Analysis of power output for piezoelectric energy harvesting systems
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INSTITUTE OFPHYSICSPUBLISHING SMARTMATERIALS ANDSTRUCTURES Smart Mater. Struct. 15 (2006) 1499–1512 doi:10.1088/0964-1726/15/6/001
Analysis of power output for piezoelectric
energy harvesting systems
Y C Shu
1and I C Lien
Institute of Applied Mechanics, National Taiwan University, Taipei 106, Taiwan, Republic of China
E-mail:yichung@spring.iam.ntu.edu.tw
Received 1 April 2006, in final form 11 August 2006
Published 25 September 2006
Online at
stacks.iop.org/SMS/15/1499
Abstract
Power harvesting refers to the practice of acquiring energy from the
environment which would be otherwise wasted and converting it into usable
electric energy. Much work has been done on studying the optimal AC power
output, while little has considered the AC–DC output. This article
investigates the optimal AC–DC power generation for a rectified piezoelectric
device. In contrast with estimates based on various degrees of approximation
in the recent literature, an analytic expression for the AC–DC power output is
derived under steady-state operation. It shows that the harvested power
depends on the input vibration characteristics (frequency and acceleration),
the mass of the generator, the electrical load, the natural frequency, the
mechanical damping ratio and the electromechanical coupling coefficient of
the system. An effective power normalization scheme is provided to compare
the relative performance and efficiency of devices. The theoretical predictions
are validated and found to be in good agreement with both experimental
observations and numerical simulations. Finally, several design guidelines
are suggested for devices with large coupling coefficient and quality factor.
1. Introduction
The development of wireless sensor and communication node networks has received a great deal of interest in research communities over the past few years. Applications envisioned from these node networks include building structural health monitoring and environmental control systems, smart homes
and tracking devices on animals in the wild [23,32]. However,
as the networks increase in number and the devices decrease in size, the proliferation of these autonomous microsensors
raises the problem of an effective power supply. The
conventional solution is to use electrochemical batteries for power. However, batteries can not only increase the size and weight of microsensors but also suffer from the limitations of a brief service life and the need for constant replacement, which is not acceptable or even possible for many practical applications.
On the other hand, simultaneous advances in low-power electronic design and fabrication have reduced low-power requirements for individual nodes. It has been predicted that
1 Author to whom any correspondence should be addressed.
power consumption could be reduced to tens to hundreds of
microwatts depending on the application [3]. This opens the
possibility for self-powered sensor nodes, and the need to power remote systems or embedded devices independently has motivated many research efforts focused on harvesting electrical energy from various ambient sources. These include
solar power, thermal gradients and vibration [37]. Among
these energy scavenging sources, mechanical vibration is a potential power source that is abundant enough to be of use, is easily accessible through microelectromechanical systems (MEMS) technology for conversion to electrical energy, and is ubiquitous in applications from small household appliances to large infrastructures [36,41].
Vibration energy can be converted into electrical energy through piezoelectric, electromagnetic and capacitive
transducers. Among them, piezoelectric
vibration-to-electricity converters have received much attention, as they have high electromechanical coupling and no external voltage source requirement, and they are particularly attractive for
use in MEMS [13, 30, 39]. As a result, the use of
piezoelectric materials for scavenging energy from ambient vibration sources has recently seen a dramatic rise for power
harvesting. For example, early work at the MIT Media Lab investigated the feasibility of harnessing energy parasitically
from various human activities [45]. It was later confirmed
that energy generated by walking can be collected using
piezoelectric ceramics [40]. Since then, piezoelectric elements
used for power harvesting in various forms of structure have
been proposed to serve specific purposes. Elvin et al [6,7]
and Ng and Liao [27] have used the piezoelectric element
simultaneously as a power generator and a sensor. They
have evaluated the performance of the piezoelectric sensor to power wireless transmission and validated the feasibility of the
self-powered sensor system. Roundy and Wright [38] have
analysed and developed a piezoelectric generator based on a two-layer bending element and used it as a basis for generator design optimization. Similar works based on cantilever-based devices using piezoelectric materials to scavenge vibration energy include [4,25,26,51,53].
Instead of 1-D design, Kim et al [16, 17] and Ericka
et al [8] have modeled and designed piezoelectric plates (membranes) to harvest energy from pulsing pressure sources. Other harvesting schemes include the use of long strips of piezoelectric polymers (energy harvesting eel) in ocean or
river-water flows [1, 46], the use of piezoelectric ‘cymbal’
transducers operated in the {3-3} mode [14, 15] and the
use of a piezoelectric windmill for generating electric power
from wind energy [31]. Jeon et al [13] have successfully
developed the first PZT MEMS power-generating device. Related works on modeling and design considerations for MEMS-scale piezoelectric-based energy harvesters can be found in [5,24,33].
Most published results have reported measurements of output voltage or power, while few have quantified the
efficiency of their devices. Umeda et al [48,49] and Goldfarb
and Jones [9] have studied the efficiency of electric power
generation with piezoelectric elements operated in the {
3-1} and {3-3} modes, respectively. Recently, Richards et al
[34] have provided an analytic formula to predict power
conversion efficiency, and showed that it depends on the electromechanical coupling coefficient and quality factor of the
device. Roundy [35] has further provided a general theory
of the effectiveness of vibration-based energy harvesting which can be applied to electromagnetic, piezoelectric, magnetostrictive and electrostatic transducer technologies. In addition, when a power harvester is applied to a system, it gives
rise to an additional damping effect. Lesieutre et al [22] have
pointed out that the damping added to a vibrating structure is due to the removal of electrical energy from the system. They have shown that the power harvesting system works similarly to a shunt damping system, except that the energy is stored instead of dissipated [20,44].
The research works cited above focus mainly on developing optimal energy harvesting structures. However, the electrical outputs of these devices in many cases are too small to power electrical devices directly. Thus, the methods of accumulating and storing parasitic energy are also the key to
developing self-powered systems. Sodano et al [42,43] have
investigated several piezoelectric power harvesting devices and the methods of accumulating energy by utilizing either
a capacitor or a rechargeable battery. Ottman et al [28,29]
have developed highly efficient electrical circuits to store the
generated charge or present it to the load circuit. They have claimed that at high levels of excitation the power output can be increased by as much as 400%. In contrast to the linear
load impedance adaptation by [28, 29], Guyomar et al [10]
and Lefeuvre et al [18,19] have developed a new power flow
optimization principle based on the extraction of the electric charge produced by a piezoelectric element, synchronized with the mechanical vibration operated at the steady state. They have claimed that the harvested electrical power may be increased by as much as 900% over the standard technique.
Badel et al [2] have extended their work to the case of pulsed
excitation.
In this paper, we propose an analysis of AC–DC power output for a rectified piezoelectric harvester. Many published results studying the conversion of energy from the oscillating mass to electricity have adopted a simple model proposed
by Williams and Yates [5, 13, 36, 39, 52]. It is based on
the assumption that the electrical damping term is linear and proportional to the velocity; however, this hypothesis may not be strictly valid in many cases. In addition, much work has been done on studying the optimal AC power flow, while little has considered the AC–DC power output. The former
includes [5, 24, 25, 30, 34, 38, 44], while the latter has
been studied recently in [10, 19,28]. As the electronic load
requires a stabilized DC voltage while a vibrating piezoelectric element generates an AC voltage, the desired output needs to be rectified, filtered and regulated to ensure electrical
compatibility. Thus, it is of importance to investigate the
optimal AC–DC power output to reflect the real electrical performance in many practical applications.
Specifically, we study the steady-state response of a piezoelectric generator connected to an AC–DC rectifier followed by a filtering capacitance and a resistor. This problem
has recently been studied by Ottman et al [28] and Guyomar
et al [10]. The former assumed that the vibration amplitude is not affected by the load resistance while the latter hypothesized that the periodic external excitation and the speed of mass are in phase. In contrast with estimates based on these two approaches, we take into account the global behaviour of the electromechanical system and derive a completely new
analytic expression of AC–DC power output in section2. We
show that the harvested power depends explicitly on a number of non-dimensionless parameters. With it, an effective power normalization scheme is provided and can be used to compare power harvesting devices of various sizes and with different
vibration inputs to estimate efficiencies. In section 3, we
derive the criterion for optimal load and power and study the asymptotic behaviour of power output for devices operated at the short and open circuit resonances, respectively. We show that selection of the correct operation frequency is important for achieving the maximal power flow, while this effect has been neglected in many other approaches. We next validate our theoretical predictions by both experimental results and
numerical simulations and find good agreement in section4.
In addition, we find that the discrepancies among these approaches become significant when the coupling coefficient and quality factor of the system are large. Finally, several design guidelines are recommended from our predictions. We
Analysis of power output for piezoelectric energy harvesting systems K Ce Piezoelectric Element u(t) Regulator F(t)
Energy Storage System η
M
Θ Cp
Figure 1. An equivalent model for a piezoelectric vibration energy
harvesting system.
2. Harvesting model
2.1. Governing equations
A piezoelectric energy harvester is often modeled as a
mass + spring + damper + piezo structure together with
an energy storage system schematically shown in figure 1
[19, 28, 34]. It consists of a piezoelectric element coupled
to a mechanical structure and is connected to a storage circuit
system. In this approach, an effective mass M subjected to
an applied forcing function F(t)is bounded on a spring of
effective stiffness K, on a damper of coefficientη, and on a
piezoelectric element characterized by effective piezoelectric
coefficient and capacitance Cp. For example, consider
a triple-layer bender mounted as a cantilever beam with polarization poled along the thickness direction as shown in
figure2. The electric field is generated through the direction
of thickness of the piezoelectric layers while strain is in
the axial direction; consequently, the transverse, or {3-1},
mode is utilized. The effective coefficients related to material constants and structural geometry can be derived using the
modal analysis [11,50] M= βM(mp+ mb) + ma, K = βKS 2 3 t3 L3+ ht2 L3 + 1 2 th2 L3 CEp11+ 1 12 h3 L3C E b11 , = βS(h + t) 2L e31, Cp= S L 2t ε S 33,
where βM, βK and β are constants derived from the
Rayleigh–Ritz approximation,e31andε33S are the piezoelectric
Piezoelectric Layers Poling Direction Strain (t) u Z(t) Vp h L S t t Base
+
-m
aFigure 2. A common piezoelectric-based power generator: a cantilever triple-layer bender operated in the{3-1} mode. The base is excited
with acceleration¨z(t).
and clamped dielectric constants, S and L are the width
and axial length of the cantilever beam, t and h, CpE11 and
CbE11, mp and mb are the thicknesses, elastic moduli and
masses of the piezoelectric and central passive layers, and
ma is the attached mass. We have performed a series of
experiments on a PZT triple-layer bender with configuration
similar to figure 2 to validate our prediction in section4.2.
Another less common piezoelectric power generator operated
in the longitudinal or{3-3}mode has been developed recently
by [13]. The advantage of utilizing this mode is that the
longitudinal piezoelectric effect is usually much larger than the transverse effect (d33> d31).
A vibrating piezoelectric element generates an AC voltage while the electrochemical battery needs a stabilized DC voltage. This requires an energy harvesting circuit to ensure
electrical compatibility. In figure 1, an AC–DC rectifier
followed by a filtering capacitanceCe is added to smooth the
DC voltage. A controller placed between the rectifier output and the battery is included to regulate the output voltage.
A simplified energy harvesting circuit shown in figure 3 is
commonly chosen for design analysis. Note that the regulation
circuit and battery are replaced with an equivalent resistor R
andVc is the rectified voltage across it. The rectifying bridge
is assumed to be perfect in the following study.
Let u be the displacement of the mass M and Vp the
voltage across the piezoelectric element. The governing
equations of the vibrator can be obtained by the conventional modal analysis [5,44]:
M¨u(t) + η ˙u(t) + K u(t) + Vp(t) = F(t), (1)
− ˙u(t) + Cp˙Vp(t) = −I (t). (2) An AC–DC harvesting circuit is connected to the power
generator, as shown in figure3,I(t)is the current flowing into
this circuit and is related to the rectified voltageVcby
I(t) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Ce ˙Vc(t) + Vc R ifVp = Vc, −Ce ˙Vc(t) − Vc R ifVp = −Vc, 0 if|Vp| < Vc. (3)
Piezoelectric Vibrator
Ce R Vc
+
_ Figure 3. A typical AC–DC harvesting circuit.
A sinusoidal mechanical excitation
F(t) = F0sinwt (4)
is applied to the system with F0the constant magnitude and
w(in rad s−1) the angular frequency of vibration. Note that in
most vibration-based power harvesting systems the source of
F(t)is due to the excitation of the base with acceleration¨z(t)
as shown in figure2.
Equation (3) is explained as follows. The rectifying bridge
is open circuited if the voltage|Vp|is smaller than the rectified
voltage Vc. As a result, the current flowing into the circuit
vanishes. On the other hand, when|Vp|reachesVc, the bridge
conducts and the piezo voltage is kept equal to the rectified
voltage; i.e.|Vp| = Vc. Finally, the conduction in the rectifier
diodes is blocked again when the absolute value of the piezo
voltage|Vp(t)|starts decreasing.
As most applications require the output DC voltageVcto
be stable, the common approach to achieving this is to assume
that the filter capacitorCe is large enough so that the output
voltage Vc is essentially constant [28]. Specifically,Vc(t) =
Vc(t) + Vripple where Vc(t) and Vripple are the average
and ripple of Vc(t), respectively. This average Vc(t) is
independent ofCeprovided that the time constantRCeis much
larger than the oscillating period of the generator [10]. The
magnitude ofVripple, however, depends onCe and is negligible
for large Ce. Under this hypothesis, Vc(t) ≈ Vc(t), and
therefore in the following, we use Vc, instead of Vc(t), to
represent the average ofVc(t)for simplicity of notation.
To solve (1)–(4) under steady-state operation, we first
determine the relation between the average value of the
rectified voltage and displacement magnitude. From (2) and (3)
the piezo voltage Vp(t)varies proportionally with respect to
the displacementu(t)if the rectifying bridge is blocked and
the outgoing piezoelectric current is zero. Therefore, solutions
ofu(t)andVp(t)are assumed to take the following forms:
u(t) = u0sin(wt − θ), Vp(t) = g(wt − θ), (5)
whereu0is the constant magnitude of displacement andg(t)
is a periodic function with period 2π and|g(t)| Vc. Let
T = 2wπ be the period of vibration, andaandbbe two time
instants (b − a = T2), such that the displacement u goes
from the minimum −u0 to the maximum u0. Assume that
˙Vp 0 during the semi-period froma tob. It follows that
b
a V˙p(t)dt= Vc− (−Vc) =2Vc. Note thatCe˙Vc(t) + Vc
R =0
fora< t < t∗during which the piezo voltage|Vp| < Vcand
the rectifier conducts whent∗ t < b. This gives from (3)
b a I(t)dt = T 2 Vc R (6) Cp Cp Θu(t) I(t) I(t) V (t)p V (t)p _ +
Figure 4. An equivalent circuit for the uncoupled model.
since the average current flowing through the capacitanceCe
is zero; i.e.abCe˙Vc(t)dt =0 for steady-state operation. The
integration of (2) from timeatobis therefore
−2u0+2CpVc= − T 2 Vc R, or Vc = wR wCpR+π2 u0. (7)
Notice that (7) is identical to that derived by [10,28].
The average harvested power can also be obtained in terms of the magnitude of displacement
P= V 2 c R = w22R wCpR+π2 2 u20. (8)
Thus, we need to find outu0to determineVc and P. There
are two approaches in the literature for estimating this [10,28].
We propose here another method for determiningu0, and show
that this new estimation is more accurate than the other two
in section4. Before showing that, we introduce the following
non-dimensionless parameters which will be used to simplify the analysis wn= K M, k 2 e = 2 K Cp , ζ = η 2√K M, = w wn, r= CpwnR, (9)
where wn is the natural frequency of short circuit,k2
e is the
alternative electromechanical coupling coefficient2, ζ is the
damping ratio and andrare the normalized frequency and
electric resistance. Finally, there are two resonances for the system since the piezoelectric structure exhibits both short circuit and open circuit stiffness. They are defined by
sc=1, oc=
1+ k2
e, (10)
where sc and oc are the frequency ratios of short circuit
and open circuit, respectively. Note that the frequency shift
is pronounced if the coupling factorke2is large.
2.2. Uncoupled analysis
Piezoelectric devices are frequently modeled as the current
source in parallel with their internal electrode capacitanceCp
as shown in figure4 [6,7,13,27,28]. This model is based on
2 The definition of k2
Analysis of power output for piezoelectric energy harvesting systems
the assumption that the internal current source of the generator is independent of the impedance of the external load. This is equivalent to assuming that the coupling is very weak and the
termVpcan be dropped from (1). As a result, the governing
equations (1) and (2) are simplified to be
M¨u(t) + η ˙u(t) + K u(t) = F(t), (11)
− ˙u(t) + Cp ˙Vp(t) = −I (t). (12)
As the displacement u(t) can be solved independently
from (11) using a simple harmonic analysis, ˙u(t) can be
treated as the known current source shown schematically in
figure4. The rectified voltageVc and the average harvested
powerPare therefore determined by (7) and (8). Finally, the
normalized displacementu0, voltageVcand powerPin terms
of non-dimensionless parameters (9) are described by
u0= u0 F0 K = 1 {4ζ2 2+ (1− 2)2}12 , (13) Vc = Vc F0 = r r +π2 k2 e {4ζ2 2+ (1− 2)2}12 , (14) P= P F2 0 wnM = 1 (r +π 2)2 ke2 2r {4ζ2 2+ (1− 2)2}. (15) 2.3. In-phase analysis
The uncoupled model assumes that the electromechanical coupling is very weak or the vibration amplitude is independent
of the equivalent resistive load R. If the coupling is not so
weak, Guyomar et al [10] have provided a new approach for
estimating the average harvested power. Indeed, they have assumed that the external forcing function and the velocity of
the mass are in phase. Precisely, (5) is changed to
F(t) = F0sinwt, ˙u(t) = u0wsinwt. (16)
As the derivation of the harvested power can be found in [10],
we here only list their main results for future comparison.
The normalized displacement u0, voltage Vc and power P
are summarized in terms of the non-dimensionless system parameters u0= u0 F0 K = 1 2ζ + 2ke2r (r +π 2)2 , (17) Vc = Vc F0 = r r +π2 k2 e 2ζ + 2k2er (r +π 2) 2 , (18) P= P F2 0 wnM = 1 (r +π 2)2 k2 er 2ζ + 2k2er (r +π 2)2 2. (19) 2.4. Analytic analysis
For a non-piezoelectric mechanical structure vibrating around
resonance, the in-phase assumption betweenF(t)and˙u(t)is a
fairly reasonable approximation in the case of low damping. However, we are not aware whether this assumption still
holds when non-small electromechanical coupling is taken into account. Hence, it is worth investigating this in detail here.
Let (1) be multiplied by ˙u(t) and (2) be multiplied by
Vp(t). Integration of the addition of these two equations from
timeatobgives the equation of the energy balance
b a F(t) ˙u(t)dt= b a η ˙u2(t)dt+ b a Vp(t)I (t)dt +1 2M˙u 2(t)|b a+12K u 2(t)|b a+ 12CpV 2 p(t)| b a. (20)
Suppose thatF(t),u(t)andVp(t)are given by (4) and (5). Let T = 2π
w anda andbbe two time instants (b− a = T2) such
that the displacementu goes from the minimum−u0 to the
maximumu0. The balance of energy (20) in this case becomes
b a F(t) ˙u(t)dt= b a η ˙u2(t)dt+ b a Vp(t)I (t)dt. (21)
We assume that ˙Vp 0 during this semi-period from a
to b. Note that Ce˙Vc(t) + VRc = 0 for a < t <
t∗ during which the piezo voltage |Vp| < Vc. This
also gives (Ce ˙Vc(t) + Vc
R)Vc = 0 for a < t < t∗.
The rectifier conducts later when the piezo voltage Vp
reaches the rectified voltage Vc, and from (3) Vp(t)I (t) =
Vc(Ce˙Vc(t) + Vc
R) during the conduction t∗ t < b.
These arguments listed above suggest
b a I(t)Vp(t)dt= V 2 c R T 2 (22)
for steady-state operation. Next, substituting (4) and (5) into
the equation of energy balance (21) results in
π 2ηwu 2 0+ π w V2 c R = π 2F0u0sinθ. (23)
Right now we have two equations (7) and (23) and three
unknownsu0, Vc andθ. We need a third one to solve them.
From (2), we have
˙Vp(t) =
Cp
[−I (t) + ˙u(t)]. (24)
Differentiating (1) with respect to timet and substituting (24)
into it, we find
Md dt¨u(t) + η d dt˙u(t) + K+ 2 Cp d dtu(t) − Cp I(t) = d dtF(t). (25)
Integrating (25) with respect to timetfromatoband using (5)
and (6) provides K − Mw2+ 2 Cp u0− π 2CpwRVc= F0cosθ. (26)
Finally, we are in a position to determineu0in terms of
system parameters. Combining both (23) and (26) gives
ηwu0+ 2 wR V2 c u0 2 + K− Mw2+ 2 Cp u0− π 2CpwRVc 2 = F2 0.(27)
As the magnitude of displacementu0is related to the rectified
voltage Vc by (7), the above equation (27) can be further
simplified to findu0. The result is
u0= F0 ηw + 2w2R (CpwR+π2)2 2 +K− w2M+ w2R CpwR+π2 2 1 2 .
The following summarizes our main findings: u= uF0 0 K = 1 2ζ + 2ke2r (r +π 2)2 2 2+1− 2+ k2er r +π2 212 , (28) Vc= Vc F0 = r r + π2 × k2e 2ζ + 2k2er (r +π 2)2 2 2+ 1− 2+ ke2r r +π2 212 , (29) P= P F2 0 wnM = 1 r +π2 2 × ke2 2r 2ζ + 2k2er (r +π 2)2 2 2+ 1− 2+ ke2r r +π2 2, (30)
where (30) is interpreted as follows. Suppose the source of
the forcing function comes from the vibration of the base
of the structure, then this gives F0 = M A where A is the
magnitude of acceleration of the exciting base. It follows that the harvested average power per unit mass is described by
P
M =
A2
wn P(r, , ke, ζ).
This shows that the harvested average power per unit mass depends on the characteristics of the input vibration (frequency
and acceleration A), the normalized electric resistancer,
the short circuit resonancewn, the mechanical damping ratio
ζ, and the overall electromechanical coupling coefficientk2
e
of the system. Thus, the scheme to optimize the power
either by tuning the electric resistance, selecting suitable operation points or adjusting the coupling coefficient by
optimal structural design can be guided completely by (30).
3. Optimal resistance and power
Suppose , ke and ζ are fixed. The design criterion for
reaching the maximal power flow under steady-state operation can be obtained by tuning the load impedance according to
∂
∂rP(r, , ke, ζ)| ,ke,ζ =0. (31)
We use the notationroptto represent the solution of (31), and
ropt = ropt( , ke, ζ)in general. Besides, the superscript ‘opt’
denotes functions evaluated at the optimal load resistanceropt.
For example,
Popt = P(ropt, , ke, ζ);etc.
The selection of the suitable operation frequency is also important to maximize the average harvested power, and we
will discuss this in section3.3.
3.1. Uncoupled analysis
Substituting (15) into (31), we find the optimal load is
ropt = π
2 or R
opt= π
2Cpw. (32)
It follows from (13), (14) and (15) that the normalized
displacement, voltage and power evaluated at the optimal load are uopt0 = u opt 0 F0 K = 1 {4ζ2 2+ (1− 2)2}12 , (33) Voptc = Vcopt F0 = 1 2 k2 e {4ζ2 2+ (1− 2)2}12 , (34) Popt = P opt F2 0 wnM = 1 2π k2 e {4ζ2 2+ (1− 2)2}. (35)
In the uncoupled model the optimal harvested power flow can be achieved by tuning the load impedance to match the internal
impedance of the piezoelectric generator, i.e. Ropt = π
2Cpw.
In addition, the rectified voltageVcopt = 12Voc where Voc is
the maximum voltage at the open circuit condition for the
uncoupled model [28].
3.2. In-phase analysis
Lefeuvre et al [19] have questioned the soundness of the
uncoupled model and proposed a modified estimation of the optimal load based on the in-phase assumption. The results are classified according to the inequality ofke2
ζ−2π and are listed
below for future comparison.
Case 1: k2e
ζ −2π 0. The optimal normalized load,
displacement, voltage and power evaluated at the optimal condition are roopt= π 2 , (36) uopt0 = u opt 0 F0 K = 1 {2ζ +k2e π} , (37) Voptc = V opt c F0 = 1 2 k2 e {2ζ + ke2 π} , (38) Popt= P opt F2 0 wnM = 1 2π k2 e {2ζ + k2e π}2 . (39)
Note that the optimal resistance (36) is the same as (32), and
the optimal power in (39) is close to (35) provided thatk2e
ζ 1
Analysis of power output for piezoelectric energy harvesting systems Case 2: ke2
ζ −2π 0. Suppose the electromechanical
coupling factor k2
e is large and the damping is small. The
optimal resistance roopt in (36) turns out to be the one
minimizing the power flow. There are two new optimal
resistances calledraopt andrbopt to maximize the power in this
case, and the corresponding normalized displacement, voltage and power are
raopt= 1 2 2 ⎧ ⎨ ⎩ ke2 ζ − π − k2 e ζ − π 2 − (π )2 ⎫ ⎬ ⎭, (40) rbopt= 1 2 2 ⎧ ⎨ ⎩ k2 e ζ − π + k2 e ζ − π 2 − (π )2 ⎫ ⎬ ⎭. (41) uopt0 |r=ropt a = uopt0 F0 K r=ropta = uopt 0 |r=ropt b = uopt0 F0 K r=rbopt = 1 4ζ , (42) Voptc |r=ropt a = Vcopt F0 r=ropt a = ke2 8ζ ⎧ ⎨ ⎩1− 1−2π k2 e ζ ⎫ ⎬ ⎭ , (43) Voptc |r=rbopt= Vcopt F0 r=rbopt = k2 e 8ζ ⎧ ⎨ ⎩1+ 1−2π k2 e ζ ⎫ ⎬ ⎭ , (44) Popt|r=ropt a = Popt F2 0 wnM r=ropt a = Popt |r=roptb = Popt F2 0 wnM r=ropt b = 1 16ζ. (45)
Note that raopt < roopt < rbopt. It is interesting to see
that the harvested average power has two identical maxima and depends only on the internal damping of the generator.
Lefeuvre et al [19] have interpreted the appearance of two
optimal resistances as characteristic of a strongly coupled system. However, this has to be taken with caution since there are always two optimal loads for each applied frequency in the
in-phase model provided that k2e
ζ −2π 0. We will discuss
it in section3.3.
3.3. Analytic analysis
The power derived from the analytic analysis is given by (30).
Although it extensively describes the characteristics of the
harvesting system, the complicated nature of (30) makes it
difficult to derive the closed form solution of the optimal
resistance from (31). Alternatively, we discuss the functional
behaviour of (30) according to the different ranges of the
parameter k2e
ζ. We study the small and medium ranges of k
2 e
ζ
in sections 4.1 and 4.2, respectively. Here we provide an
analysis to show that the harvested power can be maximized at two different electrical loads at the respective operating points provided that ke2
ζ 1. However, we are not aware under
exactly what condition there exist two optimal pairs. But if
the AC–DC harvesting circuit shown in figure3is changed to
an AC circuit, it can be shown that two optimal pairs appear whenever
k2
e
ζ 4(ζ +1).
Returning to the standard AC–DC circuit, our results of numerous numerical simulations suggest
k2
e
ζ 10 (46)
as the rule of thumb for the appearance of two optimal pairs, and we will use it as a criterion for designing a strongly coupled electromechanical system.
Case 1: Short circuit resonance. The power Pin (30) for
= sc=1 can be expressed as P F2 0 wnM = 1 ζ r x {4[(r +π2) +(r+rxπ 2)] 2+ r2x2}, (47) wherex = k2e
ζ . Suppose the parameterx 1. To optimize the
power in (47),rhas to be small or proportional to the inverse
ofx; otherwise, the power will tend to zero for non-smallr
whilexremains extremely large. This gives
P F2 0 wnM ≈ 1 ζ r x {4[π2+π2r x]2+ r2x2} (48)
for smallrandx1. The optimal power flow can be obtained
by differentiating (48) with respect tor. It follows that
rscopt≈ π2 √ 16+ π2 1 k2 e ζ ifx= k 2 e ζ 1. (49)
The corresponding normalized displacement, voltage and power are uopt0 = u opt 0 F0 K ≈ 16+ π2 32+2π2+8√16+ π2 1 2ζ, (50) Voptc = V opt c F0 ≈ π 32+2π2+8√16+ π2, (51) Popt= Popt F2 0 wnM ≈ 1 8+2√16+ π2 1 ζ. (52)
Case 2: Open circuit resonance. We set x = k2e
ζ again.
The power P in (30) for the applied frequency operated at
oc= 1+ k2 ecan be expressed as P F2 0 wnM = 1 ζ r x 2 oc 4 (r oc+π2) +(r rx oc+π2) 2 2 oc+π 2 4x2 . (53)
To estimate the optimal power in (53) for the case ofx 1,
rhas to be proportional toxby examining the termr xin the
numerator and π42x2in the denominator. This shows thatrhas
to be large to maximize (53). Hence, we may rewrite (53) as
P F2 0 wnM ≈ 1 ζ r x 2 oc 4 r oc+ xoc 2 2 oc+π 2 4x2 (54)
Table 1. The relation between the system parameters k2
eandζ and
the normalized electric resistance, displacement, voltage, current and power designed at the maximal power flow operated at the short circuit ( sc) and open circuit ( oc) resonances. The normalized current is defined as I =Vc
r . Note that the condition k2
e ζ 1 is implied in the analysis.
Optimal conditions sc oc Resistance ropt sc ∝ k12e ζ < ropt oc ∝(1+k12 e) k2 e ζ Displacement uopt0 ∝1ζ > u opt 0 ∝ζ(√1 1+k2 e) Voltage Voptc ∝ 1 < V opt c ∝√1 1+k2 e k2 e ζ Current Iopt∝ke2 ζ > I opt ∝ 1+ k2 e Power Popt∝1 ζ = P opt ∝1 ζ
providedx 1. The optimal power flow can be obtained by
differentiating (54) with respect tor. This gives
rocopt≈ √ 16+ π2 4 k2 e ζ 1+ k2 e ifx= k 2 e ζ 1. (55)
The corresponding normalized displacement, voltage and power are uopt0 =u opt 0 F0 K ≈ 16+ π2 32+2π2+8√16+ π2 1 2ζ 1+ k2 e , (56) Voptc = V opt c F0 ≈ 16+ π2 32+2π2+8√16+ π2 k2 e 2ζ 1+ k2 e , (57) Popt= P opt F2 0 wnM ≈ 1 8+2√16+ π2 1 ζ. (58)
Discussions: In the case of large k2e
ζ, we find that for each
applied frequency there is only one optimal load to maximize
the power output. In addition, we have shown that the
harvested power has two identical peaks, but is optimized at different resistances and operation frequencies; i.e.
P(rscopt, sc, ke, ζ) = P(rocopt, oc, ke, ζ)
provided thatk2e
ζ 1. These results are in contrast with those
obtained by the in-phase assumption. There always exist two
optimal resistances given by (40) and (41) for every frequency
in the in-phase model whenever ke2
ζ −2π 0 (see also
figure9(f)). Besides, the in-phase model predicts the identical
displacement evaluated at two optimal loads (see (42)) while
our analytic analysis predicts unequal peaks of displacement;
c.f. (50) and (56). This result is crucial in the design of
micro-scale power generators [5].
Next, (49) and (55) suggestrscopt ∝ (k21
e/ζ ) 1 while
rocopt ζ1. Therefore,rscopt can be made as small as possible
by increasing the electromechanical coupling coefficient k2
e
while rocopt has an upper bound. Finally, table1summarizes
the relation between the system parametersk2
e andζ and the
normalized load, displacement, voltage, current and power designed at the maximal power flow operated either at the short circuit ( sc) or open circuit ( oc) resonances.
4. Comparisons
We now show that in section 4.1 the various forms of
power derived from different approaches are almost the same
provided that the parameter k2e
ζ is small, and we will use
examples including experimental validation to demonstrate that the discrepancies among these analyses become large
when the parameter k2e
ζ is increasing. In section4.3 we find
that the average harvested power is maximized at two optimal
loads operated at different frequencies in the case of largek2e
ζ. 4.1. Smallk2e ζ Suppose thatk2 e 1 and k2 e
ζ 1. Thus, the shift in frequency
from scto ocis not pronounced for smallk2e. Let the applied
frequency ratio be operated between scand oc. We may
set
2=1+ f k2
e, 0 f 1. Setx = k2e
ζ. The power P derived from the analytic analysis
in (30) can be expressed as P F2 0 wnM = r 4ζ r +π2 2 × x 1+(r +xrπ 2) 2 2 + x2 4 2 − f + r r +π 2 2 ≈ r x 4ζ r +π2 2{1+O(x)} ≈ k2er 4ζ2 r +π 2 2 (59)
provided that x 1 and ≈ sc ≈ oc ≈ 1. The
notation O(x)denotes the higher order terms which tend to
zero asxtends to zero. Comparing (59) with the power derived
from the uncoupled assumption (15) and that from the in-phase
assumption (19) justifies our assertion. Figures5(a) and (b)
are the normalized rectified voltage and average harvested power versus the normalized resistance around resonance in
the case of ke = 0.05 and ζ = 0.03. The solid, dashed
and long-dashed lines are results derived from the analytic,
in-phase and uncoupled solutions. These three lines are
almost coincident. We therefore conclude that the conventional uncoupled solution is suitable ifke2
ζ 1.
4.2. Mediumk2e
ζ
The discrepancies among these approaches become significant
when the ratio ke2
ζ increases. For piezoelectric generators
operated in the {3-1} mode, k2
e can approach e2 31 CE 11ε S 33 if the
structure is made up entirely of piezoelectric materials [5].
Most electromechanical structures are made up of both
piezoelectric and non-piezoelectric materials. The factor k2
e is then usually less than the theoretical value. On the other
hand, the coupling coefficientk2ecan approach its upper bound
for micro-scale devices since the contribution of piezoelectric elements to the overall structural stiffness is significant in this
Analysis of power output for piezoelectric energy harvesting systems
Normalized Resistance Normalized Resistance
Uncoupled Solution In-Phase Solution Analytic Solution Normalized V oltage 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Uncoupled Solution In-Phase Solution Analytic Solution Normalized Power 0.12 0.1 0.08 0.06 0.04 0.02 0 (a) (b)
Figure 5. Comparisons in the case of smallk2e
ζ. We use ke= 0.05 and ζ = 0.03 (k 2 e
ζ = 0.083). (a) Normalized rectified voltage versus normalized resistance. (b) Normalized harvested power versus normalized resistance.
case [5]. We then investigate it in detail here. We assume
ke=0.4 andζ =0.03(k 2 e
ζ =5.33)in the following analysis.
The normalized displacement, voltage and power against
the normalized electric resistance are plotted in figures6(a)–
(c) operated at the short circuit resonance and in figures6(d)–
(f) operated at the open circuit resonance. The long-dashed, dashed and solid lines are results calculated based on the
uncoupled, in-phase and analytic solutions. As expected,
predictions from the uncoupled analysis are far away from those predicted by either analytic or in-phase solutions since the electromechanical coupling is not small in this case. We then conclude that the uncoupled solution is not suitable for
medium or even largek2e
ζ.
Figure6also reveals substantial differences between our
analytic and the in-phase analyses. We therefore use both simulation and experiment to determine which approaches predict more accurate behaviour of the energy harvesting
system. Consider the numerical simulation first. Notice that (1)
and (2) can be transformed to an equivalentRLCcircuit with
R = η2 as resistance,L =
M
2 as inductance andC =
2 K as capacitance. We use the software PSpice to simulate this equivalent circuit connected to the AC–DC harvesting circuit
shown in figure 3. The results are illustrated in figure 7
where we plot the normalized power versus electric resistance at short circuit and open circuit resonances. The numerical
results are marked in figure7using open circles. Apparently,
the numerical simulations favor results predicted based on our analytic solutions. In particular, our approach accurately predicts the optimal electric load resistance maximizing the
average harvested power. The optimal loadrscopt (analytic) =
0.45 is smaller at sc whilerocopt (analytic) = 4.64 is larger
at oc. However, the optimal resistances predicted by the
in-phase solutions at scand ocare very close:r
opt
sc (in-phase) =
1.57 androcopt(in-phase) =1.46, and therefore are not suitable
for the design.
Finally, we validate the analytic solution by our recent experiment. The specimen is a piezoelectric triple-layer bender
with the overall dimension 40 mm×20 mm×0.36 mm (L×
S× (h + 2t)) as shown similarly in figure 2. The overall
mass of the beam ismp+ mb = 2.2509 g and an attached
massma = 0.4207 g is put at the tip. The measured open
circuit and short circuit resonances are 52.9 and 53.7 Hz,
respectively. This gives a coupling factor ke around 0.17.
The mechanical damping ratio is measured at about 0.01, and therefore k2e
ζ = 2.89. The applied acceleration is around
1.856 m s−2and is slightly dependent on electrical resistances.
The power harvesting circuit is chosen using the standard AC–
DC circuit illustrated in figure3. The values of rectified voltage
Vc are measured at the open circuit resonance for various
electrical resistances and are marked in figure 8(a) using
dark circles•. The corresponding values of harvested power
are also plotted against the various electrical resistances in
figure8(b). The optimal electrical resistance can be determined
from figure8(b) and is aroundRexpopt ≈200 k . The predicted
results from the uncoupled, in-phase and analytic solutions are represented by long-dashed, dashed and solid lines in
figure8. As expected, the uncoupled solutions are not able
to reflect the electrical performances of the system. The in-phase solutions also overestimate the measured voltage and power, and underestimate the optimal load (Rinopt-phase≈88 k ). On the other hand, the analytic solutions are close to the experimental observations and the predicted optimal load is
aroundRanalyticopt ≈210 k which is pretty close to the measured
one. The deviations between the experimental results and the analytic solutions are believed to be due to diode loss which has not been incorporated in the current analysis.
4.3. Largek2e
ζ
The shift in frequency is significant if either the piezoelectric constant or the contribution of the piezoelectric element
to the overall stiffness is large; i.e. k2
e is large. In
particular, a piezoelectric power generator operated in the
{3-3}(longitudinal) mode can have high coupling coefficient
k2 e approaching e2 33 CE 33ε S 33
if the piezoelectric element constitutes
Short Circuit Resonance Open Circuit Resonance
Short Circuit Resonance Open Circuit Resonance
Short Circuit Resonance Open Circuit Resonance
Uncoupled Solution In-Phase Solution Analytic Solution Normalized Resistance Normalized Displacement Uncoupled Solution In-Phase Solution Analytic Solution Uncoupled Solution In-Phase Solution Analytic Solution Normalized Vol tage Normalized Power Normalized Vol tage Normalized Power Uncoupled Solution In-Phase Solution Analytic Solution Uncoupled Solution In-Phase Solution Analytic Solution Uncoupled Solution In-Phase Solution Analytic Solution 0 1 2 3 4 5 6 7 8 9 10 Normalized Resistance 0 1 2 3 4 5 6 7 8 9 10 Normalized Resistance 0 1 2 3 4 5 6 7 8 9 10 Normalized Resistance 0 1 2 3 4 5 6 7 8 9 10 Normalized Resistance 0 1 2 3 4 5 6 7 8 9 10 Normalized Resistance 0 1 2 3 4 5 6 7 8 9 10 17 16 15 14 13 12 11 10 9 8 7 6 Normalized Displacement 16 15 14 13 12 11 10 9 8 7 5 6 8 7 6 5 4 3 2 1 0 2.5 2.25 2 1 1.75 1.5 1.25 0.75 0.5 0.25 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0 (a) (d) (b) (e) (c) (f)
Figure 6. Comparisons for mediumk2e
ζ. We use ke= 0.4 and ζ = 0.03 (k 2 e
ζ = 5.33): (a)–(c) are the normalized displacement, voltage and power versus normalized resistance operated at scwhile (d)–(f) are those operated at oc.
ratio is small, or the factor k2e
ζ is large, the selection of the
correct operating frequency is very important for achieving the maximal power. Recently, much experimental effort has been made to fabricate small-scale piezoelectric cantilever beams with interdigitated electrodes on the beam surface to produce
the{3-3}mode using various materials [12,13,47]. Some of
their chosen materials such as PZN-PT and PMN-PT relaxor
ferroelectrics can have even higher piezoelectric constants than conventional PZT. As a result, the shift in resonance is
expected to be pronounced due to largeke2in these micro-scale
devices, and we study this effect on power harvesting now.
We assume ke = 1.14 and ζ = 0.03. This gives
sc = 1, oc = 1.52,
k2 e
ζ = 43.3. As illustrated in
Analysis of power output for piezoelectric energy harvesting systems Normalized Resistance Normalized Pow er Simulation In-Phase Solution Analytic Solution Simulation In-Phase Solution Analytic Solution 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0 0 1 2 3 4 5 6 7 8 9 10 Normalized Resistance Normalized Power 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0 0 1 2 3 4 5 6 7 8 9 10
Short Circuit Resonance Open Circuit Resonance
(a) (b)
Figure 7. Numerical validation for medium k2e
ζ. We use ke= 0.4 and ζ = 0.03 ( k2
e
ζ = 5.33). The normalized power versus normalized resistance is plotted in (a) operated at scand in (b) operated at oc.
Resistance (kΩ) 3 1 Voltage (V) Experiment Uncoupled Solution In-Phase Solution Analytic Solution Power (uW) Experiment Uncoupled Solution In-Phase Solution Analytic Solution 0 100 200 300 400 500 600 700 Resistance (kΩ) 0 100 200 300 400 500 600 700 7 6 5 4 2 0 100 90 80 70 60 50 40 30 20 (a) (b)
Figure 8. Experimental validation for medium ke2
ζ. The structure is excited at its open circuit resonance. The parameters keandζ are
measured to be 0.17 and 0.01(k2e
ζ = 2.89). (a) Rectified voltage versus resistance. (b) Harvested power versus resistance.
not realistic if k2e
ζ is not small. We then omit them here.
The normalized displacement, voltage and power are plotted against the normalized electric resistance and frequency in
figures 9(a)–(c) calculated based on the analytic solutions as
well as in figures 9(d)–(f) calculated based on the in-phase
solutions. In figure 9(c) we see clearly that the harvested
power has two optimal values of equal amount evaluated at
two different resistances and frequencies; i.e. Pis maximized
at (ropt 1 , opt 1 ) = (0.062,1.025), (ropt 2 , opt 2 ) = (17.299,1.492). (60)
Note that (60) confirms our theoretical predictions made in
section3.3which shows thatr1opt 1 and opt1 ≈ sc and
r2opt 1 and opt2 ≈ oc. However, the in-phase approach
fails to predict the optimal operating frequencies since there are always two optimal electric loads for each applied frequency
as shown in figure9(f). The effect of the optimal selection
of operating frequency in generating the desired properties is neglected in the in-phase analysis, which turns out to be important in the design criterion.
Switching between these two peaks can be achieved
by varying the electric loads along the curve ropt( , ke, ζ)
obtained from (31). The implication of this result can be
applied to enhancing the efficiency of charging a battery.
Indeed, Ottman et al [28] have shown that the efficiency of
direct charging of a battery without a suitable controller is pretty slow. The main reason for this is that the equivalent electrical resistance of a battery is much smaller than the
optimal electrical resistance. Turning the load impedance
needs a special power converter [29], which in turn may
consume additional extracted energy and make the circuitry
unrealistic (see the discussion of [43]). Alternatively, if the
(b)
(d)
(e)
(c)
Figure 9. Comparisons for large ke2
ζ. We use ke= 1.14 and ζ = 0.03 (k 2 e
ζ = 43.3). The normalized displacement, voltage and power are plotted against the normalized electric resistance and frequency in (a)–(c) calculated based on the analytic solutions as well as in (d)–(f) calculated based on the in-phase solutions.
(This figure is in colour only in the electronic version)
largek2e
ζ, the equivalent impedance of a battery can be matched
to the optimal load by selecting a suitable operating point close
to scsince the harvested power has a peak around there.
Next, unlike the power, the displacement and voltage
evaluated at these two optimal conditions (60) differ
significantly (compare figures 9(a) and (b) with (c)). The
displacement has two hills with one chain concentrated at sc
and the other at oc. But unlike the power, it is larger at sc
than at oc since the overall damping of system is higher at
the open circuit resonance (see also the predictions in table1).
The advantage of operating at the second peak close to ocis
space-saving if the smaller device is preferred.
Finally, figure 9(b) clearly demonstrates that the
normalized rectified voltage evaluated at (r2opt, opt2 ) is one
order of magnitude higher than that evaluated at(r1opt, opt1 ). It is thus advantageous to operate at the open circuit resonance to
Analysis of power output for piezoelectric energy harvesting systems
overcome the minimum voltage requirement of the rectifying
bridge in the micro-scale device. On the other hand, the
steady-state current evaluated at the first peak of power around
sc is one order of magnitude higher than that evaluated at
the second peak of power around oc, since P = I V and
the optimal power is identical at these two peaks. We may apply this result to charging batteries. Indeed, optimizing the power flowing into the battery is equivalent to maximizing the current into it as the battery voltage is essentially constant or only changes slowly. Hence, operating at the short circuit resonance together with the corresponding optimal load could enhance the efficiency of charging the battery directly without adjustable convectors.
5. Conclusions
We study the optimal AC–DC power output for a vibrating piezoelectric generator connected to an energy harvesting circuit. In contrast with estimates proposed by the uncoupled
and in-phase approaches [10, 28], we show that the power
extraction depends on the input vibration characteristics (frequency and acceleration), the mass of the generator, the electrical load, the natural frequency, the mechanical damping ratio, and the electromechanical coupling coefficient of the
system. An expression for average harvested power that
incorporates all of these factors is analytically developed
by (30). Thus, the scheme to optimize the power either
by tuning the electric resistance, selecting suitable operation points, or adjusting the system coupling coefficient by optimal
structural design can be guided completely by (30). Further,
it is also highly recommended that all these parameters be provided in all future publications to facilitate the relative comparison of various devices.
We compare our approach to others proposed based on
the uncoupled and in-phase assumptions. We show that
the conventional uncoupled solution is suitable provided that the ratio ke2
ζ 1, while the discrepancies between these
distinct approaches become significant when k2e
ζ increases.
Figures6and 9highlight the striking contrast in normalized
displacement, voltage and power output calculated based on the uncoupled, in-phase and analytic solutions for non-small ratio k2e
ζ. We perform a series of experiments and numerical
simulations to evaluate these approaches. We find our analytic solutions are in good agreement with both experiments and
simulations as shown in figures 7 and 8. The in-phase
solutions, however, overestimate the measured voltage and power and underestimate the optimal load compared with experimental observations.
We make a particular study of the important case when the shift in device natural frequency is pronounced and the quality factor of the system is large, since this has been neglected
by most current optimization schemes, as pointed out by [5].
The effect of this frequency shift is expected to be more pronounced for micro-scale harvesters, since the contribution of piezoelectric elements to the overall structural stiffness is much larger than for bulk generators. In this situation, the harvested power is shown to have two optima evaluated at (ropt 1 , opt 1 )and(r opt 2 , opt 2 ), where opt
1 is close to scand the
electric loadr1optis very small, while opt2 is close to oc and
r2opt is large. Table1sheds light on the conspicuous contrast
in the normalized displacement, electric resistance, voltage, current and power evaluated at these two power optimal pairs. Finally, for devices with strong electromechanical coupling, several design guidelines including enhancing the efficiency of charging a battery directly are recommended.
Acknowledgments
We thank Professors K C Wu and C K Lee for their constant encouragement and support. We are grateful to Professors C S Yeh, K C Chen, W J Wu and W P Shih for many helpful and stimulating discussions. We are glad to acknowledge the Ministry of Economic Affair for support under grant no 94-EC-17-A-05-S1-017 (WHAM-BioS).
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