Finding Inverse Laplace Transform via Partial Fractions
In most cases the transfer function of the circuit comes out as the ratio of a polynomial divided by another one. Each polynomial is in the s (frequency) domain. We start the chapter by practicing how to plot the transfer function, making sense of decay rates and phase changes, and introducing the decibel units. In the chapter we identify the meaning of poles and zeroes and their impact on the magnitude and phase of the transfer function. Then the bulk of the chapter deals with simplifying the transfer function in terms of series of simple fractions, each fraction having a constant in the numerator and a polynomial of order in the denominator. The numerator would have the residue, while the denominator would have the pole. Poles could be real or complex; they could also be of first order or higher ones. We outline how to calculate the residues of each simple fraction, plot the magnitude and phase, and rationalize the behavior of each. We also discuss long division and a simple method to estimate pole locations. The chapter has a lot of examples which can be used as backbone for sequel chapters in the book dealing with actual circuits.