Modeling of MEMS piezoelectric energy harvesters using electromagnetic and power system theories

This work proposes a novel methodology for estimating the power output of piezoelectric generators. An analytical model that estimates, for the first time, the loss ratio and output power of piezoelectric generators, based on the direct mechanical-to-electrical analogy, electromagnetic theory, and power system theory, is developed and takes into account the dimensions and material properties of the generator. The mechanical-to-electrical analogy and power system theory allow deriving an equivalent input impedance expression for the network. Further, electromagnetic theory allows deducing the equivalent electromechanical loss of the piezoelectric generator. By knowing the mechanical input power and the loss of the network, calculating the output power of the piezoelectric device becomes a straightforward procedure. Experimental results based on published data are also presented to validate the analytical solution. Moreover, and to fully benefit from the well established electromagnetic and electric circuit theories, further analysis efforts on the resonant frequency, bandwidth, and sensitivity are presented. Compared to conventional modeling methods currently being adopted in the literature, the proposed method provides, relatively easily, significant additional information that is crucial for enhanced device operation. Finally, and with the study provided in this paper, optimizing piezoelectric harvesters is simplified. Nomenclature α Constant relating the current density to strain rate β Constant relating the strain to output voltage specifically at resonance δ The real part of the complex frequency variable ε Dielectric constant Γ Reflection coefficient σin Input stress ω The imaginary part of the complex frequency variable ζ Unitless damping ratio a A constant being either 1 or 2 depending on the wiring of the harvester bm Damping coefficient cp Elastic Modulus of the piezoelectric material csh Elastic Modulus of the center shim Cb The capacitance of the piezoelectric bender Ck An equivalent capacitance representing mechanical stiffness d31 Piezoelectric strain coefficient g Acceleration of gravity i Electric current iin Input current j Complex number (i.e. the square root of –1) k1 A constant relating stress to force k2 A constant relating the strain to the deflection k31 Piezoelectric coupling coefficient le The length of the electrode in the piezoelectric harvester Lm Equivalent inductance representing mass m Mass n Turn ratio p Normalized complex frequency variable Pin RMS incident power PL RMS power delivered to the load PLR Power loss ratio PR RMS reflected power R Load resistance Rb A resistor representing mechanical damping S Strain tc The thickness of the ceramic/piezoelectric material tsh The thickness of the shim V Voltage w The width of the ceramic/piezoelectric material Zin Input impedance