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BELL SYSTEM TECHNICAL JOURNAL 



and applying formula (3) we have 



!X10-«,Lo=o°, 



Co = M^ = 



C2 = 



1 -pl(pl-pD(Pi-pl) 



Upl H{pi-p^{pi-Pi){pi-pi){p^-pi) 



1 _ -pm- PI) (Pi- Pi) 



^ =0.0461 XlO-«, 



L,p] II{p]-p\) iPi-pl) ipl-pl) ip^-pi) 

 1 -PI(PI-PI)(PI-PI) 



-. =0.0523X10-^, 



- = 0.0725 X10-«, 



L,pi H{p\ - PI) {p\ - Pi) ipl - PI) (/>? - Pi) 

 C8 = 0, Ls = H = 0.0om. 



These formulas gi\e the numerical values of the inductances in henries 

 and the capacities in farads. The entire set of numerical values is 

 shown in Fig. 2. It is to be noted that the anti-resonant circuit 

 corresponding to po = consists of a simple capacity since the induct- 

 ance is infinite and thus does not appear in the network, whereas for 

 pa= 0° the anti-resonant circuit consists of a simple inductance, the 

 capacity being zero and thus not appearing in the network. 



Mathematical Proof 



We shall first prove that the driving-point impedance S, as given 

 by (1), may be physically realized by either a simple parallel-series 

 or a simple series-parallel network of inductances and capacities, pro- 

 vided resistances can be made negligibly small. 



The rational function 1/5 can be expanded in partial fractions, 



J_ 

 S 



JHip JHstp iIhn-\P 



where 



II. 



P\-P 

 fpj-p 



HW 



-p- ' ■ ■ ' pi„-i- 

 (j = l,3, . . . ,2h- 



Hcncc S is ccitiai to the impedance of the parallel combinatioii nf liie ii 

 circuits ha\ing the impedances (pj — p'^)/{illjp) = illr^p + [iilljpf'')p]\ 

 that is, M simple resonant circuits in parallel, each circuit consisting 

 of an inductance and a capacity in series, with the numerical values 

 given by (2). Furthermore, these numerical \alucs of the inductances 

 and capacities gi\en by (2) are all positi\e, an even number of negative 

 factors being obtained upon substituting p = pj, since in every ca?e 

 Pi — Pj+i- Hence the network defined by (2) has llu- impedance S 

 as given by (1) and is physically realizable. 



i 



