How Magnitude and Phase Information are Separated
Start with a transfer function with an mth order numerator and and nth 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.
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).
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.
Starting from a transfer function it is possible to express both magnitude and phase as a sum of simple terms.