ELECTRICAL WAVE FILTERS 409 



Inserting these values, the element values in terms of the dimensions 

 become 



Co = 0.402 X 10-12 ij^ll^^ 



Ci = 0.289 X 10-" hnUlle, (2) 



Z/i = 1 18.2 Imle/lo- 



From these values it is seen that there is a fixed ratio between these 

 two capacitances ^ of 



r = Co/Ci = 140. (3) 



As will be evident later, this ratio limits the possibilities of the use of 

 quartz crystals in filter circuits. 



Experiments with quartz crystals, with electrodes contiguous to 

 the crystal surfaces and with the optical and electrical axes small 

 compared to the mechanical axis, show that these values are approxi- 

 mately correct. The value of Co checks the above theoretical value 

 quite closely. The value of Ci obtained by experiment is somewhat 

 larger than that given by equation (2) and the value of the inductance 

 somewhat smaller. The ratio of Co/Ci has been found as low as 115 

 to 1, but a value of 125 to 1 is about all that can be realized, when 

 account is taken of the distributed capacitance of the holder, connect- 

 ing wires, etc. 



When either of the dimensions along the electrical or optical axes 

 becomes more than a small fraction of the dimension of the mechanical 

 axis, the plane wave equations given above no longer hold accurately. 

 This is due to the fact that a coupling exists between the motion along 

 the mechanical axis and other modes of motion. For an isotropic 

 body, one is familiar with the fact that when a bar is compressed or 

 stretched it tends to stretch or compress in directions perpendicular to 

 the principal direction of motion. This state of affairs may be de- 

 scribed by saying that the modes of motion are coupled. In a crystal 

 this same relation exists and in addition, due to its crystalline form, a 

 shearing motion is set up whose shearing plane is determined by the 

 mechanical and optical axes and whpse motion is parallel to the me- 

 chanical axis. In fact the shearing motion is more closely coupled to 

 the mechanical axis motion that is the extensional motion. As long 

 as the optical axis is small compared to the mechanical axis, this coup- 

 ling action manifests itself as a decreased stiffness along the mechanical 

 axis, but if a condition of resonance is approached for the motion along 

 the optical axis, the mode of motion is entirely changed. This effect is 



^ In a paper contributed recently to the Inslilnte of Radio Engineers, it is shown 

 that this ratio limitation is a consequence of a fixed electro-mechanical coupling 

 between the electrical and mechanical systems of the crystal. 



