6 BELL SYSTEM TECHNICAL JOURNAL 



The general nature of the effect produced on the characteristic 

 impedance K by any value of the leakance G is readily seen from mere 

 inspection of equation (1), so far as regards the absolute value and 

 angle of the impedance. Thus, increasing G from any initial value 

 decreases the absolute value of the impedance and (algebraically) 

 increases the angle. Starting with C7 = 0, the angle is negative and 



has the value — \ tan - — -; increasing G decreases this negative 



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angle until G has become as large as RC/L, when the angle has become 

 zero and the impedance has become equal to the simple value V L/C, 

 and thereby equal also to V R/G. Increasing G beyond this transi- 

 tion value RC/L toward infinite values gives to the impedance a 

 positive angle which continually increases toward its limiting value 



\ tan - — , while the absolute value of the impedance goes on con- 

 R 



tinually decreasing toward its limiting value of zero. 



The statements in the foregoing paragraph hold at all frequencies, 



though the effects of leakance are usually most pronounced at low 



frequencies. In fact at zero frequency the characteristic impedance 



of a line having any finite leakance however small is merely V R/G; 



and at frequencies so low that coL is small compared with R and coC 



small compared with G, the impedance K is, approximately, 



k ;~# * i_i_; ( L C \ ' 

 Vg< + ^\2R~2g)\ 



and hence is at least roughly equal to V R/G. 



Of course, with actual lines the whole physically possible range of 

 variation of G from zero to infinite values is never traversed. On the 

 contrary the leakance G even in open-wire lines seldom reaches a 

 value as large as the transition value RC/L and hence the angle 

 seldom becomes positive; while in cables the angle probably always 

 remains negative and indeed is at least roughly equal to its limiting 



value of — -i tan -1 — - , except at very low frequencies. 

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Although, as already indicated, the effects of normal amounts of 



leakance are usually very small for both cables and open-wire lines, 



yet the effects in the two cases differ rather markedly in their nature, 



owing to the difference in the angles of the impedances of these two 



types of lines; the angle of cables begin almost —45 , while that of 



open-wire lines, though likewise negative, is much smaller (except at 



very low frequencies). 



