NETWORK SYNTHESIS 



G49 



rf the continued fraction is truncated after the term of oi'der m, and is 

 rearranged as a rational fraction N D, it will obey e(iuation (84). The 

 degrees of A^ and D will be sucii that n" + n' = m, and n" — n' = 

 or 1. The contiinied fraction may be associated with the hypotlietical 

 ladder network shown in Fig. 10, with variable impedance shunt branches 

 proportional to z'. The impedance of the (tnuicated) ladder is N/D. 



21. .VC'CUU.VCY 



The accuracy of match, a — a, may again be evaluated from the 

 final network singularities; or by (41), with H{z) as in (30), before the 

 singularities have been determined from roots of A'^ and D. A rougher 

 estimate may again be obtained from the error in the first unmatched 

 term in the Tchebycheff polynomial series. As before, (e([uation (72)), 

 this is equal to the leading term in H{z), with z "'^' replaced by — T-2m+2 • 



The rational fraction in (30) is the same as our present N/D. In terms 

 of N/D, (30) becomes: 



^R{z)Ri-z) = 1-}- H(z) 



(87) 



If (8(i), or Fig. (10), is used to determine .V and D, the leading term in 

 H(z) turns out to be: 



H{z) = 



( ^ m+l -,2m+2 



The corresponding mismatch in Tchebycheff polynomial terms is: 



(-)"' 



C-lmArl — C; 



m+2 



(aia2 • • • amf'ttrti+xKo 



(88) 



(89) 



22. ZEROS AND POLES 



When frequencies of infinite loss are to be chosen, as well as natural 

 modes, the situation in regard to | 2<, I < 1 is somewhat less favorable. 



o V\Ar 



82 



a, 



A/A — r 



zf 



33 



Fig. 10 — A ladder network, representation of the continued fraction (86). 



