. The Bell System technical journal . the same line with itsfar terminals open (Fig. 8). To obtain a network of the first kind, vvithbranches 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 occurat 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, withresidues, ^ (1-16) An — Zoip„)T{p^) as before. The network branches for the complex poles are therefore obtainedmerely by p
. The Bell System technical journal . the same line with itsfar terminals open (Fig. 8). To obtain a network of the first kind, vvithbranches 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 occurat 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, withresidues, ^ (1-16) An — Zoip„)T{p^) as before. The network branches for the complex poles are therefore obtainedmerely 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 -\- pLof the other network and the conductance branch is again found to bezero. 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 Fig. 9—Network of the first kind equivalent to the open-circuited line of Fig. 8. NETWORKS FOR IMlKDANCE FUNCTIONS 391 parallel network of Fig. 6 and the series network of Fig. 7 is the dual ofilu^ parallel network of Fig. 9. These dual i-elationships are of course aresult of the fact that the impedance of an open-circuited line is the dualof the impedance of the same line when short-circuited. Example 2: Short-circuited Concentric Line (or Toroidal Cavity withE Radial). The preceding example considered a fictitious transmissionline of invariable parameters, R, L, G, C, having a perfect short circuitat one end. The present example has to do essentially with the sameproblem but considers it from a more practical point of view. The vari-ation of R and L with frequency is taken into account and the impedanceof the short-circuit is no longer neglected. Let the line be the piece of coaxial cable plugged at both ends withconducting material as illustr
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