IMPROVED PERFORMANCE OF NONLINEAR PIEZOELECTRIC AC ENERGY SCAVENGERS

Qiliang Xu1, 3, 4, Richard M. White2, 3, 4, Igor Paprotny2, 3, 4 and Paul K. Wright1, 4

1Department of Mechanical Engineering, University of California, Berkeley, USA 2Department of Electrical Engineering & Computer Sciences, University of California, Berkeley USA

3Berkeley Sensor & Actuator Center, Berkeley, USA 4Center 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 1. INTRODUCTION

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. THEORETICAL MODELING 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,

, (1)

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.

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,

(2)

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,

(3)

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 Ti

i is the stress in the x-direction (Fig. 3). For the brass shim and the piezoelectric layer, the stress can be expressed as,

(4)

(5)

where Y11

s 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 layer.

Fig. 3: Cross-section of a bimorph piezoelectric cantilever.

Combining Eqs. 1 through. 4, the differential

equation for the dynamic response is,

(6)

where, (7)

(8)

2.2.2 Voltage response

The electric current is determined by applying Gauss’s law, expressed as follows,

(9)

where D is the electric displacement, written in the form of a linear constitutive relationship,

(10)

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).

Fig. 4: Equivalent circuit for the tuneable bimorph piezoelectric transducer. Current source:

(11)

Capacitance:

(12) 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,

(13) 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,

(14) where Xm(x) represents the mth propagating mode that is being excited. [7]. The coupled equations are then reduced to:

φm + 2ζmωm

φm +ωm2φm = −α piezo V1 t( ) −V2 t( )⎡⎣ ⎤⎦βM + F cos Ωt( )βF

CpV1 +

V1R1

= θ piezoφ

CpV2 +

V2R2

= −θ piezoφ (15,16,17)

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,

(18)

2.2.3 Resonance frequency tuning mechanism Apply Laplace transform to Eq. 15 to 17, the

voltage response is given by,

(19)

where,

(20)

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,

(21) The analytical expressions of resonance frequency when the tuning layer is short-circuited and open-circuited are derived from Eq. 21, and written as, Short-circuited:

(22) Open-circuited:

(23)

3. EXPERIMENTAL RESULTS 3.1 Nonlinear response

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, Q220-A4-203YB, 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

Fig. 6: Resonance frequency vs. resistive load on tuning layer. (Numerical simulation and experimental results.)

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.

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.

4. CONCLUSION

In this work, we show hot to optimize the performance of the piezoelectric AC energy scavenger by adjusting its resonant frequency. A new scavenger

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. 5. ACKNOWLEDGEMEMT

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.

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