4.9 BODE PLOTS

Sketching frequency response plots ( H( j ?) and ? H( j ?) versus ?) is considerably facilitated by the use of logarithmic scales. The amplitude and phase response plots as a function of ? on a logarithmic scale are known as Bode plots. By using the asymptotic behavior of the amplitude and the phase responses, we can sketch these plots with remarkable ease, even for higher-order transfer functions.

Let us consider a system with the transfer function

where the second-order factor ( s ² + b ₂ s + b ₃) is assumed to have complex conjugate roots. ^{[ ]} We shall rearrange Eq. (4.81a) in the form

and

This equation shows that H( j ?) is a complex function of ?. The amplitude response H( j ?)) and the phase response ? H( j ?) are given by

and

From Eq. (4.82b) we see that the phase function consists of the addition of only three kinds of term: (i) the phase of j ?), which is 90 for all values of ?, (ii) the phase for the first-order term of the form 1 + j ?/ a, and (iii) the phase of the second-order term

We can plot these three basic phase functions for ? in the range 0 to ? and then, using these plots, we can construct the phase...