Piezoelectric Single Crystal Power Generator for Low Frequency
Vibrating Machines and Structures
R. Shukla1,2, L. C. Lim2,3 and P. Gandhi1
Suman Mashruwala Advanced Microengineering Laboratory, Department of Mechnical Engineering,
Indian Institute of Technology Bombay, Mumbai, 400076
2
Department of Mechanical Engineering, National University of Singapore, Singapore, 119260
3
Microfine Materials Technologies Pte. Ltd., 10 Bukit Batok Crescent, # 06-02 The Spire, Singapore, 658079
1
Abstract
Relaxor-based
ferroelectric
Pb(Zn1/3Nb2/3)O3 -(6-7)%PbTiO3 single crystal of
high
d31 and
[110]L×[001]T(P) cut exhibits
E
low s11 values, making it candidate for piezoelectric
energy harvesting. In this work, a cantilever
piezoelectric power generator using this crystal cut
has been designed and fabricated.
The device is
designed for 110 Hz of resonance. The peak voltage
generated from the prototype device at resonance is 40 V/g, g being the acceleration due to gravitation.
The maximum power harvested is about 1 mW/g,
which increases with the square of the acceleration.
Both analytical and experimental results of the device are
presented and discussed.
I. INTRODUCTION
Due to recent developments in the microelectromechanical systems (MEMS) technology and the
wireless sensing networks, the demand for portable
electronics and wireless sensors is growing very fast [1].
These wireless sensor nodes require very low power
which can be supported by conventional batteries and
non-conventional means like electromagnetic and
piezoelectric power generators if the surrounding media
is vibrating in nature. Various piezoelectric power
generators are already explored by researchers using
piezoelectric ceramics [1-9] whereas rare efforts are
made with piezo single crystals [10-12].
Pb(Zn1/3Nb2/3)O3-(6-7)%PbTiO3, or PZN-(6-7)%PT,
single crystal of [110]L×[001]T cut exhibits superior
electromechanical properties, with k31 0.85 and d31 -
was evaluated at resonance. Both analytical and
experimental results are compared.
II. THE CXPG3 AND ITS ANALYTICAL
FORMULATION
A. Vibrating characteristics
A schematic of the CXPG3 is shown in Fig. 1. It
consists of a steel shim with the single crystal elements of
the same width bonded onto it in region adjacent to the
clamped end but not as part of the clamped end and an
inertial mass attached at its free end. Since the thickness
of the single crystals is significant compared to that of the
steel shim, it could produce an effect on the stiffness of
the CXPG3. Thus, the said CXPG3 is modeled as a
stepped beam with 3 segments as shown in Fig. 2 (a). An
equivalent beam of uniform cross-section is shown in Fig.
2 (b), which will be used for the approximate estimation
of the vibration characteristics of the CXPG3. In Fig. 2,
the point load (P) is applied to replicate the inertial mass
effect on the beam.
Fig. 1: A schematic diagram of the CXPG3 (not to-scale).
E
1425 pC/N, and extremely high d 31/ s 11 value of >35
C/m2 [13,14]. Its excellent elastic and transverse
piezoelectric properties make this crystal cut candidate
active materials for the development of vibration power
generator. In this work, a cantilever piezoelectric single
crystal power generator (CXPG3) of 110 Hz of resonance
is designed and built using this crystal cut. The CXPG3 is
intended for use in various industrial motors, instruments
and structures to power sensor nodes for health
monitoring purposes. The performance of the CXPG3
Fig. 2: (a) 3-section CXPG3 (b) Equivalent steel beam of uniform cross
section.
For a uniform cross-section beam with a point load (P)
applied at the free end, the moment at a distance (x) from
the free end can be represented as
M eq = − P ( L − x) .
(1)
Δ = 1 + cos( β l ) cosh( β l )
(EI )eq = (EI )s / α ,
(2a)
where
(EI )S
(EI )C
α =1+ 2
L1 L2 3 L L ,
2
L
L
L
(2b)
− 1 + 3 1 2 + 3 2 L
L
L
2
2
L1 L2 L2 L2 3
6
+
−
−
L L L L
(EI)s and (EI)c are the EI of the steel shim and that of the
composite section (i.e. segment L2 in Fig. 2(a)),
respectively. (EI)c is given by (2c) below and the
segmental lengths L1, L2 and L3 are defined in Fig. 2(a).
3 b t3
bt
eq p
+ beq t p
( EI ) c = E s s + 2
12 12
(
ts t p
+
2 2
)
2
. (2c)
The free vibration equation of the CXPG3 is described
by the expression below
( EI ) eq
∂ 4 y ( x, t )
∂ 2 y ( x, t )
+ ρAeq
= 0,
4
∂x
∂t 2
(3)
where y is the displacement in the transverse direction,
Aeq the equivalent cross-sectional area, and the mass
density per unit length. Note that negligible change in
resultant density is assumed here because both the piezo
single crystal and steel shim has nearly the same density.
In the present case, the beam is excited mechanically at
the clamped end in the y direction. Following the work of
To [15], the solution of (3), subject to the
boundary conditions of (i) y = b(t ) = Be jωt and
∂y / ∂x =0 at x = 0 , and (ii) ∂ 2 y / ∂x 2 = 0 and
(
)
(
)
( EI ) eq ∂ 3 y / ∂x 3 = −m ∂ 2 y / ∂t 2 = −mω 2 y at x = L,
where B is the amplitude of the sinusoidal displacement
at the fixed end and m the inertial mass at the free end
and ω the frequency of excitation, the response of the
system is obtained as
y ( x, t ) = Y ( x, ω )e jωt ,
(4a)
where
Y ( x, ω ) =
,
B F1 cos( βx) + F2 cosh(βx)
+ F3 [sin( βx) − sinh(βx)]
2Δ (4b)
,
m ( βl ) {sin( β l ) cosh( βl ) − cos( β l ) sinh( βl )} (4c)
−
ρA L eq By setting the end deflection of the stepped beam in Fig.
2(a) equal to that of the equivalent beam of uniform
cross-section in Fig. 2(b), the equivalent bending stiffness
of the CXPG3, (EI )eq , is obtained as
and
F1 = 1 + cos(β l ) cosh( β l ) − sin( β l ) sinh( β l ) m ( βl ) sin( β l ) cosh( βl ) − 2
ρA L eq F2 = 1 + cos(β l ) cosh( βl ) + sin( β l ) sinh( β l ) ,
m ( β l ) cos(β l ) sinh( β l ) (4d)
+ 2
ρA L eq F3 = sin( βl ) cosh(β l ) + cos(β l ) sinh( β l ) m ( β l ) cos(β l ) cosh( βl ) + 2
ρA l eq 2
ω ρAeq
.
(4e)
β4 =
( EI ) eq
The equation Δ = 0 is the characteristic equation of the
vibrating system. Its roots give the values of , each
corresponding to a given mode shape of a certain natural
frequency given by
(EI )eq
.
(5)
ωn = β 2
ρAeq
The above analysis assumes that there is no damping in
the CXPG3. Without damping, the amplitude of vibration
at resonance can be exceedingly large and may approach
infinity. In reality, damping is always present, notably
due to inter atomic interactions within the material. For
structures and devices of low damping, the damping
factor ς may be obtained from the half-power bandwidth
concept, i.e.
ωb − ω a
= 2ς ,
ωn
(6)
where ω a and ωb are the forcing frequencies on either
side of the resonant frequency ω n at which the amplitude
of vibration is 1/√2 times that at resonance (corresponding
to half the power at resonance). Typically, ς = 0.02 to
0.05 for low damping system [16]. An average value of
0.02 for ς is used in the analysis. This value was
obtained experimentally by using half power bandwidth
technique mentioned elsewhere [17]. The amplitude of
vibration at resonance was obtained by using the
approach described below.
Knowing ς and ω n , the frequency ω a (or ωb ) was
determined from (6). A new value of , hereafter referred
as β a , at ω a was determined using (4e). Substituting
the value of β a into (4), one obtained the
amplitude
response of the CXPG3 at ω a . This amplitude was
then multiplied by 2 to obtain the amplitude of
vibration at resonance. According to the above approach,
(4b) becomes
Y ' ( x, ω ) =
,
B F1 cos( β a x) + F2 cosh( β a x)
+ F3 [sin( β a x ) − sinh( β a x)]
2Δ where F1, F2, F3 and
Δ
(7)
are all modified by β a .
B. Average stresses experienced by the piezoelectric
element in CXPG3
Fig. 3: Strain and stress distributions across a cross-section (at position
x) of Section 2 of the CXPG3
The bending stresses ( σ x ) at level y and the bending
moment (Mx) and radius of curvature (Rx) over any
segment of the CXPG3 are related by the expression
below.
σx Mx E
.
(8)
=
=
y
I
Rx
Here only the amplitude values of σ x , Mx and Rx are of
The limits of integration apply over the length of the
beam where the piezoelectric single crystal starts and
ends, respectively. Lp in (11) is length of the piezoelectric
active element. Eq. (9) may thus be rewritten as
concern. Since we are interested in the average stress in
the piezoelectric element, we shall focus our discussion
on the L2-segment in Fig. 2(a), of which the moment of
inertia is given by that of the composite cross-section, i.e.,
I = (EI)c/Es where (EI)c is given by (2(c)).
Using (8) and the moment of inertia for the composite
cross-section, the bending stresses at level 1 and 2
(Fig. 3) at any cross-section (i.e. any position x) of the L2segment of the CXPG3 were evaluated. Then, by setting
the strains in the shim equal to those in the active
elements at the two said levels, the bending stresses in the
piezoelectric single crystal at the same two levels were
evaluated and their average value Tx across the thickness
of the crystal was taken. The resultant expression gives,
Ep M x
Tx ,1 =
ts + t p ,
(9)
Es 2 I
where Ep and Es are the respective elastic modulus and ts
and tp the respective thickness of the piezoelectric active
element and the steel shim. The subscript 1 denotes that
the stress is along the length (x-) direction of the CXPG3
device.
To obtain the volume average stress in the active
material, TL 2,1 , we may replace Mx in (8) by the length
(
)
average of the bending moment acting over the L2segment, M L 2 ; i.e.,
M L 2 = ( EI ) eq κ L 2 (t ) ,
(10)
where the length average curvature of the said segment,
κ L 2 (t ) , is given by
κ L 2 (t ) =
1 L
1 L ∂ 2 y ( x, t )
x
t
dx
=
(
,
)
κ
dx ,
Lp L
Lp L
∂x 2
pe
ps
pe
(11)
TL 2,1 =
(
)
E p M L2
ts + t p .
Es 2I
(12)
C. Output voltage and power generated by CXPG3
The voltage generated from the piezoelectric material
due to mechanical excitation is open circuit voltage and,
under zero applied electric field conditions, i.e., 3 = 0, is
given by the piezoelectric relationship below:
D3 = d 31TL 2,1 ,
(13)
where D3 is the electric displacement in the thickness
direction, d31 the transverse piezoelectric coefficient of
the single crystal, and TL 2,1 the volume average stress in
the active material.
Since D3 = q/A = CVs/A, where q and Vs are the charge
and voltage generated due to the bending stress
respectively, A is the electroded area of the piezo single
crystal, C is the capacitance, and since C= ε 3 A/tp, we
have
Vs =
d 31TL 2,1t p
ε3
= g 31TL 2,1t p ,
(14)
where g31 is the transverse piezoelectric voltage constant
of the active element.
The series equivalent circuit with a voltage source
from (14) was used in the present work. In such a circuit
[18], the voltage source is connected in series with the
internal impedance while the external load (RL) is
connected across the source voltage. Here the internal
impedance of the piezoelectric single crystal is modeled
by considering the capacitance and resistance of the
crystal only, its inductance being neglected due to its very
small magnitude. The electrical power output (P) is given
by the product of the voltage across the load (VL) and the
load current (IL). IL is given by
ps
where y(x,t) = Y’(x,ω)ejωt and Y’(x,ω) is given by (7).
IL =
Vs
,
Z in + RL
(15)
TABLE I DETAILED DESCRIPTION OF THE PARTS USED IN CXPG3 FABRICAION.
S.
No.
1.
Parts of CXPG
Dimensions (mm)
Steel shim
40×4.5×0.4
No. of
pieces
1
2.
Piezo single
crystal
Inertia mass
Conducting
epoxy
11×4.5×0.265
2
4×4.5×2.5
Approximate 50μm
thickness between crystal
and shim was ensured
As thin as possible
between brass and shim
15×15×2.5
2
Two
sided
3.
4.
5.
6.
Non-conducting
epoxy
Fixtures
Two
sided
2
Materials
Properties
SUS-301 (Hardened
carbon steel)
PZN-(6-7)%PT of
[110]L×[001]T cut
Brass
Circuitworks by
Chemtronics
E = 200GPa
= 7850kg/mm3
E = 25GPa
= 8350kg/mm3
= 8650kg/mm3
CW2400
Araldite
Acrylic and
polyurethin eoxy
standard
Stainless steel
where RL is the load resistance and Zin is the internal
impedance. Zin is given by Z in = Rs2 + (1 / jωC ) 2 , Rs
and C being the source resistance and capacitance of the
CXPG3. Knowing IL and Zin, we may obtain the voltage
across the load (VL) and the power output (P) of the
CXPG3 as follows:
and
VL = I L RL = Vs − I L Z in ,
(16)
P = VL I L = VL2 / RL .
(17)
Using the above expressions and taking ς = 0.02, a
Matlab code was written to estimate the natural frequency
of the CXPG3, its response to base excitation, the
induced stresses in piezo single crystals and the voltage
and power output generated from the CXPG3. The
analytical results will be presented below in conjunction
with the experimental results.
III. EXPERIMENTAL DETAILS
Table I shows the basic constituents and dimensional
details of piezoelectric single crystals, the inertial mass,
the strain-hardened steel shim, conductive epoxy, fixture
and epoxy resin used to construct the CXPG3. The
inertial brass mass was attached onto the shim surfaces at
the free end using Araldite epoxy resin. Conducting
epoxy was used to attach the single crystals onto the top
and bottom surfaces of the steel shim and to fix the
electric wires on the free side of both single crystals. Care
was exercised to ensure that there was no short-circuit
connection during the bonding. The two crystals were
connected in series electrically (in capacitance sense).
The test set-up used is shown in Fig. 4. The fixtures
were designed and constructed to ensure the clamped
boundary conditions at the fixed end. The said set-up was
used for the performance evaluation of the CXPG3.
Before performing the experiment, the accelerometer was
calibrated.
To obtain the open circuit voltage (Vs)-versus-time plot,
the CXPG3 was excited at the first fundamental
frequency for its high displacement amplitude response.
The result for 0.26g is given in Fig. (5), g being the
gravitational acceleration. For comparison, the predicted
voltage response from (14) is also given in the said figure.
Accelerometer
CXPG3
Charge
amplifier
Power
Load
Amplifier
Resistances
Shaker
Fig. 4: Photograph showing the test set-up to characterize CXPG3.
Fig. 5: Voltage vs. time response of CXPG3 at 0.26g acceleration.
TABLE II COMPARISON OF ANALYTICAL AND EXPERIMENTAL RESULTS OF CXPG3.
Specimen
No.
Input
parameters
Predicted stresses on
active element (MPa)
Average
Maximum
Maximum voltage (V)
Maximum power (μW)
Predicted
Measured
Predicted
Measured
1.
0.26g
1.7
2.36
9.93
9.24
69.2
49.3
2.
0.53g
3.45
4.82
20.24
20
288
210
3.
1g
6.5
9.10
38.2
n.m.
1,030
n.m.
n.m. = not measured.
In real applications, piezoelectric power generators
will be connected across an external load, which could be
resistive, capacitive, inductive or their combination. In
the present work, a load resistance was used. It was
connected across the CXPG3 and the voltage across the
resistor (VL) was measured. During the test, the load
resistance was varied from few hundred ohms to few
mega ohms.
much smaller than that of the steel shim. From (14), the
voltage generated is proportional to the volume average
stress induced in the piezo element, which, in turn, is
proportional to the bending moment and hence the
curvature of the beam. Since the curvature in the analysis
is an approximate one, some errors are expected. Note
that this error is expected to increase with increasing
thickness of the piezoelectric active element.
IV. RESULTS AND DISCUSSION
The test result is shown in Fig. 6, in which both
analytical and experimental peak values of the voltage at
resonance frequency are provided for easy comparison.
The predicted voltage values were obtained by first
calculating the βa at 107.5 Hz (i.e., at ωa which was
obtained by taking ς = 0.02 and ωn = 110 Hz in (6)) and
using (7) to obtain the mode shape at resonance.
Knowing the mode shape, the length average curvature,
bending moment and the induced volume average stress
in the piezoelectric active element were obtained via (11),
(10) and (12), respectively, and the induced voltage via
(14) and (16).
The output power as a function of the load resistance is
provided in Fig. 7. Such plots enable us to determine the
optimum load resistance for maximum output power,
which can be read directly from the plots. This figure
shows that the optimum load resistance value is close to
390 kΩ and that at this load resistance, the maximum
output power attainable at resonance was 50 μW (at
0.26g).
Table II compares the analytical and experimental
results obtained for the CXPG3. The variation in output
voltage is within 10% and in power is nearly 30%. In
addition, the predicted fundamental frequency of the
CXPG3 was 110Hz whereas the experimentally measured
value was 102Hz and 108Hz for the two devices made.
The observed discrepancies between the experimental
and analytical results may be explained. Firstly, the
effective bending stiffness concept used in the analysis
was based on the assumption that the mounting of the
crystal produces negligible effect on the curvature of the
elastic beam despite obvious thickness variations at the
sector joints. This was done to make the analysis tractable
since elastic modulus of the piezoelectric element is
Fig. 6: Comparison of analytical and experimental results of CXPG3 for
varying load resistance for peak voltage at 0.26g acceleration.
Fig. 7: Comparison of analytical and experimental results of CXPG3 for
varying load resistance for peak power at 0.26g acceleration.
Secondly, in the analysis, it is assumed that the epoxy
joint effects are negligible. Due to its much lower
stiffness, the epoxy layer helps moderate the transfer of
stresses from the shim to the piezoelectric active element.
This, in turn, may result in lower stresses in the latter and
hence a lower output voltage. Despite the above, it should
be mentioned that for sufficiently thin, properly prepared
epoxy joints, its effect on stress transfer is minimum and
may be neglected [19].
From Table II, it can be observed that a maximum
power of greater than 1mW is possible by the CXPG3 at
1g acceleration. Under this operating condition, the
estimated maximum tensile stress in the piezo single
crystal remains <10 MPa. This is below the maximum
tensile stress of 18 MPa which may cause the said crystal
cut to depolarize [20], thus ensuring safe operation of the
device. The present design of CXPG3 is thus very useful
for the environment where vibrational noises of about
100Hz are available and for health monitoring where
embedded motes or sensors are advantageous. The device
made has a bandwidth of 5 Hz. The design and analysis
can be easily extended to cover different surrounding (or
frequency) requirements.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
V. CONCLUSION
[9]
This work investigated a cantilever piezoelectric power
generator for harvesting vibrational energy of about 100
Hz. Relaxor ferroelectric PZN-(6-7)%PT single crystal of
[110]L×[001]T cut was chosen as the active elements due
to its high transverse piezoelectric properties and low
compliance. An analytical model was presented which
allows designing cantilever piezoelectric power
generators for different applications when working at
resonance. Physical prototypes of the devices were
fabricated and evaluated. Both the analytical and
experimental results agreed with one another fairly well.
Our results show that output power of ≈ 1 mW (at 1g
acceleration) is possible with the fabricated device, which
increases with the square of acceleration. Despite a much
smaller volume of piezoelectric materials used, the built
vibration energy harvester is very competitive in
comparison to the results obtained by earlier researchers
[11, 12] .
ACKNOWLEDGMENT
[10]
This work is supported by research grants received
from the Ministry of Education (Singapore) and National
University of Singapore, via research grant nos. R-265000-221-112, R-265-000-257-112, R-265-000-257-731,
R-265-000-261-123/490,
R-263-000-511-305
and
A*STAR SERC 072 133 0046. R.S. is thankful to
Prof Siak Piang Lim, Prof Surjya Kumar Maiti and Prof
Sanjib Kumar Panda for valuable discussions on various
aspects of the CXPG3 and also acknowledges the
research scholarship and technical support received from
IIT Bombay and NUS for him to pursue his Ph.D.
research.
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