A Generic Derivation of Magnitude and Phase Plots

How Magnitude and Phase Information are Separated

Separation of Magnitude and Phase

Start with a transfer function with an m^th order numerator and and n^th order denominator

Let us first rewrite the function so that the poles and zeros are of the form (1+s/ω₀).

The function is now a quotient of products.  For easy hand manipulation, we'd prefer to use only addition and subtraction.  To do this, let's represent the transfer function (with s=jω) as a phasor.

where

and

Calculation of magnitude

Calculation of the magnitude begins with the fact that

This is still a quotient of products.  To simplify we will express the result in deciBels.

and, voila!  We have changed the products and quotients into addition and subtraction.  As a bonus, there are only two types of terms: the constant term and the simple zeros and poles (which are added and subtracted, respectively).

Calculation of phase

The phase term is a little easier to develop, since they add and subtract naturally.  Calculation of phase begins, and ends, with the fact that

Again, there are only two types of terms: the constant term and the simple zeros and poles.

References