Gain margin and phase margin

Why it matters

A loop that is "stable" on paper can still ring badly or go unstable when a valve ages, a sensor lag grows or the process gain doubles at a new operating point. Gain margin and phase margin measure how far the loop is from instability, so they are the robustness numbers written into tuning specifications — typically a gain margin of at least 6 dB and a phase margin of 30° to 60°.

Key ideas

Crossover frequencies (for the open-loop G(jω)H(jω))

• Gain crossover frequency ωgc: where |GH| = 1 (0 dB).
• Phase crossover frequency ωpc: where ∠GH = −180°.

Gain margin (GM): the factor by which the loop gain can be multiplied before the closed loop reaches the edge of instability. Measured at ωpc. On the Nyquist plot, if the plot crosses the negative real axis at −a, then GM = 1/a. On the Bode plot it is the distance of the magnitude curve below 0 dB at ωpc.

Phase margin (PM): the extra phase lag that can be added at ωgc before instability. On the Nyquist plot, it is the angle from the negative real axis to the point where the plot crosses the unit circle. On the Bode plot it is the distance of the phase curve above −180° at ωgc.

Interpreting the signs (for minimum-phase, open-loop-stable systems): both margins positive ⇒ closed loop stable; either negative ⇒ unstable. GM = 0 dB and PM = 0° together mark marginal stability, with sustained oscillation at ωpc = ωgc. For open-loop unstable or conditionally stable systems the margins can mislead — use the Nyquist criterion.

Special values

• If the phase never reaches −180° (e.g. first- and second-order type-0 or type-1 loops), GM = ∞.
• If |GH| is always below 1, there is no gain crossover and PM is undefined (often taken as infinite).

Link to transient response: for a second-order-like loop, larger PM means more damping. A useful rule is ζ ≈ PM/100 (PM in degrees) for PM up to about 60°, so PM = 45° gives roughly 20–25 % overshoot. The gain crossover frequency sets speed: closed-loop bandwidth is usually between ωgc and 2ωgc.

Dead time eats phase margin: a delay Td adds −ωgc·Td rad at the crossover without changing |GH|, so the largest delay the loop tolerates is Td,max = PM (in rad)/ωgc. This is why process loops with transport lag are tuned conservatively.

Raising gain shifts the magnitude curve up: ωgc moves to a higher frequency where the phase is more negative, so PM falls, and GM (in dB) falls by exactly the dB added.

Formulas

• ∠G(jωpc)H(jωpc) = −180° → ωpc.
• GM = 1/|G(jωpc)H(jωpc)|; GM(dB) = −20·log10|G(jωpc)H(jωpc)|.
• |G(jωgc)H(jωgc)| = 1 → ωgc.
• PM = 180° + ∠G(jωgc)H(jωgc).
• PM = tan⁻¹[2ζ / √(√(1 + 4ζ⁴) − 2ζ²)] — exact for the standard second-order loop ωn²/[s(s + 2ζωn)].
• ζ ≈ PM/100 (approximate, PM in degrees, PM ≤ 60°).
• Td,max = (PM × π/180)/ωgc — largest added dead time for stability.
• K_max = K × GM — gain at which the loop becomes marginally stable.

Symbols: ω in rad/s; GM dimensionless (or dB); PM in degrees; Td in s; K loop gain.

Worked examples

Example 1 — both margins of a type-1 loop Given: unity feedback, G(s) = 50/[s(s + 1)(s + 10)].

1. Phase crossover: −90° − tan⁻¹ω − tan⁻¹(ω/10) = −180° → ω × ω/10 = 1 → ωpc = √10 = 3.16 rad/s.
2. |G(jωpc)| = 50/[3.162 × √11 × √110] = 50/110 = 0.455.
3. GM = 1/0.455 = 2.2, i.e. 20·log10 2.2 = 6.85 dB.
4. Gain crossover: solve ω·√(1 + ω²)·√(100 + ω²) = 50 → ωgc = 2.10 rad/s.
5. Phase there: −90° − tan⁻¹(2.10) − tan⁻¹(0.210) = −90° − 64.6° − 11.9° = −166.4°.
6. PM = 180° − 166.4° = 13.6°.

Result: GM = 2.2 (6.85 dB) at 3.16 rad/s; PM = 13.6° at 2.10 rad/s. The loop is stable but poorly damped (ζ ≈ 0.14); the gain could rise only to 50 × 2.2 = 110 before oscillation.

Example 2 — gain for a phase margin, and dead-time tolerance (GATE level) Given: unity feedback, G(s) = K/[s(s + 2)]. (a) Find K for PM = 45°. (b) Find the largest dead time that can be added before instability.

1. PM = 45° means ∠G(jωgc) = −135°: −90° − tan⁻¹(ωgc/2) = −135° → tan⁻¹(ωgc/2) = 45° → ωgc = 2 rad/s.
2. Magnitude condition: K/[ωgc·√(ωgc² + 4)] = 1 → K = 2 × √8 = 5.66.
3. Dead time adds lag ωgc·Td without changing ωgc. Instability when ωgc·Td = 45° = 0.785 rad.
4. Td,max = 0.785/2 = 0.393 s.

Result: K = 5.66 gives PM = 45° at 2 rad/s; the loop then tolerates up to 0.393 s of dead time.

Common mistakes

• Computing PM as −180° − ∠GH or ∠GH + 180° with a sign slip; PM = 180° + ∠GH(jωgc), which is positive when the phase is above −180°.
• Using a crossover frequency that is "given" without checking that |GH| = 1 there.
• Expressing GM in degrees or PM in dB.
• Assuming positive margins guarantee stability for open-loop unstable or conditionally stable systems.
• Forgetting that GM in dB drops one-for-one when gain is added in dB, while PM changes non-linearly.
• Converting PM to radians incorrectly in dead-time problems (multiply degrees by π/180).

For GATE IN

Expect numerical questions on GM (often in dB) and PM for type-1 third-order loops, the gain for a specified PM or GM, the maximum dead time a loop tolerates, and reading margins from given Bode or Nyquist data. Practise the ω1·ω2 = product of corner frequencies trick for phase crossover with two simple poles and an integrator, and quick dB conversions.

Quick check

1. The magnitude at phase crossover is −10 dB. What is the GM?
2. The phase at gain crossover is −150°. What is the PM?
3. For 20/[s(s + 3)(s + 4)], where is the phase crossover?
4. A loop has PM = 30° at ωgc = 5 rad/s. What dead time makes it marginally stable?

Answers: 1. +10 dB (a factor of 3.16); 2. 30°; 3. ω = √12 = 3.46 rad/s; 4. 0.524/5 = 0.105 s.