SMOOTH LINES AND SIMULATING NETWORKS 15 



expressed by (21), when k=0, then K! here will correspond to Kq 

 there; hence n = ik i E, and thence, when k 2 E is small compared to 

 unity, 



AK/K'=ik*E/2. 



This analysis serves to account approximately for the nature of the 

 effects of small inductance as depicted in Fig. 3. When the leakance 

 is not zero but is small, the effects of inductance are still about the 

 same. The general nature of the effect of the inductance L on the 

 characteristic impedance of any smooth line, so far as regards the 

 absolute- value and the angle of the impedance, can be readily de- 

 termined by mere inspection of equation (1), in a manner similar to 

 that already outlined regarding the effect of leakance under the head 

 ing The General Effect of Leakance. 



An alternative mode of representing the characteristic impedance of 

 cables is suggested by the fact, already mentioned, that the impedance 

 of a cable is at least roughly equal to V R/icoC, whence nts absolute 

 value is at least roughly equal to V R/coC. This suggests that we 

 study a relative impedance consisting of the ratio of K to Va/o>iC, 

 where o>i denotes any fixed value of co; and that we adopt the ratio 

 co/wi as the independent variable. In this mode of treatment it will 

 be convenient to employ the quantities w, r, F u b, b t defined by the 

 equations 



w = — — = JL, (22) 



VR/uiC \K 1 \ 



r = u/u l =f/fi, (23) F^wiL/R, (24) 



b = G/o}C, (25) b 1 = G/a> 1 C. (26) 



Thus, w denotes the relative impedance to be studied; its real and 

 imaginary components will be denoted by u and v, so that w = u-\-iv. 

 r denotes the relative frequency — relative to any fixed frequency /i. 

 Fi is one parameter. The other parameter is, respectively, b or b v 

 according as the leakance G is approximately proportional to or ap- 

 proximately independent of the frequency. 6 The corresponding 

 forms of the equation for the relative impedance w are 



,1+iFf , , ll+iFir , oa . 



Vti+Ty (27) — V-E+1P (28) 



These are seen to be of the same functional forms as (19) and (20) 

 respectively; with w corresponding to K, r to E, /q to k 2 , and &i to g 2 . 



fi It will he noted that b is the same as already denned by (7); bi is related to b; 

 and Fi is related to F, which has already been defined by (2). 



