ELECTRICAL WAVE FILTERS 



441 



shear resonance indicated by B is barely noticeable, while the z exten- 

 sional mode indicated by A is somewhat stronger although higher in 

 frequency. The frequency of the principal mode is not greatly affected 

 by an increase in the z' axis until the ratio of axes is greater than .6. 

 Another angle of some interest is = + 18.5° since there the z' ex- 

 tensional coupling disappears. The resulting resonances are shown on 

 Fig. 32. It will be noted that the z' extensional resonance curve A is 

 very weak, while the shear curve B is quite pronounced. 



A n Equivalent Electrical Circuit for a Crystal Possessing Two Degrees 



of Motion 



The above explanation accounts qualitatively for all the resonances 

 observed in the crystal and how they are varied by a rotation of the 

 crystal. It is desirable, however, to see if a quantitative check can be 

 obtained from the known elastic constants of the crystal. To obtain 

 a complete check would require a system capable of five degrees of 

 motion. However, if we take the simplest case, the — 18.5 degree cut 

 crystal, only two modes of motion have to be considered, and even for 

 the zero cut crystal, a good agreement is obtained by lumping the 

 shear and extensional mode as one mode of motion and considering its 

 reaction on the fundamental mode. Hence consideration is limited 

 in this paper to a circuit having two modes of motion. 



The properties of a single mode of motion can be represented for 

 frequencies which do not exceed the first resonant frequency of the 

 crystal, by the simple electrical circuit of Fig. ?)?)A. Here the capaci- 



I — "w^ — ^H 



""Y 



I — ^W^ 



Cy 



-Cm Cz 



i-z 



■^M^ — I 



-Cm 



A B 



Fig. 2)i — Equivalent electrical circuit of a crystal having two modes of motion. 



tance represents the mechanical compliance of the bar, the charge on the 

 condenser represents a displacement per unit length of the bar, while 

 the current flowing through the circuit represents the velocity of a 

 point on the bar. The inductance represents the mass reaction of the 

 crystal. The representation of the motion of a bar by a simple lumped 

 circuit assumes that the bar moves as a whole, that is, if a force is 

 applied to the body it contracts or expands equally at all parts of the 



