Equivalent circuit modeling of a piezo-patchenergy harvester on a thin plate with AC–DCconversion

B Bayik1, A Aghakhani1, I Basdogan1 and A Erturk2

1Department of Mechanical Engineering, College of Engineering, Koç University, 34450 Istanbul, Turkey2G. W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA30332, USA

E-mail: ibasdogan@ku.edu.tr

Received 20 October 2015, revised 1 December 2015Accepted for publication 4 December 2015Published 7 April 2016

AbstractAs an alternative to beam-like structures, piezoelectric patch-based energy harvesters attached tothin plates can be readily integrated to plate-like structures in automotive, marine, and aerospaceapplications, in order to directly exploit structural vibration modes of the host system withoutmass loading and volumetric occupancy of cantilever attachments. In this paper, a multi-modeequivalent circuit model of a piezo-patch energy harvester integrated to a thin plate is developedand coupled with a standard AC–DC conversion circuit. Equivalent circuit parameters areobtained in two different ways: (1) from the modal analysis solution of a distributed-parameteranalytical model and (2) from the finite-element numerical model of the harvester by accountingfor two-way coupling. After the analytical modeling effort, multi-mode equivalent circuitrepresentation of the harvester is obtained via electronic circuit simulation software SPICE.Using the SPICE software, electromechanical response of the piezoelectric energy harvesterconnected to linear and nonlinear circuit elements are computed. Simulation results are validatedfor the standard AC–AC and AC–DC configurations. For the AC input–AC output problem,voltage frequency response functions are calculated for various resistive loads, and they showexcellent agreement with modal analysis-based analytical closed-form solution and with thefinite-element model. For the standard ideal AC input–DC output case, a full-wave rectifier and asmoothing capacitor are added to the harvester circuit for conversion of the AC voltage to astable DC voltage, which is also validated against an existing solution by treating the single-mode plate dynamics as a single-degree-of-freedom system.

Keywords: energy harvesting, piezoelectricity, vibration, Kirchhoff plate, equivalent circuit

(Some figures may appear in colour only in the online journal)

1. Introduction

Research in the field of vibration-based energy harvesting hasreceived growing attention over the last two decades with themajor goal of powering small electronic components andthereby eliminating the need of battery replacement. Thereexist several transduction mechanisms by means of whichvibrational energy can be converted to electrical energy, suchas the electrostatic [1, 2], electromagnetic [3, 4], magnetos-trictive [5, 6] conversion methods, as well as the use of

electroactive polymers [7], electrostrictive polymers [8], andpiezoelectric transducers [9, 10]. Among these alternativeapproaches, piezoelectric energy harvesters have drawn mostattention, due to the high power density and ease of appli-cation of piezoelectric materials at different geometricscales [11].

The literature on piezoelectric energy harvesting has beenmostly concentrated on cantilevered beam harvesters due totheir simplicity [10–13]. Analytical distributed parametermodeling [14], assumed-modes modeling [15], Rayleigh–Ritz

Smart Materials and Structures

Smart Mater. Struct. 25 (2016) 055015 (10pp) doi:10.1088/0964-1726/25/5/055015

0964-1726/16/055015+10$33.00 © 2016 IOP Publishing Ltd Printed in the UK1

solutions [16, 17], and electromechanical finite-elementnumerical models [18, 19] were presented and experimentallyvalidated. Since realistic scenarios of ambient vibrations canbe of varying-frequency and random nature, numerous studiesexplored bandwidth enhancement of the conventional canti-lever design. Alternative configurations were proposed bybuilding mechanical band-pass filters with array of beams[20], adding nonlinearities with compressive axial preloadconfigurations [21, 22], attaching movable tip masses [23],introducing magneto-elastic interactions [24, 25], andmechanically stiffening structures [26] for improving theperformance of one-dimensional beam-like harvesters.

As an alternative to beam-like structures such as canti-levers, piezoelectric patches can be easily integrated to thinplates to extract vibrational energy of their host structure andconvert it into electrical energy. This implementation can beconvenient especially for thin structures employed in aero-space, automotive, and marine applications to enable compactand broadband energy harvesting. Integrating piezo-patchenergy harvesters to such host plates eliminates the massloading and volumetric occupancy of base-excited cantileverdesign. Relatively limited literature has focused on energyharvesting from two-dimensional structures, which is sum-marized next. De Marqui et al [27] presented an electro-mechanical finite element model (FEM) for a piezoelectricenergy harvester embedded in a cantilever plate, and later onextended this model to airflow excitation problems by elec-troaeroelastic coupling [28, 29] for energy harvesting fromaeroelastic flutter. Rupp et al [30] performed FEM-basedtopology optimization for cantilever plates and shells tomaximize piezoelectric power output. Erturk [31] derivedanalytical formulation for energy harvesting with piezo-electric patches from surface strain fluctuations of large andhigh impedance structures through one-way coupling. Harne[32] modeled electroelastic dynamics of a vibrating panelwith corrugated piezoelectric spring. Later, he analyzed thiscorrugated harvester device by attaching it to a panel of apublic bus [33]. More recently, Aridogan et al [34] presentedanalytical closed-form expressions and experimental valida-tions of piezoelectric patch-based energy harvesters structu-rally integrated to fully clamped plates.

From the electrical domain aspects, in modeling of bothcantilever beam-based and plate-like harvesters, usually asimple resistive load is employed to estimate AC poweroutput [34, 35]. However, practical energy harvesters requiremore complex interface circuits to provide a stable DC signalfor storage devices. A standard realistic circuit includes arectifier bridge followed by a smoothing capacitor as an AC–DC converter [36–38] and further signal regulation can beperformed. For instance, Ottman et al [37] suggested using aDC–DC converter after the AC–DC converter to maximizepower transfer to the storage device through impedancematching. Synchronized switch harvesting on inductor (SSHI)technique was proposed for increasing the power output inweakly coupled energy harvesters [38–40].

Simulation of piezoelectric energy harvesters with non-linear circuit elements (e.g. AC–DC converter and/or SSHIcircuits) requires implementing equivalent circuit models

(ECM) in circuit simulation software (e.g. SPICE) for single/multiple modes of vibration. When equivalent circuits forweakly coupled systems are built, the electromechanicalback-coupling can be ignored and the equivalent circuit canbe modeled only with the mechanical response being used asan input (i.e. driving voltage or current). However, it isknown that [41] ignoring electromechanical back-coupling,results in over estimation of generated power for stronglycoupled systems. Therefore, accurate modeling of stronglycoupled systems requires inclusion of electromechanicalback-coupling in the equivalent circuit. Yang and Tang [19]and Elvin and Elvin [16] developed equivalent circuitsincluding multiple-DOFs and electromechanical back-cou-pling. Elvin and Elvin [16] obtained equivalent circuit para-meters from the Rayleigh–Ritz method while Yang and Tang[19] used the analogy between second-order circuit equationand electromechanically coupled modal equation from thedistributed-parameter model of Erturk and Inman [14]. Elvinand Elvin [18] also developed a coupled FEM-SPICE modeland Yang and Tang [19] extracted equivalent circuit para-meters from FEM results. These studies focused on the con-ventional cantilever configuration for energy harvesting frombase excitation.

As mentioned previously, piezoelectric patch-basedenergy harvesters attached to thin plates can be readily inte-grated to plate-like structures in automotive, marine, andaerospace applications, in order to directly exploit structuralvibration modes of the host system. This approach eliminatesthe mass loading and volumetric occupancy of cantileverattachments. In the following, a multi-mode3 ECM of a piezo-patch energy harvester integrated to a thin plate is developedand coupled with a standard AC–DC conversion circuit.Equivalent circuit parameters are obtained from the modalanalysis solution of a distributed-parameter analytical model,and alternatively, from the finite-element numerical model ofthe harvester, by accounting for two-way coupling in bothcases. Both AC–AC and AC–DC problems are exploredanalytically and numerically. Specifically, an analytical sin-gle-mode model for the plate is coupled with the AC–DCconversion model of Shu and Lien [36]. Simulations of theanalytical model are compared with SPICE simulations.

2. Analytical model

In this section, a brief description of the distributed-parameterelectroelastic model of a thin plate with a piezoelectric patchharvester is presented based on the Kirchhoff plate theory[34]. The host plate with all four edges clamped (CCCC)boundary conditions and the structurally integrated piezo-patch harvester are shown in figure 1.

The host plate is excited by a transverse point force f(t)acting at the position coordinates (x0, y0). The plate isassumed to be sufficiently thin so that the effects of transverseshear deformation and rotary inertia are neglected based on

3 The term ‘multi-mode’ in the present context refers to including multiplevibration modes of the host structure plate.

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Smart Mater. Struct. 25 (2016) 055015 B Bayik et al

the Kirchhoff plate theory. A piezo-patch with a length of lpand width of wp covers a rectangular region with two cornersat (x1, y1) and (x2, y2). The length and the width of the hostplate are a and b, respectively. Thicknesses of the host plateand the piezo-patch are hs and hp, respectively. A resistiveload (Rl) is considered as an external electrical load connectedto the perfectly conductive electrode layers of negligiblethickness covering the top and bottom surfaces of the piezo-patch. It is assumed that the piezoelectric volume is muchsmaller than the host structure so the patch is coupled to thehost only electromechanically (with negligible mass andstiffness contribution) and it exhibits piezoelectricallyinduced moments at the electrode boundaries as the back-coupling effect [42].

The electromechanically coupled equation governing thetransverse vibration of the host plate with piezo-patch can bewritten as

r

qd d

d d

d d

¶¶

+¶

¶ ¶+

¶¶

+¶

¶+

¶¶

--

--

´ - - -

+-

--

´ - - -

= - -

⎛⎝⎜

⎞⎠⎟

⎧⎨⎩⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

Dw x y t

x

w x y t

x y

w x y t

y

cw x y t

th

w x y t

t

v tx x

x

x x

x

H y y H y y

y y

y

y y

y

H x x H x x

f t x x y y

, ,2

, , , ,

, , , ,

d

d

d

d

d

d

d

d

, 1

4

4

4

2 2

4

4

s s

2

2

1 2

1 2

1 2

1 2

0 0

}

[ ]

[ ]

( ) ( ) ( )

( ) ( )

( )( ) ( )

( ) ( )

( ) ( )

( ) ( )( ) ( ) ( ) ( )

where w(x, y, t) is transverse deflection of the plate. Thebending stiffness of the thin plate is n= -/D Y h 12 12 ,s s

3s

2( )where Ys and ns are its Young’s modulus and Poisson’s ratio.The mass density of the plate is ρs while c is the coefficient ofviscous damping. The Dirac delta functions are δ(x) and δ(y)along the x and y directions. H(x) and H(y) are the Heavisidefunctions and v t( ) is the electrode voltage across the externalresistive load R .l The electromechanical term θ is defined asq = e h ,31 pc¯ which is the product of the effective plane-stresspiezoelectric constant e31¯ and reference distance hpc of the

center layer of the piezo-patch from the reference surface (i.e.the neutral surface level of the plate in the piezo region) at thelocation of the patch. The governing equation of the coupledcircuit dynamics can be expressed as

ò òq+ +¶

¶ ¶

+¶

¶ ¶=

= =

⎪

⎪

⎧⎨⎩

⎡⎣⎢

⎤⎦⎥

Cv t

t

v t

R

w x y t

x t

w x y t

y tx y

d

d

, ,

, ,d d

0, 2

y y

y

x x

x

pl

3

2

3

2

1

2

1

2( ) ( ) ( )

( )

( )

where Cp is the capacitance of the piezo-patch and defined as

e= /C w l h .Sp 33 p p p( )¯ Therefore, equations (1) and (2) con-

stitute the distributed-parameter electroelastic model of piezo-patch harvester in physical coordinates. By conducting themodal analysis procedure for the two-dimensional structure[34], the electromechanically coupled ordinary differentialequations of the plate in modal coordinates can be written as

hz w

hw h

q

+ +

- =

t

t

t

tt

v t f t

d

d2

d

d

, 3

mnmn mn

mnmn mn

mn mn

2

22( ) ( )

( )

˜ ( ) ( ) ( )

å å qh

+ + ==

¥

=

¥

Cv t

t

v t

R

t

t

d

d

d

d0. 4

n mmn

mnp

l 1 1

( ) ( ) ˜ ( )( )

Here, wmn is the undamped short-circuit natural fre-quency for the mnth vibration mode and zmn is the respectivemodal damping ratio. Note that the modal forcing inequation (3) is

ò ò d d j

j

= - -

´ =

f f t x x y y x y

x y f t x y

,

d d , 5

mn

b a

mn

mn

0 00 0

0 0

( ) ( ) ( ) ( )

( ) ( ) ( )

while the electromechanical coupling term qmn˜ can be given

by

ò

ò

q qj

j

=¶

¶

+¶

¶

⎡⎣⎢⎢

⎤

⎦⎥⎥

x y

x

x y

yx

,dy

,d . 6

mny

ymn

x

x

x

xmn

y

y

1

2

1

2

1

2

1

2

˜ ( )

( )( )

3. Finite-element model

A commercial FEM package (ANSYS) is used for system-level simulations where the voltage output across the elec-trical load is calculated under the effect of point mechanicalexcitation. The model of the host structure, the piezo-patch,and the resistive load is shown in figure 2. The host plate ismeshed with 20-node structural solid elements (SOLID186)and the piezo-patch is meshed with 20-node coupled fieldsolid elements (SOLID226). The piezo-patch and the hoststructure are bonded together via CEINTF command in

Figure 1. Piezoelectric energy harvesting using a rectangular patchstructurally integrated to a thin plate that is under transverse pointforce excitation.

3

Smart Mater. Struct. 25 (2016) 055015 B Bayik et al

ANSYS, which generates constraint equations relating thenodes at the bonding interfaces. To simulate the highly con-ductive top and bottom electrode layers of the harvester patch,the electric potential degrees of freedom on the bottom andtop surfaces are separately forced to be equipotential in orderto have a single voltage output. CIRCU94 piezoelectric cir-cuit element is used for simulating the resistive load con-nected to the piezo-patch. One end of the piezoelectric circuitelement is connected to the top electrode and the other end isconnected to the grounded bottom in this model with two-wayelectromechanical coupling.

4. Equivalent circuit model

4.1. Equivalent circuit with parameters from the analyticalmodel

Although the analytical distributed-parameter model of theplate is capable of predicting the power output of the systemaccurately, its practical use is limited with linear circuitcomponents such as a simple resistive load as discussed insection 3. When a resistive load is considered for the ACinput–AC output problem, then the analytical distributed-parameter model is very efficient and accurate. However, inthe presence of nonlinear circuit components, such as diodes,another approach is required to predict the system response.To this end, an ECM of a piezo-patch harvester integrated to athin plate is presented and analyzed. This approach is basedon the superposition of infinite number of lumped-parametermodels which represent the vibration modes. For instance,figure 3 shows the ECM of the piezoelectric energy harvesterintegrated to a thin plate for the case of simple resistiveloading for AC power generation (the case of figure 2). Eachvibration mode of the harvester is represented as a second-order circuit connected to a piezo-patch capacitance (CP) anda resistive load (Rl) through an ideal transformer. By applyingKirchhoff’s voltage law to the multi-vibration mode circuitand by analogy with equation (3), system parameters of the

ECM can be determined. Table 1 gives a summary of theanalogy between the analytical distributed-parameter expres-sion and the ECM.

Note that, in figure 3, ip(t) is the dependent current sourcewhich is the sum of all currents in the branches connected inparallel to piezoelectric capacitance and resistive load.Applying Kirchhoff’s current law and by analogy withequation (4), the current flowing to piezoelectric capacitanceand the current source can be written as follows:

åånq

h= = -

=

¥

=

¥

i Ct

ti

t

t

d

d,

d

d. 7

n mmn

mnc p p

1 1

( ) ˜ ( )( )

4.2. Equivalent circuit with parameters from the finite-elementmodel

Parameters of the ECM are extracted from FEM results fol-lowing the procedure described by Yang and Tang [19] forone-dimensional cantilevers. The first step is to determine thestatic capacitance Cs and perform modal analysis to obtainshort-circuit natural frequencies w .mn The second step is toobtain the total admittance of the piezoelectric energy har-vester from FEM results with harmonic voltage input. Thetotal admittance is calculated by summing the dampedadmittance Yd with the motion admittance Ymot:

= +Y Y Y . 8d mot ( )

Therefore Ymot can be obtained by excluding Yd from thecalculated total admittance for each mode which can bedirectly extracted from FEM results as

w=Y YIm . 9mnd [ ]( ) ( )

Having determined Ymot for each vibration mode, C ,mnmot

L ,mnmot Rmn

mot and Nmn can be identified as:

w w

w

=

=-

=

=

⎡⎣ ⎤⎦

⎡⎣ ⎡⎣ ⎤⎦⎤⎦ ⎡⎣ ⎡⎣ ⎤⎦⎤⎦

RY

LR

Y Y

CL

NL

1

max Re,

min Im max Im,

1,

1,

10

mn

mnmn

mnmn mn

mnmn

mot

mot

motmot

mot mot

mot2 mot

mot

( )

( ) ( )

( )

where C ,mnmot Lmn

mot and Rmnmot represent the converted form of

C ,mn Lmn and Rmn using the following relations:

= = =LL

NR

R

NC C N . 11mn

mn

mnmn

mn

mnmn mn mn

mot2

mot2

mot 2 ( )

Figure 2. Finite-element model (in ANSYS) used for system-levelnumerical simulation of a piezo-patch integrated to a fully clampedthin plate and shunted to a resistive load.

4

Smart Mater. Struct. 25 (2016) 055015 B Bayik et al

Having determined Nmn and R ,mn the last step is todetermine Vmn using the following equation

w w=V N R j Q j , 12mn mn mn mn mn( ) ( )

where wQ j mn( ) is the charge response at each naturalfrequency wmn under short-circuit condition.

5. Model validation

In this section, ECM of a fully clamped plate with a perfectlybonded piezo-patch is constructed in SPICE software andsimulation results are validated for the standard AC–AC andAC–DC problems. ECM results for AC configuration arevalidated using the distributed parameter model given byAridogan et al [34] and also using the system-level FEMresults. A single-mode analytical AC–DC model is also

Figure 3. Equivalent circuit representation of a piezo-patch harvester shunted to a resistive load for AC power generation by accounting formultiple vibration modes (a total of ´m n modes).

Table 1. Analogy between electrical and mechanical domains of apiezo-patch harvester integrated to a thin plate.

Equivalent circuit parameters for themnth vibration mode Mechanical counterparts

Voltage source: V tmn ( ) Modal pointforce: f tmn ( )

Inductance: L tmn ( ) 1Resistance: R tmn ( ) z w2 mn mn

Capacitance: C tmn ( ) w1 mn2( )

Ideal transformer ratio: Nmn q- mn˜

Electrical charge: q tmn ( ) Modal timeresponse: h tmn ( )

Electrical current: i tmn ( ) h t td dmn ( )

Table 2. Geometric and material properties.

Aluminum Piezoceramic

Length (mm) 580 72.4Width (mm) 540 72.4Thickness (mm) 1.96 0.267Elastic modulus (GPa) 70 69Mass density (kg m−3) 2700 7800Piezoelectric constant e31 (C m−2) — −190Permittivity constant e S

33¯ (nF m−1) — 9.57Damping ratio zmn 0.01 0.01

5

Smart Mater. Struct. 25 (2016) 055015 B Bayik et al

obtained following Shu and Lien [36], and compared withnumerical simulations. The geometric and material propertiesof the aluminum plate and the piezo-patch are given in table 2.

5.1. Standard AC–AC problem

In the standard AC input–AC output problem, the energyharvesting circuit is represented by a resistor connected to thepiezo-patch as shown in figure 1. Parameters of the multi-mode ECM are identified from both the FE model and theanalytical model considering the first nine vibration modes ofthe plate [34]. Steady-state voltage frequency responsefunctions (FRFs) of the piezo-patch harvester under the effectof a harmonic point force are shown in figures 4 and 5 forvarious resistive load values (ranging from 100W to 1MW)over the frequency range of 0–200 Hz.

Figure 4 shows that voltage output FRFs of the analyticalmodel match perfectly with ECM (simulated in SPICE) fordifferent resistive load cases when parameters are identifiedfrom the analytical model. Furthermore, in figure 5, it isobserved that voltage amplitudes obtained from the system-levelFEM results are in good agreement with those obtained fromanalytical model with a slight shift in natural frequencies. Slightmismatch is potentially due to ignoring the added mass andstiffness effects of the piezo-patch in the analytical model. Thesystem-level FEM results perfectly agree with the ones obtainedfrom the ECM with parameters identified from the FE model.

Single-mode expressions of the ECM are also comparedwith the analytical multi-mode solution in the vicinity of thefirst four modes as shown in figure 6. It is observed that thesingle-mode results of the ECM are in good agreement withthe analytical multi-mode results in the neighborhood of the

Figure 4. Voltage FRFs of the piezo-patch harvester for a set of resistive loads using the modal analysis-based analytical model and theequivalent circuit model with parameters identified from analytical model.

Figure 5. Voltage FRFs of the piezo-patch harvester for a set of resistive loads using the modal analysis-based analytical model, theequivalent circuit model with parameters identified from FEM, and the system-level FEM.

6

Smart Mater. Struct. 25 (2016) 055015 B Bayik et al

resonance peaks. Since the neighboring modes are notincluded, as the excitation frequency deviates from the reso-nance condition, the single-mode solution fails to be reliableand accurate. Having validated the analytical (modal analysis)results versus numerical (FEM) results through figures 4–6(note that the analytical model was experimentally validatedpreviously [34]), next the AC–DC conversion problem isexplored by adding the standard ideal rectification circuit.

5.2. Standard AC–DC problem

Figure 7 illustrates the standard AC–DC piezoelectric energyharvesting circuit consisting of a full-wave rectifier, a

smoothing capacitor, and a resistive load. The lumped-para-meter AC–DC model of Shu and Lien considers single modeof the harvester [36] since it deals with the cantilever problem.For that reason, single-mode results of the ECM are comparedwith the analytical multi-mode solution in the vicinity of thefirst four modes in a similar way to figure 6. Having validatedthe single-mode AC input–AC output case (in the resonanceneighborhood) as well as the multi-mode cases, the rectifiedvoltage and power output of the ECM of the harvester are to bevalidated in the standard AC input–DC output problem. To thisend, using the lumped-parameters for each vibrational mode,AC–DC harvesting circuit is simulated in SPICE software and

Figure 6. Comparison of analytical multi-mode and SPICE-based single-mode voltage FRFs for the first four vibration modes (Rl=100 Ω).

Figure 7. Piezo-patch energy harvester connected to a standard AC–DC conversion circuit: a full-wave rectifier, a smoothing capacitor, and aresistive load.

7

Smart Mater. Struct. 25 (2016) 055015 B Bayik et al

validated with the analytical model proposed by Shu and Lien[36]. In their study, the authors developed an analytical for-mulation of the rectified DC voltage output for the piezoelectricenergy harvesters using a lumped-parameter approach. Byimplementing this model, the steady-state voltage (Vc ) for thepiezo-patch harvester can be written as:

ww p

=+

VL R

R Cu

2, 13c

mn l

l p0 ( )

where the displacement amplitude u0 is derived in terms ofthe point forcing amplitude and the system parameters as

j

ww

w p

www p

=

++

+ - ++

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎛⎝⎜

⎞⎠⎟

⎫

⎬⎪⎪⎪

⎭⎪⎪⎪

uF x y

RN R

C R

C LN R

C R

,

2

2

12

. 14mn

mnmn

mn mnmn

0

0 0 0

2l

p l2

2

22

l

p l

2

12

( )

( )( )

Here, the fundamental assumption is that the voltage loss inthe diode is negligible as compared to the piezoelectricvoltage output so the diode acts like a switch in Shu andLien’s model. In order to generate a stable DC voltageVc with

negligible ripple, the filter capacitor Ce must be large enoughso that a constant DC voltage output can be achieved [43].Ideal transformer and diodes are used in the ECM simulationsin SPICE to compare numerical results with Shu and Lien’sanalytical model [36]. Analysis of the harvesting circuit isperformed for each vibration mode in SPICE software. Byconducting time-domain analysis for the harvester circuit,steady-state voltage values are extracted for various values ofthe resistive loads. Figures 8 and 9 present the SPICEsimulation results for the resonant DC voltage and poweroutputs and compare the results with Shu and Lien’sanalytical model [36] for a wide range of resistive loads.The results are in excellent agreement when the assumptionsin the SPICE model agree with the analytical approach.Clearly the ECM can handle more complex circuits, such asnon-ideal diode behavior and other losses, as well asswitching and DC–DC regulation circuits, when used inconjunction with SPICE.

6. Conclusions

Piezoelectric patch-based energy harvesters attached to thinplates can be integrated to two-dimensional thin structures in

Figure 8. Rectified DC voltage output from Shu and Lien’s analytical model and from the equivalent circuit model.

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Smart Mater. Struct. 25 (2016) 055015 B Bayik et al

automotive, marine, and aerospace applications, in order todirectly exploit structural vibration modes of the host system.Based on this approach, mass loading and volumetric occu-pancy of base excited cantilever attachments can be elimi-nated. In this work, an ECM of a piezoelectric patch-basedharvester attached on a thin plate was developed and validatedfor the standard AC–AC and ideal AC–DC circuit config-urations. The equivalent circuit parameters considering mul-tiple vibration modes of the harvester were identified from theanalytical and FEMs. Then, using these system parameters,simulations were conducted in SPICE software. The simula-tion results of voltage FRFs in the AC–AC configuration werepresented for a wide range of resistive load cases and verifiedagainst both the analytical distributed-parameter and system-level FEMs. It was shown that ECM accurately represents thecoupled electromechanical behavior of the harvester.Equivalent circuit single-mode representation was also ver-ified with the analytical multi-mode solution in the vicinity ofvibration modes. Then, the single-mode ECM was connectedto a full-wave rectifier, a smoothing capacitor, and a resistiveload as a standard AC–DC conversion circuit. Steady-stateelectrical responses were investigated for the first two modesof the harvester. Resonant modal (i.e. single-mode) DCvoltage and power outputs for different resistive loads werecomputed and validated against the existing analytical single-

degree-of-freedom model proposed by Shu and Lien [36].The multi-mode ECM developed in this study enablesexploring the harvesting performance of the piezoelectricpatch-based harvesters attached to thin plates with any linearand nonlinear electrical components, including non-idealcircuit components as well as DC–DC regulation circuitswhen used along with software such as SPICE for system-level design and optimization.

Acknowledgments

The authors acknowledge support from the Koc UniversityTUPRAS Energy Center (KUTEM).

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