4 BELL SYSTEM TECHNICAL JOURNAL 



and hence by itself does not serve very suitably as a parameter. How- 

 ever, in a wide range of applications G is approximately or at least 

 roughly proportional to the frequency; and then a suitable parameter 

 is G/f or preferably G/uC. This is true of cables except at extremely 

 low frequencies. It is at least roughly true of open-wire lines at very 

 high frequencies, such as carrier frequencies, but usually not at voice 

 frequencies. For most lines the leakance G is usually approximately 

 or at least roughly a linear function of the frequency, namely, 

 G = G -{-vf, where Go is the leakance at/ = 0, and v is approximately 

 independent of the frequency. For cables, G is small compared 

 with vf except at very low values of /; but for open-wire lines Go is 

 usually not negligible except at high values of /. 



In the light of these considerations a study of equation (1) suggests 

 the employment of the quantities F, E, k, g, a, b defined by the follow- 

 ing six equations. Not all of these substitutions will be employed 

 simultaneously, but it is convenient to set them all down here together. 



F = uL/R, (2) E = uC/R, (3) 



k = VL/C, (4) g = VWR, (5) 



a = GL/RC, (6) b = G/uC. (7) 



Usually F or E will be treated as the independent variable; and 

 k, g, a, b as parameters. 



It should perhaps here be emphasized that the approximations 

 mentioned in the foregoing set of five considerations, (A) to (E), 

 are employed merely as guide in the selection of the variables and 

 parameters defined by the above equations (2) to (7), and in the 

 choice of the forms adopted below for the formula for the character- 

 istic impedance. Except where the contrary is definitely indicated, 

 the formulas that will be adopted for the characteristic impedance are 

 rigorously exact; though the variables F and E are never exactly pro- 

 portional to the frequency, and the parameters k, g, a, b are never 

 exactly independent of the frequency. If the independant variables 

 were exactly proportional to the frequency and the parameters were 

 exactly independent of the frequency, the graphs of the formulas 

 would by a mere change of scales exactly represent the impedance 

 as an explicit function of the frequency. 



With particular regard to considerations (B) and (C) it will be 

 found convenient to divide the further treatment of smooth lines 

 into two main parts, pertaining to open-wire lines and to cables 

 respectively; and then, in each of those parts, to present the impedance 

 formulas in the two forms respectively most suitable for the cases 



