An Electromechanical Representation of a Piezoelectric 

 Crystal Used as a Transducer * 



By W. P. MASON 



THE equivalent electrical representation of a piezoelectric crystal 

 when used as an element in an electrical circuit has been discussed 

 by several investigators,^ who have arrived at the circuit shown on 

 Fig. 1. Apparently, however, no circuit has been evolved for repre- 



Fig. 1 — Electrical representation of a piezoelectric crystal. 



senting a crystal when it is used as a transducer to couple electrical 

 circuits to mechanical systems. Since such crystals ^ 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. 



When piezoelectric crystals are used to drive external mechanical 

 systems, the modes of motion most often used are longitudinal vibra- 

 tions perpendicular or parallel to the applied electric field. Accord- 

 ingly, the elements of the equivalent network are derived for these cases 

 only. Thje network can, however, represent any type of motion driving 

 a load just as the network of Fig. 1 can represent the crystal for any 

 type of motion. 



Let us consider first the case of a crystal vibrating perpendicularly to 



* Digest of an article appearing in the Proc. I. R. E. for October, 1935. 



1 W. G. Cady, Phys. Rev., XXIX, 1 (1922); Proc. I. R. E., X, 83 (1922). K. S. 

 VanDyke, Ab. 52, Phys. Rev., June, 1925; Proc. I. R. E., June, 1928. D. W. Dye, 

 Proc. Phys. Soc. (London), XXXVIII, (5), pp. 399-153. P. Vigoreaux, Phil. Mag., 

 December, 1928, pp. 1140-53. 



2 A. M. Nicolson, Proc. A. I. E. £., 38, 1315-1333, 1919. E. B. Sawyer, Proc. 

 I. R. E., 19, No. 11, p. 2020, November, 1931. S. Ballantine. Proc. I. R. E., 21, 

 No. 10, p. 1399, October, 1933. 



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