12 BELL SYSTEM TECHNICAL JOURNAL 



is always smaller than — y; and is much smaller except at low values 

 of F, where the two approach equality as F approaches zero. The 

 statements regarding x and y hold also for the effect of wire resistance 

 on the characteristic resistance and the characteristic reactance, since 

 these are (approximately) proportional to x and y respectively, the 

 proportionality factor being the nominal impedance V L/C. 



Before leaving equation (12) attention will be directed to certain 

 approximate and exact forms of this equation that have been found 

 very useful in devising and proportioning networks for simulating 

 the characteristic impedance of smooth lines, as will appear more 

 fully in the latter part of this paper. At large values of F equation 

 (12) yields immediately the approximation 



whence x — 1 and y have approximately the values 



*-l=g^, (14) y= ~2F' (15) 



From equation (12) of Appendix A the exact values of x — l and y are 

 known to be 



*- 1= ^(^TTp' (16) y= ~Wx (17) 



Thus it is seen that each of the approximations (14) and (15) is always 

 somewhat larger than the exact value, since x is always greater than 

 unity. However, these two approximations are fairly good for 

 values of F as small even as unity, since there x does not exceed 1.1; 

 and they rapidly approach exactness when F is increased, since x 

 rapidly approaches unity. The exact equation for z will now be set 

 down for purposes of reference; by (16) and (17) it is 



Z = l +SF~2 (x+l)x*~ 1 2Fx- (18) 



At small values of F formula (12) shows that z is approximately 

 equal to z"=x"-{-iy", defined by the equation z" = l/viF. The 

 exact value of the fractional departure {z — z")/z" is 



z-z" iF 



z" 1 + Vl +iF' 



(18.1) 



which, at small F, is approximately equal to iF/2 merely. Thus, at 

 small F, 2 exceeds its approximate value z" by an amount which is 



