Pyroelectric Energy Conversion: Optimization Principles

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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008
Pyroelectric Energy Conversion: Optimization
Principles
Gael Sebald, Elie Lefeuvre, and Daniel Guyomar
Abstract—In the framework of microgenerators, we
present in this paper the key points for energy harvest­
ing from temperature using ferroelectric materials. Ther­
moelectric devices profit from temperature spatial gradi­
ents, whereas ferroelectric materials require temporal fluc­
tuation of temperature, thus leading to different applica­
tions targets. Ferroelectric materials may harvest perfectly
the available thermal energy whatever the materials proper­
ties (limited by Carnot conversion efficiency) whereas ther­
moelectric material’s efficiency is limited by materials prop­
erties (ZT figure of merit). However, it is shown that the
necessary electric fields for Carnot cycles are far beyond the
breakdown limit of bulk ferroelectric materials. Thin films
may be an excellent solution for rising up to ultra-high elec­
tric fields and outstanding efficiency.
Different thermodynamic cycles are presented in the
paper: principles, advantages, and drawbacks. Using the
Carnot cycle, the harvested energy would be independent
of materials properties. However, using more realistic cy­
cles, the energy conversion effectiveness remains dependent
on the materials properties as discussed in the paper. A
particular coupling factor is defined to quantify and check
the effectiveness of pyroelectric energy harvesting. It is de­
fined similarly to an electromechanical coupling factor as
� c ), where p, 90 , "� , c are pyroelectric co­
k2 = p2 90 /("33
E
E
33
efficient, maximum working temperature, dielectric permit­
tivity, and specific heat, respectively. The importance of the
electrothermal coupling factor is shown and discussed as an
energy harvesting figure of merit. It gives the effectiveness
of all techniques of energy harvesting (except the Carnot
cycle). It is finally shown that we could reach very high
efficiency using h111i0.75Pb(Mg1/3Nb2/3 )-0.25PbTiO3 sin­
gle crystals and synchronized switch harvesting on inductor
(almost 50% of Carnot efficiency). Finally, practical imple­
mentation key points of pyroelectric energy harvesting are
presented showing that the different thermodynamic cy­
cles are feasible and potentially effective, even compared to
thermoelectric devices.
I. Introduction
onstant advances in electronics push past bound­
aries of integration and functional density toward
completely autonomous microchips embedding their own
energy source. In this field, research continues to develop
higher energy-density batteries, but the amount of energy
available is finite and remains relatively weak, limiting the
C
Manuscript received June 6, 2007; accepted November 26, 2007.
This work was supported by the Agence Nationale pour la Recherche
from the French government, under grant #ANR-06-JCJC-0137.
The authors are with INSA-Lyon, Laboratoire de Génie Electrique
et de Ferroélectricité, Villeurbanne, France (e-mail: gael.sebald@insa­
lyon.fr).
Digital Object Identifier 10.1109/TUFFC.2008.680
system’s lifespan, which is paramount in portable elec­
tronics. Extended life is also particularly advantageous in
systems with limited accessibility, such as biomedical im­
plants, structure-embedded microsensors, or safety mon­
itoring devices in harsh environments and contaminated
areas. The ultimate long-lasting solution should therefore
be independent of the limited energy available in batteries
by recycling ambient energies and continually replenishing
the energy consumed by the system. Some possible am­
bient energy sources are thermal energy, light energy, or
mechanical energy. Harvesting energy from such renewable
sources has stimulated important research efforts over the
past years. Several devices from millimeter scale down to
microscale have been presented, with average powers in
the 10 µW to 10 mW range [1].
Work on vibration-powered piezoelectric electrical gen­
erators has led to new energy conversion techniques, such
as synchronized switching harvesting (SSH) techniques,
based on nonlinear processing of the piezoelectric volt­
age [2]–[7]. As a result, the mechanical-to-electrical energy
conversion capability of active materials is significantly in­
creased: typically by factors of 4 to 15, depending on the
considered technique. From the efficiency point of view it
has been shown that SSH techniques may be implemented
with electronic circuits consuming less than 5% of the en­
ergy produced by the piezoelectric element. This novel ap­
proach is very promising for improving the effectiveness
and power density of piezoelectric microgenerators. But it
can also be theoretically extended to most other energy
conversion processes (for example, strain/stress variation,
temperature variation, and other processes).
Thermoelectric modules are the main way for energy
harvesting from temperature. It is now possible to find
commercial thermoelectric generators from µW to kW
electric output energy. These are based on temperature
gradients leading to heat flow through the thermoelectric
generator, and a small percentage of the heat flow is con­
verted to electric energy. Materials properties are the key
parameter for improving both the output power (increase
of the thermal heat flow, thus making it difficult to keep
the temperature gradient) and the efficiency (improving
the Seebeck coefficient and figure of merit). However, the
possibility of harvesting thermal energy is limited in the
case of microgenerators because the temperature differen­
tials across a chip are typically low.
Pyroelectric materials may be used for thermal energy
to electric energy conversion. The pyroelectric effect was
discovered before the piezoelectric effect and is mainly used
for pyroelectric infrared temperature detectors. Contrary
c 2008 IEEE
0885–3010/$25.00 ©
sebald et al.: pyroelectric energy conversion: optimization principles
to thermoelectric generators, pyroelectric materials do not
need a temperature gradient (spatial gradient), but tempo­
ral temperature changes. This opens different applications
fields, where temperature gradients are not possible and
where temperature is not static. Small-scale microgenera­
tors with dimensions smaller than the temperature spatial
fluctuation length may with difficulty be subjected to tem­
perature gradients. Natural temperature time variations
occur due to convection process, and this thermal energy
is difficult to transform in a stable temperature gradient.
On the other hand, it is possible to transform a temper­
ature gradient into a temperature variable in time using
a caloric fluid pumping between hot and cold reservoirs.
The pumping unit may require much less energy than the
total produced energy (depending on the scale of the de­
vice) and may produce temperature variations of 1 to 20◦ C
at 2 Hz for example. To optimize energy harvesting from
temperature, the first step should be the optimization of
energy conversion. Then, the problems of electric loading
(modifying the cycles shape) should be addressed.
The aim of this paper is to present methods for optimiz­
ing energy conversion from temperature variations using
pyroelectric materials and to describe the most important
parameters in materials choice and device design. The first
part is devoted to thermodynamic cycles that could be
used for energy conversion and the second part deals with
a pyroelectric materials survey. Finally the practical ap­
plication problems of thermodynamic cycles are discussed
in the last part of the article.
539
TABLE I
Coefficients Used in the Simulations.
Coefficient
Unit
Value
εθ33
F·m−1
1000ε0 ∗
10−3
2.5 × 10−6
301
300
C·m−2 ·K−1
J·m−3 ·K−1
K
K
p
cE
θh
θc
∗ For
Carnot cycle, εθ33 = 100ε0 for the sake of clarity on the figure (to
obtain larger difference between adiabatic and isothermal dielectric
permittivity).
εθ33 =
dD
dE
, p=
θ
dD
dΓ
dU
=
, cE =
dθ
dE
dθ
.
(3)
E
In the following part, we present four different energy har­
vesting cycles. For each cycle, we give PE cycle (polar­
ization vs. electric field) and Γθ cycle (entropy vs. tem­
perature). In the two cycles, the area of the cycle is the
converted energy. It is the same area in PE cycle and in Γθ
cycle. In the PE cycle, the cycle is clockwise, meaning a
negative energy (i.e., energy given to the outer medium).
In the Γθ cycle, the cycle counter-clockwise, meaning a
positive energy (i.e., energy given by the outer medium to
the material). The coefficients defined in (3) are assumed
to be constant for the electric field and temperature ranges
considered here. Coefficients used in simulations are given
in Table I.
A. Carnot Cycle
II. Thermodynamic Cycles
When talking about energy harvesting from heat, one
should first consider classical thermodynamic cycles. We
aim to answer here several questions:
What cycles could be imagined to harvest energy from
heat?
• What is their efficiency (defined as electric work di­
vided by the heat transferred from a hot reservoir to
a cold reservoir)?
• What are the important parameters of the pyroelectric
materials for optimizing the efficiency?
• Are those cycles realistic?
•
For a given temperature variation, it is possible to con­
sider it as a static problem involving two temperature
reservoirs, which is a common interpretation in thermo­
dynamics. We need first to establish the equations of py­
roelectric materials [8].
dD = εθ33 dE + pdθ
dθ
dΓ = pdE + cE
θ
(1)
(2)
where D, E, θ, and Γ are electric displacement, electric
field, temperature, and entropy, respectively. The coeffi­
cients are defined as:
The Carnot cycle is defined as two adiabatic and two
isothermal curves on the (PE) cycle (see Fig. 1). It is con­
sidered as the optimal energy harvesting cycle whose effi­
ciency is
ηCarnot = 1 −
θc
θh
(4)
where θc and θh are cold and hot temperatures, respec­
tively.
The demonstration of that result is very interesting to
understand the underlying limitations of such cycle. In the
first adiabatic increase of the electric field (path A-B)
ln
dθ
p
= − dE
θ
cE
θh
p
= − EM
θc
cE
(5)
(6)
where EM is the maximum amplitude of the applied elec­
tric field.
In practical applications, this means that one should
know the maximum temperature variation to know what
the necessary electric field is. In the isothermal decrease of
the electric field (path B-C)
dΓ = pdE and dQ = θdS
Qh = −pEM θh
(7)
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ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008
field is limited to 2 K for bulk ceramics [12]–[18] and could
reach 5–12 K for thin films [19], [20]. A too-large tem­
perature variation will result in a degraded Carnot cycle,
because it is impossible to get enough electrocaloric effect.
Moreover, it is hardly realistic to force the electric field at
a given value without paying a lot of wasted energy (see
Section IV). As a consequence, the Carnot cycle is not
feasible at all in practical applications.
B. SECE Cycle
SECE stands for synchronized electric charge extrac­
tion. We use that acronym because of numerous papers
concerning the nonlinear switching of the piezoelectric
voltage for energy harvesting from vibrations and damping
[3], [6], [7].
From the thermodynamics point of view, this technique
is one of the most natural cycles. It consists of extracting
the electric charge stored on the active material when the
maximum temperature is reached, i.e., when the stored
electric energy is maximum, and doing it again when the
temperature is minimum (Fig. 2). This energy extracted
may be then transferred to an electrical energy storage
cell such as a capacitor or to an electrochemical battery
for future needs, using, for example, the circuit described
in [3] or the power converter detailed in Section IV.
The theoretical description of this cycle is as follows.
Along the path (C-D), the temperature is decreased re­
sulting in a decrease of the open-circuit electric field.
p
(9)
Em = − θ (θc − θh )
ε33
where Em is the minimum electric field on the sample.
During that temperature variation
Fig. 1. Thermodynamic cycles for Carnot cycle. (a) PE cycle and
(b) Γθ cycle.
where Qh is the heat taken from the hot reservoir. The
two following steps are very similar and are not detailed
here.
The resulting energy conversion ratio gives
We = (θh − θc )δΓ = −pEM
θh − θc
.
θh
(8)
The resulting conversion ratio was already given in
(4). It is very interesting to notice that this conversion
ratio does not depend on material properties. The only
restriction—and main drawback—is that one should know
first the temperature variation before starting any cy­
cle. Furthermore, (6) links electric field amplitude to the
temperatures ratio. Using realistic coefficients values (see,
for example, [9]–[11], p = 600 × 10−6 C·m−2 K−1 , and
cE = 2.5 × 106 J·m−3 ·K−1 for a 0.75Pb[Mg1/3 Nb2/3 ]O3 ­
0.25PbTiO3 ceramic), and for a temperature difference of
1◦ C around room temperature, we need an electric field of
14 kV·mm−1 , which is far beyond the electric breakdown
of bulk ceramics. The maximum temperature variation in­
duced in ferroelectric materials when applying an electric
dQ = cE dθ −
p2
θdθ
εθ33
Qc1 = cE (θc − θh ) −
p2 2
(θ − θh2 )
2εθ33 c
(10)
(11)
where Qc1 is the heat given to the cold source during the
cooling.
Then the electric field is decreased to 0 in isothermal
condition (by connecting the sample to a resistance for
example, path D-E). Due to electrocaloric activity in fer­
roelectric materials, heat is transferred from the sample to
the cold source
Qc2 = −pEm θc .
(12)
As a result, using (9), total heat Qc transferred to the cold
source is
Qc = cE (θc − θh ) +
p2
(θh − θc )2 .
2εθ33
(13)
The two other segments of the cycle are very similar. Total
heat transferred to the hot source is
Qh = cE (θh − θc ) +
p2
(θh − θc )2 .
2εθ33
(14)
sebald et al.: pyroelectric energy conversion: optimization principles
541
divided by the product of noncoupled ones). For weakly
coupled case, i.e., k 2 « 1 (most common case as shown in
Section III)
ηSECE = k 2 ηCarnot .
(19)
For a perfect coupled material (k 2 = 1), we obtain a con­
version ratio that tends to the Carnot’s one provided that
this latter is much smaller than unity.
The advantages of such energy harvesting technique are:
No control of the voltage.
No special attention to be paid to the temperature
variation; do not need to know the temperature in
advance.
• Possible whatever the material (only pyroelectric ac­
tivity is important, whatever the electrocaloric activ­
ity).
•
•
The main drawback is the poor conversion ratio compared
to Carnot cycles. In fact, the k 2 for common materials
(PZT ceramics) is around 2 × 10−3 and may reach 4.7 ×
10−2 for some single crystals (see Section III for details
about materials).
C. SSDI Cycle
Fig. 2. Thermodynamic cycles for SECE cycle. (a) PE cycle and
(b) Γθ cycle.
The total electric work is found assuming that the in­
ternal energy does not change at the end of one cycle
Qh + Qc = −WE
WE = −
(15)
2
p
(θh − θc )2
εθ33
(16)
where WE is the electric energy.
Finally, the conversion ratio gives
ηSECE =
|WE |
k2
=
ηCarnot
Qh
1 + 0.5k 2 ηCarnot
(17)
with
k2 =
p2 θh
.
εθ33 cE
(18)
Variable k 2 is a dimensionless number giving the elec­
trothermal coupling factor (at temperature θh ), similar to
the electromechanical coupling factor (coupled coefficient
SSDI stands for synchronized switch damping on induc­
tor. This technique was developed prior to the SSH tech­
niques for dissipating the mechanical energy of vibrating
structures with piezoelectric inserts to damp the structural
resonance modes [2]. Synchronized switch means that the
voltage of the ferroelectric material is switched on an in­
ductor at every maximum or minimum of the temperature,
so that the electric field polarity is quasi-instantaneously
reversed (Fig. 3). From a thermodynamics point of view,
the only difference with SECE is that the electric field is
not reduced to 0, but nearly to its opposite value. The use
of resonant circuit including an inductor is in fine an inge­
nious way to perform that operation at minimized energy
cost (due to the inductor imperfections, a small amount
of energy is lost during the electric field polarity reversal
process).
Let us start the cycle explanation from point A. The
temperature is increased in open-circuit condition. Due to
pyroelectric activity, a positive electric field appears on
the ferroelectric material. Reaching the maximum temper­
ature, the electric field is inversed from EM0 to −Em0 with
a lossy inversion ratio
Em0
=β
EM0
(20)
where β is the inversion quality. β = 1 is a perfect inver­
sion, and β = 0 is the SECE case.
Then the temperature is decreased to its minimum. The
absolute value of the electric field is increased, and then
the inversion process is repeated. This cycle is repeated
indefinitely. The maximum value of the electric field is
thus increased for every cycle and would tend to an in­
finite value for a perfect inversion process. It is assumed
542
ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008
Using (20) and (21)
Qh = cE (θh − θc ) −
p2 2
(θ − θc2 )
2εθ33 h
p2 1 + β
+ θ
ε33 1 − β
(θh2 − θh θc ). (24)
Similarly for the heat transferred to the cold source (path
C-D and D-E)
Qc = cE (θc − θh ) −
p2 2
(θ − θh2 )
2εθ33 c
p2 1 + β
+ θ
ε33 1 − β
(θc2 − θh θc ). (25)
Using (15),
WE = −
p2
εθ33
1+β
1−β
(θh − θc )2 .
(26)
Electric energy is very similar to the SECE example. The
term (1 + β)/(1 − β) shows the energy conversion magni­
fication using the SSDI energy conversion cycle.
Finally, the conversion ratio is
ηSSDI =
1 + k2
k2
2β
ηCarnot
+
1−β
2
1+β
ηCarnot .
1−β
(27)
For weakly coupled materials, (k 2 « 1), we obtain
ηSSDI = k 2
Fig. 3. Thermodynamic cycles for SSDI cycle. (a) PE cycle and (b) Γθ
cycle.
1+β
ηCarnot .
1−β
It is noticeable that the SSDI process may be seen as a cou­
pling magnification, because the apparent coupling factor
compared to SECE becomes
2
kapp
= k2
that the second principle of thermodynamics guarantees
an irreversible process due to losses during inversion. The
cycle area will increase until the electric field gain due to
temperature variation equals exactly losses.
EM − Em
p
= − θ (θh − θc ).
ε33
(21)
p2 2
(θh − θc2 ).
θ
2ε33
During isothermal voltage inversion (path F-C)
Qh1 = pθh (−Em − EM ).
1+β
.
1−β
(29)
This latter coupling may become much larger than 1. In­
deed, for excellent inversion ((1 + β)/(1 − β) » 1), the
expression may be simplified
ηSSDI =
2
kapp
ηCarnot .
2
1 + kapp
(30)
Thus, as for SECE example, the conversion ratio may tend
to Carnot’s one for excellent coupling factor (k 2 → 1) and
for excellent inversion quality (β → 1).
As noted in the beginning of this section, the SSDI tech­
nique was not designed for energy harvesting but for me­
chanical vibration damping. The so-called SSHI technique
(22) [6] was derived from SSDI for the purpose of energy har­
vesting. Energy conversion cycle of SSHI is relatively near
to that of SSDI. However, understanding of its analysis
requires a description of the associated electronic circuit
behavior. So for clarity of this paper we have chosen to
(23)
detail the SSHI cycles in Section III.
Calculation of heat transferred to hot and cold sources is
very similar to the SECE example. During temperature
increase (path E-F)
Qh1 = cE (θh − θc ) −
(28)
sebald et al.: pyroelectric energy conversion: optimization principles
543
Fig. 5. Efficiency of the conversion for different techniques as a func­
tion of the coupling factor squared. Dashed line is for Carnot cycle,
solid line is for SSDI with different inversion factors β, and dashed­
dotted line is for pure resistive load.
The external boundary condition is
E = −ρḊ
(33)
where ρ is the resistive load connected to the piezoelectric
material.
When the temperature variation is sinusoidal, there ex­
ists an optimal load depending on the frequency,
ρOPT =
Fig. 4. Thermodynamic cycles for resistive cycle. (a) PE cycle and
(b) Γθ cycle.
WE =
Ḋ = εθ33 Ė + pθ̇
(31)
Q̇ = pθĖ + cE θ̇
(32)
where dotted variables are time derivatives.
(34)
where ω is the pulsation of the temperature variation.
Electrical energy dissipated per cycle is
D. Resistive Cycle
When wondering how to consume the electricity con­
verted from heat energy, one could think “just connect a
resistor to the active material electrodes!” This is indeed
the simplest way to perform energy conversion cycles on
ferroelectric materials (which is known as “Standard AC”
in some references [5], [6]). For the sake of presenting a
comprehensive argument, we study here the correspond­
ing cycles (Fig. 4).
Lefeuvre et al. [6] gave detailed calculations for elec­
tromechanical conversion using a single resistive load. The
development presented below is an adaptation to elec­
trothermal conversion.
The starting equations become
1
ωεθ33
π p2
(θh − θc )2 .
4 εθ33
(35)
If we neglect the electrocaloric coupling,
Qh = cE (θh − θc ).
(36)
And finally,
ηResistive =
π 2
k ηCarnot .
4
(37)
E. Discussion of Cycles
We show here four different cycles with different effi­
ciencies and different principles. Table II summarizes the
results. Fig. 5 illustrates the efficiency for all techniques
as a function of the coupling factor squared. It is clear
that SSDI is much more efficient than the others (except
Carnot) and increases the overall efficiency to 50% of that
of Carnot for an inversion quality of 0.8 and a coupling
factor squared of only 10%, whereas it is limited to 5% for
other techniques.
544
ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008
TABLE II
Comparison Between Four Different Cycles for Energy Harvesting.
Cycle
Efficiency η
Carnot
ηref
SECE
SSDI
η=
Resistive
k2
ηref
1 + 0.5k 2 ηref
(
ηSSHI =
1
θc
=1−
θh
+ k2
k2
1+β
)
η
2β
ηref 1 − β ref
+
1−β
2
ηResistive =
π 2
k ηref
4
Difficulty to implement
Necessary information
High voltage amplifier
and difficulty to
really harvest energy
Full temperature
profile known
in advance.
p
(θh − θc )
εθ33
Simple and efficient
electronic circuit
No prerequisite info.
p
(θh − θc )
εθ33 (1 − β)
Simple and efficient
electronic circuit
No prerequisite info.
Very simple circuitry
Frequency information
Maximum electric field EM
EM
cE
=−
ln
p
EM = −
EM = −
(θ )
h
θc
p
EM = − √ θ (θh − θc )
2 2ε33
Fig. 6. Efficiency of the conversion as a function of temperature vari­
ation ∆θ = θh − θc , with θc = 300 K. Dashed line is for Carnot cycle
and solid line is for SECE cycle with different coupling factors.
Fig. 6 shows the efficiency of conversion as a function
of temperature difference between hot and cold reservoirs.
Compared to thermoelectric energy harvesting (using See­
beck effect), where efficiency is limited by materials prop­
erties, efficiency for pyroelectric materials may tend to
Carnot’s one. Thermoelectric conversion efficiency may be
expressed as [21], [22]
√
θh − θc
ZT + 1 − 1
η=
·√
.
(38)
θh
ZT + 1 + θh /θc
For the best thermoelectric materials, the figure of merit
ZT reaches 1 around room temperature with Bi2 Te3 ma­
terials for example [23]. As a result, the best efficiency
reaches 17% of Carnot efficiency (considering low temper­
ature differences to maximize efficiency). To get 50% of
Carnot efficiency, one should find a material having a fig­
ure of merit of 9, which is ten times higher than the best
known thermoelectric materials, and we have not consid­
ered here the large temperature differences case, which re­
sults in degrading the efficiency. Consequently, the Carnot
cycle is most interesting for energy harvesting, but full
temperature profile information is necessary before any
temperature variation occurs.
This process is indeed possible for a controlled temper­
ature variation (as for fuel engines). One could imagine a
controlled gas heater inducing temperature variations and
pyroelectric energy harvesting. Another limitation is the
necessary electric field to be applied to the pyroelectric
material. As described in Section III (for example, 1◦ C
temperature variation), using realistic materials proper­
ties, the electric field should be in the 14 kV·mm to 1·K−1
range. When not broken with electric arcs, most bulk ferro­
electric materials are highly nonlinear, either for dynamics
nonlinearities, or for static nonlinearities [24]–[27].
For noncontrolled temperature variations—i.e., imagine
a temperature perturbation in the vicinity of a pyroelec­
tric material, such as going outside by −20◦ C or opening
a door—implementing the Carnot cycle is not possible,
except if one can predict the future temperatures values.
In such cases, the resistive case is not realistic (a tem­
perature variation is rarely a sinus), since the resistor is
adapted on the frequency of excitation. On the contrary,
SECE, SSDI, and SSHI could be used. With a sinusoidal
excitation, the voltage of the ferroelectric element should
be reversed (SSDI and SSHI) or short-circuited (SECE)
at every maximum or minimum of the voltage signal. Cal­
culations given in sections B and C are suitable for any
periodic temperature signal with constant maximum am­
plitude (even if this differs from sinus). When this differs
from sinus, the necessary choice is this: when should the
switch occur? To solve this problem, a probabilistic ap­
proach [28] and similar techniques adapted to random sig­
nals are necessary to maximize energy harvesting.
III. Pyroelectric Materials
We aim in this part of the article to present a survey
on existing materials. What are the important parame­
sebald et al.: pyroelectric energy conversion: optimization principles
Fig. 7. Coupling factor squared as a function of voltage response for
different materials.
ters? For standard pyroelectric materials, different figures
of merit exist. The two following ones are dedicated to
pyroelectric sensors and are defined as [29]:
Current responsivity figure of merit:
Fi =
p
.
cE
(39)
Voltage responsivity figure of merit:
Fv =
p
.
εθ33 cE
(40)
In Section II, we showed the great importance of an­
other parameter for energy harvesting—the electrothermal
coupling factor defined in (18). This parameter may be also
named energy figure of merit FE .
For every material presented in Table III, we give two
different figures of merit at room temperature (unless oth­
erwise specified). The first one is the electrothermal cou­
pling factor. The electric field sensitivity to temperature
variation is also given (= −p/εθ33 ). Indeed, for a given tem­
perature variation, the obtained voltage that appears on
the ferroelectric material is important. Voltages that are
too high result in inherent losses of the electronic circuitry
of the harvesting devices. It is also difficult to harvest en­
ergy in the case of very small pyroelectric voltages effi­
ciently because of the voltage drop of semiconductors. One
may object that whatever the electric field sensitivity, we
can adjust the thickness of ferroelectric material to keep
the voltage in a given useful range of variation. However,
bulk ceramics below 80 µm are nearly impossible to han­
dle. Then, the technology of thick and thin films may be
used to lower thickness, but ferroelectric properties usually
decrease quickly [30], [31]. Inversely, a thick ferroelectric
material may generate high voltages for a given temper­
ature variation, but its high thermal mass opposes quick
temperature variations. Fig. 7 shows the electrothermal
coupling factor as a function of voltage response to a tem­
perature variation. Among all materials, a few exhibit a
coupling factor squared above 1%. Simple harvesting de­
vices, such as SECE, require a large coupling factor to en­
sure an effective energy harvesting. To perform the Carnot
545
cycle with a pyroelectric material, one should apply an
electric field proportional to cE /p (see Table II for de­
tails). Assuming that all the materials have very similar
heat capacity, minimizing the electric field is the same as
maximizing the pyroelectric coefficient. From that point of
view, it is highly unrealistic to think about Carnot cycles
using polymers. Composites could be interesting because
they exhibit a very high breakdown electric field while
keeping a quite high pyroelectric coefficient. Bulk materi­
als exhibit a very high pyroelectric coefficient, but usually
break above 4–6 kV·mm−1 . Additional data on electric
breakdown resistance of materials is required to get more
precise information about the feasibility of Carnot cycles.
The best materials are (1-x)Pb(Mg1/3 Nb2/3 )-xPbTiO3
(PMN-PT) single crystals, with a coupling factor as high
as 4.7%. Moreover, voltage response is quite high. How­
ever, those materials are expensive and fragile. Single crys­
tals are grown by Bridgman technique [43]–[46]. It should
thus be difficult to really think of industrial use, unless
performance is the main priority. It should be noted that
the PVDF exhibits a coupling factor as high as bulk ce­
ramics but with a voltage response much larger (six times
greater than PZT). This material seems to be very inter­
esting since it is low cost, not fragile, and stable under
large electric field or temperature variations. Another ex­
cellent material is the PZT/PVDF-HFP composite, with
very high coupling factor. This kind of material is easy to
use, because of its flexibility, but suffers large dependence
on temperature. Moreover, the value given here is only for
70◦ C, and it decreases quickly when changing the tem­
perature. Most probably, the future of pyroelectric energy
harvesting is related to composites investigations.
IV. Energy Harvesting Devices
Orders of magnitude of the powers consumed by various
CMOS electronic devices that could be powered by minia­
ture energy harvesting devices are presented in Fig. 8. As
explained in previous sections, implementation of Carnot,
SECE, SSH, or SSDI cycles require controlling the pyro­
electric voltage with the temperature variations. Practi­
cal solutions for voltage control in the cases of SSH and
SECE techniques are presented in [2]–[7]. The following
subsections concern the effective energy harvesting com­
pared to optimized energy conversion presented in Sec­
tion II. Indeed, except the pure resistive case, where the
resistance connected to the active material electrodes may
be considered as the simulation of an electric load, the con­
verted energy—that is to say, the useful or usable energy—
actually differs from harvested energy as shown below.
A. “Ideal” Power Interface
Implementation of energy harvesters using the pre­
sented energy conversion cycles requires power interfaces
for achieving the desired energy exchanges between the
active material and the electrical energy storage cell. Re­
versible voltage amplifiers could be used for this purpose,
546
ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008
TABLE III
Pyroelectric Properties for Different Class of Materials.∗
Coefficient
Unit
p
µC·m−2 ·K−1
εθ33
ε0
cE (×106 )
J·m−3 ·K−1
−p/εθ33 (×103 )
V·m−1 ·K−1
k2
%
Ref.
210
54
23
11
34
170
11
39
183
15
66
95
4.79
0.81
0.17
0.08
0.38
2.12
0.08
0.46
2.50
0.10
0.62
0.96
[9,32]
[9]
[9]
[33]
[33]
[33]
[33]
[33]
[33]
[33]
[33]
[33]
54
40
28
48
46
7
0.37
0.38
1.44
0.22
0.06
0.002
[34]
[9]
[35]
[36]
[36]
[36]
17
98
64
156
0.039
0.28
0.17
0.70
[37]
[38]
[39]
[39]
314
54
598
442
0.14
0.028
4.28
0.63
[40]
[34]
[41]
[42]
Note
PZN-PT and PMN-PT single crystals
111
110
001
001
011
111
001
011
111
001
011
111
PMN-0,25PT
PMN-0.25PT
PMN-0.25PT
PMN-0,33PT
PMN-0,33PT
PMN-0,33PT
PMN-0,28PT
PMN-0,28PT
PMN-0,28PT
PZN-0,08PT
PZN-0,08PT
PZN-0,08PT
1790
1187
603
568
883
979
550
926
1071
520
744
800
961
2500
3000
5820
2940
650
5750
2680
660
3820
1280
950
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
Bulk ceramics
PZT
PMN-0.25PT ceramic
(BaSrCa)O3
PLZT 0.5/53/47
PLZT 8/53/47
PLZT 14/53/47
533
746
4000
360
97
19
1116
2100
16000
854
238
296
(PZT)/PZT composite
PbCaTiO3
PZT 700 nm
PMZT 700 nm
180
220
211
352
1200
253
372
255
2.5
2.5
2.5
2.5
2.5
2.5
@22◦ C ± 2◦ C
Thin films
2.5
2.5
2.5
2.5
Polymers and composites
PVDF
PZT/P(VDF-TrFE) 50%
PZT/PVCD-HFP 50/50 vol%
PZT0.3/PU0.7 vol%
33
33.1
450
90
9
69.2
85
23
1.8
2
2
2
@70◦ C ± 2◦ C
∗ Values
given here may not be accurate due to the lack of precisions in references (temperatures of measurement of pyroelec­
tric coefficient especially). Moreover, the heat capacitance is most of the time estimated using similar materials. However,
the range of variation of CE being quite small and its influence is weak.
Fig. 8. Powers consumed by CMOS electronic devices.
but a critical parameter for these power interfaces is their
efficiency. The working principle of so-called “linear volt­
age amplifiers” used in audio applications limits their effi­
ciency to 50% in theory and to lower values in practice. An­
other possibility is to use switching mode power converters.
Contrary to linear amplifiers, switching mode amplifiers
may theoretically reach an efficiency of nearly 100%. In­
deed, by principle, they are exclusively made up with elec­
tronic switches (ON state = very small resistance, OFF
state = quasi-infinite resistance) associated with quasi­
lossless passive elements such as inductors, transformers,
and capacitors. The circuit presented on Fig. 9 could be the
“ideal” voltage amplifier for controlling the energy harvest­
ing voltage cycles because it may deliver a perfectly con­
trolled ac voltage and may have very weak energy losses.
This H-bridge switching mode power converter is a well­
known structure used for controlling the power exchanges
between ac and dc electrical sources [49]. The four elec­
tronic switches are controlled through a pulse width mod­
ulator (PWM) that turns ON and OFF the switches at
high frequency (much higher than the temperature varia­
sebald et al.: pyroelectric energy conversion: optimization principles
547
made up with bipolar or MOSFET transistors. Indeed, the
OFF stage resistance of such transistors is typically below
100 MΩ, so the leakage currents of the switches may reach
several microamperes, and they may dissipate most of the
power produced by the pyroelectric material. This point
becomes even more critical as the frequency of the tem­
perature cycle is low. As a conclusion, although not devel­
oped yet, it is highly feasible to think about high-efficiency
amplifiers for effective regenerative thermodynamic cycles
on ferroelectric materials.
B. Case of SSHI
Fig. 9. Circuit diagram of the energy harvesting device including the
H-bridge switching mode power interface.
tion frequency) with variable duty cycle. The duty cycle D
is the key parameter for controlling the average ac voltage
(Vac ), respectively, to the dc voltage Vdc (0 ≤ D ≤ 1):
(Vac ) = Vdc (2D − 1).
(41)
The PWM may be also disabled. In this case, all the
switches are simultaneously opened, leaving the electrodes
of the active material on an open circuit. This particular
state is necessary, for instance, in some stages of SECE or
SSDI cycles. The inductor L connected between the ac­
tive material and the electronic circuit forms a low-pass
filter with the ferroelectric material capacitor C. Thus,
the voltage V across the active material electrodes can be
considered as perfectly smoothed as long as the switching
√
frequency of the PWM remains much higher than 2π LC.
In other words, if this condition is verified, the voltage rip­
ple due to switching across the active material is negligible,
and thus:
V = Vdc (2D − 1).
(42)
Theoretical waveforms in the cases of SECE and SSDI
are presented on Fig. 10(a) and (b), respectively. In prac­
tice, energy losses due to imperfections of the real compo­
nents affect the efficiency. For instance, high power (1 kW
to 100 kW), high voltage (500 V to 5 kV) industrial switch­
ing mode amplifiers have typical efficiencies between 85%
and 95%. Low power (1 mW to 10 W), low voltage (1 V to
20 V) switching mode amplifiers commonly used in wear­
able electronic devices also have high efficiencies, typically
between 70% and 90%. However, it is important to men­
tion here that the considered energy harvesting devices are
in the microwatt range, but with relatively high voltages
(50 V to 1 kV), so the characteristics of the required power
amplifier are out of usual application domains and require
a specific design. Such switching mode interfaces have
been successfully demonstrated in the cases of vibration­
powered piezoelectric and electrostatic energy harvesting
devices. Their efficiency is typically above 80% for output
power levels in the 50 µW to 1 mW range and with me­
chanical frequencies between 10 Hz and 100 Hz [50]. One
of the critical points in this ultra low power domain is the
OFF state resistance of the electronic switches, which are
Section II explored theoretical developments for SECE
and SSDI cycles in the case of pure energy conversion
without effective energy harvesting. It was shown that the
SECE cycle is a special case of SSDI cycle where β = 0.
For the sake of presenting a comprehensive argument, we
will develop here the SSHI case.
The SSHI energy conversion cycle may be performed
with the “ideal” switching mode power interface previ­
ously presented. However, understanding of this technique
is simpler considering the circuit that was first proposed
[2]: the pyroelectric element is connected to a switched in­
ductor in parallel to the ac side of a rectifier bridge, the dc
side of the rectifier being connected to an energy storage
cell, as shown in Fig. 11(a). Typical waveforms are shown
in Fig. 11(b). After reaching a minimum temperature θc ,
the switch S1 is turned ON. An oscillating discharge of the
pyroelectric capacitor C occurs then through the
√ inductor
L. The switch is turned OFF after a time π LC corre­
sponding to half a period of the electrical oscillations, so
that the pyroelectric voltage polarity is reversed. A small
amount of energy is dissipated in the inductor during this
operation, so the reversed voltage absolute value is reduced
by a factor β compared to its value just before the switch
is turned ON (0 ≤ β < 1). Then the temperature in­
creases, resulting in an increase of the open circuit voltage.
When reaching voltage Vdc , the rectifier bridge conduct­
ing and the current flowing out the pyroelectric element
directly supply the energy storage cell until the tempera­
ture reaches its maximum value θh . We will consider here
that when the diode bridge conducts, the total electric
charge flowing out the pyroelectric element exactly equals
the charge flowing through the energy storage cell. In such
case, we have
Q̇ = CV̇ + pΦθ̇
(43)
where Q, V , θ, C, Φ are electric charge, voltage, tempera­
ture, capacitance of the pyroelectric element, and the sur­
face of its electrodes, respectively. Note that it is easier to
formulate the problem using voltage and electric charge
variables instead of electric induction and electric field
variables when the circuit used for achieving the energy
conversion cycles is included in the analysis. When tem­
perature reaches value θcond , the rectifier bridge starts to
conduct. Considering that the ON stage duration of the
548
ieee transactions on ultrasonics, ferroelectrics, and frequency control, vol. 55, no. 3, march 2008
Fig. 10. Typical waveforms of (a) SECE and (b) SSDI with the H-bridge switching mode power interface.
Fig. 11. (a) Circuit diagram of SSHI circuit and (b) typical waveforms.
switch S1 is much smaller than the temperature cycle pe­
riod, and this temperature θcond is given by
pΦ(θCond − θc ) = C(Vdc − β · Vdc ).
(44)
Then we calculate the electric charge generated by the
pyroelement between time tcond and T /2
∆Q = pΦ(θh − θcond ).
According to (47), there is an optimal value of voltage Vdc
that maximizes the harvested energy
(Vdc )opt =
WMAX =
∆W = ∆Q · Vdc .
(46)
Symmetry of the cycle and combination of (44) with (45)
and (46) lead to the expression of the energy harvested per
cycle
Wcycle = 2Vdc (pΦ(θh − θc ) − CVdc (1 − β)).
(47)
(48)
Thus, maximum energy harvested per temperature cycle
is given by
(45)
The energy received by the storage cell between time tcond
and T /2 is given by
pΦ(θh − θc )
.
2C(1 − β)
(pΦ(θh − θc ))2
.
2C(1 − β)
(49)
And the maximum harvested energy per cycle and per
volume unit of active material is given by
WMAX (J · m−3 ) =
p2 (θh − θc )2
.
2εθ33 (1 − β)
(50)
To illustrate the order of magnitude of power, energy, and
optimal dc voltage, consider a 2◦ C temperature variation
at 1 Hz (θM = 1◦ C), with an inversion ratio β of 0.8 and a
pyroelectric element of 1 nF ((111)PMN-PT single crystal,
sebald et al.: pyroelectric energy conversion: optimization principles
TABLE IV
Energy Densities for 300 to 310 K Cyclic Temperature
Variations and Number of Cycles Per Hour for Producing
30 µW/CM3.
Material
Energy
density
(J/cm3 )
Cycles per hour
for producing
30 µW/cm3
111 PMN-0,25PT Single crystal
PMN-0.25PT Ceramic
PbCaTiO3 Thin film
PVDF
0.149
0.0118
0.00855
0.00540
0.725
9.12
12.6
20.0
area of 1 cm2 and thickness of 850 µm). Those parameters
give (Vdc )OPT = 890 V, harvested power of 0.32 mW, and
harvested energy per cycle of 0.32 mJ. In addition, the re­
sults presented above show that optimizing energy conver­
sion in no-load cases is the same as optimizing the energy
harvesting, especially in terms of materials properties and
inversion ratio.
Finally, the question that comes when speaking about
pyroelectric energy harvesting is the performance compar­
ison with thermoelectric effect. Conditions for energy con­
version are not the same because energy harvesting using
pyroelectric effect requires temperature variations in time
whereas thermoelectric effect needs temperature variations
in space (temperature gradients). The proposed compar­
ison is done considering temperature variations between
300 K and 310 K at frequency F for thermoelectric energy
conversion, and pyroelectric energy conversion in the case
of 300 K and 310 K for the cold and hot sources, respec­
tively. Typical energy density of miniature thermoelectric
modules in this case of operation is near 30 µW/cm3 power
density. As a comparison, Table IV gives the number of
temperatures cycles per hour needed to get 30 µW/cm3
power density with pyroelectric effect in the case of SSHI
cycles (β = 0.6) for several materials.
V. Conclusion
Energy harvesting from heat is possible using pyro­
electric materials and may be of great interest compared
to thermoelectric conversion. Pyroelectric energy harvest­
ing requires temporal temperature variations—we may say
time gradients of temperatures—whereas thermoelectric
energy harvesting requires spatial gradients of tempera­
tures. Usual wasted heat more likely creates spatial gradi­
ents rather than time gradients. However, the conversion
ratio (defined as the ratio of net harvested energy divided
by the heat taken from the hot reservoir) could be much
larger for pyroelectric energy harvesting. In theory, it could
reach the conversion ratio of the Carnot cycle, whatever
the materials properties. However, the conversion ratio of
thermoelectric conversion is highly limited by the materi­
als properties.
We showed four different pyroelectric energy harvest­
ing cycles, having different effectiveness and advantages.
549
The simplest devices would require a very high elec­
trothermal coupling factor (k 2 = p2 θ0 /(εθ33 cE )), and
we focused the investigation on pyroelectric materials
comparing that coupling factor. We found that using
0.75Pb(Mg1/3Nb2/3 )O3 -0.25PbTiO3 single crystals ori­
ented (111) and SSHI with an inversion factor of 0.8, it
should be possible to reach a conversion of more than 50%
of the Carnot cycle ratio (14% of the Carnot cycle for ef­
fective energy harvesting). Some important problems were
pointed out that can interfere with the technical imple­
mentation of such cycles, such as the frequency problems
and efficiency optimization. Nevertheless, we can expect to
get very high harvested energies using realistic materials
compared to standard thermoelectric devices.
Finally, ferroelectric materials are both pyroelectric and
piezoelectric. When designing an electrothermal energy
harvester, it should be possible to get high sensitivity to
vibrations (when bonding a ferroelectric material on a host
structure). However, the solution will be two-fold. The fre­
quencies may be very different between vibration and tem­
perature vibration. In such cases, electronics may easily
be adapted and optimized to address only one frequency
range. On the other hand, frequencies may be close to each
other. In such cases, the resulting voltage on the active el­
ement will be the sum of both contributions. A smart con­
troller is then necessary to optimize the energy conversion,
similar to the situation illustrated by the random signals
case in electromechanical energy harvesting [18].
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sebald et al.: pyroelectric energy conversion: optimization principles
Gael Sebald was born in 1978. He graduated
from INSA Lyon in Electrical Engineering in
2001 (M.S. degree) and received a master’s de­
gree in acoustics the same year. He received
a Ph.D. degree in acoustics in 2004. He was
then a Japan Society for Promotion of Sci­
ence Awardee (2004–2005) for a post-doctoral
position in the Institute of Fluid Science of
Tohoku University, Sendai, Japan, where he
worked on ferroelectric fibers and vibration
control.
Gael Sebald is now associate professor at
INSA Lyon, Lyon, France. His main research interests are now mate­
rials characterization, hysteresis modeling, multiphysics coupling in
smart materials, and energy harvesting on vibration and heat.
Elie Lefeuvre was born in France in 1971.
He received the B.S. and M.S. degrees in elec­
trical engineering respectively from Paris-XI
University, Paris, France, in 1994 and from
Institut National Polytechnique de Toulouse,
Toulouse, France, in 1996. At the same time,
he was a student at the electrical engineer­
ing department of Ecole Normale Supérieure
de Cachan, Cachan, France. He prepared his
Ph.D. degree at Laval University of Québec,
Québec, Canada, and at the Institut National
551
Polytechnique de Toulouse, France. He received the diploma from
both universities in 2001 for his work on power electronics converters
topologies. In 2002 he got a position of assistant professor at Institut
National des Sciences Appliquées (INSA) de Lyon, Lyon, France,
and he joined the Laboratoire de Génie Electrique et Ferroélectricité.
His current research activities include piezoelectric systems, energy
harvesting, vibration control, and noise reduction.
Daniel Guyomar received a degree in me­
chanical engineering, a Doctor-engineer de­
gree in acoustics from Compiègne University,
and a Ph.D. degree in physics from Paris VII
University, Paris, France. From 1982 to 1983
he worked as a research associate in fluid dy­
namics at the University of Southern Cali­
fornia, Los Angeles, CA. He was a National
Research Council Awardee (1983–1984) de­
tached at the Naval Postgraduate School to
develop transient wave propagation modeling.
He was hired in 1984 by Schlumberger to lead
several research projects dealing with ultrasonic imaging, then he
moved to Thomson Submarine activities in 1987 to manage the
research activities in the field of underwater acoustics. Pr. Daniel
Guyomar is presently a full-time University Professor at INSA Lyon
(Lyon, France), director of the INSA-LGEF laboratory. His present
research interests are in the field of semi-passive vibration control,
energy harvesting on vibration and heat, ferroelectric modeling, elec­
trostrictive polymers, and piezoelectric devices.