442 BELL SYSTEM TECHXfC.-iL JOURNAL 



For mid-section or for mid-load termination the simulation of the 

 effects of distributed inductance described in the preceding para- 

 graph can be made exact at two different frequencies simultaneously, 

 and the requisite values of the load-inductance Lo' and section-capac- 

 ity Co of the simulating loaded line thereby determined. This simulat- 

 ing loaded line will be termed the "simulative loaded line" corre- 

 sponding to the two particular frequencies contemplated. 



In many applications a suitable simulation can be attained by 

 imposing the conditions that the simulating loaded line (Lo', Co) shall 

 have the same nominal impedance k and critical frequency fc as the 

 actual loaded line (L', L, C). The particular simulating loaded line 

 so determined will be called the "principal simulative loaded line"; 

 evidently its load-inductance W and section-capacity Co are deter- 

 mined in terms of k and Jc and also in terms of L', L, C by the pair of 

 equations 



* = \/(L'+L)/C= W/Co, (31) 



/, = p/tVi7C = 1 /ttVWCo, (32) 



of which (31) corresponds to (1), and (32) to (15) and (14) combined 

 or to (24). The solution of the pair of equations (31) and (32) is the 

 pair of values 



Lo' = L'{Vl + \)/P = k/7rfc, (33) 



Co = C/pVl + \ = l/Tfck. (34) 



In conjunciiun with (22), these formulas show iliat Lo'>L' and Co <C; 

 in fact they show that Lo'/L' = l+2X/3 and Co/C=l — X/3, as first 

 approximations; precise values of these ratios can be readily calcu- 

 lated by substituting for p the power series contained in equation (22). 

 The simulative precision of the "principal simulative loaded line" 

 depends on the value of the relative termination (o- or a'). The 

 simulation is far more precise for mid-load termination (a' = 0.5) 

 than for mid-section termination ((7 = 0.5); this can be seen by de- 

 veloping in power series the functions involved; for X = 0.12 the fact 

 is illustrated by Fig. 10 already cited. The simulative precision for 

 other terminations will not be discussed here, beyond remarking that 

 the "principal simulative loaded line" terminating at a'-load could 

 not exactly simulate the actual loaded line terminating at ff'-load, 

 even if the sinmlation were exact at 0.5-load ; for the excess-inductances 

 (ff' — 0.5)Lo' and (a' — 0.5)L' are not exactly equal, the former being 

 slightly the larger — as shown by equation (33). However, the small- 

 ness of the impedance-departure between the "principal simulative 



