178 



BELL SYSTEM TECHNICAL JOURNAL 



After substitution of these values of the admittances in (12.26) and setting 

 the real and imaginary parts equal to zero, the following two equations are 

 obtained: 



Ri 



[_ coCi w(Co + Ci)^ 



\Co + Ca/ Rg 



R. 



— fxuC-i 



1 + R 





(12.28) 



= 



and 



jLi — 



1 



1 



jCi CO (Co + 



2 r _ A _ 1 1 



1_ coCi a)(Co + C4J 



R 



C4) /Cn + C4Y 



c„ + 



(Co + C4 



C0C4 



(12.29) 



C0 + C4 



+ C3 + 



mG 



1 + i?', 



Ci - ""'J. 



Equation (12.28) gives the conditions necessary for oscillations and (12.29) 

 gives the oscillating frequency as explained below: 



12.22 Frequency of Oscillations for G-C Connection of Crystal 



Equation (12.29) for frequency is simplified by the fact that over the 

 narrow frequency range considered, the reactances of L2 and C2 do not 

 change appreciably. Also at the oscillating frequency, 



- r - — - ^ T 



[_ ^ wCi aj(Co + C4)J 

 With these approximations (12.29) may be written 



2 



1 



+ 



LiCi LiC[ 



1 



Co + Cj . Co 

 Ct C4J 



(12.30) 



where 



C, = C, + C3 + 



mCs 



1 -{-Rl(^ - C00C2) 



\CO0 i>2 / 



and coo is a constant approximating the oscillating frequency. 



