ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005
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Investigation of Electrostrictive Polymers for
Energy Harvesting
Yiming Liu, Student Member, IEEE, Kai Liang Ren, Heath F. Hofmann, Member, IEEE,
and Qiming Zhang, Senior Member, IEEE
Abstract—The recent development of electrostrictive
polymers has generated new opportunities for high-strain
actuators. At the current time, the investigation of using
electrostrictive polymer for energy harvesting, or mechanical to electrical energy conversion, is beginning to show
its potential for this application. In this paper we discuss
the mechanical and electrical boundary conditions for maximizing the energy harvesting density and mechanical-toelectrical coupling of electrostrictive materials. Mathematical models for different energy harvesting approaches were
developed under quasistatic assumptions. Energy harvesting densities then are determined for representative electrostrictive material properties using these models. Comparison with a magnetic-based energy harvesting system
suggests that electrostrictive energy harvesting systems are
preferable for “small” energy harvesting applications with
low-frequency excitation.
crystals [7]. Less known is that these materials also can be
used for mechanical-to-electrical energy harvesting. This
process is based on electric field-induced molecule motion
or phase transition, although these materials also experience an electrostatic effect, which also plays a minor role in
the energy harvesting. However, a significantly lower field
intensity is required when compared to fully electrostaticbased systems.
Electrostriction is generally defined as quadratic coupling between strain and electrical field. The strain Sij
and the electric flux density Dm are expressed as independent variables of the electric field intensity Em , En and
the stress Tij by the constitutive relation as:
Sij = sE
ijkl Tij + Mmnij En Em ,
Dm = εmn En + 2Mmnij En Tij ,
I. Introduction
iezoelectric ceramic (such as PZT)-based devices
have long been used for mechanical-to-electrical energy harvesting [1]–[4]. However, those materials tend to
be stiff and limited in mechanical strain abilities; so for
many applications in which low-frequency and large-stroke
mechanical excitations are available (such as human movement), direct coupling of the piezoelectric ceramic to the
excitation source yields a very low input mechanical energy [5]. Organic materials, however, are softer and more
flexible; therefore, the input mechanical energy is considerably higher under the same mechanical force. Piezoelectric
polymers such as PVDF, unfortunately, have a much lower
piezoelectric coefficient compared to piezoelectric ceramic
materials. A study has shown that the energy harvesting is lower than with piezoelectric ceramic bimorphs [5].
Electrostatic-based systems, such as dielectric elastomers,
typically require a very high electric field intensity (1–
6 kV on a film thickness of 50 µm, which corresponds to
20–120 MV/m) to achieve significant energy harvesting
[6]. Electrostrictive polymers recently have been discovered that generate large strain (above 5%) under moderate electric field intensity (400–800 V on a 20 µm film,
which corresponds to 20–40 MV/m), and the mechanical
energy density is comparable to that of piezoelectric single
P
Manuscript received June 30, 2004; accepted April 29, 2005. This
work was supported by Office of Naval Research under Grant No.
N00014-03-01-0569.
The authors are with the Department of Electrical Engineering,
The Pennsylvania State University, University Park, PA 16802 (email: hofmann@ee.psu.edu).
(1)
where sE
ijkl is the elastic compliance, Mnmij is known
as the electric-field-related electrostriction coefficient, and
εmn is the linear dielectric permittivity. An isotropic electrostrictive polymer film contracts along the thickness direction and expands along the film direction when an electric field is applied across the thickness, assuming that the
only nonzero stress is that applied along the length of the
film. The constitutive relation then is simplified as:
S = sT + M E 2 ,
D = εE + 2M ET,
(2)
E = E3 , D = D3 , S = S1 ,
T = T1 , s = s11 , M = M31 ,
ε = ε33 .
(3)
where:
Because our analysis is based on these simplified relations, we will no longer specify the orientation of parameters and field variables in the following. We also will
neglect the effects of dimensional changes of the material
on the electrical boundary conditions. For example, a constant voltage applied to an electrostrictive device will be
assumed to be equivalent to a constant electric field in the
material. As the strains in these materials tend to be less
than 5%, this assumption is reasonable and dramatically
simplifies the analysis.
c 2005 IEEE
0885–3010/$20.00 2412
ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005
may be difficult to achieve. In the following, we will analyze electrical boundary conditions that can be applied
to the device fairly easily with power electronic circuitry.
The concept of the coupling factor, as defined in the IEEE
standard for piezoelectric materials [9], is useful as a figure
of merit for energy harvesting, and is given below:
W1
,
(4)
k=
W1 + W2
Fig. 1. Energy harvesting cycle.
II. Energy Harvesting Cycle and Boundary
Conditions
The mechanical-to-electrical energy harvesting in electrostrictive materials is illustrated by example in the
mechanical stress/strain and electric field-intensity/fluxdensity plots shown in Fig. 1. Initially the material shown
in Fig. 1 has no applied stress, then stress is applied and
the state travels along path A. The applied stress then is
reduced. Due to a change in the electrical boundary conditions, the contraction path will not follow path A but
path B. Both in the mechanical and electrical planes, the
material state traverses a closed loop. In the mechanical
plane the rotation is counter clockwise, and in the electrical plane the rotation is clockwise. This rotation designates
that the net energy flow is from the mechanical to the electrical. The area enclosed in the loop of the mechanical and
electrical planes is equal and is the converted energy density in units of J/m3 . It has been mistakenly assumed in
other work that the converted energy is equal to the maximum stored electrical energy, although this is not true as
not all of this energy is available to be scavenged [8].
For quasistatic mechanical-to-electrical energy harvesting, one can apply various electrical boundary conditions
during the stress variation cycle to break the symmetry of
stretching and contracting of the electrostrictive polymer,
and thereby harvest energy. In order to maximize the harvesting energy density, the boundary conditions have to be
designed so that the area enclosed in the loop is as large
as possible without exceeding the limitations of the polymer material (e.g., maximum stress, breakdown field). One
approach is to adjust the mechanical boundary conditions
through the design of a mechanical transformer—such as
a cantilever, bimorph structure, or hydraulic coupler—to
modify the mechanical stresses applied to the material in
order to create a better match with the material’s properties. Another approach is to control the electrical boundary
conditions of the material. In the following we will analyze
several different methods of varying these electrical boundary conditions.
Ideally the energy harvesting cycle consists of the
largest loop possible, bounded only by the limitations of
the material. However, actual implementation of the optimal energy harvesting cycle in an energy harvesting circuit
where W1 +W2 is the input mechanical energy density, and
W1 is the output electrical energy density. The coupling
factor often is associated with a specific set of electrical
boundary conditions, as will be discussed in the following.
In this work we shall use the general definition above and
consider several different cases of electrical boundary conditions. The mechanical excitation in each case will be the
same: the application of stress from zero to a maximum
value Tmax , then returning to zero.
A. Coupling Factor Determined from Linearized Model
of Electrostriction
For many transducer and actuator applications, electrostrictive materials often are exposed to a large direct
current (DC) bias field and a smaller alternating current
(AC) perturbation. Under these conditions, local linearization, based on Taylor expansion, often is used to simplify
the quadratic nature of electrostriction, which then takes
the same form as the piezoelectric equations. Therefore,
an equivalent piezoelectric constant as a function of the
bias field can be determined for the electrostrictive material. The linearization of the constitutive relation defined
in (2) about bias values E0 and T0 can be shown to be:
S ≈ S0 + s (T − T0 ) + 2M E0 (E − E0 ) ,
D ≈ D0 + (ε + 2M T0 ) (E − E0 ) + 2M E0 (T − T0 ) . (5)
Using S0 , D0 , T0 , and E0 as the equilibrium points and
the deviations from those points as the (linearized) variables, the linearized piezoelectric constant is given by:
d = 2M E0 ,
(6)
and the linearized dielectric constant is given by:
ε = ε + 2M T0 .
(7)
The coupling coefficient, or the ratio of electrical energy
output-to-mechanical energy input, for the linearized system then can be calculated using the standard quasistatic
coupling factor for piezoelectric materials, defined in [9] as:
d
2M E0
k=√
=
.
εs
(ε + 2M T0 ) s
(8)
However, in order to maximize the energy harvested,
large variations in applied stress and electric field are required; hence, the linearized model may provide inaccurate
liu et al.: mechanical and electric boundary conditions for energy harvesting
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where γ corresponds to the relative change in dielectric
constant due to applied stress and is defined as follows:
γ=
Fig. 2. Energy harvesting cycle under constant field (1–2) and open
circuit (2–3) electrical boundary conditions.
or misleading conclusions as to how to achieve the highest
harvesting energy density and harvesting efficiency. In energy harvesting applications the quadratic model, therefore, will provide a more accurate understanding; hence,
the following analysis will be based upon the electrostrictive equations provided in (2).
B. Energy Harvesting Cycle #1: Constant Field and
Open-Circuit Electrical Boundary Conditions
The standard coupling factor expressed above for piezoelectric materials is based upon electrical boundary conditions in which the device is electrically short circuited as
stress is applied and open circuited as the stress is removed
[9]. In the case of electrostrictive materials, such boundary
conditions would not result in energy harvesting. However,
a similar excitation that would result in energy harvesting
would consist of a constant, nonzero electric field applied
to the device as stress is applied; then open-circuit conditions as the stress is removed. This electromechanical cycle
is shown in Fig. 2. The material is stress free at state 1
of the cycle, and an electric field E0 is applied and kept
constant as the stress is increased to Tmax , ending in state
2. The electrostrictive device is then open circuited when
the stress is removed, ending in state 3. From state 3 the
electric field E0 is re-established in the material, returning
to the original state 1. The total mechanical input energy
density available for the energy harvesting for such a cycle
2
can be shown to be W1 +W2 = (1/2)sTmax
, and the energy
W1 can be calculated as follows.
As the material is in open-circuit condition from state
2 to 3, the electric flux density is constant and is given by:
D = εE0 + 2M E0 Tmax .
(9)
The electric field intensity as the stress is removed, therefore, is given by:
E = E0
ε + 2M Tmax
.
ε + 2M T
(10)
This field intensity reaches its peak value Ep when the
applied stress becomes zero, or:
2M Tmax
Ep = E0 1 +
= E0 (1 + γ),
(11)
ε
2M Tmax
.
ε
(12)
For existing electrostrictive materials, γ is less than 1.
Now:
3
1 2
E2
2
W2 = −
T dS = sTmax
− 2 0 M 2 Tmax
2
ε
2
1 2
(13)
sTmax − εγ 2 E02
=
2
⇒ W1 = 2
E02 2 2
M Tmax .
ε
(14)
Therefore, the coupling factor is given by:
2M E0
k= √
.
sε
(15)
The result of (15) is the same as the linearized equivalent coupling of (8), but we should notice that the electrical
field E0 in (15) is the lowest field during the cycle, but for
(8) E0 is the middle or average field of the cycle. If E0
is chosen so that the peak electric field intensity Ep is the
maximum allowable due to material constraints, Emax , the
maximum harvesting energy density is given by:
W1 max =
γ
2
M Tmax Emax
.
(1 + γ)2
(16)
The coupling factor associated with this energy density
is given by:
k=
2M Emax
√ .
(1 + γ) εs
(17)
C. Energy Harvesting Cycle #2: Constant-Field Boundary
Conditions During Stressing and Unstressing of Material
Another possible method of imposing electrical boundary conditions is to keep the electric field constant as the
material is stressed, then change the field to a different
value that is kept constant as the stress is removed. Such
a cycle is shown in Fig. 3.
As in Section II-B, a constant electric field E0 exists
from state 1 to state 2 as the stress is increased to Tmax .
From state 2 to state 3 the electric field is increased from
E0 to E1 , then kept constant until the stress is reduced
from Tmax to 0 from state 3 to state 4. At zero stress the
electric field is reduced to E0 , returning to state 1. In the
dielectric-field plot, the path 1–4 and 2–3 are not parallel, which is due to the stress dependence of the dielectric
constant. The converted energy can be shown to be:
W1 = Tmax M E12 − E02 .
(18)
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005
Fig. 5. Passive diode circuit for energy harvesting.
Fig. 3. Energy harvesting cycle under constant electrical field conditions as the material is stressed and unstressed.
the electrical interface. We define the field at state 1 to be
E0 and the field at state 3 to be E3 .
The energy harvesting density is:
W1 =
M Tmax 2
E3 − E02 .
1+γ
(22)
Its associated coupling factor is given by:
k=
(1+γ)M E32 −E02
1
.
(1+γ)2 sTmax +(1+γ)M E32 −E02 −2M 2 Tmax Es2 ε
2
(23)
Fig. 4. Energy harvesting cycle under open-circuit conditions as
stress is applied and removed.
The maximum harvesting density then occurs when
E0 = 0 and E3 = Emax .
W1 max =
2
The input energy density W2 = (1/2)sTmax
, and the coupling factor, therefore, is given by:
M E12 − E02
k=
(19)
.
1
sTmax + M E12 − E02
2
The maximum energy harvesting density and coupling
occurs when E0 is set to be zero and E1 is set to Emax .
2
,
W1 max = Tmax M Emax
(20)
and:
k=
1
2
2
M Emax
.
(21)
2
sTmax + M Emax
D. Energy Harvesting Cycle #3: Open-Circuit Boundary
Conditions During Stressing and Unstressing of Material
Another type of electrical boundary condition is to have
open-circuit boundary conditions as the stress is applied
and removed, as shown in Fig. 4. From state 1 to state 2
and state 3 to state 4, the electric field changes automatically as the stress changes. From state 2 to state 3 and
from state 4 to state 1, the electric field is changed through
2
M Tmax Emax
.
1+γ
(24)
The coupling factor under these conditions is given by:
2
(1 + γ)M Emax
k=
.
(25)
1
2
(1 + γ)2 sTmax + M Emax
2
E. Energy Harvesting Cycle #4: Passive Diode Circuit
for Energy Harvesting
A circuit that has been proposed for electrostatic-based
energy harvesting [10] also can be used with electrostrictive materials and is shown in Fig. 5. The circuit uses highvoltage diodes for passive switching. The main advantage
of this circuit design is simplicity.
The energy harvesting cycle is shown in Fig. 6. However,
as the circuit is passive, the voltage change across the device occurs only due to the electrostrictive effect. As a
result, it can be shown that the voltages VL and VH are
constrained by the following condition:
VL < VH < γVL .
(26)
These constraints severely impair the energy harvesting
density and coupling factor when γ is small.
The harvesting energy density in this system is given by:
W1 = ε (EH − EL ) [(1 + γ)EL − EH ] ,
(27)
liu et al.: mechanical and electric boundary conditions for energy harvesting
2415
Fig. 6. Energy harvesting cycle for passive diode circuit.
where EL and EH are the electric fields associated with
VL and VH , respectively.
The coupling factor is given by:
k=
2ε (EH − EL ) [(1 + γ)EL − EH ]
2
2
sTmax
− ε (EH − EL )
.
(28)
The energy conversion density and coupling is maximized when:
EL =
2+γ
EH .
2(1 + γ)
(29)
If we set EH = Emax , the maximum energy harvesting
density is given by:
W1 max =
γ
2
M Tmax Emax
,
2(1 + γ)
(30)
and its associated coupling factor is:
k=
2
γM Emax
2
(1 + γ)sTmax + M Emax
.
γ
2(1 + γ)
(31)
III. Comparison of Different Boundary
Conditions
Inspection of the expressions for maximum energy harvesting density for different electrical boundary conditions
reveals that they can be compared directly through a normalization process based upon maximum applied stress
and electric field:
W,1 max =
W1 max
.
2
M Tmax Emax
Fig. 7. Normalized energy harvesting density for different electrical
boundary conditions.
(32)
With this definition, the normalized value of energy harvesting density becomes solely a function of the term γ.
Fig. 7 presents this comparison over a range of γ from 0
to 1, which is well above the maximum limit achievable
by existing electrostrictive materials. Inspection of Fig. 7
reveals that the constant field condition has the best energy harvesting density, with the open-circuit condition being comparable at low levels of γ. The constant-field-andopen-circuit condition and the passive-diode-circuit condition increase with γ, but they fall well below that of the
constant-field condition over the range of γ presented here.
In order to compare the effectiveness of the above electrical boundary conditions, electrostrictive material properties based on experimental data were assigned to the calculation, along with allowable electric field and stress values for safe operation. Representative material parameters
and limitations of two electrostrictive polymers, PVDFbased terpolymer and a polyurethane, are provided in Table I [7], [11].
Resulting energy harvesting densities (the maximum
energy density harvested W1 max ) and their associated coupling factors are provided in Tables II and III below. It can
be seen that, even for the same polymer, different electric
boundary conditions can result in quite different maximum
energy harvesting densities.
In previous work [5], optimistic energy harvesting densities for an optimized magnetic-field-based energy harvesting system were presented. The design optimization
was based on the analysis of a two-dimensional magnetic
circuit model of a reluctance-force-based magnetic device.
The conclusions of this analysis were that the optimum
energy harvesting density of the magnetic device was proportional to the square root of the cross-sectional area of
the device. For a small device with a cross-sectional area of
10 cm2 , the maximum energy harvesting density possible
was determined to be 0.1 J/cm3 . Hence, the results of this
paper suggest that electrostrictive materials can be significantly more effective at harvesting energy in “small”
energy harvesting systems with low-frequency excitation
in which quasistatic assumptions apply.
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 52, no. 12, december 2005
TABLE I
Electrostrictive Material Parameters and Their Operation Limitations.
Electrostrictive
Material
ε33
M13
(V/m)−2
s11
(Pa−1 )
Emax
(MV/m)
Tmax
(MPa)
γ
Terpolymer
Polyurethane
50ε0
7.5ε0
2 × 10−18
3.6 × 10−18
2.5 × 10−9
5.9 × 10−8
150
175
20
2
0.181
0.217
TABLE II
Maximum Energy Harvesting Density and Associated
Coupling Factor for Various Electrical Boundary
Conditions, Terpolymer Material.
Electrical Boundary Conditions
Cycle
Cycle
Cycle
Cycle
1,
2,
3,
4,
W1 max
Constant field and open-circuit
Constant field
Open circuit
Passive diode circuit
J
cm3
k
0.117
0.900
0.762
0.069
0.483
0.802
0.816
0.361
TABLE III
Maximum Energy Harvesting Density and Associated
Coupling Factor for Various Electrical Boundary
Conditions, Polyurethane Material.
Electrical Boundary Conditions
Cycle
Cycle
Cycle
Cycle
1,
2,
3,
4,
W1 max
Constant field and open-circuit
Constant field
Open circuit
Passive diode circuit
J
cm3
0.032
0.221
0.181
0.020
k
0.523
0.807
0.824
0.395
[3] M. Goldfarb and L. D. Jones, “On the efficiency of electric power
generation with piezoelectric ceramic,” Amer. Soc. Mech. Eng.
J. Dynamic Syst. Meas. Contr., vol. 121, pp. 566–571, 1999.
[4] H. A. Sodano, E. A. Magliula, G. Park, and D. J. Inman, “Electric power generation from piezoelectric materials,” in 13th Int.
Conf. Adaptive Struct. Technol., 2002, pp. 153–157.
[5] Y. Liu, K. Ren, H. F. Hofmann, and Q. M. Zhang, “Electrostrictive polymer for mechanical energy harvesting,” in Proc. SPIE,
Int. Soc. Opt. Eng., vol. 5385, no. 1, pp. 17–28, Mar. 2004.
[6] R. Pelrine, R. D. Kornbluh, J. Eckerle, P. Jeuck, S. Oh, Q. Pei,
and S. Stanford, “Dielectric elastomers: Generator mode fundamentals and applications,” in Proc. SPIE, Int. Soc. Opt. Eng.,
vol. 4329, pp. 148–156, Mar. 2001.
[7] Q. M. Zhang and J. Scheinbeim, “Electric EAP,” in Electroactive
Polymer [EAP] Actuators as Artificial Muscles. Y. Bar-Cohen,
Ed. 2nd ed. Bellingham: SPIE Press, 2004, pp. 95–150.
[8] J. C. Piquette, “Quasistatic coupling coefficients for electrostrictive ceramics,” J. Acoust. Soc. Amer., vol. 110, no. 1, pp. 197–
207, Jul. 2001.
[9] IEEE Standard on Piezoelectricity, ANSI/IEEE Standard 1761987, 1987.
[10] S. Ashley, “Artificial muscles,” Scientific American, vol. 289, no.
4, pp. 34–41, Oct. 2003.
[11] R. Pelrine, R. Kornbluh, and J. Joseph, “Electrostriction of polymer dielectrics with compliant electrodes as a means of actuation,” Sens. Actuators A, vol. 64, pp. 77–95, 1998.
IV. Conclusions
Electrostrictive materials have been theoretically shown
to possess significant electric energy densities that can be
harvested. Of the electrical boundary conditions investigated, the best energy harvesting density occurs when the
electric field in the material is increased from zero to its
maximum value at maximum stress, then returned to zero
at minimum stress. In the case of a small value of γ, a similar set of boundary conditions in which the device is open
circuited during the stressing and unstressing of the material provides similar energy harvesting densities. Boundary
conditions related to the standard definition of coupling
factor, and to a passive diode circuit, are shown to provide
relatively low-energy harvesting density over the range of
γ achievable by existing electrostrictive materials.
References
[1] G. A. Lesieutre, G. K. Ottman, and H. F. Hofmann, “Damping
as a result of piezoelectric energy harvesting,” J. Sound Vib.,
vol. 269, pp. 991–1001, 2004.
[2] M. Umeda, K. Nakamura, and S. Ueha, “Energy storage characteristics of a piezo-generator using impact induced vibration,” Jpn. J. Appl. Phys., vol. 36, pp. 314–315, 1997.
Yiming Liu (SM’03) was born in Lanzhou,
Gansu, China in 1972. He received the B.S.
degree in solid state physics from the Nanjing University, Nanjing, China in 1996. Between 1996 and 1999, he worked for the National Laboratory of Solid State Microstructures, Nanjing University, Nanjing, on ferroelectric thin films for nonvolatile RAM. Currently he is a Ph.D. student of the Department
of Electrical Engineering at The Pennsylvania
State University, University Park, PA.
His research interests are power electronics
and advanced materials for energy harvesting.
Kai Liang Ren was born in 1974 in Shaan
Xi, China. He received the B.Sc. degree in
electrical engineering from Tianjin University, Tianjin, China, in 1997. From 1997 to
2001, he worked in Tianjin JInke Electronic
Co. Ltd, Tianjin, China, as a software engineer. Mr. Ren is currently an M.S. student
in the Department of Electrical Engineering
of The Pennsylvania State University, University Park, PA.
His research field is advanced materials for
energy harvesting.
liu et al.: mechanical and electric boundary conditions for energy harvesting
Heath F. Hofmann (M’89) is an associate
professor of electrical engineering. He received
his B.S. degree in electrical engineering from
the University of Texas at Austin in 1992, and
his M.S. and Ph.D. degrees in electrical engineering from the University of California at
Berkeley in 1997 and 1998, respectively.
Dr. Hofmann’s research interests are in
power electronics and electromechanical systems. Specific interests are the development
and application of sensorless, field-oriented
control schemes; quiet electric drives; highspeed machine design; piezoelectric power generation; and the application of advanced numerical methods to the design and simulation of electromechanical systems, focusing on finite-element analysis
techniques.
He is the primary co-author on several journal papers on electric
machine design and control, with one prize paper award from the
Electric Machines Committee at the 1998 IEEE Industrial Applications Society Annual Meeting.
2417
Qiming Zhang (M’96–SM’00) is a professor of electrical engineering. He obtained a Ph.D. degree in 1986 from The Pennsylvania State
University, University Park, PA. The research areas in his group include fundamentals and applications of novel electronic and electroactive materials. He is one of the world’s leading experts in electroactive materials, especially electroactive polymers and their applications, including actuators and sensors, transducers, dielectrics and
charge storage devices, polymer thin film devices, and electro-optic
and photonic devices. He has 220 publications and 8 patents in these
areas. He is the recipient of the 1999 Penn State Engineering Society Outstanding Research Award and is an associate editor of IEEE
Transactions on Ultrasonics, Ferroelectrics, and Frequency Control.