Equations Governing the Electrical Behavior of an Arbitrary Piezoelectric Resonator Having N Electrodes

General formulas for the electrical admittance of a piezoelectric resonator having essentially one pair of electrodes were derived by Lewis.1 These formulas are consistent with those derived earlier for special cases such as long bars and large plates (see, for example, Mason 2 ). In Lewis' work the admittance function is expanded about its poles in an infinite series. The residue at one of these poles determines the strength of the contribution of the normal mode, associated with the pole, to the overall vibrational behavior of the resonator when it is driven at a frequency close to the natural frequency of the mode. Surprisingly, the work of Lewis seems to have seen little application, as far as can be judged, except for that of Lloyd and Redwood,3 and Byrne, et al.4 * Most of the work described here is based 011 part of the author's doctoral thesis (University of London, 1966). 1881 1882 T H E BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1967 With the current interest in multi-electroded resonators, such as the monolithic crystal filter,5 it is pertinent to consider the logical extension of the work of Lewis to the case of an arbitrary resonator having N electrodes. A discussion of this problem has previously been presented by the author,6 and also by EerNisse and Holland.7 Included in Section II of this paper are the basic equations governing the piezoelectric resonator, presented here for completeness. In Section I I I various integral relations are derived for use in Section IV where the properties of the admittance and impedance matrices are investigated.