16 BELL SYSTEM TECHNICAL JOURNAL 



For both the low pass and band pass filters the oscillations of the 

 indicial admittances are of continuously variable frequency which traverses 

 the frequency transmission band and ultimately reaches the critical fre- 

 quencies of the filter. 10 



The indicial admittances of the band pass filter, type LiL 2 C 2 are 

 shown in Figs. 14, 15, 16 for the initial, the 6th and 10th sections. 11 

 The curves show the oscillation envelopes V (j\ +J'l), whereas the 

 actual oscillations are within a constant, 



y/[A(wt/2) + J n n {wt/2) ] sin(o; m / - wtt/2 - 6„) , 



where 0„ = tan -1 (J n f /J„). For a narrow band filter the variation in 

 the phase angle 6„ is very slow. 



The principal difference between these curves and the correspond- 

 ing curves for type LiCiL 2 C 2 is that the envelope of the oscillation 

 does not go through zero as in the latter. In addition the oscilla- 

 tions are ultimately of a single frequency — (w w +w/2) while for type 



2ir 



C\L 2 C 2 the ultimate frequency is — (u m — w/2). 



Ztt 



The indicial admittance of the high pass filter, shown in the curves 

 of Figs. 17, 18, 19, 20 for the initial, the 1st, 2nd and 3rd sections, 

 differs in important respects from those of the low pass and band pass 

 filters. In the first place the current jumps instantaneously to its 

 maximum value 1/k in all sections, so that the velocity of propagation 

 is infinite. 12 After this initial jump the current oscillates with de- 

 creasing frequency and decreasing amplitude, the oscillation frequency 

 becoming ultimately the critical or cut-off frequency o)J2ir. The 

 initial frequency and the time required for the oscillation frequency 

 to reduce to w c /2n, increases, practically linearly with the number of 

 sections. The oscillation frequency varies continuously and traverses 

 the frequency transmission range of the filter from infinite frequency 

 (represented by the initial jump) down to the critical frequency of the 

 filter, below which it attenuates sinusoidal currents. 



10 From a purely mathematical viewpoint, this fact explains the transmission, 

 without attenuation, of a continuous band of frequencies. 



11 These curves are applicable to the CiL 2 C 2 type of band pass filter, due regard 

 being had to difference in phase, and to the initial jump of current. See formulas 

 (6a) and (7a). 



12 This is, of course, a consequence of the assumption of zero series inductance 

 and shunt capacity. Actually, of course, the circuit must include a finite amount 

 of both. 



