ELECTRIC WAVE.^FILTERS 2<>3 



These relations give for the derived structure in terms of its proto- 

 type and parameter s 



cosh T{s) = 1 + 2{U{s) + iV{s)), 



Wi = Wu 

 and 



W,{s) = W,[_\ + (1 - .-)(t/+ /F)], (3) 



where 



s'-iU+iV) 



U{s) + iV{s) = 



1 + (1 - s'KU+iV) 



By the above operation a new image impedance W'y{s) has been 

 obtained which is more general than the mid-shunt image impedance 

 of the prototype. 



1.22 Mid-Shunt Derivation by Operatioti D-2(s) 

 From any ladder type network Zi, S2 it is possible to derive a more 

 general one Zi"{s), z^"{s) which has the same mid-shunt image imped- 

 ance W2 as the prototype, but a transfer constant T{s) and a mid- 

 series image impedance W\{s) which are functions of an arbitrary 

 parameter s. This operation, denoted as 1^2(5), is specified by these 

 mathematical and physical relations between the derived network and 

 its prototype 



1 



and (4) 



where < 5 ^ 1 for a physical structure. At the limit 5 = 1, it 

 reduces to the prototype. (The superscript "second" refers to the 

 case of mid-shunt equivalence.) 



From these relations it follows that the derived structure has 



cosh T{s) = \ + 2{U{s) + iV{s)), 

 Wi 



Wi{s) = 



1+ (1 -s'){U+iV)' 

 and (5) 



W2 = W2, 

 where 



TU^M -irf^ S%U+iV) 



U{s) + iV{s) = 



1+ (1 -s'){U+iV) 



