SMOOTH LINES AND SIMULATING NETWORKS 33 



Actually, since x varies with frequency, Rn is limited to some com- 

 promise value. In practice Rn would usually be combined with 

 the basic resistance Ri, though the functions of the two are distinctly 

 different. (If the requisite value of Rn were negative, Ri would 

 merely be decreased by that amount.) 



Appendix C 



On the Precise Types of Excess-Simulators (Fig. 7) 



The two sets of formulas (32), (33), (34) and (35), (36), (37), rep- 

 resenting first-approximations to the proper values of the elements 

 constituting the excess-simulators in Figs. 7a and 7b respectively, 

 were originally obtained by working with values of F large or at 

 least fairly large compared with unity; for then, by (13), the excess 

 characteristic impedance K — k has approximately the value 



k k 



while, at large or fairly large values of T, the impedance J = P-\-iQ 

 of each excess-simulator in Fig. 7 can be expressed approximately 

 by the equation 



Po . (l+/) Pp fo r - 



J = Y 2 ~ — t — ' 



derived from the exact equation (16-C) below, in which /, Po, and T 

 have the values defined by the following two sets of equations (3-C), 

 (4-C), (5-C) and (6-C), (7-C), (8-C) for the excess-simulators in 

 Figs. 7a and 7b respectively : 



t = C 2 /C 3 , (3-C) t=C 6 /C t , (6-C) 



^0 = 77^, (4-C) P = P 6 , (7-C) 



(1+/) 2 ' 

 >C 2 Ri 



(5-C) T = uC b R b , (8-C) 



1+/ 



Po thus being the value of P at oo — O. Comparison of the approxi- 

 mate equations (1-C) and (2-C) gives immediately 



P /T 2 = k/8F*, (9-C) 



(l+t)P /T = k/2F, (10-C) 



as the two conditions that are necessary and sufficient for (approxi- 

 mate) equality of / and K — k at large values of F and T. This pair 



