QUARTZ CRYSTAL APPLICATIONS 213 



From these values and the elastic constants of Table III we can calculate 

 all the different forms of the piezoelectric constants. These are given in 

 Table V. 



(A.3). Derivation of Equivalent Circuit of Crystal 



The electrical impedance and electrical equivalent circuit for a fully- 

 plated crystal can be derived from the piezoelectric relations of equation 

 (A. 6) taken together with Newton's law of motion 



/^ 



Fy = ma = (pdxdydz) -y-. (A.31) 



at- 



where m is mass of an elementary volume dx dy dz, a the acceleration, and 

 ^ is the displacement of the element in the y direction. If we consider a 

 long thin X cut crystal with its length in the y direction, the piezoelectric 

 relations of interest are 



— yy = SnX^ + ^fi Yy + Sl3Z^ — sfi Yz + dnEx\ 



E,Kl (A.32) 



Qx = —. — dn Xjo + dn Yy — du Yz. 



For a long thin crystal with its long dimension in the Y direction we can set 



X^ = Z, = F. = (A.33) 



This follows since the crystal is free from external forces, and hence these 

 stresses on the edges of the crystal must be zero. On account of the small x 

 and s dimensions, the rate of change of these stresses with x or z will have 

 to be high in order that the stresses shall differ appreciably from zero, and 

 there are no mechanical strains causing a high stress gradient. Then for a 

 long thin bar the piezoelectric equations can be written 



—yy = ^fi Yy + dxiEi) 



Qx = —^ + dn Yy. 



Let us next consider a small cross section of the crystal with a dimension 

 dy along the crystal length. The total force on the section is a resultant of 

 the difference in stresses on the two faces or equal to 



dV 



LaYy, - Yy,] = -Ltt-y dy = Fy (A.35) 



oy 



where Yy the stress is considered as a compressional force acting on the faces 

 of the element. By Newton's law of motion (A.31) we have 



-L(.dy— = L (, dy f, ^ or -^^ = - p ^-, (A.36) 



