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Qiliang Xu1, 3, 4, Richard M. White2, 3, 4, Igor Paprotny2, 3, 4 and Paul K. Wright1, 4
Department of Mechanical Engineering, University of California, Berkeley, USA
Department of Electrical Engineering & Computer Sciences, University of California, Berkeley USA
Berkeley Sensor & Actuator Center, Berkeley, USA
Center for Information Technology Research in the Interest of Society (CITRIS), Berkeley, USA
Abstract: This paper reveals the nonlinear behavior of a multilayer piezoelectric cantilever used as an AC energy
scavenger. The piezoelectric material nonlinearity causes a minor shift of the resonance frequency with the varying
amplitude of the harmonic excitation, which in some instances may prevent the energy scavenger from reaching
maximum output power. To address this problem, a resonance frequency tuning mechanism employing controlled
piezoelectric stiffening has been developed. In this paper, we present a novel way of coupling the AC magnetic
field of a single current carrying conductor and an analytical model of a bimorph piezoelectric cantilever’s dynamic
and voltage response. Based on this model, we derive the relationship between the load impedance on piezoelectric
layers and the system’s resonance frequency. The enhanced magnetic coupling strength and the resonance
frequency tuning mechanism were shown to improve the overall performance of a piezoelectric scavenger.
Keywords: Energy scavenger, nonlinear modeling, piezoelectric stiffening, smart grid, wireless sensor network
The use of piezoelectric cantilevers as energy
scavengers has been extensively studied for different
ambient energy sources, such as vibration and AC
energized conductors [1,2]. In these applications, the
energy scavengers are designed such that their
resonance frequency equals the driving frequency of
the energy sources. A mismatch between the input and
the resonance frequency will significantly decrease the
overall power output, especially for those scavengers
with high quality factors.
In experiments with mesoscale AC energy
scavengers, we sometimes observe nonlinear behavior
of a bimorph piezoelectric cantilever resulting in a
resonance frequency drift with the changing magnitude
of the magnetic excitation force. The same
phenomenon has been reported by many researchers
over the past few years [3,4,5]. In these papers, the
nonlinear behavior was mostly attributed to the
inherent piezoelectric material properties, such as
creep and displacement-voltage saturation.
In this paper, we proposed a resonance frequency
tuning mechanism that is able to adjust the resonance
peak back towards the driving frequency, optimizing
the power output of the energy scavenger. The
mechanism employs the effect known as piezoelectric
stiffening, which is based on the fact that the presence
or absence of an electric field in a strained
piezoelectric crystal affects the mechanical stiffness of
the crystal. Thus, through connecting a variable
resistance between the electrodes, we can obtain a
control over the variation of resonant frequency.
Besides its capability of compensating for nonlinear
properties, this tuning mechanism also provides a
mechanism for switching off the energy scavenger by
placing it in a mode where the scavenger’s resonance
frequency is different than the driving frequency. This
feature can greatly increase the life of the AC energy
scavenger under peak current conditions. In addition,
when operating off-resonance, the voltage response of
the AC energy scavenger is proportional to the input
current, the energy scavenger can be used as an
electric current sensor.
2.1 Enhanced magnetic coupling strength for a
single current-carrying conductor
Fig. 1: Schematic of the modified design of a
piezoelectric energy scavenger coupled to an
energized conductor.
Leland et al. calculated the magnetic force on the
magnets by studying the gradient of the magnetic field
produced by the electric current. For use with single
wire, the maximum coupling strength is developed
when the magnet is oriented such that its
magnetization vector makes a 45° angle with its radial
vector [6]. However, this orientation makes the
alignment of the scavenger/sensor on a single wire
conductor very difficult. To address this problem, and
to increase the magnetic coupling strength, an
alternative way of coupling the magnetic field is
proposed, where two oppositely poled magnets are
placed above the single wire (Fig. 1). The analysis of
this new approach begins by realizing that the force on
the permanent magnets near an AC current-carrying
conductor is equal in magnitude and opposite in
direction to the force on the conductor. The force
experienced by a single wire is proportional to the
integral of the cross product of the electric current and
the magnetic flux density produced by the magnets, as
described below,
where hu,i and hl,i are the location of the upper and
lower surface of the ith layer, b is the width of the
cantilever, and Tii is the stress in the x-direction (Fig.
3). For the brass shim and the piezoelectric layer, the
stress can be expressed as,
Since the force in the z-direction is of primary
interest, the field of the y-component of the magnetic
flux density represents the ‘coupling force density’
(Fig. 2). The shade of the iso-lines’ color in Fig. 2
depicts the magnitude of By, indicating that the
scavenger’s design shown in this paper will develop
maximum response when the wire is placed
underneath the center of the two magnets.
where Y11s is the elastic modulus of the shim, c11 is the
elastic modulus of the piezoelectric layer under
constant electric field, e31 is the piezoelectric stress
coefficient and hp is the thickness of the piezoelectric
layer and Vi is the voltage across the ith piezoelectric
Fig. 3: Cross-section of a bimorph piezoelectric
Combining Eqs. 1 through. 4, the differential
equation for the dynamic response is,
Fig. 2: Plot of the y-component of the magnetic flux
density of two oppositely poled magnets, showing
enhanced magnetic coupling strength.
2.2 Modeling multilayer piezoelectric cantilevers in
use as tunable energy scavengers
The following section analytically describes the
dynamic and voltage responses of a tunable bimorph
piezoelectric energy scavenger and shows how the
presence or absence of an electric field in a strained
piezoelectric crystal affects the mechanical response of
a piezoelectric cantilever.
2.2.1 Dynamic response
The analytical model starts with the dynamic beam
bending equations,
In this differential equation, µ is the mass density
related to the beam’s length L, w is the displacement in
z-direction, F is the magnitude of the harmonic
excitation force and Ω is the excitation frequency.
Continuing, c is the damping coefficient, δ represents
the Dirac-delta function and M is the internal bending
moment expressed as,
2.2.2 Voltage response
The electric current is determined by applying
Gauss’s law, expressed as follows,
where D is the electric displacement, written in the
form of a linear constitutive relationship,
ε33S is the piezoelectric permittivity under constant
strain. Based on the above equations, an equivalent
circuit is developed to analyze the transducer’s voltage
response (Fig. 4).
2.2.3 Resonance frequency tuning mechanism
Apply Laplace transform to Eq. 15 to 17, the
voltage response is given by,
Fig. 4: Equivalent circuit for the tuneable bimorph
piezoelectric transducer.
Current source:
and j is the imaginary unit with the property j2 = -1.
The voltage response reaches its maximum when the
modulus of the denominator in Eq. 19 is minimized.
Thus, the relationship between the resonance
frequency and the resistive load connected to the
tuning layer is obtained numerically by,
Hence, the differential equations for the voltage
response are derived by applying the Kirchhoff’s law
at junctions (a) and (b) shown in Fig. 4, as follows,
The analytical expressions of resonance frequency
when the tuning layer is short-circuited and opencircuited are derived from Eq. 21, and written as,
2.2.3 Single mode analysis
The solutions to the coupled differential Eq. 6 and 13
are summations of all the vibration modes of the
energy scavenger. When the energy scavenger is
excited near one of its eigen-frequencies (Ω=ωm), the
contribution of all the other modes can be ignored.
Thus, using the separation of variables method [7], the
general solution w(x,t) can be written in the form,
3.1 Nonlinear response
where Xm(x) represents the mth propagating mode that
is being excited. [7]. The coupled equations are then
reduced to:
φm + 2ζ mω mφm + ω m2 φ m = −α piezo ⎡⎣V1 ( t ) − V2 ( t ) ⎤⎦ β M + F cos ( Ωt ) β F
C pV1 + 1 = θ piezoφ
C pV2 + 2 = −θ piezoφ
where ωm is the natural frequency of the mth mode and
ζm is the associated damping ratio. βpiezo and βF are
eigenmode related coefficients [7]. θpiezo is the
piezoelectric coupling coefficient expressed as,
Fig. 5: Nonlinear behavior displays a resonance
frequency shift with varying input current.
Experimental (Fig. 5) results show a Mesoscale
AC energy scavenger (PZT cantilever beam, Q220A4-203YB, Piezo Systems) tuned to operate at 68.4
Hz exhibits a decrease in its resonance frequency from
68.4 to 64.6 Hz as the driving current increases from 1
Arms to 15 Arms. The decrease of resonance frequency
with increasing amplitude of harmonic excitation
shows a softening type of nonlinearity.
3.2 Resonance frequency tuning
design has been developed to enhance the magnetic
coupling strength. We have also developed a
resonance frequency tuning mechanism, which allows
us to gain useful control over the energy scavenger’s
response by shifting its resonance peak towards or
away from the driving frequency by 3-5%. This tuning
concept is especially useful for tuning the resonance
frequency of narrow-band energy scavengers in order
to optimize their performance due to nonlinear
properties or minor driving frequency fluctuations.
Fig. 6: Resonance frequency vs. resistive load on
tuning layer. (Numerical simulation and experimental
Using the same bimorph PZT cantilever, we have
observed 2.7% of tuning (1.8 Hz near 67 Hz) when the
resistive load on one layer was increased from 4 Ω to
1.3 MΩ while the other layer was scavenging energy
at its optimal load. The experimental results agree with
our numerical simulation. Fig. 5 also indicates that the
resonance frequency become most sensitive to the
tuning resistance when the impedance of the tuning
resistor matches to the impedance of the capacitance of
the tuning layer.
This work was supported by grants from the
California Energy Commission (CEC), contract
numbers 500-01-43, 500-02-004 and POB219-B, as
well as research and infrastructural grants from the
Berkeley Sensor & Actuator Center (BSAC) and the
Center for Information Technology Research in the
Interest of Society (CITRIS), at UC Berkeley.
This report was prepared as a result of work
sponsored by the California Energy Commission. It
does not necessarily represent the views of the Energy
Commission, its employees, or the State of California.
The Energy Commission, the State of California, its
employees, contractors, and subcontractors make no
warranty, express or implied, and assume no legal
liability for the information in this report; nor does any
party represent that the use of this information will not
infringe upon privately owned rights.
Fig. 7: Shift of resonance peak in response to a
decrease of driving current.
In another experiment, the resonance frequency of
our energy scavenger was initially tuned mechanically
by moving the position of the magnets in order to
match the driving frequency of 56.4 Hz at 10 Arms.
When the driving current decreased to 5 Arms, the
resonance frequency increased to 58 Hz. We
demonstrated that using our tuning mechanism, one
could shift the resonance peak of the energy scavenger
back to the driving frequency (Fig 7). Experimental
results show a 160% increase in output power was
achieved after decreasing the resistive load on the
tuning layer from 500 kΩ to 20 kΩ. The amount of
increased power will become more significant if the
device has a higher quality factor.
In this work, we show hot to optimize the
performance of the piezoelectric AC energy scavenger
by adjusting its resonant frequency. A new scavenger
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