Bode Plots and Stability Margins
The one method you can run on hardware nobody has modelled — and the only one that says HOW stable.
Skip to the animationSweeping a system with sinusoids and recording gain and phase gives a description that can be measured on hardware nobody has modelled — and the two margins read off it answer the question Routh-Hurwitz could not: not whether the loop is stable, but how close to oscillating it is.
Why frequency response
A sinusoid into an LTI system comes out at the same frequency, changed only in amplitude and phase. Recording those two numbers across frequency characterises the system completely — and it can be done to real hardware with no model at all, which is this method's practical advantage over root locus.
Decibels and log frequency turn products into sums, so a complicated G(s) is sketched by adding straight-line asymptotes: −20 dB/decade and −90° per pole.
The two margins
Sustained oscillation needs loop gain 1 and phase −180° at the same frequency. The margins measure how far the loop is from meeting both at once.
- Phase margin
- At the gain crossover (where |G| = 1), how much more lag the loop tolerates before reaching −180°. Target 45–60°, corresponding roughly to ζ ≈ PM/100.
- Gain margin
- At the phase crossover (where ∠G = −180°), how many dB the gain may rise before reaching 0 dB. Target 6–12 dB, so the loop tolerates 2–4 times its design gain.
Both are needed. A system can have a comfortable margin of one kind and a dangerous one of the other.
The trade-off, made explicit
Raising the gain moves the crossover to a higher frequency where the phase is already more negative, so speed is bought with margin. Escaping that requires reshaping the loop: a lead compensator adds phase near the crossover, buying margin back at the frequency that matters.
Why it survives simulation
- It is measurable — sweep a real plant, no model needed.
- It handles delay exactly —
e^(−sT)is pure phase lag, where Routh needed a polynomial approximation. - It localises the problem to a frequency, so it says what to change rather than only that something is wrong.
- Nyquist generalises it to open-loop-unstable plants, where the Bode margin reading can mislead.
Margins on a Bode plot are the standard acceptance criteria in real control specifications. A simulation says whether a design works; margins say how much reality it will tolerate.
The numbers you will be asked for
- Gain in decibels
20·log₁₀|G(jω)|
So cascaded blocks add.
- Phase margin
PM = 180° + ∠G(jω_gc)
At the gain crossover frequency.
- Gain margin
GM = −20·log₁₀|G(jω_pc)| dB
At the phase crossover frequency.
- Damping estimate
ζ ≈ PM / 100
Rough, and good enough for design.
- Pure delay
∠ = −ωT radians
Unbounded phase lag, exactly represented.
Advantages and disadvantages
Advantages
- Measurable on hardware with no model.
- Gives how much margin, not merely whether any.
- Represents transport delay exactly.
- Localises the problem to a frequency, guiding compensation.
Disadvantages
- Straight-line asymptotes are approximations near corner frequencies.
- Margins can mislead for open-loop-unstable or non-minimum-phase systems — Nyquist is needed.
- Requires a frequency sweep, which not every plant tolerates.
- Says less about transient shape than a time-domain plot.
Watch it work
Check yourself
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One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.