Bode Plot

It is the log scale that can cover wide range of Frequency. The total response of cascade system is the sum of individual response

Let us consider,

\[
T(s) = \frac{s + Z_1}{s + P_1} = \frac{Z_1(s Z_1 + 1)}{P_1(s P_1 + 1)}
\]

Magnitude and Phase Plot

For Magnitude Plot

\[
T(s) = K(s + Z_1)(s + P_1)
\]

\[
\text{For } s = j\omega:
\quad T(j\omega) = K(j\omega + Z_1)(j\omega + P_1)
\]

\[
|T(j\omega)| = K \cdot \sqrt{\omega^2 + Z_1^2} \cdot \sqrt{\omega^2 + P_1^2}
\]

For Phase Plot

\[
\angle T(j\omega) = \text{Phase of } K + \angle(j\omega + Z_1) - \angle(j\omega + P_1)
\]

Case 1: K > 0
\[
\text{Phase of } K = 0^\circ
\]
\[
\angle T(j\omega) = 0 + \tan^{-1}\left( \frac{\omega}{Z_1} \right) - \tan^{-1}\left( \frac{\omega}{P_1} \right)
\]

Case 2:  K < 0
\[
\text{Phase of } K = 180^\circ
\]
\[
\angle T(j\omega) = 180 + \tan^{-1}\left( \frac{\omega}{Z_1} \right) - \tan^{-1}\left( \frac{\omega}{P_1} \right)
\]

fig: Phase and Magnitude plor based on Zeros and Poles