Question 10.1: Find the analytical expression for the magnitude frequency r...
Find the analytical expression for the magnitude frequency response and the phase frequency response for a system G(s) = 1/(s + 2). Also, plot both the separate magnitude and phase diagrams and the polar plot.
First substitute s = jω in the system function and obtain G(jω) = 1/(jω + 2) = (2 − jω)/(ω² + 4). The magnitude of this complex number, |G ( jω)| = M (ω) = 1/\sqrt{(ω² + 4)} , is the magnitude frequency response. The phase angle of G(jω), ϕ(ω) = − tan^{-1} (ω/2), is the phase frequency response.
G(jω) can be plotted in two ways: (1) in separate magnitude and phase plots and (2) in a polar plot. Figure 10.4 shows separate magnitude and phase diagrams, where the magnitude diagram is 20 log M (ω) = 20 log (1/\sqrt{ω² + 4} ) s. log ω, and the phase diagram is ϕ(ω) = − tan^{-1} (ω/2) vs. log ω. The polar plot, shown in Figure 10.5, is a plot of M (ω) < ϕ (ω) = 1 / \sqrt{ω² + 4} < − tan^{-1} (ω/2) for different ω.
