COAXIAL AND BALANCED TRANSMISSION LINES 



291 



For this case the image transfer constant and the impedance ratio of 

 the transforming filter are given by the equations 



cosh 6 = 



2(jili 2co/i 

 o 7 / 7 j^ 7 \ SI" cos 



„ Iwli . / Zoi + Z03 \ V V 



cos^ h 



V 



IZ 



02 



tan 



w/2 



V 



+ sin 



2 2co/i 



Z01Z03 (i6oi Z03) 



4Zo22tan2^^ 



4Zn,Z 



oi^oa 



. co/i co/i 

 , ^ , ^ sm — cos — 



COh Zoi . 2^^! 1 -^Oi Z^ t' 



COS'' -^ sm^ h V 



V Z03 Z' Z02 



w/2 

 tan — 



V 



Q.O'sr 



h Z03 . 9C0/1 Z03 z' 

 — — -;^sm^ — + 



. co/i call 

 sm — cos — 



C0/2 

 tan — 



(39) 



When we solve the expression for cosh d for the cut-off frequencies, we 

 find that one of them is given by the expression 



and the other by 



tan 



lull 



- 2 



Zoj + Zo 

 Z01Z03 



Z02 tan 



CO/ 2 



(Zoi + Zo3)^ 



Wi2 



(40) 



(41) 



^ 7 27 2 — ^02 Lan — 



Zoi Z03 ?> 



If we consider the special case Z02 = ZoiZoJ^Zq^ + Z03) equation (41) 

 reduces to the simple form 



tan 



2(jili 



tan 



2o)U 



or for narrow bands 

 2aji/i 



2^1/2 



and /i = 



4(/i + /a) 



(42) 



(43) 



in agreement with equation (17). 



At the two cut-off frequencies, it can be shown that 



7^ 2 



(44) 



