Analysis of a micromachined piezoelectric energy harvester

Miniaturization is a continuing trend in the electronics industry. The power required by smaller devices has decreased, but the challenge remains to find energy-efficient power sources for them. Battery technology has not been able to keep pace with the extremely rapid developments in microelectronics, drawing attention to energy-harvesting MEMS (microelectromechanical systems) as a viable and clean energy source.¹ Autonomous energy harvesters vis-à-vis creating self-powered 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 battery-free 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. h₀: Thickness of the seismic mass. L: Length of the cantilever. L₀: 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 first-order 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=α₁F where α₁ 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=α₂S, where α₂ 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. : First-order 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 h³₀ 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 d₃₁ is the piezoelectric strain constant in the 31-mode. 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=α₁α₂b_m. The inductance L_m that relates the mechanical stress to the second-order derivative of the strain S is similarly related, so that L_m=α₁α₂m.

It is then possible to state:


where L₀ 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 findings⁴ 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 self-powered miniaturized devices such as self-powered 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.