790 BELL SYSTEM TECHNICAL JOURNAL 



The distortion at x produces a third harmonic current at the receiving 

 end 



2Zo 



The total third harmonic current at the receiving end may now be 

 obtained by integrating (81) over the length of the line /, which gives us 



'' 2Zo [(2a -7) +7(3/3 -5)]^' ^- ^^'^ 



Inasmuch as we are concerned with the output amplitude, the above 

 expression may be put in a somewhat more convenient form: 



-iyl 



„ / arj. J. I) \ t 



h^3 



2Zo / (2« - t)- + (3/3 - 5)2 



X [1 + e-^(2a-7)Z _ 26-(2«-V)i COF (3^ - 5)/]. (83) 



Thus when I is zero the harmonic vanishes as it should, and as / 

 increases the current passes through maxima and minima determined 

 by the cosine term. If the attenuation is not very great, the maxima 

 occur approximately at the line lengths 



(2« - l)7r 



/ 



3/3 - 5 



where w is a positive integer. These distances correspond to odd half 

 wave-lengths, as is true of the optical case. As / is increased the 

 bracketed expression approaches unity and the current falls expo- 

 nentially. Before this point is reached, however, the current increases 

 in certain regions as the line length is increased. The fact that in an 

 actual line the parameters vary with frequency means that these 

 maxima and minima will vary in position according to the frequency, 

 so that a maximum for one frequency may well coincide with a mini- 

 mum for another frequency. 



In the case of lumped loading, the integrations used for continuous 

 loading are replaced by summations, as was done by Mason in the 

 unpublished investigation previously cited."* If the spacing between 

 coils is Xx, and if we have w coils for which nxi = y, the received third 

 harmonic current at the end of a properly terminated line may be 

 written 



*■'* ~ 2Zo (el2«-7+;(3(3-«)]x. _ 1) ' ^^'^^ 



which is to be compared with (82) for the continuously loaded line. 



