204 BELL SYSTEM TECHNICAL JOURNAL 



difficult to evaluate. Re is most readily eliminated from the equation by 

 revising the picture slightly. With reference to the simplified oscillator 

 circuit, Fig. 12.22B, it is apparent that the static crystal capacitance Co and 

 the circuit capacitance Ct may be combined. This leaves for the crystal 

 branch the inductance Lc (different from Lc) which is a function of frequency 

 and the resonant resistance of the crystal Ri which is not a function of 

 frequency. Now Ri may be assumed small compared to p with considerable 

 accuracy. It is only necessary, then, to replace Ct in equation (12.88) with 

 (Co + Ct) and Re by Ri . This equation then becomes 



PI = J, 2./. ^x2 (12.89) 



Rio) (Co + Ct) 



An exact equation for PI is derived in section 12.83 and it is shown that the 

 error in the simple expression above will in most cases be very small. 



An approximate equation for the relation between i?i and Re is obtained 

 by dividing ( 1 2 .88) by ( 1 2 .89) . We thus find 



, = ^^^1^' (12.90) 



or the effective resistance of the crystal at the operating frequency is 



Re = Ri (~^ + 1 J (12.91) 



Because of the approximation in equation (12.88) the equation for Re above 

 is accurate only when ( tt + 1 ) <^ M^ as will be shown in section 12.83. 

 The expression for PI as given by (12.89) may be written 



.'d;..(i + 



PI = 2^2 „ A , CX (12.92) 



which is the most convenient form for calculating PI from the constants 

 of the crystal and the oscillator circuit. 



12.82 Relation Between M and PI 

 It was found that 



M = -pr^ (12.93) 



coi Co Ai 



and this is essentially equal to „ „ over the narrow frequency range 



coCo-fvi 



considered. Therefore, 



_ M 



PI = / ^^2 (J2.94) 



(aCt 



(-gj 



I 



