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BELL SYSTEM TECHNICAL JOURNAL 



1.4 Results for Any Uniform Line — Impressed Field 



The differential equations to be satisfied in this case are given by (1.5). 

 A solution which reduces to v{o) and i{o) at .v = is 



v{x) = cosh xV v(o) — sinh xT ZoUp) 



+ f cosh {x - k)vmdi - [ sinh (x - |)rZ„/(^)^^ 



•'0 Jo 



(1.32) 

 i{x) = -sinh xT' Z^ v(o) + cosh aT' i(o) 



- f sinh (.V - ^)T'Z7UU)d^ + [ cosh (x - ^)r't(^)d^ 

 Jo Jo 



The matrices cosh aT, sinh xF and Zg are the same as the ones discussed in 

 §1.2 and §1.3. The elements of the integral of a matrix U {U is not neces- 

 sarily a square matrix) are given by the integrals of the corresponding 

 elements of U. 



In many cases of practical interest the impressed field varies exponentially 

 with respect to .v. The column matrices l(x) and t(x) may then be ex- 

 pressed as 



l(x) = e 



Aft 



iix) = e-'' 



Ta 

 Tb 

 Tc 



(1.33) 



(1.34) 



where the X's and r's are constants and 6 is the propagation constant of 

 the impressed field in the direction of the line. The integrations in the 

 expressions (1.32) may be performed with the result 



v(x) — cosh xT v(o) — sinh xT Zoi(o) 



+ i (e^' - .-^V) (r + dir\\ - Zot) 

 - 1 (e-^' - g-^V) (r - drr\\ + Zot) 



i(x) = —sinh aT' Zg v(o) + cosh .vF' i(o) 



+ i (e^"' - e-'V) (r' + dir'ir - z-\) 



- \ ie-'^' - e-"V) (r' - Qir\r + Z-\) 



provided that the inverse matrices exist. The matrix {e^ — e''' I){T' + 

 dl)~^ is the transposed of (/^ — e~'''^I){T + dl)~^, etc. If one of these 

 matrices, say F — 61, has no inverse then it is necessary to evaluate the 



fi F.D.C. §2.10. 



