Electric Circuit Theory and the 

 Operational Calculus' 



By JOHN R. CARSON 



CHAPTER VI 



Propagation of Current and Voltage Along the 

 Non-Inductive Cable 



THE principal practical applications of the operational calculus in 

 electrotechnics are to the theory of the propagation of current 

 and voltage along transmission systems. Of such transmission sys- 

 tems the simplest is the non-inductive cable. The theory of the 

 non-inductive cable is not only of great historic interest, relating as it 

 does to Kelvin's early work on the possibility of transatlantic teleg- 

 raphy, but is also of very considerable practical importance today, 

 and serves as a basis for the theory of submarine telegraphy over long 

 distances. We shall therefore consider the propagation phenomena 

 in the non-inductive cable in some detail. 



The propagation phenomena in any type of transmission system 

 are isolated and exhibited in the clearest possible manner when we 

 confine attention to the infinitely long line, with voltage applied 

 directly to the line terminals. Furthermore, as we shall see later, 

 the solution for the infinitely long line is fundamental and can be 

 extended to the more practical case of the finite line with terminal 

 impedances. We therefore, in this chapter, shall confine our atten- 

 tion to the case of the infinitely long cable with voltage applied directly 

 to the cable terminals. 



Consider a cable of distributed resistance R and capacity C per 

 unit length, extending from x = along the positive x axis. From a 

 previous chapter (see equations (64) and (65) ), we are in possession 

 of the operational equations of voltage and current; they are, for the 

 infinitely long line, 



V=e-^'^p Vo, _ (162) 



^ = ^x^^P ^"^^ ^°= ^|^-^"^''^ ^'o, (163) 



where a = x-RC, and Vo is the terminal cable voltage at .v = 0. Let 

 us now assume that the terminal voltage Vo is a "unit e.m.f."; then 



F = g-V^, (164) 



/ = -^.\/a^ e-^'«^. (165) 



1 Continued from tlie October, 1925, issue. 



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