20 BELL SYSTEM TECHNICAL JOURNAL 



proximate steady state in the nth section is given approximately by the 

 formula 



An 1 



T = 



[}-<^T] 



1/2 



for type LxCiLid and one half this amount for the other types of band 

 pass filters discussed in this paper. 



These curves show the envelope of the oscillations with fidelity but 

 are not well adapted to exhibit the actual frequencies. These are 

 given by the formula 



VC 2 + S 2 sin [w^ + e + tan-KS/C)] 

 where C and 5 denote the definite integrals 



„ fwt 2{oi — oi m )\ j fwt 2(co — u m )\ 



The envelope is therefore substantially independent of the phase 

 angle of the applied e.m.f. The frequency is ultimately the applied 

 frequency u/2r. The transient distortion is analyzable into two 

 frequencies 



^(^ + f\' 1_(4w/w/)2 ) and h( Um ~5^ l ~~^ n/wt) *)' 



and its envelope is ultimately 



\ w ) 



\ W J 



4 



'<j0 — OJ m \ 2 \ TTWt 



The building-up of alternating currents in the high pass filter ha8 

 been investigated only qualitatively owing to the extremely laborious 

 computations required. The process is essentially different from that 

 in the low pass and band pass filters. When an e.m.f. sin (a^+0) is 

 applied the current in all sections jumps instantly to_the value 



^-sin (w/ + 6). 

 k 



Therefore the process depends on the applied frequency. If the 

 applied frequency is within the transmission band (co>co c ), the current 

 builds up to its ultimate frequency, the time required being given 

 approximately by the formula 



2nu c 1 



T = 



\ 



-(?) 



(by the principle of stationary phase; see footnote 31). 



