SMOOTH LINES AND SIMULATING NETWORKS 13 



proportional to iz" , through a proportionality factor (F/2) which is 

 real and positive; since the angle of z" is —45° it follows that x is 

 greater than x" by about the same amount that — y is less than —y". 

 This analysis serves to account for the shape of the curves of x and y 

 at small values of F and no leakance (the curves 6 = in Fig. 1, and 

 a = in Fig. 2). The shape of the curves at any value of F can be 

 accounted for by means of the exact formula (18.1), or a suitable 

 approximation thereof. In fact formula (18.1) shows immediately 

 that 



x -*">(-/')-(-y) 



and that this inequality increases with F. 



Part III 

 Impedance of Cables 



It will be recalled that the impedance of an ordinary cable depends 

 chiefly on its capacity and resistance, relatively little on its inductance, 

 and far less still on its leakance; and hence that its impedance is at 

 least roughly equal toV R/iuC = {1- i)V R/2uC. 



Of the quantities defined by equations (2), . . . (7), the four most 

 suitable for describing cables are E, k, b and g. E is suitable as the in- 

 dependent variable, approximately proportional to the frequency, k is 

 suitable as one parameter. For the other parameter, which evidently 

 must involve the leakance, b or g respectively is the most suitable ac- 

 cording as the leakance G is approximately proportional to or approxi- 

 mately independent of the frequency. The corresponding suitable 

 forms of the equation for the impedance are then 



K l l+UPE (19) K= K ll±H 2 Z (20) 



These two formulas (19) and (20) for cables are less simple than 

 the corresponding formulas (8) and (9) for open-wire lines, because 

 in (19) and (20) neither of the two parameters enters as a mere factor, 

 and hence the number of effective parameters cannot be reduced to 

 less than two. For purposes of mere specific computations this is 

 not much of a complication; but in graphical representation it is 

 enough to prevent the desired simplicity and compactness, if the 

 representation is required to be exact and comprehensive. (Explicit 

 formulas for the two components M and N of K are included in 

 Appendix A.) 



