Optimum Values of Load Resistance at the Short-Circuit and Open-Circuit Resonance Frequencies of the Voltage FRF

Equation (5.39) can be used to obtain the optimum load resistance for the maximum electrical power output at a given excitation frequency & (around the respective resonance frequency). The problem of interest is the resonance excitation and one can use Equation (5.40) for excitation at & = 1 to obtain

which is the optimum load resistance for excitation at the short-circuit resonance frequency of the voltage FRF. A similar approach can be followed for estimating the optimum load resistance for excitation at the open-circuit resonance frequency of the voltage FRF as

For excitations at the short-circuit and open-circuit resonance frequencies, the optimal values of load resistance are inversely proportional to the capacitance and the undamped natural frequency. The electromechanical coupling and the mechanical damping ratio also affect the optimal load resistance. The optimal resistive loads obtained in Equations (5.42) and (5.43) can be back substituted into Equations (5.40) and (5.41) to obtain the maximum power expressions for excitations at these two frequencies.

Recall that the short-circuit and open-circuit resonance frequencies defined here (based on the voltage FRF) are not necessarily the frequencies of maximum power generation. One can first obtain Rlp>(&) from Equation (5.39) by setting д |П (&)| /дRl = 0 and then use it in Equation (5.39) to solve д |П (&)| /д& = 0 for the frequencies of the maximum power output (see Renno et al. [3] for a detailed analysis of this problem based on lumped-parameter modeling for a piezo-stack).