ELECTRO-MECHANICAL REPRESENTATION OF CRYSTAL 719 



the direction of the applied field. Two subdivisions of this case are 

 usually of interest, the first when the crystal is supported at its center 

 and drives two symmetrical loads, and the second when the crystal is 

 supported on one end and drives a load on the other end. The sym- 

 metrical case is considered first. 



By employing the well known analogies between electrical and me- 

 chanical systems, it is possible to obtain a simple network, expressed in 

 terms of electrical symbols, which represents the properties of a piezo- 

 electric crystal. In this representation, force is the analogue of voltage, 

 mechanical displacement of electrical charge, and velocity of electrical 

 current. When the electrodes are attached to the crystal faces, the 

 equivalent network of the crystal is shown by Fig. 2, The voltage E 



Fig. 2 — Electromechanical representation of a symmetrical 

 piezoelectric crystal. 



is the voltage applied across the plates of the crystal, the force F is 

 the force applied to each end of the symmetrical crystal, Qi is the elec- 

 trical charge flowing in the wires connected to the crystal and Q2 and Qz 

 are the mechanical displacements of the ends of the crystal which are 

 equal on account of the symmetry of the crystal. 



The constants of the crystal can be evaluated by considering limiting 

 cases. The capacitance Co is the electrostatic capacitance of the 

 crystal clamped. The compliance Ce is the mechanical compliance of 

 the crystal. In c.g.s. electrostatic and mechanical units, these have 

 the values 



— —, — f- ; C-B — TT . K'^) 



where K is the dielectric constant of the crystal clamped, h the dimen- 

 sion of the crystal in centimeters perpendicular to the surfaces of the 

 electrodes, Im the length of the crystal in the direction of vibration, /o the 

 length of the third axis, and 5 the modulus of compliance of the crystal 

 (the inverse of Young's modulus) along the axis of vibration. The 

 inductance Le represents the mass reaction of half the crystal. At low 



