Wideband energy harvesting for piezoelectric devices ... - IEEE Xplore

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IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control ,

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Wideband Energy Harvesting for Piezoelectric Devices With Linear Resonant Behavior Cheng Luo, Student Member, IEEE, and Heath F. Hofmann, Member, IEEE Abstract—In this paper, an active energy harvesting technique for a spring-mass-damper mechanical resonator with piezoelectric electromechanical coupling is investigated. This technique applies a square-wave voltage to the terminals of the device at the same frequency as the mechanical excitation. By controlling the magnitude and phase angle of this voltage, an effective impedance matching can be achieved which maximizes the amount of power extracted from the device. Theoretically, the harvested power can be the maximum possible value, even at off-resonance frequencies. However, in actual implementation, the efficiency of the power electronic circuit limits the amount of power harvested. A power electronic fullbridge converter is built to implement the technique. Experimental results show that the active technique can increase the effective bandwidth by a factor of more than 2, and harvests significantly higher power than rectifier-based circuits at offresonance frequencies.

I. Introduction

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nterest in the use of piezoelectric materials for energy generation has increased over the years. With the simultaneous decrease in size and power requirements for microelectronics, it is conceivable that some circuits could be powered by extracting energy directly from the environment in which the circuit operates. Researchers and engineers have made enormous progress in energy harvesting technologies in the last decade, including new active materials, better device design, new circuits and controls, and new energy storage devices. However, current technologies still do not meet the requirements of many demanding applications, such as harvesting power from small mechanical vibrations. A common approach to harvesting power from mechanical vibrations is to excite the mechanical resonance of a spring-mass system, in which the spring either consists of, or is mechanically attached to, a piezoelectric device. When the device is excited at its resonance frequency, the original mechanical displacements are amplified so that the piezoelectric device is more heavily excited, thereby increasing the amount of power harvested. However, the power harvesting bandwidth of these devices is usually

Manuscript received December 14, 2010; accepted April 26, 2011. The authors thank the National Science Foundation for their support of this work under Grant No. 0529029. C. Luo is with General Electric Company, Salem, VA. H. F. Hofmann is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI (e-mail: [email protected]). Digital Object Identifier 10.1109/TUFFC.2011.1949 0885–3010/$25.00

relatively small, and a slight frequency difference between excitation and resonant frequency typically results in a steep reduction in power output. Even in applications for which the excitation frequency is well-known, it is difficult to design a spring-mass device for which the resonance frequency can be determined with a high degree of accuracy, and which is insensitive to variations in material properties caused by temperature or manufacturing tolerances. Attempts to address this issue have been made by designing mechanical structures with nonlinear properties [1], [2]. The interface circuit between the piezoelectric device and electrical load plays an important role in the energy harvesting process. The predominant energy harvesting circuits under investigation are based on the passive acdc diode-rectifier circuit [3], [4] and the semi-active SSHI (synchronized switch harvesting on inductor) circuit [5]– [7]. Recently, so-called active energy harvesting techniques have been pursued, in which the voltage across the device is actively controlled [8]–[10]. The advantages of active energy harvesting are related to the fact that it can push the harvested energy to the limits of the piezoelectric device by applying appropriate electrical boundary conditions. As will be seen in this paper, the benefits are even more apparent when the mechanical system is driven at an excitation frequency that differs from its resonance frequency. In this paper, a model of a linear dynamic spring-massdamper piezoelectric energy harvesting device is first developed and analyzed. A wide-band active energy harvesting technique for this dynamic system is then proposed. A power electronic full-bridge converter with control circuitry is built to implement the technique. Finally, experimental results are shown. II. Piezoelectric Dynamic Model A typical dynamic piezoelectric energy harvesting system is a cantilever beam with tip mass, as shown in Fig. 1(a). A piezoelectric bimorph is often chosen for the beam, so that the vibration of the end mass will cause deformation to the bimorph, which will then generate an alternating voltage at the output. The system can also be modeled as a simple spring, mass, and damper system, as shown in Fig. 1(b), where m is the mass, b is the damping coefficient, the spring coefficient k is the device stiffness, y is the base displacement, and x is the free displacement of the mass. The spring represents the ideal compliance, the

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luo and hofmann: wideband energy harvesting for piezoelectric devices

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Fig. 1. (a) Piezoelectric dynamic energy harvesting system, (b) spring-mass-damper model.

Fig. 2. Circuit model for piezoelectric dynamic system.

mass is an energy storage element, and the damper accounts for internal mechanical losses. The constitutive equations for the piezoelectric dynamic system are

I = −Q = −C ′V − kdδ, (1) −my = mδ + bδ + k δ − kdV ,

where C ′ = C − kd2 is the piezoelectric capacitance under fixed displacement, C is the piezoelectric capacitance under constant force, δ = x − y is the relative displacement of the mass with respect to the base. The base acceleration can be treated as a force excitation to the dynamic system F = −my. If we do not consider the last term of the second equation in (1), −kdV, it becomes the standard dynamic equation for a spring-mass-damper system. However, for systems with a piezoelectric device as the beam, the application of an electrical voltage to the device terminals will affect the mechanical system, which will then in turn affect the device current. It is this coupling which can be used to harvest electrical energy from the mechanical system. The equivalent circuit model for the dynamic electromechanical system can be established as in Fig. 2 [11], [12]. The circuit inside the dashed-line box can be thought of as a load on the mechanical system. If we assume the excitation force is sinusoidal, the internal mechanical source impedance is given by

(

)

k Z m = b + j ωm − , (2) ω

where the notation ~ denotes complex phasor variables. It can be shown that the magnitude of Z m will be minimized

at the frequency ω = k/m, which is known as the natural resonance frequency of the mechanical system. Maximum power transfer theory states that the load which would maximize power extracted from the mechanical system corresponds to the complex conjugate of the mechanical source impedance. Therefore, to achieve maximum energy harvesting, the optimal mechanical load impedance should be b + j(k/ω − ωm). Assuming the sinusoidal excitation force has a constant magnitude of Fm, the maximum harvested power is then given by Pmax =

Fm2 . (3) 8b

Note that this value is the maximum available power from the mechanical system, which is independent of the excitation frequency, independent of the electromechanical coupling coefficient, and only depends on the magnitude of the excitation force and damping coefficient. III. Active Energy Harvesting To harvest the maximum amount of energy from the mechanical system, the piezoelectric device must emulate the optimal mechanical load. One approach to achieving this would be to attach an electrical impedance at the device terminals that would generate the desired overall mechanical load impedance. It can be readily shown that this electrical impedance is given by

Z opt =

ω 2m − k + j ωb . (4) ω C ′b − j ω(ω 2C ′m − Ck) 2

The implementation of such an impedance would be difficult, however, and so we instead think of applying a voltage to the piezoelectric device terminals that corresponds to the voltage which would be across the optimal electrical impedance. With the expression of the optimal electrical load impedance in (4), the corresponding optimal control voltage can be derived from Fig. 2 in terms of excitation force:

−ωb − j(k − ω 2m) Vopt = F . (5) 2kd ωb

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TABLE I. System Parameters of Resonant Energy Harvesting Device. m k b d C C ′ ς k e2

Fig. 3. Maximum harvested power of the active technique for different power electronic circuit efficiencies, simulation results.

The optimal magnitude of the control voltage can be calculated as

V mag =

ω 2b 2 + (k − ω 2m)2

Fm , (6) 2kd ωb

and the optimal phase angle with respect to the excitation force is given by

φ = 180 − tan −1

( k −ωωb m ). (7) 2

When this optimal voltage is applied to the piezoelectric terminals, it can be shown that the harvested power is Fm2 /8b, the same as in (3). Theoretically, as long as the voltage rating of the piezoelectric device is higher than this optimal voltage, the maximum power can always be harvested. The voltage applied to the piezoelectric terminals must be generated by a power electronic circuit whose operation will be less than 100% efficient. This inefficiency will decrease the actual energy harvested from the theoretical maximum value. This will become particularly pronounced as the excitation frequency moves away from the resonance frequency. This is because the optimal electrical impedance under these conditions becomes more reactive in nature, and hence the power factor of the circuit is reduced. Because the losses in a power electronic circuit depend more upon apparent power than real power, the effective efficiency of the circuit is reduced. It is difficult to incorporate the effects of circuit inefficiency in analysis, and so simulations were performed. Fig. 3 shows the results, clarifying the effect of the efficiency on energy harvesting. The parameters used in the simulation can be found in Table I. The magnitude and phase of the applied voltage were manually determined in these simulations to optimize the amount of power harvested. As can be seen, the harvested power will not be a constant under

7 g 4046 N/m 0.4635 N·s/m 1.86 μm/V 0.18 μF 0.166 μF 0.0435 0.844

different excitation frequencies for non-unity efficiencies, and the harvested power decreases as the excitation frequency moves away from the resonance frequency. The simulation result for a standard ac-dc rectifier energy harvesting circuit with 100% efficiency is also plotted in Fig. 3 as the dashed curve. As can be seen, the active method harvests much more energy than the rectifier circuit at offresonance frequencies, even at a relatively low efficiency. Because of the complex expression for the optimal voltage, the strong dependence of this expression on mechanical parameters, and the effects of circuit efficiency on energy harvesting, a control scheme is needed to adaptively achieve the optimal voltage in practical implementation. The goal is to build a control algorithm to find the optimal voltage that results in maximum energy harvesting with information that can be easily obtained from sensors. Such information includes the excitation force (which can be determined via an accelerometer) and the average harvested power. Assuming the harvested power is eventually used to charge a battery or power a dc circuit, this power can readily be determined through the measurement of the dc current provided by the energy harvesting system. The proposed adaptive algorithm is a two-step gradientbased algorithm, in which one first chooses initial values for the voltage magnitude and phase angle Vm1 and ϕ1, and then records the resulting harvested power P1. Next, the voltage magnitude is increased from Vm1 to Vm1 + Vstep, and the harvested power P2 is compared with P1. If P2 > P1, the new voltage magnitude is chosen; otherwise, it is set to Vm1 − Vstep, and the power under the new magnitude and original phase angle is recorded as P3. Next, the phase angle is increased to ϕ1 + ϕstep with the voltage magnitude unchanged. The power is recorded as P4 and compared with P3. Again, if P4 > P3, the new phase angle is kept; otherwise, the phase angle is set to be ϕ1 − ϕstep. By repeating this process, the optimal operating point can be reached. This algorithm was simulated and is compared with a contour plot of the harvested power in Fig. 4. To illustrate the convergence of the method, we chose an initial condition far away from the optimal point. The dark line in the figure represents the searching path to the optimal control voltage. In this simulation, the excitation is at the mechanical resonance frequency (121 Hz) with parameters in Table I, which yields an optimal voltage magnitude and phase angle of 24.18 V and 180° by (6) and (7), respectively, agreeing with the optimal control voltage shown in Fig. 4. This algorithm is simple enough to be implemented


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Fig. 5. Simulated waveforms of piezoelectric voltage, current, instantaneous power, and force. Fig. 4. Searching path of the adaptive method on contour plot of harvested power as a function of voltage magnitude and phase.

on a low-power microcontroller. Assuming the excitation magnitude and frequency are essentially constant, this process does not need to proceed very quickly, and therefore a low clock frequency may be used for the microcontroller, which would further lower the power consumption. IV. Circuit Implementation The optimal voltage under a sinusoidal excitation force is also sinusoidal. However, the generation of a sinusoidal voltage using switch-mode power electronic circuitry is undesirable in actual implementation, because it requires continuous pulse-width modulation (PWM) of the power transistors. At the low power levels associated with energy harvesting, the resulting switching losses are significant and reduce the effectiveness of the active technique. Instead, we propose applying a square-wave voltage whose fundamental frequency component has the magnitude and phase angle of the optimal voltage. Because the magnitude of the fundamental component is larger than that of a square-wave, the actual voltage magnitude can be reduced to π/4 times the original voltage magnitude. The phase angle of the fundamental component is controlled by the voltage transition instant of the square-wave voltage. This voltage waveform can be easily generated by a full-bridge converter. During the period in which the voltage is constant, we do not need to perform switching, and so the circuit becomes much more efficient. During the transitions between the positive and negative voltages of the square wave, a current regulator is used to limit the peak current through the device and switches. This regulator limits the peak current, and hence the peak I2R losses, during the transition. However, the resulting PWM results in additional switching losses. Typical waveforms, determined by simulation, are shown in Fig. 5.

Fig. 6. Bidirectional full-bridge inverter with flyback converter.

The complete energy harvesting circuit is shown in Fig. 6. A full-bridge converter is used to apply the square-wave voltage to the piezoelectric device, and a filter inductor is placed in series with the piezoelectric device to smooth the current. MOSFETs were chosen as the switches (ZVN3320 N-channel MOSFETs, Zetex Semiconductors plc, Oldham, UK). The four MOSFETs in the full-bridge converter are controlled in such a way as to achieve the optimal phase angle determined by the adaptive technique, as discussed previously. The flyback converter is used to control power flow in such a way that the bus voltage of the full-bridge converter, and therefore the magnitude of the square-wave voltage applied to the device, can be prescribed. The optimal voltage magnitude determined by the adaptive algorithm can therefore be achieved through adjustment of the duty cycle of transistor T. Under startup conditions, the MOSFETs are off, and so the full-bridge converter acts as a diode rectifier, drawing power from the piezoelectric device in a passive fashion. Once sufficient energy has been stored in capacitor Cf, the active approach is initiated, thereby drawing power from the device to charge the bus capacitance until the optimal voltage has been reached. The circuit does not draw any power from the battery.

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Fig. 8. Voltage regulation circuit.

Fig. 7. Hysteresis current regulator.

During the voltage transition periods, the lower MOSFETs use PWM to limit the peak current. A simple hysteresis current regulator was developed to achieve this, as shown in Fig. 7. The threshold voltage of the comparator is determined by the combination of resistors R1, R2, and R3. When the output is 0 V, the threshold voltage V+ is

V +1 =

R 2 || R 3 V C, (8) R1 + R 2 || R 3

Fig. 9. Dead-time generation circuit.

and when the output is VC, then the threshold voltage is

V +2 =

R2 V C. (9) R2 + R1 || R 3

When the piezoelectric voltage is held constant, the output current is usually very low, and so the comparator inherently outputs a high-level signal and ensures that the low-side MOSFETs are always on, thus clamping the voltage across the piezoelectric device. During the voltage transition period, once the actual current signal reaches the threshold V+2, the comparator outputs a “low” signal, and the corresponding low-side MOSFET will be turned off, resulting in a decrease of the current until the current signal hits the lower threshold V+1. The output of the comparator will then be high again, and the MOSFET is turned on. This circuit has very low power consumption. When using a TS3022 by ST Microelectonics (Geneva, Switzerland) as the comparator, the power consumption was measured to be 0.7 mW in the experimental tests discussed in the following section. A drive signal also must be generated to control the upper MOSFETs. During the period of a voltage transition in which the electrode capacitance of the piezoelectric device is discharged, both upper MOSFETs must be turned off. The appropriate upper MOSFET is then turned on during the charging period. A comparator (TS862, ST Microelectronics, Geneva, Switzerland) can be used to compare the piezoelectric voltage to a reference signal (close to zero), as shown in Fig. 8, to generate the upper MOS-

FET control signals. The measured power consumption of this circuit was 0.3 mW. To avoid the simultaneous turn-on of two switches in the same bridge arm, an extra circuit is needed to provide dead-time insertion, as shown in Fig. 9. The diode-R-C circuit at the first stage of the circuit is inserted to provide a time delay for the input signals. The time interval between the input signal and the output signal is determined by the RC time constant and the threshold voltage of the Schmitt trigger inverter, noting that the output voltage of the Schmitt trigger inverter is established after the voltage of the capacitor reaches the threshold value. A PCB board was fabricated to implement the power electronic interface circuit and the associated control circuits, as shown in Fig. 10. In the following experiments, power for the control and gate drive circuitry was provided by an external power supply. V. Experimental Results In the experimental setup, the energy harvesting device was constructed using a commercial piezoelectric bimorph (QuickPack QP20W, Mide Technology, Medford, MA) driven by a mini-shaker (ES020, KCF Technologies, State College, PA), as shown in Fig. 11. One end of the piezoelectric device was clamped to a shaker to form a basedriven cantilever, and an evenly-distributed proof mass (m


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Fig. 12. Passive energy harvesting circuit used for comparison.

Fig. 10. PCB board for active energy harvesting.

the alternative electromechanical coupling coefficient of the system. The theoretical maximum power that can be harvested from the system is therefore estimated to be Fm2 /8b = 35.7 mW. The ratio k e2/ς is about 2, which means this system is a relatively strongly coupled system [13]– [15]. Next, the piezoelectric bimorph was connected to both the proposed active energy harvesting circuit in Fig. 6, and a passive circuit, a diode bridge rectifier with a flyback converter prescribing the rectifier voltage, as shown in Fig. 12, to compare the harvested power under different excitation frequencies. In the experiment, the battery is a Li-ion rechargeable battery (ER-C520 by Energizer Battery Co., St. Louis, MO) with voltage Vbat = 7.7 V. For both the active and passive circuits, a 100-Ω sense resistor is placed in series with the battery to measure the current flowing into the battery; the power obtained by the battery can be calculated by

Fig. 11. Resonant piezoelectric device: experimental setup.

= 7 g) was mounted at the other end of the bimorph. To minimize the end effect of the clamp, the clamp piece was made much more massive than the end mass. A sinusoidal voltage generated by a stereo amplifier was fed into the shaker to generate the excitation force. Three MEMS accelerometers (ADXL78 by Analog Devices, Norwood, MA) were attached to the shaker base, the clamp, and the end mass to measure accelerations. The MEMS accelerometer is better than other types of measuring instruments because it is very light and inexpensive. The natural frequency of the system was tuned to be near 120 Hz, and the magnitude of the base acceleration was maintained at ab = 52 m/s2 for all the excitation frequencies in all the experiments to emulate a sinusoidal force with constantmagnitude Fm = mab = 0.364 N. Open-circuit and short-circuit tests were performed to obtain the parameters of the system, and the corresponding resonance frequencies were measured to be 126 Hz and 121 Hz, respectively. The parameters of the system can be measured and calculated, as listed in Table I, where ς = b/2 mk is the damping ratio, and k e2 = kd2/(C − kd2) is

Pharvested = V bat

V dc − V bat . (10) R sense

In the following experiment, the duty cycle of MOSFET T in the flyback converter and the transition instants of the full-bridge converter in the active circuit were manually tuned to find the maximum power at different excitation frequencies. The typical waveforms of the piezoelectric voltage and output current obtained by the active circuit are shown in Fig. 13. It can be seen that the current is well-regulated during the voltage transition. The maximum harvested power at different excitation frequencies for the active and passive circuits are shown in Fig. 14. The maximum power harvested by the active circuit is 27.13 mW at 124 Hz, compared with the theoretical maximum value of 35.7 mW. The ratio of harvested power from the active and passive techniques is shown in Fig. 15. It can be seen that the maximum harvested power by the active technique is 25% larger than the passive technique at the resonance frequency. However, the power harvested by the active technique is up to 6 times larger than that obtained by the passive technique at off-resonance frequencies. The proposed adaptive technique was also implemented. The microcontroller ATmega48 by Atmel Corp. (San Jose, CA) was chosen for our application because of its

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Fig. 13. Typical waveforms of piezoelectric voltage and current for active circuit.

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Fig. 15. Ratio of harvested power by active technique compared with passive technique.

TABLE II. Typical Experimental Results of Adaptive Control Based on ATmega48 Microcontroller. Phase angle (°) 111.7 114.6 117.5 120.0 122.9 125.8 128.6 131.2 128.6

Fig. 14. Harvested power of active and passive techniques.

high performance and low power consumption (250 µA active current at 1.8 V and 1 MHz clock frequency). In the implementation, the microcontroller generates the PWM signal to the MOSFET of the flyback converter to control voltage magnitude, and also a control signal to a digitalto-analog converter (DAC) to control the phase angle of the control voltage. The measured power signal is fed into the microcontroller through the on-chip analog-to-digital converter (ADC). A counter running from 0 to 255 is used as the main time sequence to apply the different voltage waveforms and record the corresponding power. The counter frequency was set to 14.3 Hz, and so it takes about 18 s to run one period. The internal clock frequency of the microcontroller is 8 MHz, and the PWM signal was set to 10-bit resolution, so the switching frequency was 8 kHz. In the experiment, depending on the initial operating point, it takes several minutes to converge to the optimal operat-

Bus voltage (V)

Power (mW)

5.3 6.2 7.5 8.2 10.7 12.1 13.5 16.6 19.4

7.40 8.42 11.20 13.84 16.67 19.77 21.99 25.05 25.55

ing point. A typical experimental result is shown in Table II. The excitation frequency was 120 Hz, and the initial values were 4% duty cycle and 112° phase angle. The optimal operating point can be reached after 9 steps, which takes about 3 minutes VI. Discussion Despite the differences between different energy harvesting techniques, the energy harvesting ability of a piezoelectric dynamic system strongly depends upon the characteristics of the system itself, especially the electromechanical coupling coefficient k e2 and the mechanical damping ratio ς [13]–[15]. For a weakly coupled system, the active technique shows more obvious advantages over the passive technique because the source impedance is mainly reactive. For a strongly coupled system operating near resonance frequency, the source impedance is mainly resistive, and so the passive technique is often good enough for most applications. However, at off-resonance frequencies, the power harvested by the passive technique will drop significantly, and the active technique shows its advantage.


VII. Conclusions The active energy harvesting technique allows the effective application of a controllable electrical impedance to the piezoelectric device, which can be prescribed so as to maximize the power harvested from the device as a function of frequency. A full-bridge converter with control circuitry was built to implement the technique, and the experimental results show that the active technique can significantly improve the energy harvesting at off-resonance frequencies compared with the passive approach. The amount of energy harvested at off-resonance frequencies is a strong function of the efficiency of the power electronic circuitry. Furthermore, in the experimental results presented the control and gate driver circuitry was externally powered. Future work will therefore consist of investigating high-efficiency, standalone converter designs for this application. With a low-power-consumption microcontroller (recently microcontrollers with nanowatt power consumption have become commercially available), and using integrated circuit technology for the control and drive circuit, it is expected that the overall power consumption of the active circuit design can be limited to a very small amount, and therefore the actual harvested power by the active technique should still be promising even at very low vibration levels. References [1] M. Ferrari, C. Trigona, M. Guizzetti, B. Anda, S. Baglio, and V. Ferrari, “Improved energy harvesting from wideband vibrations by nonlinear piezoelectric converters,” Sens. Actuators A, vol. 162, no. 2, pp. 425–431, 2009. [2] C. C. McGehee, S.C. Stanton, and B.P. Mann, “Nonlinear dynamics for broadband energy harvesting: Investigation of a bistable piezoelectric inertial generator,” Physica D, vol. 239, no. 10, pp. 640–653. [3] G. K. Ottman, H. F. Hofmann, A. C. Bhatt, and G. A. Lesieutre, “Adaptive piezoelectric energy harvesting circuit for wireless remote power supply,” IEEE Trans. Power Electron., vol. 17, no. 5, pp. 669–676, Sep. 2002. [4] G. K. Ottman, H. F. Hofmann, and G. A. Lesieutre, “Optimized piezoelectric energy harvesting circuit using step-down converter in discontinuous conduction mode,” IEEE Trans. Power Electron., vol. 18, no. 2, pp. 696–703, Mar. 2003. [5] A. Badel, A. Benayad, E. Lefeuvre, L. Lebrun, C. Richard, and D. Guyomar, “Single crystals and nonlinear process for outstanding

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vibration-powered electrical generators,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 53, no. 4, pp. 673–684, Apr. 2006. [6] D. Guyomar, A. Badel, E. Lefeuvre, and C. Richard, “Towards energy harvesting using active materials and conversion improvement by nonlinear processing,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 52, no. 4, pp. 584–595, Apr. 2005. [7] E. Lefeuvre, A. Badel, A. Benayad, L. Lebrun, C. Richard, and D. Guyomar, “A comparison between several approaches of piezoelectric energy harvesting,” J. Phys. IV France, vol. 128, pp. 177–186, Sep. 2005. [8] Y. Liu, “Active energy harvesting,” Ph. D. dissertation, Dept. of Electrical Engineering, The Pennsylvania State University, University Park, PA, 2006. [9] G. Tian, “Active energy harvesting on piezoelectric materials: Experimental demonstration and standalone circuit implementation,” Master’s thesis, Dept. of Electrical Engineering, The Pennsylvania State University, University Park, PA, 2008. [10] Y. Liu, G. Tian, Y. Wang, J. Lin, Q. Zhang, and H. F. Hofmann, “Active piezoelectric energy harvesting: General principle and experimental demonstration,” J. Intell. Mater. Syst. Struct., vol. 20, no. 5, pp. 575–585, Mar. 2009. [11] Y. C. Shu and I. C. Lien, “Analysis of power output for piezoelectric energy harvesting systems,” Smart Mater. Struct., vol. 15, pp. 1499–1512, Sep. 2006. [12] Y. C. Shu, I. C. Lien, and W. J. Wu, “An improved analysis of the SSHI interface in piezoelectric energy harvesting,” Smart Mater. Struct., vol. 16, pp. 2253–2264, Oct. 2007. [13] Y. C. Shu and I. C. Lien, “Efficiency of energy conversion for a piezoelectric power harvesting system,” J. Micromech. Microeng., vol. 16, pp. 2429–2438, Sep. 2006. [14] S. Roundy and P. K. Wright, “A piezoelectric vibration based generator for wireless electronics,” Smart Mater. Struct., vol. 13, pp. 1131–1142, Aug. 2004. [15] S. Roundy, P. K. Wright, and J. M. Rabaey, Energy Scavenging for Wireless Sensor Networks. Norwell, MA: Kluwer Academic, 2004.

Cheng Luo was born in Nanchang, China, in 1981. He received his bachelor’s and master’s degrees, both from Tsinghua University, Beijing, China, and both in electrical engineering. He received his Ph.D. degree in electrical engineering from The Pennsylvania State University, University Park, PA, in 2010. He is now a lead engineer at General Electric Company, Salem, VA. His research interests are power electronics design, energy harvesting, and motor drive and control.

Heath Hofmann received his Ph.D. degree in electrical engineering and computer science from the University of California at Berkeley in 1998. He is currently an Associate Professor at the University of Michigan. Dr. Hofmann’s research area is power electronics, specializing in the design, analysis, and control of electromechanical systems. Specific interests are propulsion drives for electric and hybrid electric vehicles, energy harvesting, flywheel energy storage systems, and numerical analysis techniques for electromechanical devices.