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THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



similar means. Since, however, it is a dual of the parallel network of Fig. 

 9 for the open-circuited line, next to be discussed, it can be drawn im- 

 mediately, without further calculation, once the latter has been found. 



Example lb: We now calculate a network for the same line with its 

 far terminals open (Fig. 8). To obtain a network of the first kind, vvith 

 branches in parallel, we deal with the admittance function. 



Y = Yo tanh r 



(1-14) 



The singularities of Y are found among the zeros of coth r, which occur 

 at 



r = iirin + i), n = 0, ± 1, ± 2, ± 3, • • 



(1-15) 



The points p = —R/L and —G/C are both regular points tliis time. 

 { — G/C is a zero of F.) The singularities are simple poles, as before, with 

 residues, 



^ (1-16) 



An — 



Zoip„)T'{p^) 



as before. 



The network branches for the complex poles are therefore obtained 

 merely by putting n + | in place of the n in all formulas of the short- 

 circuit network. There is no branch corresponding to the branch R -\- pL 

 of the other network and the conductance branch is again found to be 

 zero. The complete parallel network is drawn in Fig. 9 and the series net- 

 work, in Fig. 10. 



It will be observed that the series network of Fig. 10 is the dual of the 



R,L,G,C 



Z=- 



Fig. 8 — Open-circuited transmission line. 



Fig. 9 — Network of the first kind equivalent to the open-circuited line of Fig. 8. 



