equivalent circuit modeling of piezoelectric energy harvesters_published

Equivalent Circuit Modeling of Piezoelectric Energy Harvesters

YAOWEN YANG* AND LIHUA TANG

School of Civil and Environmental Engineering, Nanyang Technological University, 50 Nanyang Avenue,639798 Singapore

ABSTRACT: Last decade has seen growing research interest in vibration energy harvestingusing piezoelectric materials. When developing piezoelectric energy harvesting systems, it isadvantageous to establish certain analytical or numerical model to predict the system perfor-mance. In the last few years, researchers from mechanical engineering established distributedmodels for energy harvester but simplified the energy harvesting circuit in the analytical deriva-tion. While, researchers from electrical engineering concerned the modeling of practical energyharvesting circuit but tended to simplify the structural and mechanical conditions. The chal-lenges for accurate modeling of such electromechanical coupling systems remain when compli-cated mechanical conditions and practical energy harvesting circuit are considered in systemdesign. In this article, the aforementioned problem is addressed by employing an equivalentcircuit model, which bridges structural modeling and electrical simulation. First, the parametersin the equivalent circuit model are identified from theoretical analysis and finite element analysisfor simple and complex structures, respectively. Subsequently, the equivalent circuit model con-sidering multiple modes of the system is established and simulated in the SPICE software. Twovalidation examples are given to verify the accuracy of the proposed method, and one furtherexample illustrates its capability of dealing with complicated structures and non-linear circuits.

Key Words: energy harvesting, piezoelectric materials, finite element analysis, equivalentcircuit model.

INTRODUCTION

CURRENT wireless sensing applications or portableelectronics are designed to include external power

supply, which requires periodical maintenance forlong-term operation. With advancement of low-powerelectronics, continuously self-powered wireless sensingsystems are coming from concept to practice. As oneubiquitous energy form in our daily life, vibration is apromising ambient source for energy harvesting. Thereexist several basic mechanisms for vibration-to-electricalenergy conversion. Using piezoelectric material is one ofthe most popular ways, attractive for its high energydensity. In the past few years, significant research effortshave been devoted to vibration energy harvesting usingpiezoelectric materials (Anton and Sodano, 2007).During the design stage of a piezoelectric energy

harvester, it is important to establish certain analyticalor numerical model to estimate the output power of thesystem. Some analytical models are already available inthe literature. Erturk and Inman (2008a) established ananalytical distributed parameter model for cantileveredpiezoelectric energy harvesters. Liao and Sodano (2008)

developed a theoretical model of piezoelectric energyharvesting system to predict the power around a singlevibration mode based on the Rayleigh�Ritz approach.However, these researches focused on the maximumpower that can be achieved by simplifying the energyharvesting circuit as a resistor or some combination oflinear electrical elements (Liao and Sodano, 2009). Inreality, the circuit attached to an energy harvester ismore complicated than a resistor for current regulationand power management. Theoretical analysis is very dif-ficult when some non-linear electrical components areincluded. On the other hand, some researchers, usuallyfrom electrical engineering, focused on developing modelswith non-linear circuit techniques, such as synchronizedswitch harvesting on inductor technique (Badel et al.,2005; Lefeuvre et al., 2005), and simplified the structureas a single degree of freedom system. However, thesemodels cannot account for higher vibration modes ofthe energy harvester. It is obvious that all the modelsaforementioned can only be applied for evaluation ofsystem performance in some specific conditions, whereeither the mechanical or electrical condition is much sim-plified. The challenge remains to establish a more generalmodeling approach to account for both complex mechan-ical conditions and practical energy harvesting circuit.

Finite element analysis (FEA) and SPICE software arepowerful tools in structural and electrical engineering,

*Author to whom correspondence should be addressed.E-mail: [email protected] 2, 4�8 and 10�15 appear in color online: http://jim.sagepub.com

JOURNAL OF INTELLIGENT MATERIAL SYSTEMS AND STRUCTURES, Vol. 20—December 2009 2223

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respectively. Using FEA, it is easy to estimate the open-circuit voltage or short-circuit current of the piezoelectricenergy harvester. Unfortunately, the calculated resultscannot be applied in the SPICE simulation to evaluatethe system performance unless the electromechanical cou-pling is weak and negligible. When the electromechanicalcoupling is strong, the backward coupling force will affectthe mechanical vibration and hence the voltage or currentsource is not constant with various electric loads.Elvin and Elvin (2009a) proposed one general equivalentcircuit model for piezoelectric energy harvesters, whichconsidered the backward coupling effect in the mechani-cal domain and multiple vibration modes. The param-eters of equivalent circuit were derived using theRayleigh�Ritz approach. However, this approachrequired accurate assumption of mode shapes and usuallyconsidered a large number of assumed modes, which werechallenging for complicated structures. To avoid this dif-ficulty and utilize the available powerful tools in struc-tural and electrical modeling, Elvin and Elvin (2009b)developed a coupled FEA�SPICE simulation model foranalyzing piezoelectric energy generators. However, inthis model, an automation program is needed to extractnodal displacements from the FEA output and couplingvoltage from the SPICE output, and then transfer thesedata between the FEA and SPICE solvers at each itera-tion. However, a new difficulty accompanied is that for acomplicated model, the post-process of data extractionand transfer is quite difficult to implement and tediouseven for one iteration, which limits the applicability ofthis simulation model. The challenge now is to enableefficient collaboration of robust FEA and SPICE solversand exert their respective advantages, and meanwhiletake into account the backward coupling effect.In this article, an efficient method based on the equiv-

alent circuit model (ECM) is developed to bridgethe FEA and SPICE simulation for accurate estimationof performance of piezoelectric energy harvesters.According to different mechanical conditions, thesystem parameters used in the equivalent circuit modelare determined by theoretical analysis or FEA.Numerical examples are presented to validate the accu-racy of the developed equivalent circuit model and toillustrate its capability of dealing with complicated struc-tures and non-linear circuits. Finally, together with theproposed ECM-based modeling method, other modelingtechniques for piezoelectric energy harvesters and theirapplicability are summarized.

ANALYTICAL MODELING

A reliable analytical model provides useful insightinto characteristics of piezoelectric energy harvestingsystem. Several mathematical models have been estab-lished in the past few years, including uncoupled andcoupled single degree of freedom (SDOF) models,

uncoupled and coupled distributed parameter modelsas well as approximate distributed parameter modelsderived by the Rayleigh�Ritz approach. Erturk andInman (2008b) reviewed these models and clarifiedsome misleading modeling issues on piezoelectricenergy harvesting. Here we give a brief introduction ofthe distributed parameter model for a simple cantilev-ered unimorph harvester, established by Erturk andInman (2008a). The energy harvester with bimorph con-figuration can be derived similarly.

The unimorph cantilever beam is shown in Figure 1.The profile of energy harvester is rectangular and the den-sity of beam along the longitudinal direction is uniform.The analytical model is derived based on the constitutiverelations of piezoelectricity and the following assumptions:(a) Euler�Bernouli beam assumption; (b) negligible exter-nal excitation from air damping; (c) proportional damping(i.e., the strain rate damping and viscous air damping areassumed to be proportional to the bending stiffness andmass per length of the beam); and (d) uniform electric fieldthrough the piezoelectric thickness.

The e-form piezoelectric constitutive relations are:

T1 ¼ YE1S1 � e31E3,

D3 ¼ e31S1 þ "S33E3,

ð1Þ

where T1 and S1 are the stress and strain components inthe piezoelectric layer along 1-direction (longitudinaldirection), respectively; D3 and E3 are the electric dis-placement and electric field along 3-direction (thicknessdirection), respectively; YE

1 and "S33 are the Young’smodulus and electric permittivity of the piezoelectriclayer, respectively; the superscripts E and S indicatethat the parameters are measured at constant electricfield (short-circuit) and at constant strain (beam istotally clamped), respectively; and e31 is the piezoelectricconstant. Based on assumptions (a) and (b), the govern-ing equation of mechanical motion is:

@2M x,tð Þ

@x2þ csI

@5wrel x,tð Þ

@x4@tþ ca

@wrel x,tð Þ

@tþm

@2wrel x,tð Þ

@t2

¼ �m@2wb x,tð Þ

@t2, ð2Þ

Piezoelectric beam

R

L

xüg

Figure 1. Rectangular unimorph energy harvester subjected tobase excitation.

2224 Y. YANG AND L. TANG

where M(x,t) is the internal bending moment of thebeam; I is the equivalent area moment of inertia ofthe composite cross section; wb(x,t) and wrel(x,t) arethe base excitation and the deflection relative to thebase motion, respectively; cs and ca are the strain ratedamping coefficient and viscous air damping coefficient,respectively; m is the mass per unit length; x is the lon-gitudinal coordinate and t is time. Applying the firstconstitutive equation and expressing the moment byintegrating stress distribution over the composite crosssection of the beam, and considering assumption (d),we obtain:

YI@4wrel x,tð Þ

@x4þ csI

@5wrel x,tð Þ

@x4@tþ ca

@wrel x,tð Þ

@tþm

@2wrel x,tð Þ

@t2

þ �VðtÞd� xð Þ

dx�d� x� Lð Þ

dx

� �¼ �m

@2wb x,tð Þ

@t2, ð3Þ

where YI is the average bending stiffness; � is the elec-tromechanical coupling coefficient; V(t) is the outputvoltage of the energy harvester; and �(x) is the Diracdelta function. Integrating the second constitutive equa-tion over the area of piezoelectric layer and differentiat-ing it with respect to t, we obtain:

VðtÞ

Rlþ CS dV tð Þ

dtþ

Z L

0

e31hpcb@3wrel x,tð Þ

@x2@tdx ¼ 0, ð4Þ

where CS is the capacitance measured at constantstrain, corresponding to the clamped electric permittiv-ity "S33; Rl is the resistance of the attached resistiveload; b is the width of the beam; hpc is the distancefrom the center of the piezoelectric layer to neutralaxis of the beam. Based on assumption (c), the vibrationresponse relative to the base can be represented asan absolute and uniform convergent series of the eigen-functions as:

wrel x,tð Þ ¼X1r¼1

�r xð Þ�r tð Þ, ð5Þ

where ur(x) and gr(t) are the mass normalized eigenfunc-tions and the modal coordinate of the r-th mode, respec-tively. Substituting Equation (5) into Equations (3) and(4), the electromechanical coupled ordinary differentialequations for the modal response of the beam can beobtained as:

d2�r tð Þ

dt2þ 2�r!r

d�r tð Þ

dtþ !2

r�r tð Þ þ �rV tð Þ ¼ �fr €ug tð Þ, ð6Þ

VðtÞ

Rlþ CS dV tð Þ

dt�X1r¼1

�rd�r tð Þ

dt¼ 0, ð7Þ

where xr and �r are the natural frequency and dampingratio of the r-th mode; �r is the modal electromechanicalcoupling coefficient; and �fr €ugðtÞ is the modal mechan-ical forcing function.

�r ¼

Z L

0

�d� xð Þ

dx�d� x� Lð Þ

dx

� ��r xð Þdx

¼

Z L

0

�d2�r xð Þ

dx2dx ¼ �

d�r xð Þ

dx

��x¼L, ð8Þ

fr ¼

Z L

0

m�r xð Þ dx: ð9Þ

When the harmonic base excitation ugðtÞ ¼ Aej!t isapplied (j is the unit imaginary number and x is theexcitation frequency), the steady state voltage responseacross the resistive load can be expressed as:

VðtÞ ¼ i tð Þ � Rl

¼

P1r¼1

j!�rfr=ð!2r � !

2 þ j2�r!r!Þ

P1r¼1

j!�2r=ð!2r � !

2 þ j2�r!r!Þ þ j!CS þ 1Rl

A!2ej!t,

ð10Þ

where i(t) is the current through the resistive load.

SYSTEM-LEVEL FINITE ELEMENT MODELING

To achieve optimal system performance, multilayerpiezoelectric energy harvesters are usually used, andthe profile of energy harvester is not necessarily designedas shown in Figure 1. Roundy et al. (2005) pointed outthat with the same volume of PZT, some alternativemechanical structures can significantly increase energyharvesting performance. For example, a trapezoidalcantilever beam, which distributes the strain moreevenly such that maximum strain is attained at eachpoint on the surface of the beam, can supply morethan twice the amount of energy than a rectangularbeam. In such case, the parameters of beam, i.e., YI,m, �, hpc and b in Equations (3) and (4) should bereplaced by YI(x), m(x), �(x), hpc(x), and b(x), respec-tively. Hence, �r and fr should be written as:

�r ¼

Z L

0

� xð Þd2�r xð Þ

dx2dx, ð11Þ

fr ¼

Z L

0

mðxÞ�rðxÞ dx: ð12Þ

Equivalent Circuit Modeling of Piezoelectric Energy Harvesters 2225

However, in such case, theoretical derivation of modeshape functions and parameters �r and fr at each reso-nance is a tough task. To avoid this difficulty, FEA canbe used to estimate the response of the piezoelectricenergy harvester. Several robust commercial codessuch as ABAQUS and ANSYS allow for the model-ing of piezoelectric transducer as both actuator and gen-erator (energy harvester). Although various circuitelements from resistors to diodes are available in theelement library of ANSYS, only the basic linear circuitelements such as capacitor, inductor and resistor arecompatible with the piezoelectric elements. This limitsthe applicability of system-level FEA. However,if one’s concern is to estimate the maximumachievable power from the energy harvester, linear cir-cuit elements are adequate for system-level simulation inANSYS.A finite element model for system-level simulation is

shown in Figure 2. In this model, the voltage degreesof freedom on the top and bottom surfaces of the piezo-electric layer are coupled separately to implement uni-form electrical potentials on the top and bottomelectrodes. It is to be noted that Figure 2 is just an illus-tration on how to model a piezoelectric energy harvester

in ANSYS. In the later part of this article, the mesh ofthe finite element model is further refined.

EQUIVALENT CIRCUIT MODELING

Generally, the analytical models and system-levelFEA are only applicable to estimate the maximumpower available to be extracted from a piezoelectricenergy harvester by connecting a resistive load. In prac-tice, however, an energy harvesting circuit is required toinclude some non-linear electric components such as rec-tifier and regulator as well as an energy storage module.A practical energy harvesting circuit may look like theone shown in Figure 3. Furthermore, if the geometry ofenergy harvester is complicated so as to achieve optimalsystem performance (as mentioned in the previous sec-tion), modeling of the energy harvesting system will bemore challenging. Neither the analytical models nor thesystem-level FEA is able to address these issues. Anequivalent circuit modeling method is thus proposed inthis work to solve these problems. However, differentfrom the approach of Elvin and Elvin (2009a), the para-meters of equivalent circuit in the current methodcan be conveniently extracted from FEA results for

Figure 2. Finite element model for system-level simulation.

Piezoelectric beamRegulator

Electric load/Energy storage

module

RectifierL

xüg

Figure 3. Practical energy harvesting circuit.


complicated structures, avoiding the tedious calculationby the Rayleigh�Ritz approach.In Equations (6) and (7), it is observed that an anal-

ogy exists between the mechanical and electricaldomains of the piezoelectric coupling system, as illu-strated in Table 1. Actually, the equivalent circuit repre-sentations of electromechanical transducers havebeen studied for a long time. Tilmans (1996, 1997)have summarized the equivalent circuit models for var-ious electromechanical transducers, including the piezo-electric transducer. In this article, a method based onequivalent circuit model is developed to address thechallenges of traditional approaches. Once the equiva-lent circuit parameters are determined, the performanceof energy harvesting system can be evaluated by theSPICE simulation. The remaining issue is thus identifi-cation of the required parameters.Here, we consider two cases, i.e., a simple and a com-

plex mechanical condition of the energy harvester. Thesimple mechanical condition refers to simple geometricconfiguration and simple mechanical boundaries. Thecomplex mechanical condition refers to non-uniformbeam configuration and complex mechanical bound-aries. For the first case, the parameters in the equivalentcircuit model can be directly obtained by analogizingEquations (6) and (7) with circuitry differential equa-tions. For the second case, a parameter identificationmethod by FEA will be proposed. It should be men-tioned that no matter in which case, the basis for param-eter identification is the analogy between the electricaland mechanical domains. Since the mechanical motionEquation (3) implies the small deformation of energyharvester, non-linear structural behavior cannot betaken into account by the proposed equivalent circuitmodeling method.

Equivalent Circuit Model with Theoretically

Derived Parameters

After theoretical modal analysis, we can determinethe corresponding parameters Cr, Lr, Rr, Nr, and Vr(t)

which will be used in the equivalent circuit model,as listed in Table 1, by analogizing Equations (6)and (7) with the differential equations of a circuitnetwork. The circuit network consists of infinitebranches, each composed of an inductor, a capacitor,a resistor, an ideal voltage source and an ideal trans-former, as shown in Figure 4. The r-th circuitbranch represents the r-th vibration mode of thesystem. It is noted that Equations (6) and (7) satisfyKirchoff’s voltage and current laws respectively,after equivalent circuit representation. In Equations (8)and (9), the integrals may give positive or nega-tive values. Since the SPICE software only accepts pos-itive input value, we can change the wire connectionpattern when the ideal voltage magnitude or transformerratio is negative from the integrals (8) and (9).For example, if the transformer ratio N2 is negative inFigure 4, the terminals of capacitor CS and transformershould be connected with the pattern: ‘þ’!‘�’,‘�’ !‘þ’.

The procedure to evaluate the system performanceincludes the following steps:

(A) Theoretical modal analysis to obtain the naturalfrequencies and mode shape functions of theenergy harvester;

(B) Determining the equivalent circuit parametersthrough analogies; and

(C) System modeling in the SPICE software with thedetermined parameters.

Equivalent Circuit Model with Parameters Identified

by FEA

As mentioned in the previous section to achieveoptimal system performance, the geometry of theenergy harvester could be more complicated and theharvester may have complex mechanical boundaries.In such case, it is difficult to theoretically derive theparameters of equivalent circuit model. Here we proposea parameter identification method based on FEA. Theparameters to be identified are Cr, Lr, Rr, Nr, and Vr(t).The first four parameters will be identified from theadmittance of piezoelectric energy harvester, andsubsequently, the last parameter Vr(t) will be determinedfrom the short-circuit charge response with baseexcitation.

IDENTIFICATION OF Cr, Lr, Rr, AND Nr

To obtain the admittance of piezoelectric transducer,a harmonic voltage V(t) is applied to the energy har-vester. Hence, replacing V(t)/Rl with �i(t) in Equation(7), setting �fr €ugðtÞ to zero in Equation (6) and trans-forming the two equations into frequency domain,

Table 1. Analogy between electrical and mechanicaldomains.

Equivalent circuitparameters at r-th mode

Mechanicalcounterparts

Charge: qr (t) Modal coordinate: gr (t)Current: ir (t) Modal velocity: dgr (t)/dtInductance: Lr 1Resistance: Rr 2�rxr

Capacitance: Cr 1=!2r

Voltage source: Vr (t) �fr €ugðtÞIdeal transformer ratio: Nr Electromechanical

coupling:�r


we can obtain the admittance Y as:

Y ¼i

V¼

j!Q

V¼ j! CS þ

X1r¼1

�2r!2r � !

2 þ j2�r!r!

!: ð13Þ

Here, the negative sign of �i(t) means that the currentflows into the system, as shown in Figure 5, since nowthe piezoelectric transducer works as an actuator. This isdifferent from the current flow direction in Figure 4,in which the transducer works as a generator (energyharvester). Applying the analogies list in Table 1 toEquation (13), we obtain:

Y ¼ j! CS þX1r¼1

N2r

1Cr� Lr!2 þ jRr!

!: ð14Þ

According to the properties of an ideal transformer, theimpedance in the circuit is transformed by the square ofturn ratio. Hence, Lr, Rr, and Cr can be converted fromthe left side of the transformer to the right side using thefollowing relations, as shown in Figure 5.

Lmr ¼Lr

N2r

, Rmr ¼Rr

N2r

, Cmr ¼ CrN2r ð15Þ

Rearranging Equation (14), we obtain:

Y ¼ j! CS þX1r¼1

Cmr

1� LmrCmr!2 þ jRmrCmr!

!: ð16Þ

The resonance frequency and quality factor (Ikeda,1990) can be written as:

!r ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

LmrCmr

p , Qmr ¼!r

!1 � !2¼

1

Rmr

ffiffiffiffiffiffiffiffiLmr

Cmr

r

¼1

RmrCmr!r¼!rLmr

Rmr, ð17Þ

where x1 and x2 are the half-width frequencies near ther-th resonance.

Consider the following three cases:

(1) x! 0, from Equation (16), we obtain:

Y! j! CS þX1r¼1

Cmr

!¼ j!CT, ð18Þ

where CT is the static capacitance measured when noexternal mechanical force is applied on the energyharvester.

(2) !!1, from Equation (16), we obtain:

Y! j!CS, ð19Þ

where CS is the static clamped capacitance. The result isunderstandable because when the frequency of appliedvoltage approaches infinity, the structure deformationcannot accompany with the alternating electric field.Hence the boundary condition is similar to the casethat the harvester is totally clamped.

C1

Cs

C2

Cr Lr Rr

L1

L2

R1

R2

N1

N2

Nr

V1 V(t)

i (t )

Load

V2

Vr

GND

GND

GND

+

–

+

–

+

–

Figure 4. Multi-mode equivalent circuit model.


(3) x!xr, we can derive the admittance as follows:

Y¼ j!

CSþCm1

1�Lm1Cm1!2þ jRm1Cm1!

þCm2

1�Lm2Cm2!2þ jRm2Cm2!þ

�� þCmr

1�LmrCmr!2þ jRmrCmr!þ��

0BBBBBBB@

1CCCCCCCA

� j! CSþ0 � � �0|fflffl{zfflffl}r�1

þCmr

1�LmrCmr!2þ jRmrCmr!þX1i¼rþ1

Cmi

!

¼ j! CT�Xri¼1

CmiþCmr

1�LmrCmr!2þ jRmrCmr!

!

¼ j! CdrþCmr

1�LmrCmr!2þ jRmrCmr!

� �¼YdþYmot;

ð20Þ

where Yd and Ymot are termed damped admittance andmotional admittance (Ikeda, 1990), respectively.

Yd ¼ j!Cdr ð21Þ

Ymot ¼j!Cmr

1� LmrCmr!2 þ jRmrCmr!ð22Þ

The approximation in Equation (20) can be understoodby substituting Equation (17) into it and examining thedenominators term by term. The locus of Ymot in thecomplex plane near the r-th resonance is an approximate

circle whose equation can be written as:

Re Ymotð Þ �1

2Rmr

� �2þ Im Ymotð Þ½ �

2¼

1

2Rmr

� �2

: ð23Þ

Cdr is termed damped capacitance (Ikeda, 1990) at ther-th vibration mode. It should be noted that Cdr is notequal to CS as it also includes the contribution from thehigher modes (rþ 1 and above). Hence, when we drawthe approximate circle of Ymot of each mode, differentYd should be excluded from the total admittance Y,according to different Cdr:

Cdr ¼ CT �Xri¼1

Cmi: ð24Þ

From Equation (20), we know that Cdr can be directlyextracted from FEA results according to:

Cdr ¼ Im Y !rð Þ½ �: ð25Þ

Then draw the admittance locus of Ymot of each vibra-tion mode, which is an approximate circle as describedby Equation (23), and consider Equation (17), we canidentify parameters Cmr, Lmr, Rmr, and Nr as:

Rmr ¼1

2 � raduis¼

1

max Re Ymotð Þ½ �

Lmr ¼Rmr

! min Im Ymotð Þ½ �½ � � ! max Im Ymotð Þ½ �½ �

Cmr ¼1

!2rLmr

Nr ¼

ffiffiffiffiffiffiffiffi1

Lmr

r

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð26Þ

Cr

Cs

Cs

RrLr

Nr

Vr=0 V(t)

V(t)

i (t )

i (t )

Lmr

Cmr

Rmr

GND

GND

Figure 5. The r-th branch of equivalent circuit model of piezoelectric transducer in actuator mode.


Parameter Nr is obtained in the above equation.According to Equation (15), we can determine theother three parameters Cr, Lr, and Rr. Here, we haveno concern for the sign of Nr, which will be clarified inthe following section.

IDENTIFICATION OF Vr(t)The last undetermined parameter in the equivalent

circuit model is Vr(t). If a unit harmonic base accelera-tion is applied to the system, the magnitude of Vr(t) is fr.Given short-circuit condition, according to Equation(10), we can obtain from FEA the charge response ateach resonance frequency xr as:

Q j!rð Þ ¼ �r�r j!rð Þ

¼�rfr

!2r � !

2 þ j2�r!r!

��!¼!r

¼�rfr

�2rRmrj!r¼

frNrRmrj!r

: ð27Þ

Hence, with the obtained Nr and Rr, we can calculatefr as:

fr ¼ NrRmrj!rQ j!rð Þ: ð28Þ

The sign of �r fr can be determined according to the phaseangle of the charge response by FEA. Furthermore, inEquation (10), it should be noted that it is the sign of �r frthat affects the response of the system rather than the signof �r or fr alone. Hence, we need not determine the sign of�r or fr alone. Actually, from the viewpoint of circuit,there is no change in the flow direction of the outputcurrent if we change at the same time the terminalwiring pattern of the ideal transformer and the ideal volt-age source of one circuit branch in Figure 4.

GENERAL PROCEDURE OF EQUIVALENTCIRCUIT MODELINGIn summary, the general procedure of equivalent cir-

cuit modeling to evaluate the performance of the energyharvester is as follows:

(A) Finite element static analysis to determine the staticcapacitance CT and CS;

(B) Finite element modal analysis to determine theshort-circuit resonance frequency of each vibrationmode;

(C) Finite element harmonic analysis to obtain thecharge response and then the admittance with aharmonic alternating voltage input. Identify theparameters Cr, Lr, Rr, and Nr from each admittancecircle of Ymot;

(D) Finite element harmonic analysis to obtain thecharge response at each resonance, with base exci-tation applied, to determine Vr(t);

(E) Circuit modeling and simulation in the SPICE soft-ware with the parameters identified from FEA toevaluate the performance of energy harvester.

MODEL VALIDATION

Two examples are investigated in this section to vali-date the equivalent circuit modeling method presented inthe section ‘Equivalent Circuit Modeling’. For Examples1 and 2, the parameters used in the equivalent circuitmodels are identified by theoretical analysis and FEA,respectively. A further example is also presented to showthe capability and applicability of the proposed ECM-based method for the case in which complicatedmechanical condition and practical energy harvestingcircuit are considered.

Validation Example 1

The first validation example considers a simple rect-angular piezoelectric beam with unimorph configura-tion. A resistor is attached to the beam to representthe energy harvesting circuit, as shown in Figure 1.Since the geometry of the energy harvester is simple,the parameters of the equivalent circuit model arederived by theoretical analysis, as depicted in the section‘Equivalent Circuit Model with Theoretically DerivedParameters’. Although the equivalent circuit model canaccount for more complicated practical energy harvest-ing circuit, only a resistor is considered here so that theresults can be compared with those obtained by the ana-lytical model and system-level FEA. All geometric andmaterial properties of the energy harvester are listed inTable 2.

We consider the first three vibration modes of thesystem in the frequency range of 0�1000Hz. A multi-mode equivalent circuit model is established with the

Table 2. Properties of uniform beam with unimorphpiezoelectric layer.

Item Value

Young’s modulus of piezoelectric layer 66 GPaYoung’s modulus of substrate layer 100 GPaDensity of piezoelectric layer 7800 kg/m3

Density of substrate layer 7165 kg/m3

Clamped permittivity of piezoelectriclayer "S

33

1.593e�8 F/m

Piezoelectric constant e31 �12.54 C/m2

Rayleigh damping constant a 4.894Rayleigh damping constant � 1.2349e�5Beam length L 100 mmBeam width b 20 mmThickness of piezoelectric layer hp 0.4 mmThickness of substrate layer hs 0.5 mm


parameters determined from theoretical analysis aslisted in Table 3. Figures 6�8 compare the magnitudeand phase of the output voltage and power of the energyharvesting system from the equivalent circuit model,analytical model and system-level FEA for five differentresistive loads. It is observed that the results of equiva-lent circuit model match perfectly with those from the

analytical model and system-level FEA. In Figure 8, itshould be noted that the maximum power is obtained atdifferent frequency with different electric loads, which iscaused by the backward electromechanical couplingeffect on vibration. Hence, it validates that the equiva-lent circuit modeling is capable of accounting for thisbackward coupling effect as the system-level FEA andanalytical model (Erturk and Inman, 2008a) do.

Validation Example 2

The second validation example considers an isoscelestrapezoidal piezoelectric beam with unimorph config-uration, as shown in Figure 9(a). Again, a single resistorof 1MX is attached for the convenience of comparingthe equivalent circuit model with the system-level FEA.

000100101–200

–150

–100

–50

0

50

100

150

200

ECM (R=1e2) System–level FEA (R=1e2) Analytic model (R=1e2)ECM (R=1e3) System–level FEA (R=1e3) Analytic model (R=1e3)ECM (R=1e4) System–level FEA (R=1e4) Analytic model (R=1e4)ECM (R=1e5) System–level FEA (R=1e5) Analytic model (R=1e5)ECM (R=1e6) System–level FEA (R=1e6) Analytic model (R=1e6)

Vol

tage

pha

se

Frequency (Hz)

Figure 7. Phase of output voltage from analytical model, system-level FEA, and equivalent circuit model with various resistive loads.

1000100101E–4

1E–3

0.01

0.1

1

10

ECM (R=1e2) System–level FEA (R=1e2) Analytic model (R=1e2)





|V/w

2 A)|

(V

s2 /

m)

Frequency (Hz)

Figure 6. Magnitude of output voltage from analytical model, system-level FEA, and equivalent circuit model with various resistive loads.

Table 3. Parameters of equivalent circuit model deter-mined from theoretical analysis.

r-th mode Nr Lr Rr Cr fr

1 0.017563 1 6.00833 1.1082E�05 0.0906552 �0.06010 1 48.6582 2.82171E�07 0.0502413 0.10014 1 348.013 3.59905E�08 0.029457


The geometry of energy harvester is more complicatedthan that in the first validation example. The two par-allel sides of the isosceles trapezoid of the beam are oflength 15 and 5mm, respectively. The other geometricand material parameters are the same as those listed inTable 2.For this example, there is no analytical model avail-

able to identify the parameters of the equivalent circuit.Thus, they are identified by the FEA method proposedin the section ‘Equivalent Circuit Model withParameters Identified by FEA’. Again, we consider thefirst three modes of the system. Figure 10 shows thelocus of motional admittance Ymot near resonancesobtained from FEA. Applying Equations (15) and(26), we can obtain Cr, Lr, Rr, and Nr from eachadmittance circle of Ymot in Figure 10. According toEquation (28), we can determine fr, which is the magni-tude of Vr(t), from the charge response at each reso-nance frequency from FEA. Table 4 lists all theparameters identified from FEA.

1E-10

1E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0.01

461E-6

1E-5

1E-4

ECM (R=1e2) System-level FEA (R=1e2) Analytic model (R=1e2) ECM (R=1e3) System-level FEA (R=1e3) Analytic model (R=1e3) ECM (R=1e4) System-level FEA (R=1e4) Analytic model (R=1e4) ECM (R=1e5) System-level FEA (R=1e5) Analytic model (R=1e5) ECM (R=1e6) System-level FEA (R=1e6) Analytic model (R=1e6)

Frequency (Hz)

|P/w

2 A)2 |

(W

s4 /

m2 )

100010010

48 50 52

Figure 8. Output power from analytical model, system-level FEA, and equivalent circuit model with various loads.

Piezoelectric beam Piezoelectric beamR

L L

CL

x xüg üg

(a) (b)

MDA2500

Figure 9. Isosceles trapezoidal unimorph energy harvester subjected to base excitation with (a) a purely resistive load and (b) a practicalenergy storage circuit attached.

–6E–5

–3E–5

0E+0

3E–5

6E–5

0E+0 3E–5 6E–5 9E–5 1E–4

Re (Ymot)

I m (

Ym

ot)

Mode 1

Mode 2

Mode 3

Ymot

Figure 10. Locus of motional admittance Ymot.


Finally, we model the equivalent circuit of the energyharvesting system in the SPICE software, as shown inFigure 11. The magnitude of voltage and the power onthe electric load by equivalent circuit modeling andsystem-level FEA are shown in Figures 12 and 13,respectively. Again, excellent agreement in the resultsof equivalent circuit modeling and system-level FEAis observed, validating the accuracy of the equivalentcircuit model with the parameters identified fromFEA. Here, FEA is a pre-process to determine the para-meters of SPICE simulation. Compared with thecoupled FEA�SPICE model given by Elvin and Elvin(2009b), which involves the alternate FEA andSPICE simulation till convergence in each time incre-ment and the computationally expensive pre-processand post-process at each iteration, the proposedapproach based on the equivalent circuit model ismuch simpler and more efficient. It should be empha-sized that although the FEA and SPICE simulation areseparately conducted, the equivalent circuit model isable to account for the backward coupling effect in themechanical domain.

Further Example

As validated in the previous section, the proposedECM-based modeling method is applied to a more gen-eral case considering complicated mechanical conditionand practical circuit including non-linear electrical ele-ments, which the analytical model and system-level FEAcannot deal with. Here, we consider the same structureof the unimorph energy harvester as that in validationexample 2 but replace the purely resistive load with apractical energy storage circuit composed of a rectifierand a energy storage capacitor, as shown in Figure 9(b).Hence the equivalent circuit parameters of the energyharvester are the same as those identified in validationexample 2, as listed in Table 4. The full-wave bridgemodel MDA2500 (manufactured by Motorola) ischosen as the rectifier in the SPICE simulation and sixdifferent capacitances of the capacitor CL are consideredfor energy storage. The energy accumulated on thestorage capacitor can be calculated by E ¼ CLV

2=2and the instant energy harvesting power is approximatedby P ¼ �E=�t for a small variation time �t.

V3

C3

V2

C2

L1 R1

N1

GND

GND

GND

C1 Cs

Lr

L3 R3

V1

L2 R2

N2

N3

80.08nF

Rload1M Ω

1H

1H

1H

6.9366 Ω

57.868 Ω

343.56 Ω

0.0267198Vpeak=8.30E-2

Vpeak=5.23E-2

Vpeak=3.05E-2

5.874uF

227.4nF

32.96nF

0.0541673

0.0817594

+–

+–

+–

2

8

7

12

11

13 14

9 10

1

3 45

6

Figure 11. Multi-mode equivalent circuit model with parameters identified from FEA.

Table 4. Parameters of equivalent circuit model identified from FEA.

r-th mode Cdr Lmr Rmr Cmr Nr Lr Rr Cr fr

1 8.5134E�08 1400.662 9715.873 4.19E�9 0.02672 1 6.9366 5.874E�6 8.30E�22 8.4306E�08 340.8209 19722.66 6.67E�10 0.054167 1 57.868 2.274E�7 �5.23E�23 8.413E�08 149.5977 51396.43 2.203E�10 0.081759 1 343.56 3.296E�08 3.05E�2


Figures 14 and 15 show the energy accumulation proce-dure and the instant power on various storage capacitorsduring one second time, respectively, when the energyharvester is excited at the first natural frequency. It isnoted that small capacitors such as 1, 3.3, and 10 mF arequickly charged to saturation, as shown in Figure 14,which means that it is favorable to use up the energyinstantly rather than to store it if the attached capacitoris small. While, a large capacitor such as 330 mF is favor-able to store the energy for later use because of its largerenergy capacity but smaller instant power comparedwith small capacitors, as shown in Figure 15.However, we will not go further on capacitor selection

and circuit design. The purpose of this example is toillustrate the capability and applicability of the proposedgeneric ECM-based method. Issues on how to optimizethe energy harvesting system with a practical energystorage circuit will be investigated using the proposedECM-based method in future.

CONCLUSIONS

In summary, various methodologies towards model-ing piezoelectric energy harvesters are compared inTable 5. If the maximum achievable power of thesystem is the concern, the analytical models and

200

0.01

0.1

1

10

Frequency (Hz)

ECM System–level FEA

0 1000800600400

|V/w

2 A| (

V s

2 /m

)

Figure 12. Magnitude of voltage on the resistor R¼1 MX bysystem-level FEA and equivalent circuit model.

0

1E–11

1E–10

1E–9

1E–8

1E–7

1E–6

1E–5

1E–4

Frequency (Hz)

ECMSystem–level FEA

|P/(

w2 A

)2 | (

W s

4 /m

2 )

1000800600400200

Figure 13. Power on the resistor R¼ 1 MX by system-level FEA andequivalent circuit model.

0.00.0

5.0×10–6

1.0×10–5

1.5×10–5

2.0×10–5

2.5×10–5

3.0×10–5

3.5×10–5

4.0×10–5

Time (s)

CL=1 μF

CL=3.3 μF

CL=10 μF

CL=33 μF

CL=100 μF

CL=330 μF

f = fr =65.669 Hz

E/(

Aw

2 )2

(J s

4 /m

2 )

1.00.80.60.40.2

Figure 14. Energy accumulation procedure on the storage capac-itor when the harvester is excited at the first natural frequency.

f = fr =65.669 Hz

0.0

CL=1 μF

CL=3.3 μF

CL=10 μF

CL=33 μF

CL=100 μF

CL=330 μF

1.0×10–5

2.0×10–5

3.0×10–5

4.0×10–5

5.0×10–5

6.0×10–5

7.0×10–5

8.0×10–5

Time (s)

P/(

Aw

2 )2

(W s

4 /m

2 )

0.0 1.00.80.60.40.2

Figure 15. Instant energy harvesting power when the harvester isexcited at the first natural frequency.

Table 5. Methodologies for modeling of piezoelectric energy harvesters.

Mechanical conditionEnergyharvesting circuit

Simple (e.g., uniform beam withunimorph/bimorph configuration,

simple boundaries)

Complex (e.g., non-uniform beam withcomplex profile or multilayers,

complex boundaries)

Only linear elements included (resistor,capacitor, and inductor)

Analytical model System-level FEA

Non-linear elements (e.g., rectifier,DC�DC converter, and energy storagemodule) included

Equivalent circuit model (withparameters determined by theoretical

modal analysis)

Equivalent circuit model (withparameters identified by FEA)


system-level FEA can be employed for system evalua-tion. When the practical energy harvesting circuit isconsidered in system design, the equivalent circuitmodel-based method proposed in this article is recom-mended to address the challenges of accurate modelingof the electromechanical coupling system. The para-meters used in the equivalent circuit model can beidentified by theoretical analysis or FEA according todifferent mechanical conditions. Two validation exam-ples are investigated and the results demonstrate theaccuracy of the proposed equivalent circuit modelingmethod and its capability of accounting for the back-ward coupling effect in the piezoelectric energy harvest-ing system.Although we consider a single resistor as the energy

harvesting circuit in the two validation examples, for thepurpose of comparison with the analytical model andsystem-level FEA, the method developed is applicablefor complicated practical circuit including non-linearelectric components, as shown in the last example.For piezoelectric energy harvesters with linear structuralbehavior, the ECM-based method proposed in this arti-cle supplies a generic solution for system modeling andevaluation.

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