Frequency Response Analysis

Frequency response

For a stable LTI system in sinusoidal steady state, G(jω) describes the output/input amplitude ratio and phase shift at angular frequency ω. For a feedback loop use the loop transfer L = GH when determining classical stability margins. A Bode plot shows 20 log10|L(jω)| in dB and phase versus logarithmic frequency. A polar or Nyquist plot shows the complex locus. Frequencies in rad/s and hertz differ by 2π.

Margins and Nyquist

At a gain crossover |L| = 1, phase margin is 180° + arg L under the conventional negative-feedback phase branch. At a −180° phase crossover, gain margin is 1/|L|, or its 20 log10 value in dB. Multiple crossovers and unstable open-loop poles require the full stability analysis; a single positive margin is not a universal proof. For a clockwise contour enclosing the right half-plane, let N be net clockwise encirclements of −1 by L, P the number of enclosed open-loop poles and Z the number of closed-loop right-half-plane poles. Then N = Z − P. Stable closure requires Z = 0; imaginary-axis singularities require contour indentation.

Worked example

For unity negative feedback with L(s) = 10/[s(s + 1)], magnitude is 10/[ω√(1 + ω²)]. Gain crossover satisfies ω²(1 + ω²) = 100, giving ωgc = 3.084 rad/s. Phase is −90° − arctanω. At crossover it is about −162.0°, so phase margin is about 18.0°. There is no finite −180° phase crossover, giving infinite classical gain margin. Nevertheless the small phase margin indicates limited robustness to added delay and dynamics. The characteristic polynomial is s² + s + 10, whose poles have negative real parts, consistent with stable closure.

Common mistakes

Use 20 log10 for amplitude ratios, not 10 log10. Do not infer closed-loop bandwidth directly from every open-loop corner. Keep feedback sign and Nyquist encirclement convention explicit.

Quick check

1. What is 0 dB magnitude? A ratio of 1.
2. Does a time delay affect magnitude? An ideal delay has unity magnitude but adds phase −ωT.
3. Why inspect all crossovers? A single margin can miss an adverse crossing.