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BELL SYSTEM TECHNICAL JOURNAL 



E. Simpler Form for Replacing an Impedance 

 In the preceding section, the transient solution of an inductance 

 and resistance in series was obtained by replacing the impedance 

 R + joiL by the expression 



i?o tanh P, where R^P = R -\- jo^L and i?o -^ «^ ; P -> 0. 



Tanh P has a numerator and a denominator both of which must be 

 expanded in order to obtain the reflections. If a single term can be 

 used, the expansion becomes simpler. In order to effect such a 

 simplification, it is necessary to find a physical structure, which has 

 only one term in its impedance expression and which approaches a 

 resistance and inductance as a limit. 



Ro-*oo p-»o 

 Ro P = R+j(jJL 



Fig. 2 — Short circuited line and shunt resistance. 



Such a structure is shown on Fig. 2. It consists of a short circuited 

 line shunted by a resistance Rq. The current into the combination is 



t = 



E 



E 



Ro X Ro tanh P Ro{l - e'^^) 



(30) 



Ro + Ro tanh P 2 



If now in the short circuited line, we let R and L be finite and G -> 0, 

 C — > 0, the combination obviously reduces to a resistance and induc- 

 tance in series, since the infinite shunt will not affect the result. 

 Hence the replacement of a resistance and inductance in the equation 



by the expression in (30), is justified. 



The solution of (30) is gotten by expanding the expression and is 



2F 



i^^ [l + e-2P_^,-4P + 



Ro 



_1_ g-2(n-l)P_^ •• •] = 



2£[1 - e-2"-P] 



Upon substituting in the values RoP 

 letting i?o — > «^ ; P — > 0, we have 



Roll - e-^^2 

 R + jcoL, n = tj2D, and 



E 



I _ g-t[RlL + ioi] 



R + jcoL 



in agreement with equation (26). 



