18 BELL SYSTEM TECHNICAL JOURNAL 



to an e.m.f. sin (a^+9), which is removed at time / = 0. We can 

 represent this condition correctly by regarding the e.m.f. sin («/+9) 

 as continuing, while a negative e.m.f., — sin (coJ-f 0), is applied at time 

 / = 0. The resultant current for f^.0, is then 



a»(w) sin (co/ + 6) +j3„(co) cos (w/ + 0) 

 — -r [sin 9C 2 n(^;X)+cos 0S 2 n(x; X)] 



for the low pass filter with a corresponding expression for the band 

 pass. oc n (oi) and /3„(co) are the real and imaginary parts of the steady 

 state admittances of the filter at frequency w/2t. 



Figs. 21-32 exhibit the phenomena attending the building-up 

 of alternating currents in the low pass filter for a sufficient number of 

 representative cases to show the effects of the length of the filter and 

 the applied frequency. For w c />25, the curves represent the transient 

 distortion, that is the difference between the final steady state and 

 actual current. For co c /<24 the actual current is shown. An im- 

 portant outstanding result which follows from a study of these curves 

 and the formulas of Appendix III may be stated as follows: 



The time T required for an alternating current of frequency u/2ir to 

 build up to its proximate steady state in the nth section of a low pass wave- 

 filter is given approximately by the formula 



T = — * 



«< Vl-(co/w c ) 2 ' 



The first factor 2n/co c represents the delay due to the apparent finite 

 velocity of propagation, while the second factor represents the effect of 

 the applied frequency in its relation to the cut-off frequency of the filter. 

 This formula is a rather rough approximation when the number of 

 sections n is small. Furthermore the time at which the current reaches 

 its proximate steady state does not admit of precise definition. 14 

 Nevertheless the formula is in substantial agreement with the facts as 

 regards the effect of a number of filter sections, cut-off frequency and 

 applied frequency on the phenomena, and is of great practical im- 

 portance. 16 



14 Actually the time T corresponds to a singularity in the mathematical formulas. 

 See Appendix III. 



15 This formula has been applied in the design of periodically loaded cable circuits, 

 which are of such length in the Bell System as to make transient phenomena a 

 factor which must be taken into account. The formula is in close agreement with 

 a large amount of experimental evidence. 



