SMOOTH LINES AND SIMULATING NETWORKS 9 



have respectively the values M—k and N, since k is real; M and N 

 denoting the resistance and reactance components of K. The concept 

 " excess impedance " will be found convenient in various connections, 

 particularly in the description of networks for simulating the impedance 

 of smooth lines. 



The ratio K/k of the characteristic impedance K to the nominal 

 impedance k will be termed the " relative impedance " and will be 

 denoted by z = x-\-iy, whence x — M/k will be termed the " relative 

 resistance," and y = N/k the " relative reactance." This complex 

 number z is roughly equal to unity over most of the voice frequency 

 range, and approaches unity as a limit when F is indefinitely in- 

 creased. Its exact value, written in the two forms corresponding to 

 (8) and (9) respectively, is 



z =ll±il_, (io) -J!±£ (ID 



\(b+i)F \a+iF 



Thus z, which is proportional to the characteristic impedance K 

 (except for the fact that the proportionality factor k is not strictly 

 independent of the frequency), depends merely on the two quantities 

 F and b, or F and a, and hence can be readily represented by tables 

 or graphs. 



When z has once been tabulated or graphed the value of K in any 

 specific case (R, G, L, C specified) is readily obtained therefrom by 

 entering such tables or graphs of z with the values of the arguments 

 F = wL/R and b = G/u>C of (10) or the arguments F = uL/R and 

 a = GL/RC of (11), and then multiplying the value of z there found by 

 k=vL/C. (Graphically this would amount merely to a change of 

 scales if the parameters employed were strictly independent of the 

 frequency.) Thus the function 



z = V(l+iF)/{b+i)F=V(l+iF)/(a+iF) 



represents simply and comprehensively the properties of the char- 

 acteristic impedance of all smooth lines, though it is more suitable 

 for representing open-wire lines than cables. 



The two components x and y of z are represented as functions of 

 F by the curves in Figs. 1 and 2 with b and a respectively as para- 

 meters. (Explicit formulas for x and y are included in Appendix A.) 



The effects produced on z = x-\-iy by the leakance G are exhibited, 

 in Figs. 1 and 2, through the parameters b and a. These effects may 

 be conveniently represented analytically in a manner formally the 

 same as that already outlined in connection with equations (7.1) and 



