QLARTZ CRYSTAL APPLICATIONS 



189 



or X axis. Conversely, a voltage applied along the X axis will produce a 

 charge displacement and consequent!}- a mechanical stress along the Y axis 

 which will set up a longitudinal wave along the mechanical axis. As shown 

 by Fig. 1.8, the type of motion resulting when the crystal is free to move on 

 the ends is one in which the center is stationary and the ends move in and 

 out. The crystal can then be clamped at its center or mounted from leads 

 soldered to electrodes deposited on the surface. 



In using a crystal in an electrical circuit it is desirable to have an electrical 

 equivalent circuit which will represent the electrical impedance as measured 

 from the terminals of the crystal. Such a circuit^ is shown in Fig. 1.8. In 



+ 00 



L, ^ 



^R- 2\\PS2Z 21 ' ^22 Y 

 f^ GIVEN BY SOLVING 



^o — v~^°^^7^- I"l2j KS22 ~ 



Fig. 1.8— Longitudinally vibrating crystal and electrical equivalent circuit 



this representation Co is the static capacity of the crystal which would be 

 measured if the crystal were held from moving. Ci is the stiflfness of the 

 crystal transformed into electrical terms through the piezoelectric effect of 

 the crystal, while Li is the effective mass of the crystal also transformed into 

 electrical terms. The resonant frequency of the crystal is determined by 

 the Young's modulus and density of the bar according to the usual formula: 



/« 



1 



2? 



^0 

 P 



(1.5) 



^ Circuits of this type for representing the electrical impedance of a crystal were first 

 derived by Van D>ke; see reference (7). The method of deriving them from Voigt's 

 equations is discussed in the appendix. 



