14 BELL SYSTEM TECHNICAL JOURNAL 



The first term in A n {t) exists in consequence of the fact that at the 

 instant the voltage is applied the filter behaves like a pure capacity 

 network. For narrow band filters the factor (w/2co m ) n is small so 

 that this term does not contribute appreciably to the steady state. 

 As a matter of fact in actual filters which necessarily have some series 

 resistance, it does not exist. 



The approximate formulas for y>n are 



2w I o 

 A n {t)= — ihn+l — [cos (q„y — 6») sin (x + nir/2) — 



U>m& \iry 



q n sin (q n y — Qn) cos (x+mr/2)] (7b) 



w | 9 



~ — l hn \\ — [(1+ffn) sin (x-q„y+Q n +nir/2) + 

 ' * *« \ ry 



w« 



(l-g„) sin + g n y-e n + «7r/2)] (7c) 



and ultimately 



4 -»"3\^"»(*-»+- r^*)- (7d) 



8. Discussion of Indicial Admittances. 



The indicial admittances for the low pass filter, that is, the current 

 in response to a steady unit e.m.f. applied at time / = 0, are shown 

 in the curves of Figs. 8, 9 and 10, for the initial or zero-th, the 3rd 

 and the 5th sections. These curves together with the exact- and ap- 

 proximate formulas given above are sufficient to give a reasonably 

 comprehensive idea of the general character of these oscillations and 

 their dependence on the number of sections and the constants of the 

 filter. 



It will be observed that the current is small until a time approxi- 

 mately equal to 2w/w c = wvIiC 2 has elapsed after the voltage is ap- 

 plied. Consequently the low pass filter behaves as though currents 

 were transmitted with a finite velocity of propagation « c /2 = l/ V L\Cz 

 sections per second. This velocity is, however, only apparent or 

 virtual since in every section the currents are actually finite for all 

 values of time>0. 



After time / = wvLiC 2 has elapsed the current oscillates about the 

 value 1/k with increasing frequency and diminishing amplitude. 

 The amplitude of these oscillations is approximately 



Vl-(2»/W) 2 V*w 



