PERFORMANCE INDEX OF QUARTZ PLATES 239 



The basis for this analysis depends upon the ability to utilize the following 

 equation to represent any impedance whose frequency of operation is con- 

 trolled by a crystal. This equation is known as a linear fractional trans- 

 formation 



W = ^^±11 (15.52) 



The terms a, /3, 7, and 5 represent complex constants and Z represents a 

 complex variable later to be chosen to represent a linear function of fre- 

 quency. Since TI^ and Z represent the dependent and independent variable, 

 they may also be considered as representing two separate planes. The 

 abscissa and ordinate of these two planes represent their real and imaginary 

 components respectively. The planes are linked by (15.52), that is, this 

 equation will transform a specific pomt from one plane to the other. 



The Constantsa, /3,7 and 5 for the equivalent crystal circuit are determined 

 by writing the expression for Zc in the form of (15.52). For example 

 (neglecting Rl), Zc from Fig. 15.1 may be written as 





Zc= r^ ^ L \ ^^^^^/^^^_, (15.53) 



By substituting (15.48) and (15.49) in (15.53), this impedance may be writ- 

 ten as 



jcoCo [wi?! -f _7Zi(co — C02)(C0 + CO2)] 



Since the operating frequency, w, represents some frequency between coi 

 and C02 , and co ;:^ coo — wi , we can make the following approximations 

 in this operating range. The symbol coo is defined as the average operatmg 

 radian frequency. 



"' ~ ^ I (15.55) 



COe = CO j 



If (15.55) is substituted in (15.54), factor We out, add and subtract C02 from the 

 imaginary component in both numera tor and denominator, we may write Z^ as 



r 1 , . 1 2Li . n1 I r 1 1 • 2^1 r ^ 



— TT + J —^ -^ (C02 - COl) + —- ; ^- (cO - CO2) 



J<1 



