pKii i\i:i'oiM i.\ii'i:i>.i.\\i: oh ino-Mnsii cikciiis 



671 



It is loini-iiicnl to limit this i!ivfstit(ali()ii to tlif (Jtti'riuiiiiitii)n of 

 itiosf networks whirli, without supcrtluous i-lctnents, realize aiu' 

 tlri\in^-point iin|)e(laiue ha\ing arbitrarily assigned roots. A net- 

 work is considereii to li.ivf sii[HTtliious cli'nu-nts if tlit-re exist other 



M 

 L, 



M23 ho 



M,3 kOQfi^ — WlArHH 

 Li R3 C3 



l-'ig. 7 — Most Ki'i'iT-'l lU'tworIc ohtaiiUHi l)y iipoiiinK one liranch of a two-mesli rirriiil 



networks with fewer elements wliich. iiuii\ ic!u,ill\ or (•olle('ti\ei\', 

 realize the same range of possible impedances. Impedances with 

 zero, pure imaginary, or infinite roots can be realized by the limiting 

 cases of these networks. 



.A network realizing an impedance with .irl)ilrarii\' assigned root> 

 must consist of at least fi\e elements, — one resistance, two cajxicilies, 



L, R, C, 



^im^ WW — Ih 



M,2 



■o 



r-^my^ WW — \\- 



L2 r\2 ^2 



Fig. 7a — Special case of Fij;- ", ol)tained by replacing the elements of one br.inr 

 l)v a short circuit 



and two self-inductances, in order that the numerator of the im|x.'dance 

 expression (lb) may contain odd [lowers of X, a constant term, and a 

 term in X\ resjiectively. 



Since the general expression for the driving-point imix-ilance cf>n- 

 lains essentially seven constants which may be assigned arbitrarily, 

 subject to the restrictions already eslablisheil. it is to l)e exjjected 

 that the entire range of |K)ssible iin|H-dances ran lie realized by one 

 or more networks consisting of seven elements only. This proves 

 to lie the case Hence all networks with more than se\en elements 



