30 BELL SYSTEM TECHNICAL JOURNAL 



and then converting into a definite integral, is to employ the integral 

 equation II. If Z n (p) denote the transfer operational impedance of 

 the nth. section of the infinitely long low pass filter, we have 



COc 



I ( V¥+^-P \ 2n (i.2) 



z*(p) k Vp 2 +o>A u c J 



and writing x=u c t, F„(x) =kA n (t), the integral equation II becomes 

 fF^)e-d X = ^=( V p^l- p y: (1.3) 



The solution of this integral equation is known 26 ; it is 

 F n (x) = / J 2n (xi)dxi 



which agrees with the preceding. 



The "mid-series" termination is chosen not only for its importance 

 in practical applications but because in general the indicial admittance 

 has been found to take the simplest form when the voltage is ap- 

 plied at this position. This is not always the case v however. For 

 example in the low pass filter if the e.m.f. is applied, not directly at 

 mid-series but through a terminal inductance L=L\/2 = k/o} c , the 

 integral equation becomes 



f™F„(x)e->*dx=(l-p(Vpt+i-p)) J- L [V¥+l~p)T 

 whence 



F n (x)= J2n(Xi)dXi — J2n + l(x). (1.4) 



Unless, however, the terminal impedance is related in some simple 

 manner to the constants of the filter, the resulting formula is neces- 

 sarily complicated. 



2. High Pass Wave-Filter, Type GL 2 . 



For this type of filter it can be shown, by the first method discussed 

 above in connection with the low pass filter, that the indicial ad- 

 mittance is expressible as the definite integral 



o /»x> cos(2« sin -1 —) sin x\ d\ 



'*•'' Vx 2 -i 



26 Nielsen, Cylinderfunktionen, page 186, formula 13. 



