32 BELL SYSTEM TECHNICAL JOURNAL 



This represents K as having a resistance component k that is inde- 

 pendent of frequency, and a reactance component —k/2F that is 

 negative and inversely proportional to the frequency/ (since F = wL/R) 

 and thus leads exactly to the values of Ri and C\ expressed by (30) 

 and (31), whence the impedance of this network is exactly equal to 

 the approximate value of the line impedance expressed by (1-B). 



To obtain more precise and comprehensive knowledge regarding 

 the simulative precision of this network its exact impedance k — ik/2F 

 will here be compared with the exact value of the line impedance 

 (when leakance is neglected). For this purpose it is convenient to 

 employ the line impedance in the form 



K = xk-ik/2xF, (2-B) 



obtained by means of the relation y= —l/2Fx found under equation 

 (12) in Appendix A. The equation (2-B) shows that to exactly simu- 

 late the line impedance by a resistance Ri and capacity C~Y in series 

 with each other these would have to possess the values 



R,'=x\/LjC, (3-B) 



c j = 2xVLC i (4 . B) 



which differ only by the factor x from the values of Ri and C\ ex- 

 pressed by (30) and (31). Thus the ideal resistance R\ and capacity 

 C\ for exactly simulating the line impedance would vary with F in 

 precisely the same way as x varies with F. Moreover the ratio of 

 these ideal values to the fixed values of R\ and C\ expressed by (30) 

 and (31) is merely x. By reference to Fig. 1 (with b = 0) it will be 

 seen that, except at small values of F, the factor x is nearly inde- 

 pendent of F and is only slightly greater than unity. Thus the values 

 of Ri and C\ determined by means of equations (30) and (31) are 

 slightly too small at all frequencies; while the values determined by 

 means of equations (3-B) and (4-B), for any specified frequency 

 (by inserting the appropriate value of x), are slightly too small at 

 lower frequencies and slightly too large at higher frequencies. 

 Since (3-B) can, by (30), be written in the form 



R x ' = R x + {x-lWLFc: (5-B) 



and since x is always greater than unity, it is seen that the simulation 

 can be somewhat improved by supplementing the excess-simulator 

 with a small series resistance element Ru, the ideal value of which 

 would be 



R n = (x-l)\/L/C. (6-B) 



