674 BELL SYSTEM TECHXICAL JOURNAL 



iuinil)ers 1, 2 10; and seven critical ijoiiiis by tin- letters a, 



h, . . . , g, as illustrated by Figs. 4 and 5. 



Networks with superfluous elements are indicated by placinij 

 parentheses around the corresponding reference mark, single paren- 

 theses for one superfluous element and double parentheses for two. 

 In order that a seven-element network may contain no superfluous 

 elements it must give the entire domain or a region in it, a six-element 

 network a critical line, and a fi\e-element network a critical point. 



That is, an impedance with arbitrarily assigned roots, and with a 

 pole chosen arbitrarily in the domain corresponding to these assigned 

 roots, can be realized with the minimum number of elements only 

 by a seven-eleinent network. If the pole is chosen so as to satisf\- 

 one additional contiition, namely, chosen at a point on one of the 

 critical lines of the domain (including the boundary curve), the 

 impedance can be realized by the six-element network giving that 

 line. If the pole is chosen so as to satisfy two additional conditions, 

 nameK', chosen at one of the critical points, the inipcdaiuf cm be 

 realized b>' the corresponding five-element network. 



The conditions for ihc critical lines and for the critical points are 

 given by Tables I\' and \ , respectively, in terms of the coefficients 

 of I lie inii)c(!ance. 



lAHl.l-: W 

 Critical Lines 



r. alb\ + (iaiat-2a~<Ji)bibi-2aiaJ}ibi-{-^aoai-a\)bl + {ia,fl3-2a,a-i)h2bi-{-a\b\ = 0. 



1. {'&aia\ — Aaifiiai-\-a\)b\ — (\(>atfl\-'r2a^a2a^—^a2a^-\-a^^a\)b\bi 

 + (%a^ia^ — Aalata^+a^al)b\bz-\-(9:alflia^ — ^a^a■!a^-\-alal}b^b\ 



— (>(a^\ — a\at)bibib3 — (?>a(fl\ai — ia^!fi3-\-a\a3)bibl 



— (aoaj — a?U4)62 — (8(1 (pfl 1(14 — 4<io<'2«3-l-a ia.0''2''3 + ( 16(1 Ju)-t-2iioiii<'j 



— 4(J 0(12 -I-" 1(1 2)62*]- (8a 2(i3- 4(2 oaia2-}-(/ 1)63 = 0. 



2. 2aJbM,-aJ>ibl-aJb\-lrath\bi-aibJbl+axb\=Q. 



3. atb\ — aib\bi — a\b\bi-\-a\btb\-\-2aiji)ibibi — atib\=0. 



4. a,b\b.-a,b\hi-\-a,b,hl-athJb\ = 0. 



5. (ij(i,iJ/>3 -l-(i.t''i/'a -(- (did* —a^i)b\b\ — (a lU, — (i:.U3)6i/>26i 



— 2a i(ij6;/i,t — (1 lUii i/<2 4- (1 i"aii/'2''3 + '"""a — c iC 2)6 1626J 

 +a\bib\-\-a^tJb\ — atiiiib\bi-\-aifltb\b\ — ataibib\=0. 



6. ii>tjl)\b-, — a\b\bs — a-tiijb\b\ — (a \ai —aiai)b\b-.bi + 2a \azb\b\ 

 +a ia4/)|/;2 — 11 iU3/»i62''3+((J«<'3 —aia-i)b\bih\ - affciAj — Oi/jjfcJ 

 +a,flib2b,-(,<i,^i,-ii[)blbl-aoii,btbl=0. 



7. a|^l^b]bl — alb\b,—a:a^blbl + (a|a^-^a^a3)b]b3bs — 2ala3b^hj 

 ■\-ii,aJ>,b] — a,ii,btb2b3-\-{ai^ii+aia])bibjb\ — a\bib] — ai,'i,b2 

 +a(i(iifrjfri — aoflifriftj -l-a<)<i lij* j = 0. 



