BELL SYSTEM TECIIXIC.IL JOVRKAL 



the received power by a ratio of r- of that obtained before tlie two 

 miles were inserted. A number of miles of cable, w, inserted will 

 reduce the received power to a ratio r". Thus the power ratio cor- 

 responding to any given number of miles of cable is an exponential 

 function of the ratio corresponding to one mile, the exponent being 

 the length in miles. The length in miles is, therefore, a logarithmic 

 function of the power ratio. 



In an infinite length of uniforni liiu- ha\ing resistance, inductance, 

 capacity and conductance of R, L, C and G per unit length, the at- 

 tenuation a per unit length, of a current of frequency/ flowing aJong 

 the line can be shown to be equal to the real part of the exprt'ssion 



a +j b = V(R+J2TrfL) (G+j 2 Tvf C). 



For the standard cal)le line, since L and 6' are zero 



a = Virf R C, 



and since R = SH oiiins aiul (' = (). ().")t micrDfarad per mile the curri'iit 

 attenuation per mile of standard cable is 



a = 0.00386 VTT 



If It' and 1/ are tlu' ciurents, res|X'cti\el\', at the beginiiini; and 

 end of a mile of line, then 



// /.;'— -f" or a=logr // I/. 



Similarly if I , and /•.■ arc the currents, at jxiints 1 ,ind 2. ri'>i)ecti\'el\-, 

 at the beginnint; and end of a section of / niili's 



I [ I" — ^" and / a 



/, /- 



I'Or this case, the effi'Ct of iiisiTtiiig the section of / miles into the 

 line on the current at point 2, or at any point be\ond 2, is that the 

 currents at the jioint before and after the insertion are in the same 

 ratio as I\ /-j. Furthennore, .since the impedance of the line looking 

 toward the receiving end is the same at points 1 and 2 (aiul at any 

 other points), then the ratio of the powers at the two poiiits is (-([ual 

 to the stpiare of the current ratio. 



Tluis ilic ]ii)wer attenuation is repre.scnted In' 



Simil.irU lor a line, lenninalcd in a lixed iniped.mce which ma\' 

 be dilTerent from the characteristic impedance of the line, the ratio 

 of the powers received before and after a change in the length of the 



