CIRCUIT THEORY AND Ol'liRATtONAl. CALCULUS 7J.? 



Thesf «.-i|uati<>iis have llu- m-ntT.il solutions 



r=r,f-^'+lV (59) 



I=^[VxC-"-\\e-''\ (60) 



where 



y = VjRC. (()!) 



The term in e'"" represents the direct wave and the term in e" the 

 reHectetl wave. I'l and l\ are constants wliich must he so chosen 

 as tu satisfy the im(>osed lioniular\- conditions at the terminals of 

 the cable. 



For the present we shall assume that tiic line is infinitely long so 

 that the reflected wave is absent. We shall also assume thai a voltage 

 E is impressed directly on the cable at x = o: we ha\e then, 



V' = £e-»v'pCR=£e-Vo7 (02) 



I = yj^Ee-''^-^=yj^Ec-^^^ (63) 



where a denotes x-RC. 



To con\ert these to operational cc|uations let us suppose that E 

 is a "unit e.m.f." (zero before, unity after time t = o). We have 

 then, in operational notation 



p'=e-Vip (64) 



I = ^l^e-^rp. (65) 



Now suppose that x = o so that a = o, in other wcjrds consider a 

 point at the cable terminals. Then 



(66) 



'^'4 



The first of these equations means that l' is simply the impressed 

 voltage, zero before, unity after time t = o, as of course, it should be 

 from physical considerations. 



Corresponding to the operational equation 



/ = ^. (66) 



