2'>2 BELL SYSTEM TECHNICAL JOURNAL 



A mid-half section is that dissymmetrical part between the mid- 

 point of one series impedance and the mid-point of the next shunt 

 admittance, or vice versa. The image impedances at the two ends 

 are, respectively, W\ and Wi, or vice versa. Its transfer constant is 

 one-half that of a full section, mid-series or mid-shunt. Obviously, 

 two mid-half sections when connected with like image impedances, 

 Wz or Wi, adjacent, will form a mid-series or mid-shunt section, 

 respectively. 



Well-known formulas for the transfer constant, T, of a full section 

 and for the mid-series and mid-shunt image impedances, Wi and W^, 

 are 



cosh T - cosh (^ + i5) = 1 -f 1^ = 1 + 2{U+ iV), 



Wi = VziZ2 + W = VsiZaVl + U+iV, 

 and 



Z1Z2 VZiZ2 21Z2 



VziZ2 + W Vl+ U+iV w, ^ ^ 



where 



4Z2 



Such a general structure is illustrated in the upper part of Fig. 1. 



1.2 Fundamental Derivations 



1.21 Mid-Series Derivation by Operation Di{s) 



From any ladder type network Zi, z-2 it is possible to derive a more 

 general one Zi{s), z^'is) which has the same mid-series image impedance 

 Wi as the prototype, but a transfer constant T{s) and a mid-shunt 

 image impedance W-iis) which are functions of an arbitrary parameter 

 5. This operation, denoted as Di{s), is specified by the mathematical 

 and physical relations between the series and shunt impedances of 

 the derived network and those of the prototype, namely,^ 



Zx\s) = SZu 



and (2) 



where < .y ^ 1 for a physical structure. At the limit s -= \, it 

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

 case of mid-series equivalence.) 



6 See footnote 3. Also U. S. Patent No. 1,538,964 to O. J. Zobel, dated May 26, 

 1925. 



