Computer optimization of transducer transfer functions using constraints on bandwidth, ripple, and loss

Transducers, having one piezoelectric layer near its half-wave resonance and N quarter-wave layers, are designed using computer optimization to adjust the thicknesses and impedances of the various layers so as to fit the resulting transfer function to a target function. An augmented Mason model is used to evaluate the transducer. Optimization of fit is by a steepest descent algorithm. Essentially error-free fits are achieved for target functions that match the underlying dynamics. By applying classical filter theory to a lumped-element transducer model, the transducers dynamics are identified as all-pole filters, which are characterized by polynomials of order N to N+1. The design methodology is tested by designing a series of low-loss transducers that explore fractional bandwidths from 45 to 116%. From these studies there appears to be constraints on the minimum Q of the poles, and other properties. Typical power transfer efficiencies of -1 dB are achieved by impedance scale matching. Using a second-order Fano bound, it is shown that the matching layers function as an optimal compensation network for low-loss flat bandpass transducers. Finally, by the inclusion of loss, lower Q poles are demonstrated with a Bessel transducer.