232 BELL SYSTEM TECHNICAL JOURNAL 



The quantity Qi makes possible the calculation of Qc where Qc is defined 

 as 



wLc Xt 



Qc- ^ = 



(-8)' 



^c - ^ /, , CoV (15.37)^ 



It can be shown that Qc in terms of Qi is given by the following equation if 

 RL»Xt. 



Rl Rl 



The Figure of Merit, M, defined by (15.3) at the series resonant frequency 

 of the crystal, coi, becomes 



M ^ -^ = ^ (15.39) 



The measurement of M can be determined from Qc provided Ct and Co 

 are known. M may be determined from the following expression 



M - ^' ^' {^ 4- ^«Y 



^ . _ PJ. Co \ "^ cj (15.40) 



Rl 



If Ct is selected such that C< ^ Co , then for most cases Rl is large com- 

 pared to QiXt . This means that Qi = Qc and R = Rc . With this approxi- 

 mation, (15.40) becomes 



The relationship between i?c and Ri as a function of frequency may be 

 expressed directly from the input impedance expression of the equivalent 

 circuit. This is given as 



„ Rl 



^. - r- 2 ,-,2 . y5^2) 



[0)2 — W , 1 



where w is the unity power factor frequency in the P.I. meter, neglecting Rl 

 If 



j. fco; - con 



'•H) 



The relationship i?c = i?i ( 1 + — ) was derived in Fair's paper*^ 



