An electromechanical representation of a piezoelectric crystal used as a transducer

The equivalent electrical representation of a piezoelectric crystal when used as an element in an electrical circuit has been discussed by several investigators.
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who have arrived at the circuit shown on Fig. 1. Apparently, however, no circuit has been evolved for representing a crystal when it is used as a transducer to couple electrical circuits to mechanical systems. Since such crystals 2 are used in loud speakers, microphones, supersonic radiators, and other apparatus, it is a matter of importance to obtain such a representation. This paper discusses such an equivalent circuit and relates the elements to the mechanical, electrical, and piezoelectric constants of the material. When used as a purely electrical circuit, this representation reduces to that of Fig. 1.