On the efficiency of piezoelectric energy harvesters

Accepted Manuscript On the efficiency of piezoelectric energy harvesters Zhengbao Yang, Alper Erturk, Jean Zu PII: DOI: Reference:

S2352-4316(17)30048-2 http://dx.doi.org/10.1016/j.eml.2017.05.002 EML 283

To appear in:

Extreme Mechanics Letters

Received date: 14 March 2017 Revised date: 5 May 2017 Accepted date: 9 May 2017 Please cite this article as: Z. Yang, A. Erturk, J. Zu, On the efficiency of piezoelectric energy harvesters, Extreme Mechanics Letters (2017), http://dx.doi.org/10.1016/j.eml.2017.05.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

On the Efficiency of Piezoelectric Energy Harvesters Zhengbao Yang *1, Alper Erturk2 and Jean Zu1 1

Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, M5S 3G8, Canada 2 G.W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, USA

Abstract Energy harvesting is an essential technology for enabling low-power, maintenance-free electronic devices, and thus has attracted a great deal of attention in recent years. A variety of designs and approaches have been proposed to harvest ambient vibration energy, but crucial questions remain regarding figures of merit characterizing the performance of energy harvesters. Of primary importance is the energy conversion efficiency. There are large discrepancies in the definition and tested values of efficiency in the literature. This study is intended to answer the fundamental question for energy harvesters: how to define and calculate the energy conversion efficiency. We first review studies on efficiency and analyze the energy flow in an energy harvesting system. Based on the analysis, we derive an efficiency expression for linear cantilever energy harvesters. The developed efficiency expression transparently and quantitatively reveals the relationship between efficiency and key parameters. Experiments are performed to validate the efficiency expression. Furthermore, nonlinear energy harvesters are tested in both on-resonance and offresonance conditions. Both experimental and theoretical studies manifest that the energy conversion efficiency tends to decrease as the excitation frequency rises and its value is related to the phase difference between excitations and responses. Around resonance states where the phase difference of both linear and nonlinear energy harvesters is about 90 degrees, the efficiency calculation is much simplified.

Keywords:  energy harvesting; piezoelectric, efficiency; energy conversion; vibration   1. Introduction The dramatic decrease in power consumption of electronic components sets a stage for autonomous operation by using the energy harvesting technology [1-3]. Harvesting mechanical energy from ambient vibration and deformation via the piezoelectric effect has been intensively studied in the past decades. Researchers have recently focused on improving the performance of piezoelectric energy harvesters (PEHs) via high-performance piezoelectric materials [4-8], structure & manufacturing process innovation [9-14] and optimization of dynamic characteristics [15-17]. In order to optimize PEHs’ performance, appropriate performance metrics need to be defined first. Taking various structural, material and vibration parameters into account, researchers have proposed several comprehensive figures of merit including normalized power density (NPD) [18], effectiveness [19, 20], volume figure of merit (FoMv) [21] and systematic figure of merit with 1

bandwidth information (SFoMBW) [22]. Although these metrics, developed under some assumptions, provide comprehensive information on PEHs’ performance, basic metrics such as power output and efficiency are still preferred by both researchers and end users. The ultimate goal of research on PEHs is to generate as much electrical energy as possible from a mechanical vibration source, that is, high power output and high efficiency. While the electrical power output can be measured in experiments and calculated in theoretical models, the mechanical power input cannot be easily monitored and there is no unified calculation process. As a consequence, a large discrepancy exists in the reported efficiency values. As Table 1 lists, it has been estimated that efficiency can be over 80% [23, 24]; it has also been argued that efficiency cannot exceed 50% no matter how well an energy harvester is optimized [25]. Some researchers claimed that efficiency around the resonance state is much higher than that in off-resonance states [26], while others believed that efficiency decreases monotonically as the excitation frequency increases [27]. Efficiency is not only a critical metric for the development and optimization of PEHs, but also has been widely used to compare PEHs with other power generation methods (e.g., solar panels, thermal energy harvesters, triboelectric nano-generators, etc.). Therefore, it is essential to understand efficiency, the fundamental performance metric of energy harvesters. In this paper, we present an insight into the PEHs’ efficiency in the full frequency range both theoretically and experimentally. Table 1 Reported efficiency values of various piezoelectric energy harvesters Reference [23] [24] [28] [29] [30]

Efficiency ~50-90% >80% 2.56% 21.8% 7.5%

[31] [32] [33] [4] [34] [35] [36] [26]

>80% ~7% 0.72% 5.4%/14.9 %/27.5% <44% 3.1% 1.2% 26% / <2%

[37]

12.47%

PZT; bimorph cantilever; vibration; theoretical estimation PZT sandwiched between two Terfenol-D discs PZT; Impact-type using a rotational flywheel; experimental data PZT cantilever beam; vibration; on-resonance/ off-resonance; experimental data PVDF/AlO-rGO beam; direct deformation; experimental data

[38]

80.3%/35.1 %/15.4% 10% 5-18%

Stack/ Membrane/ Cantilever; ball drop impact; experimental data PZT fixed-fixed beam; ball drop impact; experimental data Piezoelectric nanowires: direct deformation; experimental data

[39] [40]

Note PZT; bimorph cantilever; vibration; theoretical estimation PZT; tube; flow-induced vibration; theoretical estimation PVDF film; rainbow bimorph; theoretical estimation PVDF nanofiber; direct deformation; experimental data PZT; flextensional structure; direct deformation; experimental data PZT; fixed-fixed bimorph plate; theoretical estimation PZT; cantilever; vibration; experimental data PZT; cantilever; fluid flows experimental data PZT/PMN-PT/PZN-PT; cantilever; vibration; experimental

Efficiency, also called energy conversion efficiency, generally refers to the ratio of the output energy to the input energy of a system. To maintain consistency, it makes sense to define efficiency of PEHs in the same way, i.e., the ratio between the output electrical energy Eout and

2

the input mechanical energy Ein (   Eout Ein ). This definition is in a form similar to the electromechanical coupling factor squared kij2, where kij is the material coupling factor (electrical field in direction i, stress in direction j). However, the value of kij from piezoelectric material suppliers represents how efficient the employed piezoelectric element (piezoelectric material alone) is in terms of converting mechanical energy into electrical energy and it does not account for the structural design and electrical circuit aspects. Therefore, the material coupling factor kij cannot be applied to the entire structure. The overall harvester efficiency η is usually much smaller than the material coupling factor kij. The concept of coupling factor, or the coupling coefficient, has also been defined at the system level [41] for the piezoelectric structure as 2 k sys  (o2   s2 ) / o2 , where ωo and ωs are, respectively, the open-circuit and short-circuit natural

2 2 frequencies of the vibration mode of interest (note that ksys   2 / (1   2 ) , where  is defined as

an electromechanical coupling coefficient in the following sections. It is widely used in the energy harvesting literature [27, 42, 43]). The system coupling coefficient k sys is a measure of mechanical to electrical energy conversion within the lossless structure and is not equal to the efficiency η either since it does not account for the presence of an electrical load as well as mechanical and dielectric losses. While most researchers have reached an agreement on the definition of efficiency (   Eout Ein ), the expressions of the input and output energy vary, which leads to a large discrepancy in the reported efficiency values. In this paper, we analyze the dynamic characteristics of the input and output power of both linear and nonlinear PEHs in detail. It is found that the phase difference between the excitation and the response significantly affects the input energy, and further the efficiency of the system. In the following, we first review some representative work on efficiency. We then analyze the energy flow of the energy harvesting process and derive an algebraic efficiency expression based on a universal SDOF model. Following that, an experiment is conducted to validate the developed efficiency expression. Via the developed model, we discuss the effect of different parameters on the energy conversion efficiency. Finally, the phase responses of nonlinear PEHs are studied experimentally.

2. Literature Review Piezoelectric energy conversion efficiency has been discussed for a long time. In 1990s, Goldfarb and Jones [44] studied the efficiency of a piezoelectric stack energy harvester under a steady-state sinusoidal compressive force. They proposed a linear circuit model without accounting the hysteresis phenomenon to analyze the effect of different parameters on efficiency. The experiments indicated that large-amplitude and low-frequency force with a high load resistance tends to achieve high efficiency. A maximum efficiency around 10% was obtained at approximately 5 Hz (several orders of magnitude below the resonance frequency of the piezoelectric stack). It is noteworthy that piezoelectric stacks are not practical to be used directly as energy harvesters without any auxiliary structures due to their ultra-high stiffness. By contrast, beam + inertial mass configurations, thanks to their flexible characteristics, receive the most attention. Most recently proposed models are based on the beam configurations.

3

In 2004, Richards et al. [42] derived an exact efficiency formula based on a simplified SDOF model. 

1 2 2 1  2

 1 1 2  .   2   Q 2 1  

(1)

In Eq. 1, efficiency η depends only upon the quality factor Q and the electromechanical coupling coefficient κ2 of the whole system. This definition only works at the resonance state with a matching resistance. The theoretical study estimated that the efficiency value can be over 90% with the assumed weak damping and strong coupling effects, which high value has seldom been achieved in experiments. Nevertheless, the exact formula reveals the effects of damping and electromechanical coupling on efficiency. In 2006, Shu and Lien [27] theoretically analyzed the energy conversion efficiency of a cantilever PEH coupled with a full-bridge rectifier around resonance states. They assumed that the input mechanical energy was the sum of extracted electrical energy and the energy dissipated by the structure damping. The efficiency expression was  2 . (2)  2     2    2 Efficiency is dependent upon the frequency ratio (response frequency/natural frequency), the normalized resistance α (load resistance/matching resistance), the electromechanical coupling coefficient κ2 and the mechanical damping ratio ζ. In general, the conversion efficiency can be improved with a large coupling coefficient and a small damping ratio. It is known that, due to the stiffness difference of piezoelectric elements between the short-circuit condition (natural frequency=1) and the open-circuit condition (natural frequency= √1 ), the frequency to obtain the maximum power output shifts along the variation of the coupling factor. Therefore, the analysis was classified into two categories: weak coupling and strong coupling. Maximum efficiency of 46% and >80% were estimated to be reached by the defined weak and strong coupling systems, respectively. It is worth mentioning that the assumed strong coupling system has seldom been made in experiments. Most experimental efficiency values are much lower than the estimated value [32]. The same expression has also been derived in different ways [25, 45]. In 2009, Liao and Sodano [46] argued that the classic definition of efficiency did not provide information on the capacity of a PEH for converting mechanical energy to electrical energy. All systems with low damping ratios showed high efficiency, irrespective of the mechanical-toelectrical energy conversion performance. Therefore they proposed to redefine the efficiency of PEHs as the ratio of power output to strain energy over each cycle. The energy conversion efficiency was redefined as  2  PH  . (3) 1  (1   2 )( ) 2 In this definition, the efficiency value is not related to the structural damping. The numerical study estimated a maximum efficiency of ~2.5%, attained around the system resonance frequency. Off-resonance efficiency is way smaller than the on-resonance efficiency. There is no experimental validation in the study. The authors did not give an explicit way to measure the defined strain energy in experiments. Hence this definition has not been widely used. Furthermore, it is inconvenient to compare PEHs with other energy harvesting methods by the redefined metric  PH . The metric  PH , similar to the loss factor used for structural damping, reflects 4

the mechanical-to-electrical energy conversion of PEHs in a certain aspect, but cannot replace the role of efficiency. In 2014, Shafer and Garcia [25] presented a theoretical analysis on the efficiency of a baseexcited bimorph cantilever coupled with a full-bridge rectifier. They found that energy was transferred back and forth between the energy harvester and excitation source over each cycle. For some systems (e.g., backpacks, shoes), energy transferred back to the excitations was thought to be impossible to be recovered. The authors thus defined two cases: conservative and nonconservative. For the conservative case, the defined efficiency was the same as that in Eq. 2; for the non-conservative case, a complex expression was derived nc 

 2

   2 

1 2

1

  ( )

,

(4)

where ψ and ξ(θ) account for information of the modal response and phase difference, respectively. The efficiency vs. frequency response curve exhibited three peaks and, and at the resonance states efficiency had an upper limit 44%. The three-peak response has seldom been found in experiments and is different from that in Ref. [27] where efficiency is defined in the same way. The difference is deemed to be from that, in the theoretical study herein, the efficiency value at each frequency was calculated with a load resistance actively changing to maximize power output. In 2015, Kim et al. [23] presented another efficiency expression for a base-excited cantilever PEH with a resistive load.  2  . (5) 2 1   2 2    2 They argued that efficiency has no limit and can be over 80% under certain circumstances. There is no experimental validation in the paper. The developed model indicated that the optimal resistance to get the maximum power output was different from that to obtain the highest efficiency, which was also discussed in Ref. [25, 46].

3. Theoretical Analysis In terms of the working principle, PEHs can be classified into two types: inertial energy harvesters and non-inertial energy harvesters. For non-inertial energy harvesters, force directly applies to the systems and causes active materials to expand or shrink. For inertial energy harvesters, excitations do not deform the active materials directly, but induce inertial force in the systems. Such energy harvesters are usually fixed on a vibration base such as human bodies, animals, vehicles and buildings. In this paper we mainly focus on inertial energy harvesters.

5

3.1. Energy Floow Analysis

Fig.1 Energgy flow in an piezoelectricc energy harvvesting system m Befoore modellingg an energy harvesting ssystem, we fiirst need to ccomprehend the energy fflow in the system. s Physsically, an energy harvesting system m involves thhree parts: ann excitation source suchh as human m motion or vehhicle vibratioon, an energyy harvesting device and an a interface circuit. Overrall, there aree four energyy reservoirs aand three steeps of energyy transformattions in the pprocess as shhown in Fig. 1. Step 1. Mechhanical energgy is capturedd and transferrred from thee excitation ssource to the PEH. S S 2. Mechhanical energgy is converteed to the elecctrical energyy in the PEH.. Step S 3. Electtrical energy is extracted from Step f the PEH H to the exterrnal load. Eachh energy reseervoir couplees with its neighbors in booth forward and a reverse ddirections. Thhe first reserrvoir refers too the mechannical energy in the host. T The effect off adding a PE EH should bee trivial to thhe dynamic ccharacteristiccs of the hosst. (e.g., vehiicles and buiilding sensorrs). That meaans the excittation energyy overall is innfinite relativve to the enerrgy used by the t followingg energy harvvesting systeem. The seccond reservooir contains oscillating kkinetic energgy of the ineertial mass, elastic potenntial energyy of the struucture and sstructural daamping energgy. The thirrd reservoir stores electtrical energyy in a PEH, w which cannoot be fully exxtracted by thhe external eelectrical loaad. The fourtth reservoir rrepresents thhe electrical eenergy used bby the electrrical load. In the whole process, a paart of energyy is inevitablly transferredd to heat duue to a varietty of effectss such as stru ructural dampping, dielectrric loss and ccurrent leakaage, etc. As a dynamic system, a PE EH experiencces an oscilllating energyy flow. Theerefore, we should conssider the stabble state andd the net ennergy transfoormation in each step w while analyziing the energy transmisssion process.. Following the classic eefficiency definition, efficiency oughht to be equaal to the ratio of the net innput energy too the 4th reseervoir and thaat to the 2nd reservoir.

3.2. Electromechanically Coupled M Model

6

Fig. 2 A cantileveer piezoelecttric energy haarvester undeer a base excitation.

Fig. 3 Scheematic of a piezoelectric energy e harveesting system m. most widelyy used cantilever PEH ass the study object. The PEH consistts of a We choose the m morph beam with one ennd fixed at thhe excitationn source, andd the other eend attached with a unim masss block (Fig. 2). This elecctromechaniccal coupling system is moodeled as an equivalent luumpedparam meter SDOF F dynamic system (Fig. 3)) with the dynnamic equatiions mX X  cX  kX  V  mZ , (6) 1 C pV  V   X , R

(7)

wherre m, c, k, Cp, R and θ are the equuivalent masss, damping, stiffness, pieezoelectric innternal capaacitance, exteernal resistannce and piezzoelectric couupling factorr, respectiveely. V is the output voltaage over thee piezo elem ment. X is thhe relative displacement d and Z is thhe base exciitation. reepresents diffferentiation tto time “t”. The value of the couplinng factor θ ccan be identiffied by eitheer theoreticall analysis [344, 47, 48] or experimental method [499]. The goverrning equatioons can be dderived from the piezoeleectric constittutive equatiions and Euller–Bernoullii beam theorry, and havee been used for f analyzingg different PE EHs [34, 47, 50-53]. Notee that the devveloped modeel does not aaccount for thhe dielectric loss in piezooelectric matterials, and itt does not woork under ulttra-low frequuency condittions. A vaariety of elecctrical circuitts have been proposed to treat the AC C current from PEHs, inccluding varioous DC-DC converter toopologies [544, 55], SSHI [56], SSHI--MR [57], SECE [58], O OSECE [59],, bias-flip innterface circuuit [60], singgle-supply prre-biasing ciircuit [61] annd so on. E Each of them m shows its oown characteeristics and hhas different effects on thhe energy floow. Without loss of geneerality, we heere consider the simplestt scenario: coonnecting PE EHs with a reesistor. This is also the m most commoonly discusseed circuit in tthe literaturee. Efficiencyy of differentt energy harvvesting circuuits can be foound in the boook [62]. Wee would not ttalk about it here. h To simplify s the solving proccess, we introoduce the foollowing nonn-dimensionaal variables innto the goveerning equatiions,    n t ,  

 c ,  , 2m n n

2 

2 kC p

, 

1 , RC p n

wherre  n is natuural angular frequency (  n  k / m ));  is freqquency ratio;  is the daamping ratioo,  2 is the cooupling coeffficient. It is increasingly recognized that most practical appliccations 7

of PEHs are with small coupling coefficients. Here we also introduce r  1 RCp    , named resistance ratio, into the system. The maximum power is extracted by the external electrical load at the impedance matching condition, that is, r=1 (weak coupling systems). The solutions below can be greatly simplified with the resistance ratio ‘r’. The nondimensional governing equations are x  2 x  x   2 v   z  , v   v  x where x  X , z  Z , v  CpV /  , and ()  is differentiation to the nondimensional time “τ”.

(8)

The harmonic base excitation is expressed as

 ) z   A cos(

,

(9)

2 which can also be expressed as Z  n A cos(t ) .

Assume responses are also harmonic at the same frequency as that of the excitation.  )  x2 cos(  ) x  x1 sin(  )  x2 sin(  ) x   x1 cos(

,

(10)

 )  x2 cos(  ) x    x1 sin(   v  v1 sin( )  v2 cos( ) .  )  v2 sin(  ) v   v1 cos( 2

2

(11)

Submitting Eq. (9)-(11) into Eq. (8), we get  )  x2 2 cos(  )  2  x1 cos(  )  x2 sin(  )  x1 2 sin(  )  x2 cos(  )   2  v1 sin(  )  v2 cos(  )   A cos(  ) x1 sin(  )  v2 sin(  )    v1 sin(  )  v2 cos(  )  v1 cos(

.

(12)

 )  x2 sin(  )  x1 cos(

Then, equating the coefficients of sin( t) and cos( t) on both sides, we have x1 2  x2 2  x1   2 v1  0 x2 2  x1 2  x2   2 v2  A . v2   v1  x2

(13)

v1   v2  x1

The solutions are

x1  

2 

 2  2 2 2    1   2   2 A; x   2   2 A , 2

 v1  2

(14)



  (  2 ) 3  4  (1  2   2 ) 2 A; v2  A 2 2  (   )  ( 2   2 ) 2

2

,

(15) 2

  2 2    2   2 2    2r    2  2     1    2  where     2  1  2  . 2  2  2  1 r   1 r2            

3.3. Mechanical Response The displacement response is

8

  x ) x  X ap cos(

,

(16)

wherre the displaccement amplitude and thee phase differrence are X ap 



x  tan 1

x1  tan 1 x2

  2  

1

x12  x22 

 2    2   2 

 2 2  1   2  2   2

A ,



(17)



  2   tan 1



2

1 r2

 2r 

 1 r2 

(18)

.

 1   2

The magnificatioon factor is M

X ap Dbase

 2



2

 2 2    2r      2    1   2  1 r   1 r2  

2

,

(19)

mplitude of tthe base excittation. wherre Dbase  A /  2 is the dispplacement am

→ 0, thenn we have

For the t short-circcuit conditionn, we let the load resistor  2 M R 0 

 2 1   2  2

(20)

. 2

M R  0 reaches th he maximum m value wheen   1  2 2  1 , whhich is usuallly defined as the

resonnance frequeency. For the open-circuitt condition, w we let the load resistor  2 M R 



2

1   2

   2  2

.

→ ∞, then w we have (21)

2

2 2  M R  1   2 , which is known as th he anti reaches the maximum value when   1    2

resonnance frequeency. Resonaance and antti-resonance points are veery close to each other iin lowfrequuency and weeak-couplingg circumstancces.

Fig. 4 Variationns of phase angle  2  0.0078 .

w with the freqquency ratioo . The cooupling coeffficient

Fig. 4 shows the phase anglee at different excitation frrequency poiints. Before tthe resonancee state, r tendds to keep a ssame pace with the excitaation. After thhe resonancee state, the reesponse the response 9

tends to keep an opposite pace with the excitation. Around the resonance point, a phase shift of 90 degree occurs no matter what the damping ratio is. is in the range [0 180 ] . If the coupling coefficient   0 , the phase difference is the same as the expression for the mechanical SDOF systems (Eq. 3.77 in Ref. [63]).

3.4. Electrical Response We express the voltage response as  )  v2 cos(  )  Vap cos(   v ) . v  v1 sin(

(22)

From Eq. (15), we get the response amplitude and phase difference Vap 



X ap 



1

A

    v   (  2 ) 3 v  tan 1 1  tan 1 4   (1  2   2 ) 2 v2 The power consumed by the external resistor is calculated as Pout

 v / C   p

R

2



   2

2

2

2

.

(23)

2

 2 2 1 2 2    v ) . X X ap2 cos 2 ( cos ( )     ap v 2 2 2 2 C p R    Cp R 1 r2

(24)

The frequency of the power response is upgraded to 2ω and thus the cycle period becomes T  2 2 . The electrical energy generated per cycle is 2

Eout   2 Pout dt  0

2  VRMS  2 r X ap2   R 2 Cp 1 r2

,

(25)

where VRMS refers to the root mean square voltage.

3.5. Input Mechanical Energy Unlike the calculation process of the output electrical energy, the input mechanical energy does not have a uniform definition. A typical wrong definition is . This definition goes against the facts in two ways. First, the excitation force from the host to the PEH cannot be expressed as mass multiplying acceleration. Supposing that the excitation acceleration is kept constant in a full frequency range, the acting force between the PEH and the host structure should keep constant based on this definition. However, the fact is that the acting force increases significantly around the resonance state. Secondly, velocity in the power definition should refer to the velocity of the point the force applied to, and it must be in the direction of the force. The velocity of tip mass in a PEH is not the one of the excitation point. Energy input to the PEH is from the point fixed on the excitation. The velocity of the point is, therefore, the velocity of the excitation source . There are three ways to define the input mechanical power. Case 1: Pin  F  Z  m  Z  Z Some assume that the whole structure vibrates at a same acceleration as the host’s. This assumption is not appropriate because there are relative motions in the system. Case 2: Pin  F  Z  m  X  Z 10

As the model in Fig. 2 indicates, a spring, a damper and a piezo element are connected with the excitation base. Thus the force of the load point is assumed to be equal to the damping force + the spring force + the piezo force, which is also equally applied to the tip mass, that is, m  X . Therefore, the input power can be expressed as 1 1 Pin   mn2 AX ap sin(2 t  x )  mn2 AX ap sin(x ) 2 2 . (26) 1    m X Z   sin(2 t  x )  sin(x )  amp amp 2 If there is no relative displacement (X=0), the tip mass moves along with the base. The acceleration of the tip mass is equal to that of the base. That means the tip mass experiences a periodic force. However, no relative displacement means no spring, damping or piezo force, that is, no force applied to the mass block. The self-contradiction indicates that this equation does not reflect the real situation.

   





Case 3: Pin  F  Z  m  X  Z  Z In this case, we do not face the self-contradiction issue in Case 2. If there is no relative motion, the tip mass moves along with the base at an acceleration of Z . The applied force is equal to the mass times the absolute acceleration.  4 A2 1 1  Pin   mn2 AX ap sin(2 t  x )   mn2 AX ap sin(x )   m n sin(2t ) 2 2 2  .

(27)

n4 A2 1 Z m     m X     sin(2 t ) sin( ) sin(2t )  x x  amp amp 2 2 There are sinusoidal functions in the power response equations, of which average value is zero per period. The effective value is only from the constant term sin . Comparing Case 2 and Case 3, we can find that the net input power is same, although the Case 3 shows a higher oscillation amplitude.

   

Using Eq. 27, we examine the power flow in both on- and off- resonance conditions. Three 0.5, case 2: 1, case 3: 1.2 . The phase shift conditions are considered here, case 1: is clearly illustrated by the Lissajous curves.

11

m. The red rregion denottes that Fig. 5 Phase shiift illustration and energgy flow of a PEH system med “positivee work” herre; the blue region energy flows to the energy harvesting system, nam denootes that enerrgy flow is frrom the energgy harvestingg system to thhe outside, nnamed “nativve work” here. ( = 0.05663; 1.57; 2.91189).



At low fr frequency

0.5)

The positive work region is slightly largerr than the negative n woork region. As A the excitationn frequency goes smalleer, the net innput tends too be naught. The displaccement response and the basee vibration haave a phase ddifference off ~ 0 degree.



At high ffrequency

1.2

The net m mechanical eenergy that innputs to the mechanical system tendss to be naughht, too. The displacement ressponse and thhe base vibraation have a pphase difference of ~180 degree.



At resonance frequenncy

1

The maxximum input power happeens while thee base velocitty and the ressponse acceleration are in phhase. That iss the displaceement responnse and the base vibratioon has a 90 degree phase diffference. Theere is almost no negative work occurreed in this situuation. The excitation ddoes positive work to thee energy harrvesting systeem (W+); it also does neegative W-). That meaans actually nnot all mechaanical energyy input workk to the enerrgy harvesting system (W to thhe system is utilized (dam mping heat & converted electrical ennergy), but soome of them m flows backk the excitatioon source. Thhe net input m mechanical energy e is W+-W - -. m which is ddetermined by b the phasee difference. At the The net power iis from the cconstant term resonnance state ((phase angle 90o), almost no energy flows back tto the excitaation source and all 12

mechhanical energgy is used bby the PEH. When the eexcitation freequency is faar from the natural frequuency of the PEH, the neegative and ppositive workk cancels each other and there t is a verry little amouunt of poweer input to thhe system, for f compensaating a varieety of dampiing loss. Forr baseexcitted vibrationn systems, it iis inevitable to do negativve work for a certain periiod in each ccycle in off-rresonance coonditions. Neevertheless w ways to sustaain a 90o phaase gap betw ween the exccitation and rresponse are worth explooring. Worrk done by thhe external exxcitation on the t system peer cycle T  2

W   2 Pin dt  0

 2

mn2 AX ap sinn(x ) 

2 can be deerived as 2

 1 m   X    Z  sin(x ) , amp amp 2

(28)

 180 ] . The w wherre x [0 1 work done by the external excitation iss always posiitive.

The input mechhanical energgy is proporrtional to thhe mass, ressponse accelleration, exccitation velocity, and is directly rellated to the phase diffeerence. The phase term sin peeaks at m resonance,, as shown in i Fig. 6. At the resonnance and pplunges once excitations deviate from resonnance state sin =1, we can eassily calculatee the input energy by just j measuriing the excittation and response veloccity/accelerattion.

Figg. 6 Variatioons of sin

at differennt excitation frequency f pooints.

3.6. Efficiency As aaforementioned, efficiencyy is the ratio of the net ouutput electriccal energy (Eq. 25) to the net inpuut mechanicall energy (Eq.. 28). E   out  1 W m  X 2

 

2 VRMS S

amp

R  Z

 

. amp

(29)

sin(x )

Eq. 229 interprets the definitioon of efficiency, and it cann be applied to different eenergy harvesters undeer base excitaations. For the t defined liinear system, we have 

2 1 2 (  r )   2 r

2

 2

 2   2   2 

.

(30)

13

Efficciency is maainly related tto the electroomechanical coupling efffect, dampinng effect, exccitation frequuency and eelectrical loaad. It is notteworthy thaat here iss also relatedd to the stru ructural stiffnness and thee capacitancee of the empployed piezoelectric elem ment. Therefo fore, to gain a high efficciency, we neeed take full considerationn of the struccture, materiaal, excitationn and electriccal load. The maximum eefficiency occurs while thhe resistancee ratio

1

1. However, tto keep

r=1 at different frequency pooints, we havve to activelyy adjust the load resistannce, which reequires greatt efforts andd may consuume energy. Usually in experiments, a fixed-value load resistance 1 is selected. s If more m complex conditionning circuits are added, the expresssion of efficciency and opptimum impeedance will bbe changed. For examplee, if a full brridge rectifierr and a smooothing capaccitor are addeed between thhe PEH and the load resiistor, the resiistance ratio for the 1 maximum efficieency and pow wer output is .

4. E Experimentaal Validatioon

Fig. 7 Experimental E setup and thhe fabricated prototype. T Table 2 Geom metric and m material propeerties of the P PEH prototyppe. Name Substruccture length Substruccture width Substruccture thickneess Substruccture Young’s modulus Substruccture densityy Piezo pllate length Piezo pllate width Piezo pllate thicknesss Tip masss Dielectrric constant Piezoeleectric charge constant, d311 Piezoeleectric voltagee constant, g331

Value 73 mm 21 mm 0.5 mm 69 GPa 2.73 g/cm3 37.6 mm m 15.1 mm m 0.50 mm m 15.6 graams 1680 -180*100-12 C/N -11.3*100-3 Vm/N 14

To validate the defined efficiency, we have fabricated a prototype as shown in Fig. 7, which specifications are listed in Table 2. The active material used here is soft piezoceramic PZT-5A. The static capacitance Cp and loss tanδ were measured to be 13.18 nF and 0.015, respectively. Fig. 7 shows the experimental platform. The PEH prototype was mounted vertically on an electromagnetic shaker that supplied a harmonic excitation while operating. Excitation parameters were set via a PC interface and loaded to a controller. The controller regulated the vibration of the shaker via a signal amplifier based on the instructions from the PC and the feedback signal from an accelerometer. The electrical and mechanical responses were monitored using an oscilloscope and a laser Doppler vibrometer, respectively. All experiments were conducted on a vibration isolation platform to reduce unwanted interference. The excitation acceleration was maintained constant at 0.3 g (g=9. 8 m/s2) in the testing. To calculate the efficiency of the PEH using Eq. 29, we first need to identify the involved parameters. The equivalent mass m of the structure can be roughly calculated with m

33 M beam  M tip . 140

(31)

The equivalent mass of the fabricated prototype is m =16.4 grams. The coupling coefficient is 0.0883 , decided using the experimental method. The damping ratio of the system is   0.021, which was determined using the logarithmic decrement method [49]. By doing the frequencysweep testing, we find that there is not a big difference between the open-circuit resonance frequency and the short-circuit one of the constructed system; both are around 28.4 Hz ( 178.6). The matching resistance R=400 KΩ is identified following the method in Ref. [49]. The resistance was kept constant in the whole experiments. With these parameters in hand, we can calculate the efficiency value of the system via Eq. 30. We got the voltage data from the oscilloscope and used Eq. 25 to calculate the electrical energy per cycle. To get input mechanical energy, the relative acceleration and base velocity are needed. We used the laser vibrometer to measure the absolute velocity of the tip mass and base velocity, and derived the input mechanical energy at each frequency point. The tests have been repeatedly conducted for many times to ensure the validity and accuracy of the measured data. Fig. 8 shows the measured efficiency compared with the estimated value. The comparison indicates that the developed theory can estimate efficiency very well. The efficiency of the fabricated PEH is about 6-12%. Both experimental and theoretical studies manifest that the energy conversion efficiency of energy harvesters tends to decrease as the excitation frequency rises. Efficiency values of off-resonance states are not always much lower than those of the onresonance states. For example, the efficiency is about 11% at 20 Hz in this case as shown in Fig. 8, which is higher than that (< 10%) at the resonant state (28 Hz). Also, one should note that conditions to attain the maximum power transfer (around resonant points) do not coincide with conditions to achieve the highest energy conversion efficiency.

15

Fig. 8 Efficienccy at differeent frequenccy points coollocated froom the expeerimental daata and mated by the developed m model. The reesonance freqquency is aboout 28 Hz. estim

Fig. 9 Velocity reesponses of tthe tip end annd the base oof the PEH unnder differennt excitations. Black line: base velocitty; red line: tiip velocity. The phase differeence betweenn the excitatiion and the ssystem response is observved in experiiments, whicch matches thhe model esttimation (Figg. 4). As shoown in Fig. 9, 9 in low-freequency condditions, the rresponse tends to keep a same pace with the basse excitationn. When it coomes to resoonance, rougghly a 90 deggree phase lagg appears. Inn high-frequenncy conditionns, the respoonse lags behhind the excittation about 180 degree. That means when the hoost goes to onne direction, the tip masss of the PEH H, instead of ffollowing thee excitation, m moves to thee opposite dirrection.

5. L Linear Systeems 5.1. Validation n via the En nergy Flow Analysis From m the perspeective of eneergy flow, thhe mechanical energy inn the energy harvesting system finallly is consum med by the damping efffects and exttracted by thhe external lload. Therefo fore we shouuld have W  Eout Wdampingg . This forrmula has beeen used to analyze eff fficiency of energy harvvesters as afoorementionedd. Here, we examine e if thhe derived eefficiency exppression meeets this deduuction. 2

Eout   2 Pouut dt  0

 2

r X ap2 2 Cp 1 r2

(32)

2

 Wdamping   2 cX 2 dt  ccX ap 2 0 2

(33)

16

W

 2

mn2 A AX ap sin(x ) 

 2

mn2 AX ap

1 



  2  

 2    2  2 2   m   2       X ap n  2   2  2  2   2  

  r  2 2   m 2  m  n X ap 2 1 r2  2   2 r  2   c   X ap 2 2 C p 1  r 2    Wdamping  Eout 2

(34)

The derived efficciency expresssion is validdated from thhe perspectivee of the energgy flow.

5.2. Parametric Study Accoording to E Eq. (30), effi ficiency is m mainly affected by the load resistaance, dampinng and electtromechanicaal coupling effect. Next, we talk about a the efffects of theese parametters on efficciency. As illustrated i inn Fig. 10, looad resistance has a siggnificant efffect on the energy convversion efficciency. As thhe load resisttance increasses, the efficiiency peaks around the m matching resistance poinnts (resistancee ratio r=1), and then droops dramaticaally. With a same resistannce, a PEH uunder a low--frequency exxcitation shows high efficciency.

Fig.110 Effect of the t load resisstance on efficciency. ( 0.0078, ζ=0.021).

Fig. 11. Dampinng effect on eefficiency ( 0.00 078, α=1).

Fig. 11 shows thhe effect of tthe structuraal damping oon efficiencyy in both on-- and off-resoonance statees. Efficciency variattion becomess increasinglyy small as damping d asceends. That means m the effiiciency value tends to keeep constant, uncorrelatedd to damping variation onnce damping tturns strong. Alsoo, the couplinng coefficiennt plays a decisive rolee in the exhiibited efficienncy of a PEH H. Fig. 12 illlustrates thee effect of thhe coupling coefficient on efficiencyy. As expected, the highher the couppling coefficiient is, the more m efficienttly the PEH behaves. Thhe slopes of tthe response curves get iincreasingly larger as wee increase k2. That meanns for system ms with a weeak coupling effect, efficciency is pronne to keep unnchanged as the t excitationn frequency vvaries. In conntrast, system ms with a rellatively stronng coupling effect, e the effficiency valuue experiencces a big diff fference. It iss worth notinng that the ccoupling coeefficient is innversely propportional to the structuraal stiffness and a the interrnal piezoeleectric capaccitance. Thuus, designs with soft structures aand small internal capaacitance tend to exhibit hiigh efficiencyy. 17

Fig.. 12. Effect oof the couplinng coefficient on efficienccy (α=1, ζ=00.021).

6. N Nonlinear Syystems As discussed d beffore, linear systems s expeerience a 90 degree phasee difference at resonancee states and shift phase nnearly 180 deegree when fr frequency is much m higher than the resoonant one. We W can utilizze this characcteristic to obbtain the effiiciency valuee of linear ennergy harvesters. To broadden the frequuency bandw width, researrchers have introduced nonlinear n vibbration into energy harvvesting systeems [15, 43, 64, 65]. It is unclear wheether the phasse differencee of these nonnlinear system ms will matcch the linear ones. To study the phase shift phennomenon in nnonlinear PEH Hs, we consttructed two pprototypes as shown n maagnetic forcee was utilizedd to provokee nonlinear ddynamic respponses. in Fiig. 13. The nonlinear By aadjusting the magnet layoout, we achieeved both haardening andd softening nonlinear n respponses. Wheen there is a ssoft repulsivee force betweeen the moviing and fixedd magnets, thhe system shoows the harddening nonlinnearity; wheen there is an a attractive force betweeen the movving and thee fixed magnnets, the systtem shows thhe softening nonlinearity. n The prototype waas fabricatedd with a high carbon sprinng steel beam m of 70×7×0..25 mm3. As shown in Fiig. 13, a smalll piezoelectrric element w was attached near to the fiixed end of thhe steel beam m (13.7 mm to the end). The piezoellectric elemeent was madee of PZT-5A A and its dim mension is 7× ×8×0.2 mm3. A Neodym mium magnett ( 8.31 2 2 ) (Gradde N42) was attached at the free endd of the steell beam, faciing a pair oof same-sizee magnets ffixed on thee base. The experimentss were condducted on thee platform deescribed befoore, and the excitation e freequency wass swept upwaard and downnward with a constant ratte of 0.1 Hz/ss.

Fig.13 Nonlinear PEHs: P (a) hardening, (b) ssoftening. 18

mental amplituude and phaa se responsees of the harddening and softening s nonlinear Fig. 14 Experim Hs. Both expperiments weere done witth a constannt excitation of 0.2 g acceleration. (a) The PEH harddening system m is under an excitationn of 10-30 Hz; (b) the softening system is unnder an excittation of 20-440 Hz. Takee the upwardd frequency--sweep case of the harddening nonlinnear PEH foor example, as the excittation frequeency rises coonstantly from 10 Hz, w we can see a slight increase in the vvelocity respoonse. When the excitatioon frequencyy approachess to the resoonance (20.44 Hz), the reesponse starts to increasees quickly. Once it passses the resonnant frequenncy (i.e., jum mp frequency), the respoonse jumps down and ddecreases sloowly ever sinnce. The phase differencce, likewise,, keeps nearly zero at frequency poinnts before thhe resonancee and quicklyy passes 90 ddegree and rrises to ~1800 degree whiile the frequeency is sweppt through thhe resonance. For the dow wnward freqquencysweeep case, the pphase differeence stays arround 180 deegree and droops quickly to 0 degree around the jjump-up freqquency pointt (~18.5 Hz).. It is noticeaable that a ddistinct hysteeresis loop exxists in the nnonlinear syystems, whicch is not observed in thhe linear sysstems. The ssoftening nonnlinear systeem shows a ssame type off phase shift as a the hardenning nonlineaar system. Coomparing thee phase angle figures of linear system ms (Fig. 4) and a nonlinearr systems (F Fig. 14), we find f that lineear and nonllinear system ms act in a very v similar way and alll reverse aroound the resoonance pointts. The form mer energy floow analysis ccan also be ap applied to thee nonlinear syystems. For both hardeniing and softeening nonlinnear PEHs, thhere exists a roughly 90 degree phasse shift betw ween excitatioons and respoonses aroundd the jump pooints. Conseqquently, we can c easily caalculate the input mechannical energy and a further efficiency usiing Eq. 29.

7. D Discussions We have presennted a comprrehensive fraamework forr discussing the efficiency of piezoeelectric energy harvesterrs. Parameteers that affecct the harvesters’ efficieency have bbeen identifieed and

19

quantitatively analyzed. It is noted that efficiency is directly related to the phase difference between excitation and response, which complicates the efficiency calculation. In order to take advantage of the unique 90 degree phase shift at resonance, we suggest comparing different PEHs at the resonance states while evaluating their efficiency values. The reasons are as follows. 1. The on-resonance efficiency is easy to be calculated experimentally and theoretically. 2. To generate high-power output, most PEHs are designed to work around the resonance states. Therefore, the on-resonance efficiency reflects the real working performance well. 3. Efficiency has an approximate monotonically decreasing relationship with the excitation frequency. Consequently, efficiency values at other points can be extrapolated with the on-resonance efficiency in hand. Note that the resonance state here means the point of the maximum power output. It is known that optimal power output conditions (frequency, resistance, damping, etc.) do not coincide with those of the optimal efficiency, especially for strongly-coupling systems. As the analysis and experiments indicate, systems with weak damping effects and low-frequency excitations always show high efficiency, regardless the energy harvesting capability. Therefore, it is more reasonable to discuss efficiency values at the maximum power output conditions, not at the optimal efficiency points. It is worth noting that solely pursuing maximum efficiency may result in a significant drop on power output, and misguide the development of new PEHs. When evaluating different PEHs, we suggest to use a set of figures of merit, including efficiency, power density, excitation-normalized power density, frequency bandwidth, etc., instead of one performance metric. Meanwhile, the working conditions such as volume, mass and excitation strength ought to be considered seriously.

8. Summary In this paper, we theoretically and experimentally studied PEHs’ efficiency of converting mechanical input energy to electrical energy dissipated by a load resistance. We reviewed representative work on efficiency and analyzed the energy flow in the energy harvesting process. We derived an analytical efficiency expression based on a SDOF model for the most commonly used cantilever PEHs, and validated the efficiency expression experimentally. The study indicates that efficiency is mainly related to the electromechanical coupling effect, damping effect, excitation frequency and electrical load. Light damping and strong coupling effects help improve efficiency. In contrast, both small and large resistors lead to a significant drop in efficiency; efficiency peaks with a modest matching impedance. Both experimental and theoretical studies show that the energy conversion efficiency decreases as the excitation frequency increases. Depending on the period of time in each cycle, the work done by the excitations on the PEHs can either be positive or negative, which is characterized by the phase difference between excitations and responses. The phase shifts nearly 180 degree as the excitation frequency ramps up and down. Around the resonance frequency, a phase difference is 90 degree for both linear PEHs and nonlinear PEHs. A 90 degree phase difference means the excitation always does positive work in each working period. This characteristic yields exceptional convenience in calculating the efficiency value of the energy conversion process. Efficiency at resonance states reflects the real

20

working performance well and is easily calculated. Therefore, while discussing different PEHs we suggest to use the efficiency at the resonance condition.

Acknowledgements This work was supported by the Natural Science and Engineering Research Council of Canada.

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