SMOOTH LINES AND SIMULATING NETWORKS 3 



Part I 



General Considerations Pertaining to Smooth Lines 



The exact formula for the characteristic impedance K of any smooth 

 line is usually written in the form 



\ U-\-1(joC 



R, G, L and C denoting, as usual, the fundamental line constants, 1 

 namely, the resistance, leakance (leakage conductance), inductance, 

 and capacity', per unit length; w denoting 2ir times the frequency/; 

 and i the imaginary operator V — 1. 



However, this form is neither the simplest nor the most significant. 

 For it involves separately the four quantities R, G, L, and C and is 

 thus a function of not less than four 3 variables, whereas its value 

 evidently depends on only the relative values of these quantities and 

 hence must be expressible as a function of only three independent 

 variables — namely the ratios of any three of them to the fourth. 



In deciding just what form or forms of expression to adopt for K 

 we shall here be guided by the following practical considerations: 



(A) In telephony we are chiefly interested in the dependence of 

 K on the frequency/; or, stated more generally, in the dependence of 

 some quantity that is approximately proportional to K on some 

 quantity that is approximately proportional to /. 



The class of smooth lines is comprised between the following two 

 rather wide extremes, having very different characteristics: 



(B) At one extreme are the large gauge open-wire lines, particularly 

 when used at high frequencies. For them R is small relatively to wL, 

 and G relatively to coC; and hence K is approximately or at least 

 roughly equal to v L C. 



(C) At the other extreme are the small gauge cables, particularly 

 when used at low frequencies. For them R is large relatively to «L, 

 though G is small relatively to a>C; and hence K is approximately or at 

 least roughly equal to X R icoC. 



(D) The line constants R, L, C do not change much with frequency 

 oxer at least the voice frequency range; and hence they, or combina- 

 tions of them, serve suitably as parameters. 



(E) The leakance G, which is nearly always the least important 

 of the tour line constants, usually varies greatly with the frequency 



2 Constants as to current and voltage. 



3 Five if io is regarded separately from L and ('. 



