Unit Step Response as Figured from Inverse Transform
The unit step response of a system is second in order of importance after the impulse response. Before, we figured the unit step response either by directly solving the relevant differential equations, or by simply integrating the impulse response. Here we figure the unit step response by starting with the system transfer function, dividing by s, and then by finding the inverse Laplace transform of the resulting function. The division by s is needed since 1/s is the Laplace transform of the unit step function! Notice that the frequency spectrum of the unit step function samples all frequencies, but more strongly at DC and low frequencies; this is in contrast to the impulse which samples all frequencies uniformly. By doing frequency multiplication the only remaining step is to do the inverse transform (which is frequency integration); but in the impulse response integration approach we have to first do the frequency integration (to get impulse response) and then do time integration (to get step response). We illustrate the unit step response derivation via a handful of examples where we picked the impedance as our transfer function. We apply the method on various RLC networks starting with the transfer function, dividing by s, and then doing inverse transform. Along the way we study some interesting details of those circuits and do some variations on them.