Miniaturization is a continuing trend in the electronics industry. The power required by smaller devices has decreased, but the challenge remains to find energyefficient power sources for them. Battery technology has not been able to keep pace with the extremely rapid developments in microelectronics, drawing attention to energyharvesting MEMS (microelectromechanical systems) as a viable and clean energy source.^{1} Autonomous energy harvesters visàvis creating selfpowered miniaturized devices is highly sought after in many applications, such as remote supervision of patients' vital signs and health management, gas and liquid flow control and handling, environmental observation, and security and defense. These demands for miniaturization and autonomy call for the ability to convert mechanical vibrations into electric energy, thus offering batteryfree operation.^{1, 2}
Mechanical energy from vibrating sources in the surrounding environment is pervasive and accessible, found in such places as automobile engines, rotating equipment, and the human body, in all instances translating vibration into electrical energy via the deformation of a piezoelectric material (see Figure 1). Harvesting this energy is one of the most promising transduction techniques owing to the high energy density and ample power it provides (on the order of 100μW). The minimal microfabrication restrictions on such devices, and their amenability to both indoor and outdoor applications, make them useful in a wide range of situations.
Figure 1. Illustration of the piezoelectric (P) transduction technique.
A piezoelectric energy harvester consists of a piezoelectric film on a silicon cantilever with a seismic mass, sandwiched between electrodes that are used to collect the generated power (see Figure 2). When the piezoelectric material is subjected to mechanical vibrations, stress is induced within the material, thus giving rise to an electromotive force that generates an electrical impulse. We are interested in optimizing the design of piezoelectric energy harvesters to increase performance efficiency. We have analyzed, modeled, and benchmarked a piezoelectric energy harvester stimulated by random vibrations.
Figure 2. Schematic of a piezoelectric energy harvester. AlN: Aluminum nitride. h: Thickness of the cantilever. h_{0}: Thickness of the seismic mass. L: Length of the cantilever. L_{0}: Length of the seismic mass. Si: Silicon. t: AlN thickness.
Electromechanical analysis and modeling are based on using the equivalent electrical circuit representation of the piezoelectric harvester. Applying Kirchhoff's current (i) and voltage (V) laws to the harvester equivalent circuit (see Figure 3), the generated stress σ within the cantilever beam can be expressed as:
and the generated current can be represented by:
where C_{m} is the equivalent capacitance of the generator, is the capacitance of the piezoelectric thin film (ε is the permittivity of the piezoelectric film, w is the width of the piezoelectric cantilever, L is the length of the cantilever, and t is the thickness of the piezoelectric film), L_{m} is the equivalent inductance of the generator mass, n is the voltage ratio, and R_{m} is the equivalent resistance of the mechanical damping.^{3} and are the second and firstorder derivatives of the strain S. R_{L} is the resistance of the maximum power transfer. The average stress generated within the beam is directly proportional to the vibrational force F, so that σ=α_{1}F where α_{1} is a proportionality constant. The generated strain S within the beam meanwhile is directly proportional to the deflection of the piezoelectric beam z, so that z=α_{2}S, where α_{2} is a geometric constant. Considering the 31 operation mode of the device, the stress along this axis is defined as:
where z ′ is the distance from the neutral axis to the center of the piezoelectric thin film, E is Young's modulus of the piezoelectric film, and ρ(x) is the radius of curvature.
Figure 3. Equivalent circuit of the harvester. C
_{m}: Equivalent capacitance of the generator. C
_{p}: Capacitance of the piezoelectric thin film. i: Current. L
_{m}: Equivalent inductance of the generator mass. n: Voltage ratio. R
_{L}: Resistance of the maximum power transfer. σ: Generated stress.
: Firstorder derivative of the strain S. V
_{out}: Voltage out.
Figure 4. Cross sectional view of the device. L_{e}: Length of the electrode.
Since the thickness of the seismic mass h^{3}_{0} is much greater than the corresponding value of the cantilever beam (see Figure 4), and the width w is uniform along the device structure, the moment of inertia can be approximated as:
Thus, the output current i and voltage ratio n can be calculated as follows:
where L_{e} and w are the length and width of the electrode and d_{31} is the piezoelectric strain constant in the 31mode. The damping coefficient can be described as:
where w_{n} is the natural frequency, ζ is the damping ratio, and m is the proof mass. The damping b_{m} is modeled by the resistor R_{m} and the two are related by the product of the proportional and geometric constants, so that R_{m}=α_{1}α_{2}b_{m}. The inductance L_{m} that relates the mechanical stress to the secondorder derivative of the strain S is similarly related, so that L_{m}=α_{1}α_{2}m.
Table 1.Benchmarking of modeled and experimental results for three different device dimensions using an acceleration of 0.2g (m/s2).
Specifications  Device 1  Device 2 
Beam width (mm) 
3.0 
5.0 
Beam length (mm) 
1.31 
1.01 
Beam thickness (μm) 
45 ± 5 
45 ± 5 
Mass width (mm) 
3.0 
5.0 
Mass length (mm) 
3.0 
5.0 
Mass thickness (μm) 
675 
675 
Experimental resonant frequency (Hz) 
1050 
573 
Modeled resonant frequency (Hz) 
1050.6 
573.1 

It is then possible to state:
where L_{0} is the length of the seismic mass. Substituting back into Equations (1) and (2) results in:
These are the analytical equations that we model in MATLAB. The modeling results were validated against the available experimental findings^{4} using a thin film of aluminum nitride of 1μm thickness and various device dimensions (see Table 1).
Figure 5. Comparison of modeled and experimental results of the generated power as a function of frequency for the unpackaged device 1 at 0.2g and in air and vacuum.
Figure 6. Comparison of modeled and experimental results of the generated power as a function of frequency for the unpackaged device 2 at 0.2g and for air and vacuum ambients.
Figures 5 and 6 show how closely our model agrees with experimentally derived values for generated output power against resonant frequencies typical for two different device geometries in air and vacuum. The optimal load resistance for maximum power transfer was calculated and found to be in very good agreement with experimental findings (see Figure 7). Furthermore, the generated output power level of the device increases significantly with increasing acceleration values (see Figure 8). The latter result shows the level of power that can be harvested corresponding to available acceleration sources.
Figure 7. Comparison of modeled and experimental results of the generated power as a function of load resistance for the unpackaged device 2 at 0.25g in air.
Figure 8. Modeled results of the generated power as a function of frequency for the unpackaged device 2 at various accelerations in air.
In summary, we have presented an effective analytical tool for the optimization of piezoelectric energy harvesters. We have shown that our model accurately describes the function of such devices as benchmarked against experimental measurements. The accuracy of the model permits dimensional scaling of the device and thus precise prediction of generated power and resonant frequency. This model will serve to advance research capabilities in the field of piezoelectric energy harvesters and opens the way for designing integration of manifold selfpowered miniaturized devices such as selfpowered sensors and actuators for a variety of applications.
This work has been supported by the Ontario Research Fund–Research Excellence and the Natural Sciences and Engineering Research Council Canada.
Ali Badar Alamin Dow, Nazir P. Kherani
University of Toronto
Toronto, Canada
Ali B. Alamin Dow graduated with a PhD from the University of Bremen, Germany (2009), while working as a research scientist at IMSAS. Currently, he is a postdoctoral associate and research scientist at the University of Toronto, pursuing design, optimization, and development of MEMS and NEMS (nanoelectromechanical systems) devices.
Nazir P. Kherani graduated with a PhD in physics from the University of Toronto (1994). Currently, he is an associate professor at the University of Toronto in the Departments of Electrical and Computer Engineering and Materials Science and Engineering, leading the Advanced Photovoltaics and Devices research group.
Ulrich Schmid
Vienna University of Technology
Department for Microsystems Technology, Austria
Ulrich Schmid studied physics and mathematics and received his PhD from the Technical University of Munich, Germany (2003), while working in the research laboratories of DaimlerChrysler AG (now EADS Deutschland GmbH). Since October 2008 he has been a full professor at the Vienna University of Technology, heading the Department for Microsystems Technology.
References:
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2. G. Despesse, T. Jager, J. Chaillout, J. Leger, A. Vassilev, S. Basrour, B. Chalot, Fabrication and characterisation of high damping electrostatic micro devices for vibration energy scavenging, Proc. Design Test Integrat. Packag. MEMS MOEMS, pp. 386390, 2005.
3. S. Roundy, P. K. Wright, A piezoelectric vibration based generator for wireless electronics, Smart Mater. Struct. 13, no. 5, pp. 11311142, 2004.
4. T. M. Kamel, R. Elfrink, D. Hohlfeld, M. Goedbloed, Y. van Andel, C. de Nooijer, M. Jambunathan, R. van Schaijk, MEMSbased aluminum nitride piezoelectric energy harvesting module, Smart Syst. Integrat. Conf., 2009.