Wave-propagation analysis of the monolithic-crystal filter

The monolithic-crystal filter has two or more pairs of electrodes on the same piezoelectric plate; mechanical-wave propagation in the regions between electrodes takes place in an evanescent mode. A method of analysis, based on solutions to the wave equation, is first developed in general terms and then applied specifically to AT-cut quartz. Boundary conditions are used to determine resonant frequencies and vibration patterns under short-circuit conditions; from these, the electrical equivalent circuit is developed. Experimental results confirm the validity of the method. The theoretical results presented describe the bandwidth and motional capacitances of 2- and 3-resonator filters as functions of their geometry (electrode width, thickness and spacing). A method of predicting unwanted resonances is developed. A 3-resonator filter, with a centre resonator of different geometry, is considered. Some results relevant to multiple-resonator filters (up to ten) are described. Extension of the theory to overtone-mode devices is also discussed.