on the top plate, V is the voltage across thecapacitor, ae is the electrode area, "0 is the permit-tivity constant, and C(z) is the capacitance of thestructure.In operation, the capacitive device is charged at

some input voltage Vin in the maximum capacitanceposition (Cmax). The charge (q), which is constantafter the instantaneous charge is supplied from Vin, isgiven by

q ¼ VinCmax ð40Þ

As the structure is being used as an electricalgenerator, the stored energy at the output is the electricalenergy stored in the capacitive device, and is given by

Eelec ¼q2

2CðzÞð41Þ

The stored mechanical energy is given by

Emech ¼1

2kspz

2 ð42Þ

The coupling coefficient is then given by

k2 ¼Eelec

Eelec þ Emech¼

1

1þ ðEmech=EelecÞð43Þ

Substituting ksp¼m!2 into Equation (42), Equations(41) and (42) into (43), and simplifying yields thefollowing expression for the coupling coefficient

k2 ¼V2

inC2max

V2inC

2max þm!2z2CðzÞ

ð44Þ

Note that, as the electrostatic force is constant withrespect to z in the constant charge case, there is nospring softening effect. However, the spring softeningeffect must be considered in the constant voltage case(see Yaralioglu et al., 2003).As the capacitance (C(z)) is a function of posi-

tion, the coupling coefficient changes continuouslythroughout the operation of the generator. Althoughthe expression derived here is valid only for parallelplate generators operating in a constant charge mode,it is usually the case that the coupling coefficientfor electrostatic converters is dependent on position.The instantaneous maximum power transfer to theload can be calculated using Equation (26). Themaximum energy density output per cycle can thenbe calculated as:

uout ¼�ðQAÞ2

4!2

Z t2

t1

kðtÞ2 dt ð45Þ

Finally, the output power density, averaged over time,for the load condition that maximizes the transfercoefficient is

pave ¼ uout f ð46Þ

where uout is given by Equation (45) and f is thefrequency in Hz (!/2�).

COMMON VIBRATION SOURCES

One of the aims of this study is to develop a simplemeans of calculating the maximum power that can begenerated from a given vibration source. Thus, it isreasonable at this stage to review some characteristicsof commonly occurring vibrations. The focus here is onrelatively low-level vibrations that occur in commonenvironments to make the discussion as broadlyapplicable as possible.

Data summarizing the results of measurements takenin from several vibration sources are shown in Table 1.The table lists the vibration source, the accelerationmagnitude of the dominant vibration frequency, and thedominant (or peak) vibration frequency. Note that thedominant frequency is generally quite low, between 60and 200Hz for most cases. Note also that the accelera-tion magnitudes range from about 0.1 to 10m/s2 or10mg to 1 g. We will, therefore, focus on vibrations inthis range.

Figure 6 shows the vibration spectra of a few of thesources listed in Table 1. Acceleration magnitude versusfrequency is shown. The spectra shown in Figure 6highlight two important points. First, as is the casewith all sources listed in Table 1, the vibrations areconcentrated at a single frequency and its harmonics.They are not broadband vibrations. Second, theacceleration magnitudes of the harmonics are signifi-cantly lower than those of the fundamental mode.As clearly shown by Equations (25) and (26), designsshould target the vibration mode with the highest

Table 1. Summary of several vibration sources.

Vibration sourcePeak acceleration

(m/s2)Frequency

(Hz)

Base of 3-axis machine tool 10 70Kitchen blender casing 6.4 121Clothes dryer 3.5 121Door frame just as door closes 3 125Small microwave oven 2.25 121HVAC vents in office building 0.2–1.5 60Wooden deck with foot traffic 1.3 385Breadmaker 1.03 121External windows (2 ft�3 ft)

next to a busy street0.7 100

Notebook computer while CDis being read

0.6 75

Washing machine 0.5 109Second story floor of a wood

frame office building0.2 100

Refrigerator 0.1 240




acceleration magnitude, which is generally the lowestfrequency mode.

COMPARISON OF DIFFERENT TECHNOLOGIES

The preceding theory is meant to provide a simplebasis both for comparing different approaches tovibration-based generation and for estimating themaximum possible power density that a given vibra-tion source can theoretically supply. In the precedingdiscussion, the only parameter that distinguishes dif-ferent technological (i.e., electromagnetic, piezoelectric,etc.) approaches is the coupling coefficient (k). Thequality factor (Q) and density (�) are design dependent,with physical limits of course. The acceleration magni-tude (A) and frequency (!) of the input vibrations areoutside the control of the designer. The purpose of thissection is to compare different technologies, or in otherwords, to compare the coupling coefficients (and byextension the maximum transmission coefficient) ofdifferent technological approaches.The coupling coefficient expressions in Equations

(28) and (36) for piezoelectric and magnetostrictivematerials are completely dependent on materialproperties. It should be noted that the overall couplingcoefficient of a structure will differ from that ofthe underlying material. In the case of piezoelectricmaterials, the material coupling coefficient is depen-dent on the elastic and dielectric properties of thematerial. A structure containing piezoelectric materialwill likely have non-piezoelectric elastic elements andpossibly dielectric elements. Therefore, the coupling of

the entire structure will differ from that of thepiezoelectric material, which will probably make upa portion of the structure. Furthermore, supple-mentary loads to the structure may alter its elasticproperties, which again will alter the overall couplingcoefficient of the structure (see Lesieutre and Davis,1997). While it may be possible that the couplingcoefficient of a piezoelectric structure can exceed thatof the material under certain circumstances, it isgenerally the case that the coupling coefficient of thematerial will exceed that of the structure. Table 2shows the coupling and maximum transmissioncoefficients for some common piezoelectric materials,and Table 3 shows the coupling and maximumtransmission coefficients for some magnetostrictivematerials (Cullen et al., 1997).

Figure 6. Vibration spectra from the casing of a microwave oven, an HVAC duct in an office building, and the base of a milling machine.

Table 2. Coupling and transmission coefficients forpiezoelectric materials.

Material k33 k31 �33 �31

PZT-5A 0.72 0.32 0.175 0.027PZT-5H 0.75 0.44 0.196 0.054PVDF 0.16 0.11 0.006 0.003PZN-PT 0.91 0.5 0.353 0.071

Table 3. Coupling and transmission coefficients formagnetostrictive materials.

Material k33 �33

Terfenol-D 0.6–0.85 0.11–0.28Terzinol 0.95 0.411

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The coupling coefficient equation for electromagneticconverters (Equation (17)) is dependent on designparameters. If we assume that the device consists of acylindrical coil of radius r and length h, then we note thefollowing relationships: l¼ 2N�r and L¼��0N

2r2/hwhere N is the number of turns in the coil and �0 isthe permeability of free space. We also note thatksp¼ �v!

2 where � is the density of the proof massand v is the volume of the proof mass. We also assumethat volume is roughly equal to �r2h. Substituting theserelationships into Equation (17) yields the followingexpression for the coupling coefficient.

k2 ¼4B2

�!2�0r2ð47Þ

Figure 7 shows the coupling coefficient contours andmaximum transmission coefficient contours for electro-magnetic generators versus frequency and the areaenclosed by the conductor loop (�r2 in this case). Thecalculations were made assuming a density of 7.5 g/cm3

and a magnetic flux density (B) of 0.1 T. Possiblecoupling coefficients for electromagnetic generators arecomparable to those for piezoelectric and magnetostric-tive technology converters.As discussed in the previous section, the coupling

coefficient for electrostatic devices is a function ofposition, and thus changes over time. Figure 8 showsthe instantaneous coupling coefficient versus positionof the top capacitor electrode relative to the bottomelectrode. Notice that while the coupling coefficienthas a maximum value of 1.0, the average value is farbelow 1.0.An average coupling coefficient can be determined by

simply calculating the average value of the coupling

coefficient across the whole operating range. Thecoupling coefficient is very sensitive to the minimumallowable capacitor gap. Typically, this minimum gap isenforced with a mechanical stop. Figure 9 shows theaverage coupling coefficient for electrostatic generatorsversus operating frequency and minimum gap.

As with previous calculations, a quality factor (Q)of 30 was used to generate the data in Figures 8 and 9.The nominal capacitor gap (d ) was determined as themagnitude of the oscillations that the top electrodewould undergo at a Q of 30. So, d¼Q�Ain/!

2þ gmin,

or 195 mm, where gmin is the minimum allowablecapacitor gap, 0.5 mm in this case. Together with thedata shown in Figures 8 and 9, these numbers highlightone of the issues with electrostatic generators. To designa generator with high power output, the range of motionof the generator must be hundreds of times greater thanthe minimum capacitor gap. This represents a practicalimplementation difficulty. If the nominal capacitorgap (d ) is designed to be much less than QAin/!

2, theeffective quality factor (Qeff) will be much lower(Qeff¼ d!2/Ain). While the coupling coefficient wouldactually go up slightly in this case, the power outputwould go down (see Equations (44) and (46)). The firstgraph in Figure 10 shows the power output versusoperating frequency and nominal capacitor gap for aminimum capacitor gap of 0.5 mm. The second graphin Figure 10 shows the effective quality factor versusthe same parameters. Note that for reasonable qualityfactors (10–50), the power output values are in the rangeof 0.5–5mW/cm3. Furthermore, note that by decreasingd, both the power and effective quality factor decreasedramatically.

With the exception of electrostatic converters, thepower expression given in Equation (26) applies to alltechnologies discussed. The only differentiator betweentechnologies is the coupling coefficient. In fact, if the

Figure 7. Coupling coefficient and maximum transmission coeffi-cient contours for an electromagnetic converter vs frequencyand area of the conductor (wire) loop. Density (�) is 7.5 g/cm3 andmagnetic flux density (B) is 0.1 T.

Figure 8. Coupling coefficient vs top electrode position for a parallelplate electrostatic generator for three different input voltages.Electrode area is 1 cm2, minimum capacitor gap is 0.5�m, density is7.5 g/cm3, frequency is 100 Hz, and Q is 30. Average power outputscorresponding to the three curves are 3.8, 5.9, and 7.5 mW/cm3.




average coupling coefficient for electrostatic convertersis used with Equation (26), the power estimates matchthose obtained with the technology-specific equations(44)–(46). Therefore, given the acceleration amplitudeand frequency ranges of commonly occurring vibrationsas shown in Table 1, and the coupling coefficientsalready presented, maximum power output values canbe computed using Equation (26). Figure 11 shows themaximum power contours for a realistic input vibrationrange for three different coupling coefficients.

A density (�) of 7.5 g/cm3 was used for all cal-culations. This is roughly the density of steel. Note thatthe density figure in Equation (26) can be interpreted intwo ways. First, it could represent the density of theproof mass. In this case, the power density numberreflects on only the proof mass, not on the rest of thestructure. Second, and perhaps more useful, the densitycould represent the proof mass divided by the entireconverter volume. In this case, the power density takesinto account the entire converter structure, and thedesigner can increase the density by minimizing emptyspace within the converter volume. In either case, powerscales directly with density, and so the data shown inFigure 11 can easily be adjusted for different overalldensities. A quality factor of 30 was assumed for thesecalculations.

While potential power density and efficiency are theprimary subjects of this study, output voltage andcurrent levels are also important in practical application,and can be a key differentiator between technologies.

Figure 11. Power density (mW/cm3) contours vs frequency andacceleration amplitude of the input vibrations for three differentcoupling coefficients. Density (�) is 7.5 g/cm3 and quality factor (Q)is 30.

Figure 10. Average power output density contours (mW/cm3) andthe necessary quality factor to obtain that power density vs operatingfrequency and nominal capacitor gap. The minimum capacitor gap is0.5�m.

Figure 9. Average coupling coefficient contours vs frequency andminimum allowable capacitor gap for an electrostatic generator.Electrode area is 1 cm2, overall density is 7.5 g/cm3, and Q is 30.

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Voltage contours versus frequency and area of theconductor loop for electromagnetic converters areshown in Figure 12. As demonstrated by the contoursin Figure 12, low voltage output is frequently a problemfor small electromagnetic converters.As with all graphs in this section, Figure 12 is

normalized per cubic centimeter. Therefore, as the areaof the conductor loop increases, the length of the coil (h)must decrease to maintain constant volume. As thelength of the coil decreases, the number of turns alsodecreases. Since V¼BlN _zz where l is the length of oneloop and N is the number of loops, the decrease in Ntranslates to a decrease in the voltage. It is true thatas area (al) increases, l will also increase, but as l isproportional to

ffiffiffiffialp

, and N is proportional to 1/al, thenet effect of increasing the area of the conductor loop isto decrease voltage. In Figure 12, N was calculated using200 turns per centimeter. If the volume were allowed toincrease, keeping the coil length (h) constant as areaincreased, the voltage would increase with increasingarea.Note that voltage decreases as frequency increases.

This non-intuitive fact results from the assumption thatthe acceleration magnitude is fixed (2.5m/s2) across thewhole frequency range, which is a reasonable assump-tion given the many vibration spectra measured asalready reported. Thus, the velocity magnitude of thevibrations decreases as 1/!, because of which the voltagealso decreases as 1/!.In general, the voltage magnitude of electromagnetic

converters increases with device volume. An increase involume will increase the potential number of turns whilekeeping the area of each coil constant, thus increasingvoltage. Piezoelectric devices suffer from the opposite

problem, usually exhibiting high voltages and lowcurrents. The actual voltage and current outputs for agiven power output depend on the type of converterstructure used. Increasing the volume of piezoelectricmaterial can either increase the voltage or the currentproduced depending on the way in which the volume ofmaterial is increased. However, as low voltage is rarelya problem for piezoelectric converters, we can say thatincreasing the volume will generally increase the currentfrom the device.

The voltage output of electrostatic devices can bedetermined in a fairly arbitrary manner by specifying theinitial charge up voltage (electrostatic converters needto be charged up initially). However, the current out-put of electrostatic devices depends on the capacitance.Therefore, larger devices, with larger capacitancesprovide higher currents. Thus, in general, electro-magnetic converters can be scaled down in size at thecost of output voltage while piezoelectric and electro-static converters can be scaled down only at the cost ofcurrent output.

VIBRATION GENERATOR EFFECTIVENESS

It is often useful to have a figure of merit that canbe used to compare dramatically different designs orapproaches to vibration generator effectiveness. Inmany cases, efficiency is an appropriate figure ofmerit. If efficiency is defined synonymously withtransmission coefficient, as seems natural, it is prob-lematic for use as a figure of merit regarding vibration-based generators. First, the actual transmissioncoefficient (see Equation (7)) is dependent on theoperating load condition. While the maximum transmis-sion coefficient, as defined in Equation (13), is onlydependent on the coupling coefficient, it does not takeinto consideration the quality factor (Q) of the design,nor the overall density (�) of the design, both of whichplay an important role in determining the output powerdensity.

To address this problem, we will define a dimen-sionless figure of merit that we will call ‘effectiveness’.First, given the theory presented already, considerhow a designer could maximize power output for agiven generator size. The first would be to designthe generator with the maximum possible systemcoupling coefficient. The second would be to minimizethe mechanical loss in the system, which is equivalentto maximizing the quality factor (Q). A thirdconsideration is the overall density (�) of the design.By increasing the density of the proof mass, anddecreasing the empty space within the generator, theoverall density is improved. Finally, the designershould design the load to maximize power transfer,

Figure 12. AC voltage magnitude contours for an electromagneticconverter vs frequency and area of the conductor loop. Accelerationof input vibrations is 2.5 m/s2 and number of turns per centimeteris 200.




which is the same as maximizing the transmissioncoefficient. Research on power circuit designs thatmaximize power output (Ottman et al., 2003) can beclassified here as efforts to improve the actual(not maximum) transmission coefficient. In terms ofthe discussion here, it does not matter if suchpower circuits are linear or not as long as thetransducer itself exhibits linear behavior. The im-proved power circuit will then improve the trans-mission coefficient ratio (�/�max). Therefore, thefollowing dimensionless term for ‘effectiveness’ isdefined.

e ¼ k2Q2 �

�0

�

�maxð48Þ

where e is the effectiveness, �0 is a baseline density(perhaps 7.5 g/cm3 as used in this article), � is the actualdensity of the design, � is the actual transmissioncoefficient, and �max is the maximum transmissioncoefficient as defined in Equation (14).The general power expression shown in Equation

(26) was derived under the assumption that thegenerator was operating at its maximum transmissioncoefficient. Noting that power output is directlyproportional to the transmission coefficient (seeEquation (15)), the actual power density ( p) can beexpressed as:

p ¼ pmax�

�max¼

k2� QAð Þ2

4!

�

�maxð49Þ

Substituting Equation (48) into Equation (49) yields thefollowing expression for power output:

p ¼e�04

A2

!ð50Þ

Thus, the power is dependent only on the designeffectiveness, the baseline density, and the parametersof the input vibrations. Therefore, dramatically differentdesigns can easily be compared by simply comparingtheir effectiveness assuming that the same baselinedensity is used in those calculations.

COMPARISON TO EXPERIMENTAL RESULTS

In general, published experimental results forvibration-based generators indicate power levels signi-ficantly below the maximum power output that thesimplified theory presented in this article would predict.First, it should be noted that often the couplingcoefficient of the overall device is quite low and theoverall density of the device is significantly lower thanthe 7.5 g/cm3 used herein. However, recall that the theorypresented here assumes that the loading is such that thetransmission coefficient, and thus power output, are at amaximum. It is generally not the case that generators areoperating at this optimal level. The circuitry that loadsthe generator often is designed more to condition thepower for use by an electronic device than to optimizethe power transfer from the generator. Thus, the actualtransmission coefficient is often well below the maximumtransmission coefficient as given in Equation (14).

The concepts developed in this study are appliedto the author’s experimental results using piezoelectricgenerators, some of which have been previouslypublished (Roundy and Wright, 2004). The twogenerators consisting of piezoelectric bimorphs mountedas cantilever beams are shown in Figure 13.

The generators were driven at a vibration magnitudeof 2.25m/s2 at 85Hz for Design 1 and 60Hz forDesign 2. Power was dissipated through a simpleresistive load. The measured power output versus loadresistance for the two generators is shown in Figure 14.

The coupling coefficient and quality factor wereexperimentally measured for each design. The couplingcoefficient was determined by measuring the opencircuit and short circuit natural frequencies andapplying Equation (51) (Lesieutre, 1998). The dampingratio was measured by applying an impulse input tothe system and measuring the resulting dampedharmonic oscillation (James et al., 1994). Once thedamping ratio is known, the quality factor is justQ¼ 1/(2�). The maximum transmission coefficient(�max) was then calculated from the measured coupling

Figure 13. Two piezoelectric generators consisting of piezoelectric bimorphs mounted as cantilever beams.

820 S. ROUNDY



coefficient using Equation (14). The overall size(consisting of a rectangular cube placed around thebeam and proof mass) of each design is 1 cm3. The k,Q, �, and �max values for each design are shown inTable 4.

k2sys ¼!2oc � !

2sc

!2oc

ð51Þ

The relationship for power dissipated through a loadresistor for the devices shown in Figure 13 has beenpreviously published (Roundy and Wright, 2004). FromEquation (49), �/�max¼ p/pmax. pmax can be obtainedby using the optimal load resistance. Assuming a smallvalue for k2, as is the case for the designs presented here,the relationship between the transmission coefficientratio (�/�max) and the load resistance can be expressedanalytically by the following equation:

�

�max¼

4R!Cp

1þ R!Cp

� �2 ð52Þ

where Cp is the capacitance of the piezoelectric device.Using Equation (52) and the data in Table 4, the

theoretical effectiveness versus the load resistance foreach design is shown in Figure 15.Finally, the theoretical power versus load resistance

can be calculated using Equation (50) and the effective-ness values in Figure 15. The theoretical and measuredpowers versus load resistance are shown in Figure 16for each design.

Although the match between the theoretical andmeasured values is not perfect, the predictions arereasonably close and show the right trend. Thetheoretical predictions for Design 1 are about 30%too high, and predictions for Design 2 are about10% too high. There are several possible reasons forthe discrepancies. They may be due to unaccountedfor losses. For example, first piezoelectric materialshave an effective series resistance that was notmodeled. Second, multiple measurements were takenfor both the coupling coefficients and dampingratios. There was some variation in these measurements.A simple average value was used for theoreticalpredictions, which may also account for some of theerror.

Figure 14. Measured power output vs load resistance for the two generators shown in Figure 13.

Figure 15. Theoretical effectiveness vs load resistance for eachdesign shown in Figure 13.

Table 4. Parameter values for the two piezoelectricgenerators shown in Figure 13.

k Q � (g/cm3) �max

Design 1 0.12 17 7.45 0.004Design 2 0.14 17 8.15 0.005




CONCLUSIONS

The purpose of this study is to provide a generaltheory upon which dramatically different types ofvibration-based generators could be compared. Thetheory demonstrates that for any type of generator,the power output depends on the system couplingcoefficient, the quality factor of the device, the densityof the generator defined as the proof mass divided bythe entire generator size, and the degree to whichthe electrical load maximizes power transmission (thetransmission coefficient ratio). An expression for the‘effectiveness’ of a vibration-based generator has beendeveloped that incorporates all of these elements. Thepower output is, of course, also dependent on themagnitude and frequency of the input vibrations.Estimates of potential power output based on thetheory presented have been given for a range ofcommonly occurring vibration sources. Estimates ofmaximum potential power density range from �0.5 to100mW/cm3 for the range of vibrations ranging from 1to 10m/s2 at frequencies from 50 to 350Hz.The general theory was applied to electromagnetic,

piezoelectric, magnetostrictive, and electrostatic trans-ducer technologies. The primary differentiator, in termsof power output, between different technologies is thecoupling coefficient. Calculations show that, dependingon material types and operation parameters, relativelyhigh coupling coefficients (0.6–0.8) are possible for eachof these technologies. The most suitable technology,therefore, needs to be decided upon based on theoperating environment and the constraints of the designproblem.Some of the considerations are as follows.

Electromagnetic generators tend to produce very lowAC voltages. Furthermore, the voltage output scalesdown as the size scales down. Piezoelectric generatorstend to produce high voltages and lower currents. Forboth piezoelectric and electrostatic generators, current,not voltage, will scale down with size because the

capacitance of device in general decreases with decreas-ing size. Finally, generating a level of power comparableto other technologies with electrostatic generatorsrequires that the device oscillates at a magnitude ofhundreds of microns while maintaining a minimumcapacitive air gap of �0.5 mm or less. This situationpresents practical implementation and stability issues.

The concept of design ‘effectiveness’ was applied totwo piezoelectric generators, and used to calculatetheoretical power outputs. The power predicted by theeffectiveness theory was about 30% more than themeasured power output for one design and 10% morefor the other design. Possible reasons for the discrepan-cies are unmodeled series resistance and variability inthe measured coupling coefficients and damping ratios.Additional experiments need to be conducted to morecarefully identify the discrepancy.

While the general theory presented contains manysimplifications that may be unrealistic in certain cases,the author believes that the theory is useful primarilybecause of its generality. It provides a starting pointfrom which to develop an overall theory of effective-ness for vibration-based generators, and a method forcomparing dramatically different design concepts.

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