196 



BELL SYSTEM TECHNICAL JOURNAL 



By substitution of the relations 



Cl C02 — COl 1 J ^ 



- = 2— and - -— = Xo 



Co cci '*^^o 



into (12.67) we obtained 



Xc = Xo 1 — -;:; ^ ^ 



L Co C02 — CO J 



and by differentiation 



Xq r Cl COl "1 V ^1 r *'i^" 1 

 ~77 M " r ^ 2 ~ ° r r^ v 



C'' L Co C02 — OJ J Co L(C02 — CO ) J 



(12.68) 



dX, 

 do3 



,2 2.2 . (12.69) 



(C02 — 



Multiply by ~ to obtain the stability coefficient 



Xc 



L Co C02 — CO J 



Xc do 



Xc 



Xo Cl CO 2coi 



Ac Co COl {032 ~ CO j 



(12.70) 



and eliminate co by substituting in (12.70) the relations obtained from 

 equation (12.68). These are 



2 



2 

 C02 



h _ ( 1 _ ^ ] ^ 



- co^ V ^Yo/ Cl 



and 



-. = ^' + 1 



COl ^0 



Cl _^ 



X, 



Xo 



(12.71) 



Thus 



F{Xc) = -1 -2^" (1 - 9-' 



X. 



(■ - s)' 



1 + ^^- 



Li 



1 - 



X, 



o_ 



(12.72) 



which may be written 

 Fix.) = -^ 



^0 



The stability coefficient of a coil and condenser F{Xy may be obtained 

 from (12.73) by letting Ci = oo. Then 



^(X)' = -I [.+(,- I)] (12.74) 



