250 BELL SYSTEM TECHNICAL JOURNAL 



From Fig. 15.2 



Zi = ~ -\-^ F ^-— F ^ . /.'''^'^:!xnn (15-90) 



By substituting (15.48) and (15.49) in (15.90), we find 



jco \_Ct Co L^'^Ai + jLi{co- — CO.]) J J 



This may be expressed in the form, 



Zi 





CO^l 



If ' '^'i Ct ' Co J (15.92) 



— coZi(aj- — CO2) +i<^^-^i 

 where 



CoC< 



Z? 



Co + C, 



Now adding and subtracting wo to the — — term we have 



Co 



z, = 



'1 I T r^"^^ ~ "2) I (w^ — CO?) (aj2 — C'Ji)"! 



— wLiioo — CO?) + joo Rl 



Rationalizing (15.93) and equating it to Re and substituting in Lx 

 ^ (obtained from (15.50)), we find 



Co(co2 — col 



Rl 



Re- ^ . ... ^ . .. ^j5^^^ 



|_co| — COt J l_coi Mj 



15.92 Derivation of Equations (15.51) c//(i (15.44) 



Equation (15.51) makes possible the theoretical computation of the fre- 

 quency difference between the generalized oscillator and the minimum im- 

 pedance frequency adjustment of the PI meter. The derivation assumes 

 that i^L is negligible and that the total capacitance across the crystal ter- 

 minals is lumped in series with the crystal. 



The impedance, Zab , in the generalized oscillator. Fig. 15.2, was given by 

 (15.77). This equation may be expressed as follows: 



H +jM] + Z 



ZABOOeiCo + Ct) = 



[-^.-^]-^^ 



(15.95) 



