Transients in the low-pass filter

Abstract

The paper gives methods for calculating the transient performance of the low-pass filter. It is shown that the current Im in the termination of an m-section filter whose critical frequency is ?c/(2?) is given by Im=1/?c??ct0g(?)E(?ct??)d? where E(?ct) is the voltage applied to the filter at t=0, and g(?ct) is the response of the filter to a unit impulse. Results are given in the cases for which g(?) can be calculated, which include filters terminated in a resistance of ?(L/C), and short- and open-circuits (in this last case the voltage across the open-circuit is found). The method also gives solutions for the uniformly dissipative filter with these terminations. For all these cases g(?) is expressed in terms of the repeated integrals of Bessel functions, and Im can be found in terms of these functions by, at most, one numerical integration of the above equation. These solutions prove very suitable for numerical computation, especially at the start of the transient. The method has been applied to work out in detail the transient performance of 3-, 4-, and 5-section filters terminated in a resistance of ?(L/C). The concept of delay time, based on a treatment using ?ideal? filters or ?smooth? lines, is examined and shown to be very approximate. It is pointed out that the solutions obtained can be interpreted in terms of waves propagated to and fro through the filter, but that this is an approximate conception, for the ?wave? starts to arrive immediately any voltage is applied.