Sketching Bode plots

We can use MATLAB’s bode command to create Bode plots from LTI system models. However, we must understand how these plots relate to their transfer functions. In this section, we learn to sketch Bode plots in order to deepen our intuition of this relationship.

Let \(H(s) = \prod_i H_i(s)\); that is, let \(H(s)\) be the product of several factors \(H_i(s)\). The magnitude and phase are \[\begin{aligned} \abs{H(s)} = \prod_i \abs{H_i(s)} && \text{and} && \angle{H(s)} = \sum_i \angle{H_i(s)}. \end{aligned}\] The Bode plot consists of plots of \(20 \log_{10} \abs{H(s)}\) and \(\angle{H(s)}\) with \(s\mapsto j\omega\). The magnitude and phase expressions, become \[\begin{aligned} 20 \log_{10} \abs{H(j\omega)} = \sum_i 20 \log_{10} \abs{H_i(j\omega)} && \text{and} && \angle{H(j\omega)} = \sum_i \angle{H_i(j\omega)}. \end{aligned}\] This result means we can graphically sum both the magnitude and phase Bode plots of the individual factors of \(H(s)\), as long as we are adding magnitudes in dB.

sys = 2e5*...
tf(...
  [1,1],...
  [1,110,11000,1e5]...
);
bode(sys);