Lesson 2

Bode Plots and Phase Margin: Stability Criterion in Frequency Domain

AC sweep the LC filter to understand -40dB/dec slope, 180° phase shift, and phase margin.

L1 showed oscillation, but 'why at 100x gain' needs a frequency-domain perspective. This lesson uses AC analysis (small-signal frequency sweep) to directly observe the magnitude and phase response of the Buck's LC output filter (L=1mH, C=100μF, R_load=10Ω).

Key parameters: resonant frequency fc = 1/(2π√(LC)) ≈ 504Hz. Below fc, gain ~0dB and phase ~0°. Above fc, gain drops at -40dB/dec and phase approaches -180°. Q (quality factor) ≈ R√(C/L) = 3.2, producing a ~10dB gain peak at fc.

Stability criterion: at the frequency where gain crosses 0dB (crossover frequency f_cross), measure how far phase is from -180° — this is Phase Margin (PM). PM > 45° minimum, PM ≈ 60° is optimal (fast response + no ringing), PM > 75° is overcautious with sluggish response.

Gain Margin (GM) is another metric: at the frequency where phase crosses -180°, how far below 0dB must the gain be. GM > 10dB is generally safe. PM and GM together determine closed-loop stability and transient behavior.

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LC Filter AC Sweep: Bode Plot

AC sweep 10Hz-100kHz, L=1mH, C=100μF. Resonant peak at fc=504Hz, -40dB/dec above.

Open in standalone page

After reading this section, run the simulation and observe the waveforms. To explore further, open the example in a standalone page.

Key Takeaways

fc = 1/(2π√(LC)) ≈ 504Hz, where LC filter has maximum phase transition
Phase margin PM = 180° + ∠T(f_cross), where T is loop gain
PM ≈ 60° optimal compromise, < 45° causes ringing/oscillation
In L1 with gain=100, loop gain still > 0dB at ~500Hz while phase ≈ -180° → oscillation

Watch Items

AC sweep result: magnitude shows resonant peak near 504Hz
Phase curve: transitions smoothly from 0° to -180° (double-pole signature)
High-frequency slope: -40dB/dec (40dB drop per decade)