SMOOTH LINES AND SIMULATING NETWORKS 7 



To formulate analytically the effects of any value of the leakance G, 

 let A denote the value of K when G = 0, and let AA denote the incre- 

 ment A — A due to the presence of the leakance G. A suitable meas- 

 ure of the effect of the leakance is then the ratio AA/A . In order 

 to obtain for the value of this ratio a formula which will be convenient 

 for use with the formula for A (which involves a radical) it is ad- 

 vantageous to write AA in the form AA = (A 2 - K 2 )/(K+K ), and 

 to introduce for brevity the quantity /z defined by the equation 



H=K 2 /K 2 -1. (7.1) 



This procedure leads readily to the following simple identity for 

 AA" A , namely 



AA a 



Ko 1 + Vl+i 



Vl+M-1. (7.2) 



In particular, when this is applied to the formula (1) for A, the value 

 of ju is found to be 



(7.3) 



1—iG/wC 



Equations (7.2) and (7.3) enable the exact value of AA/A to be cal- 

 culated for any value of G/uC. For small values of G/oiC, the formula 

 for AA' A'o takes the very simple approximate form 



AK/K =iG/2coC; (7.4) 



and this shows that, to the degree of approximation involved, AA~ 

 is proportional to iK Q through a proportionality factor (G/2wC) which 

 is real and positive. Now, for cables, A has an angle of nearly 

 — 45°; and hence, by (7.4), it is seen that the addition of small leakance 

 increases the resistance component and decreases the negative-re- 

 actance component of the impedance by about equal amounts. For 

 open-wire lines, on the other hand, the angle of Ao is much smaller, 

 though negative, and hence a small increase in the leakance changes the 

 reactance component of the impedance much more than it does the 

 resistance component; evidently, the change in the negative-reactance 

 component is a decrease, but the change in the resistance component 

 may be of either sign, depending on the frequency. This fact regard- 

 ing the effect of leakance on the resistance component of the impedance 

 is not completely represented by the approximation (7.4) — which 

 indicates the change as being always an increase — but it can be in- 

 ferred from a study of the exact formulas (7.2) and (7.3). These 

 would have to be employed also if the effects of large leakance were 



