Original Article

Journal of Intelligent Material Systemsand Structures2019, Vol. 30(2) 213–227� The Author(s) 2018Article reuse guidelines:sagepub.com/journals-permissionsDOI: 10.1177/1045389X18808390journals.sagepub.com/home/jim

Electrical power management andoptimization with nonlinear energyharvesting structures

Wen Cai and Ryan L Harne

AbstractIn recent years, great advances in understanding the opportunities for nonlinear vibration energy harvesting systemshave been achieved giving attention to either the structural or electrical subsystems. Yet, a notable disconnect appears inthe knowledge on optimal means to integrate nonlinear energy harvesting structures with effective nonlinear rectifyingand power management circuits for practical applications. Motivated to fill this knowledge gap, this research employsimpedance principles to investigate power optimization strategies for a nonlinear vibration energy harvester interfacedwith a bridge rectifier and a buck-boost converter. The frequency and amplitude dependence of the internal impedanceof the harvester structure challenges the conventional impedance matching concepts. Instead, a system-level optimiza-tion strategy is established and validated through simulations and experiments. Through careful studies, the means tooptimize the electrical power with partial information of the electrical load is revealed and verified in comparison to thefull analysis. These results suggest that future study and implementation of optimal nonlinear energy harvesting systemsmay find effective guidance through power flow concepts built on linear theories despite the presence of nonlinearitiesin structures and circuits.

KeywordsNonlinear energy harvesting, DC–DC converter, impedance analysis, optimal power delivery

Introduction

The possibilities for wireless, self-sufficient electricalsupplies to advance myriad applications in an emerging‘‘Internet of things’’ are recently well established (Beebyet al., 2006; Mitcheson et al., 2008). For instance, self-sufficient power resources would propel new and futurepractices of structural health and condition monitoring,wildlife tracking, and precision agriculture (Beeby et al.,2006; Jawad et al., 2017; Shafer et al., 2015). In addi-tion, achievements in energy-efficient circuit design(Chandrakasan et al., 1998; Davis et al., 2002) suggestthat the possibilities for a transformative ‘‘energy har-vesting’’ infrastructure are closer at hand than ever.

Among all kinds of ambient energy sources, vibra-tional energy is a desirable choice to harvest for theabundance and persistence of kinetic energy in applica-tions. Because many energy harvesting devices are com-posed from structural oscillator designs, the narrowfrequency of resonance for linear harvester systems isunfavorable for robustness in electrical power delivery.As a result, a wide variety of methods have been intro-duced to increase the frequency range of effectiveenergy capture in vibration energy harvesters. For

example, self-tuning (Roundy and Zhang, 2005), har-vester arrays (Xiao et al., 2014), mechanical stoppers(Wu et al., 2014), nonlinearity (Gu and Livermore,2011; Harne and Wang, 2013; Sebald et al., 2011), andother concepts have been investigated. Bistable nonli-nearities have attracted broad attention due to the non-resonant nature of oscillation that is less susceptible toperformance deterioration in low-level vibration envir-onments (Erturk et al., 2009; Ferrari et al., 2010; Harneet al., 2013; Scarselli et al., 2016). A bistable oscillatorhas two statically stable equilibria, which makes it pos-sible to exhibit three classes of vibration in consequenceto single-frequency excitation: snap-through, quasi-lin-ear, and chaotic vibration (Harne and Wang, 2013).The non-resonant nature of snap-through enables it to

Department of Mechanical and Aerospace Engineering, The Ohio State

University, Columbus, OH, USA

Corresponding author:

Ryan L Harne, Department of Mechanical and Aerospace Engineering,

The Ohio State University, 201 W. 19th Avenue, Columbus, OH 43210,

USA.

Email: harne.3@osu.edu

be triggered in a diverse range of excitation environ-ments, especially at low frequencies characteristic ofpractical ambient vibrations (Cottone et al., 2009). Inaddition, snap-through may yield large velocities ofharvester motion that corresponds to high outputpower. For example, Erturk et al. (2009) demonstrateda 200% increase in the open-circuit voltage amplitudewith harmonic excitation by leveraging snap-throughvibrations of a bistable energy harvester, while Ferrariet al. (2010) found an 80% increase in the root meansquare (RMS) voltage with white noise excitation.

Since microelectronic devices require a regulateddirect current (DC) input voltage, a rectifying circuit isnecessary for energy harvesters to convert the oscillat-ing alternating current (AC) signals to DC power.There are two classes of rectification: passive and activerectification (Szarka et al., 2012). Comparing with thepassive methods, active methods may improve thepower conversion efficiency at low power levels (Leet al., 2006). On the other hand, active rectificationmay require careful design to overcome challenges suchas ‘‘cold start’’ (Lam et al., 2006). Therefore, despitethe possible improvements from active rectificationmethods, passive rectification such as full-wave diodebridges are widely employed for rectification needs invibration energy harvesting investigations. To improvepower capture, such rectifiers may be modified viaswitching strategies. For instance, the parallel synchro-nized switching harvesting on inductor (SSHI) pre-sented by Guyomar et al. (2005) adds an inductor anda switch in parallel with the linear piezoelectric elementelectrodes to the input of the rectifier bridge so as toenhance power potentially by 900% according to theswitch-controlled current flow (Guyomar et al., 2005).Other types of nonlinear circuit like series-SSHI andoptimized series-SSHI may improve the output poweras well (Garbuio et al., 2009; Lefeuvre et al., 2006;Makihara et al., 2006).

Yet, for a selected circuit, the optimal output poweris only achieved when the load impedance matches thesource impedance. Liang and Liao (2012) characterizedsuch influences for a linear piezoelectric energy har-vester to reveal optimal working condition for a varietyof nonlinear rectification circuits (Liang and Liao,2012). On the other hand, rectified voltages from piezo-electric energy harvesters may significantly exceed therequired operational voltages of rechargeable batteriesand microelectronics. As such, a DC–DC converter isintroduced after the piezoelectric voltage rectificationstage to provide the needed power management(Ottman et al., 2002). Lefeuvre et al. (2007) present theroles of a buck-boost converter working in the discon-tinuous current mode (DCM), revealing that the non-linear converter appears as a resistive load impedancewith respect to the linear piezoelectric energy harvesterplatform regardless of the real load resistance bearingthe DC power. These findings demonstrate that the

nonlinear rectification and power management are cru-cial to understand in order to effectively implement lin-ear piezoelectric energy harvesters in the conventionalresonant vibration mode.

Considering the state of the art, nonlinearities in thevibration energy harvesting structure or nonlinearitiesin the rectification and power management stages maybe leveraged for effective DC power delivery. On theother hand, understanding of the suitable integration ofthese subsystems is lacking so that a disconnect ofknowledge exists on the system-level harvester imple-mentation. In order to bridge the gap in the under-standing of optimal use of nonlinear energy harvesterswith nonlinear rectification and power management cir-cuits, this research takes a first step forward to studythe structural–electrical interactions manifest in a non-linear energy harvester coupled with a passive diodebridge rectifier and a buck-boost DC–DC converter. Byharnessing the principles of impedance, new insights arerevealed on practical strategies for system optimization.

This report is organized as follows. The next sectionstudies the internal impedance of the nonlinear har-vester structural platform. Then, rectification andbuck-boost converter stages are taken into detailedconsideration. Following validation efforts, the impe-dance changes observed at the system level are revealedand the optimal working conditions for a buck-boostDC–DC converter integrated with the nonlinear har-vester are uncovered through a rigorous impedance-based analysis. Finally, a summary of key insights fromthis work concludes the report.

Source/load modeling of nonlinear energyharvesting system

Energy harvester platform

The structural platform of the experimental energy har-vesting system considered in this work is shown inFigure 1(a). A piezoelectric cantilever (PPA-2014; MideTechnology) has the clamped end affixed to an electro-dynamic shaker (APS Dynamics 400). The shaker isdriven by a controller (Vibration Research ControllerVR9500) and amplifier (Crown XLS 2500). Laser dis-placement sensors (Micro-Epsilon ILD-1420) are usedto measure the absolute displacements of the beam tipand shaker table. An accelerometer (PCB Piezotronics333B40) is used to provide feedback for the shaker con-troller. The nonlinearity exploited in this research is abistability realized by magnetoelastic effects.Specifically, at the free end of the piezoelectric cantile-ver a steel extension is added to decrease the lowestorder natural frequency of the beam and introducenonlinearity by magnetic forces from an adjacent pairof neodymium magnets. The total mass of the extensionis 9 g. The length L of the cantilever beam and steelextension is 60.8 mm. With careful adjustment of the

214 Journal of Intelligent Material Systems and Structures 30(2)

distances d and D shown in Figure 1(b), the bistablenonlinearity is realized (Erturk and Inman, 2011; Feenyet al., 2001; Hikihara and Kawagoshi, 1996; Moon andHolmes, 1979). Based on the survey of state-of-the-artdevelopments in section ‘‘Introduction,’’ the advance-ments provided by a bistable vibration energy harvesterare associated with the snap-through behavior.Therefore, in the following investigations the studyplaces attention on scrutinizing the DC power deliveryfrom the snap-through response.

Governing equations of motion for the energyharvesting system

In this work, the frequencies of base acceleration arearound the lowest order linear natural frequency of thepiezoelectric cantilever, so that the deflection of thecantilever is primarily in the first vibration mode(Erturk and Inman, 2008). Therefore, a single-degree-of-freedom model is adopted to study the nonlinear

energy harvesting system. The governing equations areexpressed as

m€x+ d _x+ k1 1� pð Þx+ k3x3 +avp = � m€z ð1aÞ

Cp _vp + ip =a _x ð1bÞ

where x is the beam tip displacement relative to themotion of the base displacement z; m, d, k1, and k3 arethe equivalent lumped mass, viscous damping, linearstiffness, and nonlinear stiffness corresponding to thefirst vibration mode, respectively; p is the load para-meter, which is used to indicate the influence of mag-netic forces on reducing the linear stiffness; a is theelectromechanical coupling constant; Cp is the internalcapacitance of the piezoelectric beam; vp is the voltageacross the piezoelectric beam electrodes; and ip is thecorresponding current that passes into the harvestingcircuit impedance ZL as exemplified in Figure 1(b). Theoverdot operator indicates differentiation with respectto time t.

Studies have shown the analogy between mechanicaland electrical systems in the energy harvesting literature(Kong et al., 2010; Liang and Liao, 2012; Yang andTang, 2009). Here, such concept is employed to identifyan equivalent electrical system shown in Figure 2(a)that represents the vibration energy harvesting system.The electromechanical coupling constant a is inter-preted as a transformer turn ratio. In Figure 2(a), vs1

corresponds to the base acceleration via vs1 =� m€z.The inductance Ls1, resistance Rs1, and capacitance Cs1

in the circuit, respectively, relate to the mass Ls1 =m,viscous damping Rs1 = d, and compliance Cs1 = 1=k1.For the nonlinearity in the energy harvesting systemcharacterized by the negative linear stiffness �pk1 and

Figure 1. (a) Photograph and (b) schematic of theexperimental setup.

Figure 2. (a) Equivalent circuit of a base-excited nonlinearenergy harvesting system and (b) equivalent circuit for theinternal source impedance determination.

Cai and Harne 215

nonlinear stiffness k3, a general impedance NL block isshown in Figure 2(b) that will be clearly defined in thesubsequent modeling of this work. In the followingstudy, the components in the red dashed box are collec-tively considered to be the source impedance Zs of thenonlinear energy harvesting system. The currentthrough this source impedance is related to the relativevelocity of the beam tip via i=a _x.

Experimental system identification

To identify the parameters of the nonlinear cantilever,an impulsive ring-down test is first conducted to ensurethe symmetry of the two static equilibria via identicallinear natural frequencies of oscillation around eachstable equilibrium. The equivalent lumped mass m andthe linear stiffness k1 are calculated by classical rela-tions (Erturk and Inman, 2011). After determining thenatural frequency vn from the ring-down experimentsand measuring the static equilibria x�, the load para-meter p and nonlinear stiffness k3 are found by

p= 1+v2

nm

2k1

, k3 =k1 p� 1ð Þ

x�ð Þ2ð2Þ

The identified parameters of the nonlinear energyharvester platform are shown in Table 1.

Source impedance characterization

The original source vs1 in Figure 2(a) is removed tostudy the source impedance Zs of the harvester itself.Instead, an AC voltage v is connected to the output ter-minal, which is referred to as the driven voltage shownin Figure 2(b). Then, the governing equation is

m€x+ d _x+ k1 1� pð Þx+ k3x3 =av ð3Þ

The driven voltage is

v= � V cos vtð Þ ð4Þ

The governing equation (3) after non-dimensionalization is

x00+hx0+ 1� pð Þx+bx3 = kv ð5Þ

The corresponding non-dimensional parameters aredefined as follows

x=x

x0

, V =V

V0

, t =v0t

b=k3x2

0

k1

, k=aV0

k1x0

v0 =

ffiffiffiffiffik1

m

r, v=

v

v0

, h=d

mv0

ð6Þ

where x0 and V0 are the characteristic length and vol-tage, respectively, that are defined to be 1 mm and 1 V,respectively. These selections are made to keep the non-dimensional response values around the order of 1.

Using the principles of harmonic balance (Harneand Wang, 2017), the solution to equation (5) may beapproximated as

x tð Þ= c tð Þ+ a tð Þ sin vtð Þ+ b tð Þ cos vtð Þ ð7Þ

The coefficients of the constant, sine, and cosineterms are collected after substituting equation (7) intoequation (5). Since only the fundamental frequency ofvibration is considered in equation (7), the higher orderharmonics are neglected. In addition, the coefficientsare assumed to vary slowly in non-dimensional time.The resulting system of equations collecting togetherconstant and sinusoidal terms is

� hc0= c 1� p+3

2br2 +bc2

� �ð8aÞ

� ha0+ 2vb0= aL� bX ð8bÞ

� 2va0 � hb0= aX + bL+Vk ð8cÞ

The following terms are identified

r2 = a2 + b2, L= 1� p� v2 +3

4br2 + 3bc2, X =hv

ð9Þ

The steady-state behavior of the nonlinear equiva-lent source is determined by first solving equation (8a)to yield

c= 0 or c2 =p� 1

b� 3

2r2 ð10Þ

Here, the snap-through behavior is related to c= 0

so that the displacements of the piezoelectric cantileverundergo zero-mean oscillations. Therefore, only c= 0

is considered in the following evaluations.Then, combining equations (8b) and (8c), the snap-

through response is studied through solutions to theroots of the polynomial (equation (11))

L2 +X2� �

r2 = Vkð Þ2 ð11Þ

Table 1. Experimentally identified system parameters.

m (g) d (N s/m) k1 (N/m) p (dim)

9.45 0.194 722 1.27

k3 (MN/m3) Cp (nF) a (mN/V)

86 88 1.4

216 Journal of Intelligent Material Systems and Structures 30(2)

Equation (11) is a cubic polynomial in terms of r2.Complex and negative solutions are not physicallymeaningful. Stability of the roots is assessed accordingto the eigenvalues of the associated Jacobian fromequation (8) (Harne and Wang, 2017).

For the circuit shown in Figure 2(b), the voltage andcurrent are expressed in a complex exponential notationto characterize the source impedance Zs

Zs =v

i=�Vejvt

a _x=�V0

ax0v0

Vejvt

vr � ej vt�uið Þ =�V0Veui

ax0v0vr

ð12Þ

where

tanui =�b

að13Þ

Figure 3 presents the source impedance results thatcharacterize the impedance change for a change in thedriven voltage. The experimentally identified systemparameters from Table 1 are used to generate Figure 3using the aforementioned model. The fixed amplitudeof the driven voltage for Figure 3(a) is 51 V and thefixed driven voltage frequency for Figure 3(b) is 22 Hz.Since attention is only on the snap-through behavior,the source impedance does not exist in the whole rangeof the driven voltage frequency and amplitude forFigure 3. In addition, it is observed that the reactance isfrequency and amplitude dependent. Such dependenciesare evidence of the cubic nonlinearity and the presenceof negative linear stiffness via the bistable nonlinearity.

In the steady-state analysis, only the fundamentalfrequency vibration is considered. Therefore, a linear-ized version of the system (5) is

x00+hx0+ 1� p+3b

4r2

� �x= kv ð14Þ

From equation (14), the roles that the nonlinear andnegative stiffness terms play to culminate in the systembehaviors become clear. The terms modeled as the gen-eric NL impedance in Figure 2 are therefore analogousto a capacitor that has a response-dependent capaci-tance value. This also explains the amplitude-dependentsource reactance in Figure 3. In contrast, the resistancein Figure 3 is independent of the frequency and ampli-tude of the driven voltage, due to the nature of the vis-cous damping–based resistance Rs1 = d in Figure 2.

For any harmonically driven electrical system, opti-mal power delivery to a load is achieved when the loadimpedance is the complex conjugate of the source impe-dance. A suboptimal approach is resistive impedancematching that may be relevant for vibration energy har-vesting systems (Kong et al., 2010). Figure 4(a) shows asimplified circuit schematic for a vibration energy har-vesting system. The source voltage amplitude is V s,characteristic of the base acceleration. The correspond-ing voltages across the source impedance and loadimpedance are, respectively, V 1 and V 2. Figure 4(b) pre-sents these voltages in the complex plane. ByKirchhoff’s law, the combination of V 1 and V 2 is equal

Figure 3. (a) Driven voltage frequency and (b) amplitude influences on the source impedance.

Figure 4. (a) The simplified circuit schematic for a nonlinearharvesting system and (b) voltage shown in the complex plane.

Cai and Harne 217

to V s. For the nonlinear system examined here, if theload impedance is changed to match the source impe-dance, the voltage across the load impedance is changeddue to the amplitude-dependent nature of the system.This correspondingly changes V 1. As a result, a stan-dard practice of impedance matching is significantlychallenged. Therefore, in this research, the investigationof optimal power delivery from the nonlinear energyharvesting system with power management circuitryrequires studious consideration of the platform–circuitsystem in operation, rather than sole attention to onesubcomponent.

In the following section, a rectifier and a buck-boostconverter are introduced to the load impedance modelof Figure 4(a) to examine the system-level behaviorsand optimal DC power delivery conditions.

Analytical model formulation and solutionfor the nonlinear energy harvestingsystem with DC power management

System model and approximate solution to thegoverning equations

The governing equations of motion (equation (1)) arenon-dimensionalized to be

x00+hx0+ 1� pð Þx+bx3 + kvp = � z00 ð15aÞ

v0p + ip = ux0 ð15bÞ

The non-dimensional base acceleration is

� z00= a cosvt ð16Þ

where a is the normalized base excitation of a.Additional non-dimensional parameters are defined

u=ax0=ðCpV0Þ, a=am

k1x0

, Ip = Ip=ðCpV0v0Þ, Vp =V p

V0

ð17Þ

The capital terms Ip, Ip, Vp, and V p are the ampli-tudes of the corresponding variables ip, ip, vp, and vp.

Assuming that the nonlinear energy harvesterresponds with oscillatory displacements with the samefrequency as the base acceleration, the non-dimensionaldisplacement is given by

x tð Þ= k tð Þ+ h tð Þ sin vtð Þ+ g tð Þ cos vtð Þ ð18Þ

Prior to having means to approximately solve forthe system response, the characteristics of the non-dimensional voltage vp and current i must be identified.

Figure 5 presents the interface circuit to which thenonlinear piezoelectric cantilever is attached. The cir-cuit includes a rectifying bridge, smoothing capacitorC1, and a buck-boost converter. The components in thegreen dashed box constitute the buck-boost converter

that modulates the voltage across the load R so as topermit practical implementation of the DC power.Such power management is required because piezoelec-tric energy harvesters may generate high rectified vol-tages (.20 V), whereas many rechargeable batteries ormicroelectronics use low, standardized voltage levelssuch as 3 and 9 V. The inductor, capacitor, resistor,and diode in the buck-boost converter are labeled L,C2, R, and D, respectively. The n-type metal-oxide-semiconductor field-effect transistor (MOSFET) MN inthe converter acts like a switch. A pulse width modula-tion (PWM) signal in red is sent to the gate of theMOSFET to control the open state.

In general, a buck-boost converter may run in twomodes: DCM and continuous current mode (CCM).Compared to CCM, the switching losses are much lessfor DCM (Szarka et al., 2012), so that attention in thisresearch is solely on the DCM operation. A buck-boostconverter running in DCM will realize an impedance ana-logous to an equivalent resistance (Lefeuvre et al., 2007)

Req =2Lfs

D2ð19Þ

where L represents the inductance in the circuit; theswitching frequency fs and the duty cycle D are proper-ties of the PWM signal. Since the behavior of the con-verter is not influenced by the real load R in Figure 5,the optimal working condition for a buck-boost con-verter will not change for different loads R. Therefore,the DCM operation is the preferred choice for energyharvesting applications.

Since the buck-boost converter in DCM can bereplaced by an equivalent resistance Req (equation(19)), the interface circuit is reduced to a rectifier andan ReqC1 circuit. Based on previous studies (Dai andHarne, 2017; Liang and Liao, 2012), the non-dimensional voltage vp across the piezoelectric beamcapacitance Cp is

vp =�g tð Þ

pu sin2 Y+

h tð Þ2p

2Y� sin 2Yð Þ� �

sinvt

+h tð Þ

pu sin2 Y+

g tð Þu2p

2Y� sin 2Yð Þ� �

cosvt

ð20Þ

Figure 5. The nonlinear vibration energy harvester with abuck-boost converter.

218 Journal of Intelligent Material Systems and Structures 30(2)

In equation (20), only the fundamental harmonic ofvoltage is considered on the basis that the fundamentalcontributes a significant proportion of the Fourierseries reconstruction of the actual voltage signal (Daiand Harne, 2017; Liang and Liao, 2012). The para-meters in equation (20) are

Y= arccospr � 2v

pr + 2v

� �, r =

1

ReqCpv0

ð21Þ

Then, equations (18) and (20) are substituted intoequation (15a). Using the method of harmonic balance,equation (22) is then obtained. The roots of the polyno-mial (equation (22)) are the squared amplitudes of thenon-dimensional displacement n2. The voltage vp isthereafter determined from equation (20)

L21 +X2

1

� �n2 = a2 ð22Þ

The terms in equation (22) are

n2 = g2 + h2, L1 = 1� p+Bk� v2 +3bn2

4+ 3bk2,

X1 =Ak+hv, A=usin2Y

p, B=

u 2Y� sin 2Yð Þ2p

ð23Þ

When a buck-boost converter operates in DCM, thevoltage across the resistor R in Figure 5 is calculated by(Rogers, 1999)

V O =VpV0D

ffiffiffiffiffiffiffiffiR

2Lfs

sð24Þ

Then the DC output power is

PO =V 2

O

R=

V 2p V 2

0 D2

2Lfsð25Þ

where Vp is the amplitude of vp, V O is the DC voltageacross the resistor R, and PO is the corresponding DCoutput power.

Determination of load impedance

The components in Figure 5 in the red dash-dot boxare taken to be the load impedance ZL for the nonlinearenergy harvesting system. vp is defined by equation(20). By equation (15b), the non-dimensional currentthat flows into the load is

i= ux0=vu h cos vtð Þ � g sin vtð Þ½ � ð26Þ

In the following calculation, current and voltage areexpressed in the complex exponential form for loadimpedance calculation

ZL =vp

i=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 +B2p

ej ui�uvð Þ

vuð27Þ

The phase angles are

tanuv =�gA+ hB

hA+ gB, tanui = � g

hð28Þ

The corresponding dimensioned load impedance isthus

ZL =V0u

ax0v0

ZL ð29Þ

which yields the load resistance RL and load reactanceXL

RL =V0A

ax0v0v, XL =

�V0B

ax0v0vð30Þ

Validation of the analytical modelformulation and solution

Power management circuit design

The specific circuit components used for electricalpower management are listed in Table 2. In the experi-mental validation, the PWM signal is provided by afunction generator (Siglent SDG 1025) instead of a self-powered oscillator (Kong et al., 2010). This approachto PWM signal generation inhibits the complication ofthe assessment of the principles studied here regardingthe analytical prediction of optimal DC power deliveryconditions. Future work will consider the methods ofself-powering the power management circuit.

Experimental validation and comparison to theanalytical and numerical results

The experimental system is used to validate the pro-posed analytical formulation for DC power optimiza-tion. In addition, verification of the accuracy of theapproximate analytical solution is made by direct

Table 2. Circuit components used for rectifier and buck-boostconverter.

Component Value or part

Rectifierdiodes

1N4148 (Vd1 = 1 V; Rd1 = 100O)

Diode D Schottky 1N5820G (Vd2 = 0:37 V;Rd2 = 0:37O)

C1 (C2) 10 mF (470 mF)L 1 mH (RL = 0:216O)MN IRLZ44 N (Vds = 1:3 V; Rds = 0:022O)R 2 kO

Cai and Harne 219

consideration of the equations of motion via aMATLAB/Simulink model of the equivalent circuit, asshown in Figure 4(a), where Zs is given by equation(12) and ZL is given by equation (29). Runge–Kuttanumerical integration is employed for any given baseacceleration combination of amplitude and frequency.For such combination, eight randomly distributed setsof initial conditions are prescribed for the currents inorder to ensure that the simulations identify all possiblesteady states of response in the nonlinear harvesterstructure and circuit system. The calculation time foreach harmonic excitation condition is over 200 periodsto ensure that the steady state is reached. The needs toutilize such long simulation times and multiple initialconditions make the simulation significantly morecostly to compute than the analysis.

In order to take the losses of the electrical compo-nents into account, the respective forward voltage dropsand the respective on-resistances for diodes and theMOSFET are employed in the analytical model andSimulink model. The values for the parameters aregiven in Table 2.

In the analysis, the output voltage with electricallosses can be written as (Rogers, 1999)

V O =Vp1V0D

D2

� Vd2 � iLRL �D

D2

iL RL +Rdsð Þ ð31Þ

where D2 relates to the time that the current throughthe inductor drops to zero; Vd2 is the voltage drop ofthe diode; RL and Rds are the on-resistances of theinductor and MOSFET, respectively; iL is the currentthrough the inductor; and Vp1 is the input voltage ofthe buck-boost converter, which is the voltage acrossthe capacitor C1. Considering the losses caused by therectifier bridge, Vp1 can be expressed as

Vp1 =Vp � 2Vd1 � 2ipRd1 ð32Þ

where Vd1 and Rd1 are the voltage drop and on-resistance of the diode in the rectifier bridge,respectively.

Since the currents iL, ip and the on-resistances RL, Rds

are relatively small values, equations (31) and (32) areapproximated by

V O =Vp1V0D

D2

� Vd2 ð33Þ

Vp1 =Vp � 2Vd1 ð34Þ

The expression for D2 adopted is

D2 =

ffiffiffiffiffiffiffiffi2Lfs

R

rð35Þ

Figure 6(a) presents the amplitude of the harvesterdisplacement determined by analysis and simulation.The base acceleration amplitude is 9.8 m/s2 and the

excitation frequency is around the linearized naturalfrequency of 30 Hz. Since greater values of the buck-boost converter switching frequency result in greaterlosses, in this work the switching frequency remains at1 kHz in all evaluations. As a result of the switchingfrequency selection, it is recognized that the preferredrange of duty cycle may be less than 1% (Kong et al.,2010). For the specific result presented in Figure 6(a),the duty cycle is 0.1%. Based on the analytical andsimulation results, the large-amplitude snap-throughresponses and small-amplitude intrawell responsescoexist in the lower frequency range, such around 10–17 Hz. In the frequency range around 20 Hz, onlysnap-through responses occur. The simulation predictsa narrower range of frequencies across which onlysnap-through exists compared to analysis (Harne andWang, 2017). At higher frequencies such as 30 Hz,both simulation and analysis agree that an intrawellresponse alone may occur. In the range from 23 to30 Hz, the simulations yield aperiodic oscillations thatresult in different amplitudes of displacement based onthe initial conditions of the simulation. In contrast, theanalysis, based on the method of harmonic balance andsteady-state assumptions, predicts that either an intra-well or a snap-through behavior may occur. Thesetrends are seen elsewhere in the literature that com-pares analytical predictions to numerical simulations ofnonlinear dynamics of vibration energy harvestersaround frequency bands of coexistent responses (Daiand Harne, 2017; Panyam et al., 2014).

Figure 6(b) shows the corresponding experimentalresults. Compared to the model results, the large-amplitude snap-through dynamic occurs only in thefrequency band from 20 to 25 Hz and the displacementamplitude of intrawell oscillations at higher frequenciesis larger. One explanation for this discrepancy is aminor imperfection between the two stable equilibria.Asymmetry of stable equilibria is known to result indisplacement amplitudes among the dynamic steadystates that are nearer in value than in the case of perfectsymmetry (Goodpaster and Harne, 2018; Kovacicet al., 2008). In addition, the piezoelectric cantilevercontains layers of FR4 glass–reinforced epoxy laminatesubstrate, piezoelectric PZT-5H, and copper electrodes.The FR4 layers may possess viscoelastic, and thustime-dependent, material properties that cause moreintricate damping influences than the viscous dampingmodel currently employed in the model formulation.These details of experimental system design and imple-mentation may be introduced in a refined model for-mulation in future investigations. Yet, despite suchdeviation associated with inevitable limitations toachieve perfect configurations in experiments, theexperimental results are in good qualitative and quanti-tative agreement with the analytical and simulationresults, giving validation support to the model formula-tion and solution efforts.

220 Journal of Intelligent Material Systems and Structures 30(2)

Snap-through behavior is predicted by the modelingand experiment in the frequency range from 21 to24 Hz. To validate the model predictions of the DCpower delivery from the buck-boost converter, the fre-quencies of 22, 23, and 24 Hz are considered. The out-put power shown in Figure 6(c) and (d) corresponds tothe power delivered to the load R. The analysis inFigure 6(c) predicts the greatest DC power deliveryamong all the results. Because the analysis idealizes theresponse in steady state as being from a single harmo-nic, no higher order losses are incurred in the analyticalprediction that otherwise occur for the simulation andexperiment. In addition, both the simulation and theexperimental system are influenced by much moredetailed losses of the electrical components, like the on-resistances of the diodes and MOSFET that contributeto smaller output powers in the simulation and experi-ment shown in Figure 6(c) and (d), respectively. Despitethis discrepancy, all three methods of assessment iden-tify the same strategy for optimization using the non-linear energy harvester with buck-boost converter.Specifically, the optimal duty cycles at the three

frequencies are all around 0.4%–0.5%. In addition, agreater reduction of the DC power occurs for dutycycles less than this optimal range than above thisrange, which is manifest in all of the results.

Overall, the good qualitative and quantitative agree-ment among all the analytical, simulation, and experi-mental results establishes the efficacy of the analyticalmodel formulation to characterize the mechanical andelectrical behavior of the nonlinear energy harvestingsystem.

An impedance-based assessment foroptimal DC power delivery

After validating the analytical method, the same para-meters shown in Table 2 are considered to conduct athorough analysis to reveal the relationships betweenimpedance change and optimal output power. For theproposed harvesting system, the structural dynamicsx(t) and electrical voltage across the piezoelectric beamvp(t) are obtained by solving the cubic polynomialequation (22). The corresponding output power PO is

Figure 6. Mechanical and electrical responses of (a) beam tip displacement amplitude predicted by analysis and simulation and (c)output power at different excitation frequencies given by analysis and simulation; (b) and (d) the corresponding results fromexperiments.

Cai and Harne 221

then determined by equation (25). Figure 7 shows theoutput power across the load R at different base accel-eration frequencies and buck-boost converter dutycycles. The white dashed line displays the duty cycle ateach frequency to deliver the maximum DC power. Aclear frequency dependence of the duty cycle to maxi-mize the DC power is observed in the analytical resultsshown in Figure 7.

With equations (12) and (29), the source and loadimpedances are, respectively, obtained. Figure 8 pre-sents the corresponding impedance results using thesame x and vp values that are computed for the resultsshown in Figure 7. Figure 8(a) and (b) shows the influ-ence of change in excitation frequency and duty cycleon the source resistance and reactance, respectively.The source resistance Rs in Figure 8(b) is constant at99 kO, as described in section ‘‘Source impedance char-acterization.’’ The absolute value of the source reac-tance Xs decreases with increase in the excitationfrequency and increases with increase in the duty cycle(Figure 8(a)). When the source reactance is zero, snap-through behavior is no longer possible. Figure 8(c)and (d) presents the load impedance results. Although

the buck-boost converter displays a pure resistanceproperty, combined with the rectifier bridge and

Figure 8. Impedance results at the base acceleration amplitude of 9.8 m/s2 of (a) source reactance and (b) source resistance and(c) load reactance and (d) load resistance.

Figure 7. Analytical prediction of DC power delivery at thebase acceleration amplitude of 9.8 m/s2 considering the entirenonlinear energy harvester and power management circuit. Thewhite dashed line shows the duty cycle at each frequency togenerate the maximum DC power.

222 Journal of Intelligent Material Systems and Structures 30(2)

piezoelectric capacitance, the load reactance XL is influ-enced by duty cycle and base acceleration frequency. Inaddition, from Figure 8(d), the load resistance RL pre-sents a maximum value with changing duty cycle ateach frequency.

Based on conservation of energy, the power deliv-ered to the load R in Figure 5 is equal to the poweracross ZL in Figure 4(a). Consequently, this outputpower is

Po =V 2

s RL

RS +RLð Þ2 + Xs +XLð Þ2ð36Þ

By conventional impedance matching theory, theoptimal output power is delivered when the load impe-dance is the complex conjugate of the source impe-dance, in other words, when RL =RS and XL = � XS .Yet, in the case of this study, the source reactance var-ies based on the base acceleration frequency and thebuck-boost converter duty cycle, as shown in Figure8(a). In addition, the load resistance and reactance are

unable to realize the same reactance and resistance val-ues as a conjugate of the source impedance. This isobserved by comparing Figure 8(c) to (a) and Figure8(d) to (b) wherein it is found that the load reactance inFigure 8(c) takes on the same sign as the source reac-tance in Figure 8(a). Also, the maximum load resis-tance in Figure 8(d) is less than the source resistance inFigure 8(b). In other words, the conventional impe-dance matching theory is not able to be employed.

Yet, despite this limitation, the results of Figure 8indicate that a suboptimal set of working conditions inthe duty cycle and base acceleration frequency are iden-tified that still yield peak output power. The whitedashed lines in Figure 8 correspond to the full systemanalysis from Figure 7 that results in the optimal work-ing conditions. For Figure 8(d), the white dashed linefollows the peak value of resistance for the given dutycycle and excitation frequency. Although this value stillfalls short of the source resistance 99 kO, the load resis-tance is maximized and thus nearest to the idealcondition.

Figure 9. (a) Optimal duty cycle of the nonlinear energy harvesting system at different excitation frequencies with a baseacceleration amplitude of 9.8 m/s2; (b) the optimal output power corresponding to (a); (c) optimal duty cycle at different baseacceleration amplitudes with an excitation frequency of 22 Hz; (d) the optimal output power corresponding to (c).

Cai and Harne 223

Because the phase of the load impedance is influ-enced by an intricate combination of power manage-ment circuit characteristics (see equations (21), (23),and (30)), the load reactance and resistance are unableto be individually selected for the sake of maximizingoutput power. Based on equation (36), the influences ofload resistance and reactance on the output power areexpressed in equation (37). Here the sum of source andload reactance is taken as a new variable K=XS +XL

∂Po

∂RL

=V 2s �

Xs +XLð Þ2 � R2L +R2

S

RS +RLð Þ2 + Xs +XLð Þ2h i2

=V 2s �

K2 � R2L +R2

S

RS +RLð Þ2 +K2h i2

ð37aÞ

∂Po

∂K=V 2

s ��2 Xs +XLð ÞRL

RS +RLð Þ2 + Xs +XLð Þ2h i2

=V 2s �

�2KRL

RS +RLð Þ2 +K2h i2

ð37bÞ

By setting equations (37a) and (37b) to zero and sol-ving for RL and K, the impedance matching condition isobtained. Considering the parameters of the energy har-vesting platform studied here, the comparison of loadand source resistance reveals

RS.2RL ð38Þ

Therefore, the absolute difference of the two gradi-ents is

∂Po

∂RL

� ∂Po

∂K

.V 2

s �K2 +R2

L � 2 Kj jRL

RS +RLð Þ2 +K2h i2

.0 ð39Þ

Equation (39) reveals that the change in outputpower caused by resistance change is more influentialthan the change in output power that results from thechange in reactance. Specifically, when RS.2RL theresistance becomes a more influential factor toward theoptimal conditions of duty cycle at different base exci-tation conditions for greater peak output power. Thisresult explains the fact that the load resistance maximain Figure 8(d) correlate to the peak DC power operat-ing conditions computed from the full analysis (see thewhite dashed curve in Figure 8(d) that is obtained fromthe maxima in Figure 7).

Optimal duty cycle predicted byimpedance analysis

From the analysis of the previous section, although theconventional approach to impedance matching is notpossible using the nonlinear energy harvesting system

considered here, a suboptimal matching condition stillmaximizes the output power. To investigate this subop-timal approach to optimization in further detail, theinsight that resistance matching is more influential inthe output power is studied further. Equation (36) istaken omitting the influence of reactance

P=V 2

s RL

RL +Rsð Þ2ð40Þ

The equivalent voltage V s and source resistance Rs

are constants. In other words, the only variable in equa-tion (40) is RL, which is related to circuit parametersaccording to equation (30).

The red dotted curves with circular data points inFigure 9 represent the results from the analysis consid-ering both the harvester structure and buck-boost con-verter, termed the ‘‘buck-boost analysis.’’ The resultsfrom the buck-boost analysis are derived from the pro-cedures described in section ‘‘System model andapproximate solution to the governing equations.’’ Toobtain the results shown in Figure 9, using the buck-boost analysis, equation (22) is solved to identify baseacceleration excitation frequency, excitation amplitude,and the duty cycle that lead to optimal working condi-tions. The green solid curves with square data points inFigure 9 correspond to the ‘‘impedance analysis’’ thatis computed using equations (21), (23), (30), and (40).From 10 Hz, for an increase in the excitation fre-quency, the optimal duty cycle slightly increases fromaround 0.3% to 0.5% according to the impedance anal-ysis. Yet, for an increase in the excitation frequencyaround 25 Hz, an abrupt increase in optimal duty cycleis observed in the buck-boost analysis. This large quali-tative change in the optimal working conditions of theconverter is due to the loss of snap-through dynamics,shown in Figure 10. Figure 10(a) and (b) presents theamplitude of displacement and the resulting outputpower for the base acceleration frequencies of 22, 26,and 27.3 Hz when the base acceleration amplitude is9.8 m/s2, respectively. As the frequency increases, theresponses separate into two branches. As observed inFigure 10(b), the peak power condition therefore shiftsto conditions of higher duty cycle. This trend disagreeswith the impedance matching–based results in Figure9(a) that presumes the snap-through behavior may per-sist for all frequencies. Yet, despite this nuance of pre-diction capability, the output DC powers predicted bythe buck-boost and impedance analyses are in goodquantitative agreement verifying the viability of theimpedance matching–based analysis to characterize theprimary trends.

A similar analysis is conducted by studying the roleof change in the base acceleration amplitude in Figure9(c) and (d). For amplitudes of the base accelerationfrom 7 to 9 m/s2, the snap-through dynamic is not pos-sible for all duty cycles for the 22 Hz frequency. This

224 Journal of Intelligent Material Systems and Structures 30(2)

explains the sudden variation in the buck-boost analy-sis optimal duty cycle that is not evident by the impe-dance analysis. For base acceleration amplitude greaterthan 9 m/s2, the optimal duty cycle changes slightly inboth analytical approaches. On the other hand, sincethe base acceleration amplitude does not affect the loadresistance, the change of load resistance is the same ateach of the base acceleration amplitude, which explainsthe constant optimal duty cycle determined by theimpedance analysis. These results establish confidencein the source impedance analysis.

Based on the results shown in Figure 9, the optimalworking condition for the nonlinear vibration energyharvester interfaced with a buck-boost converter maybe determined by the impedance analysis. In contrastto the buck-boost analysis itself, the impedance analysisonly requires information regarding the source andload resistances. Since both resistances are derived insections ‘‘Source/load modeling of nonlinear energyharvesting system’’ and ‘‘Analytical model formulationand solution for the nonlinear energy harvesting systemwith DC power management,’’ this research identifiesan efficient means by which to evaluate the optimalworking condition of the energy harvesting system.

Conclusion

This research exploits the perspective of impedanceto examine the optimal DC power operation of anonlinear energy harvester interfaced with a buck-boost converter for practical electrical power man-agement. The internal, or source, impedance of thenonlinear harvester platform is shown to be influ-enced by change in the driven conditions of fre-quency and amplitude. Because such changeschallenge the conventional impedance matching con-cepts, a rigorous analytical study is undertaken to

uncover means to maximize DC power delivery fromthe system despite the inability to obtain conventionalcomplex conjugate impedance matching. After vali-dating the analysis, the output power results andimpedance results are shown to demonstrate a subop-timal condition for maximum DC power deliverythrough the buck-boost converter. By studying thesuboptimal conditions in detail, it is explicitlyrevealed that resistance of the load plays a more influ-ential role for the system optimization. With thisinsight, a simplified method of DC power maximiza-tion is created that considers only the load and sourceresistances to optimize the working conditions. Boththe predicted duty cycle and output power from suchreduced modeling strategy show good agreement withthe whole system analysis, exemplifying the strengthof the analytical formulation established here uponimpedance principles.

Acknowledgements

The authors are grateful to Dr Matilde D’Arpino for helpfuldiscussions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest withrespect to the research, authorship, and/or publication of thisarticle.

Funding

The author(s) disclosed receipt of the following financial sup-port for the research, authorship, and/or publication of thisarticle: This research was supported by the National ScienceFoundation under Award No. 1661572.

ORCID iD

Ryan L Harne https://orcid.org/0000-0003-3124-9258

Figure 10. (a) Amplitude of beam tip displacement at a base acceleration amplitude of 9.8 m/s2 and (b) the corresponding outputpower as a function of duty cycle at a base acceleration amplitude of 9.8 m/s2.

Cai and Harne 225

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