22 BELL SYSTEM TECHNICAL JOURNAL 



By (14) we may write for the general series impedance 



m 2 rz L z c , Q1 v 



Zn = tniz L -\ : — , (31) 



rz L + z c 



in which the coefficients mi and ra 2 are to be determined. Substitu- 

 tion of these relations in (15) gives one expression for the shunt 

 impedance 



L*C 



~(l-mi)z 3 L + r 2 \ l+-fr (l+r-m x (rai + ra 2 r)j|2i 

 + r[2+^((l + r) 2 -(w 1 + m 2 r) 2 )lz z .2°c + z? : 



* _ /oo\ 



m 1 r 3 sz' 2 L ^-r 2 s(2mi-\-m2r)z L z c -{-rs('mi-{-m2r)zc 



Also, since the series impedance has three elements, the most general 

 structure for z 2 i is three resonant components in parallel. Letting 

 these components be az L -\-bz c , cz L -\-dzc, and ez L -\-jzc, as in Fig. 8, 

 the corresponding total impedance expression is 



2 2 1 = 



ace (ac ae ce\ , / a c e\ 



m* L+ Xbd+w+ d f H Zc + vt + ^ + 7)^+4 (33) 



bdf Zl+ bdf— - ZlZc +\J+J+t) Zc 



Equality between (32) and (33) at all frequencies requires that the 

 following relations be satisfied: 



ace r 3 s ... ,. 

 _= T (1-^), 



Td + Yf + |='fi+|( 1 +'-^(™i+™^)]' 



\ + j+j=r[2+|((l+r) 2 -(m 1 + m 2 r) 2 )], 



G)r+®J+(5)7-^- < 34 > 



G+f)ita+7)s + (? + D7-'" <!, -' + - :) ' 



and r -f -r + 7 =r5 (m 1 + m 2 r), 



o d J 



where r and 5 are given in (28) and (30). 



