Engineering · Electrical engineering and control

Bode plot: gain and phase margins

Phase and gain margins measure, on a Bode plot, the distance of the open loop from critical feedback conditions.

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Formula

PM = 18 0^{\circ} + ar g L (j ω_{g c}), GM = 1/∣ L (j ω_{p c}) ∣

Frequency in rad/s; phase in degrees; GM dimensionless or dB = 20 log₁₀(GM)

Variables & units

Variables & units
Symbol	Meaning	Unit
$L(jω)$	Open-loop transfer function at angular frequency ω	dimensionless transfer ratio
$ωgc$	Gain-crossover frequency where \|L\|=1 (0 dB)	rad/s
$ωpc$	Phase-crossover frequency where phase is −180°	rad/s
$PM$	Phase margin at ωgc	°
$GM$	Gain factor to the critical magnitude at ωpc	dimensionless or dB

LaTeX

\mathrm{PM}=180^\circ+\arg L(j\omega_{gc}),\quad \mathrm{GM}=1/|L(j\omega_{pc})|

QUANTA STUDYquanta-study.de

Two logarithmic-frequency plots. Above, magnitude falls through 0 dB; a marker shows gain crossover and the vertical distance to critical phase. Below, phase falls through −180 degrees; horizontal distance to the 0 dB level marks gain margin. — Schematic Bode response: phase margin at gain crossover, gain margin at phase crossover. Curves and margins are not to scale.

Derivation & background

For the usual unity-feedback loop, phase margin at each 0-dB crossing is 180° plus open-loop phase; gain margin is the reciprocal magnitude at a −180° crossing. Positive distance is a robustness indicator in the usual case, but multiple crossings, unstable open-loop poles, delays and nonminimum-phase behavior require careful Nyquist/stability analysis. Margins are local frequency-domain measures, not a replacement for Routh pole counting; they also depend on loop break and model uncertainty. Report consistently unwrapped phase in degrees.

Exam blueprint

Validity range

Continuous linear unity-feedback loop; inspect every relevant 0-dB and −180° crossing. With unstable open-loop poles or multiple crossings, use a full Nyquist analysis.

Derivation steps

A margin measures the additional phase or gain distance to the critical feedback condition.

1At |L|=1, determine phase φ: PM=180°+φ.
2At φ=−180°, determine magnitude: GM=1/|L|.
3Express GM in dB as 20log₁₀(GM); document convention and all crossings.

Rearrangements

Critical loop magnitude

|L(j\omega_{pc})|=10^{-GM_{dB}/20}

GMdB is gain margin in decibels at phase crossover.

Phase from margin

\varphi(\omega_{gc})=PM-180^\circ

Applies at gain crossover under the stated convention.

Task variant

For L(s)=20/((s+1)(s+2)(s+5)), calculate margins given ωgc≈1.286 rad/s with φ≈−99.30° and ωpc=√17 with |L|=10/63.

PM=180°−99.30°=80.70°.
GM=63/10=6.30; GMdB=20log₁₀(6.30)≈15.99 dB.

Check: 10^(15.99/20)≈6.30.

Common mistakes

Report linear GM as a dB value or inspect only one crossing.

Report GM as both factor and dB, and inspect all critical cases when multiple crossings exist.

Exam context

Feedback-controller frequency response and robustness assessment.

Formula cluster

Electrical engineering and control

Bode margins complement pole-count tests with frequency-domain robustness distances.

For a cubic: a₂>0, a₁>0, a₀>0 and a₂a₁>a₀

Routh–Hurwitz stability criterion

C(s) = KP + KI/s + KDs

Ideal PID controller transfer function

T = G/(1+GH)

Closed-loop transfer function

T = GH/(1+GH)

Complementary sensitivity function

Worked example

For L(s)=20/((s+1)(s+2)(s+5)), the 0-dB crossing is ωgc≈1.286 rad/s with phase −99.30°, giving PM≈80.70°. The −180° crossing is ωpc=√17≈4.123 rad/s; |L|=20/126=10/63, so GM=6.30≈15.99 dB. Check: the three pole phases sum to −180° at ωpc.

Applications

Assess frequency-domain robustness margins of continuous unity-feedback systems and compare controller settings.

How do you calculate with Bode plot: gain and phase margins?

Here is how to work through a typical Bode plot: gain and phase margins (PM = 180° + φ bei |L|=1; GM = 1/|L| bei φ=−180°) task step by step:

1
Task
For L(s)=20/((s+1)(s+2)(s+5)), calculate margins given ωgc≈1.286 rad/s with φ≈−99.30° and ωpc=√17 with |L|=10/63.
Solution path
1. PM=180°−99.30°=80.70°.
2. GM=63/10=6.30; GMdB=20log₁₀(6.30)≈15.99 dB.
Check: 10^(15.99/20)≈6.30.

Frequently asked questions about Bode plot: gain and phase margins

Where is phase margin read?+

Find the frequency where open-loop magnitude is one, or 0 dB. Read phase φ there and calculate PM=180°+φ when phase is near −180°. In the example φ≈−99.30°, giving PM≈80.70°. Unwrap phase consistently across frequency. If there are multiple gain crossings, inspect every critical case rather than reporting only a favorable value. Assume a specified open-loop transfer function with consistent unity feedback. With multiple crossings or unstable open-loop poles, simple read-off rules are insufficient; assess Nyquist stability and model assumptions. Application: Assess frequency-domain robustness margins of continuous unity-feedback systems and compare controller settings. The numerical example For L(s)=20/((s+1)(s+2)(s+5)), the 0-dB crossing is ωgc≈1.286 rad/s with phase −99.30°, giving PM≈80.70°. The −180° crossing is ωpc=√17≈4.123 rad/s; |L|=20/126=10/63, so GM=6.30≈15.99 dB. Check: the three pole phases sum to −180° at ωpc anchors the idea in a concrete measurement case.

How is gain margin obtained?+

Find the frequency where open-loop phase reaches −180°, then evaluate |L| there. The factor up to critical magnitude one is GM=1/|L|; in decibels use 20log₁₀(GM). Here |L|=10/63, so GM=6.30 or about 15.99 dB. Do not confuse the linear factor with its decibel value. Assume a specified open-loop transfer function with consistent unity feedback. With multiple crossings or unstable open-loop poles, simple read-off rules are insufficient; assess Nyquist stability and model assumptions. The corresponding units are Frequency in rad/s; phase in degrees; GM dimensionless or dB = 20 log₁₀(GM). For L(s)=20/((s+1)(s+2)(s+5)), the 0-dB crossing is ωgc≈1.286 rad/s with phase −99.30°, giving PM≈80.70°. The −180° crossing is ωpc=√17≈4.123 rad/s; |L|=20/126=10/63, so GM=6.30≈15.99 dB. Check: the three pole phases sum to −180° at ωpc. This checks that the reference quantities are compatible.

What do positive margins indicate?+

For a usual stable open-loop plant with unity feedback, positive PM and GM indicate distance from critical phase and gain. This is a useful robustness indicator for changes in gain or dynamics. It is not a universal stability guarantee when open-loop poles lie in the right half-plane, multiple crossings occur, or delay and uncertainty are unaccounted for. Assume a specified open-loop transfer function with consistent unity feedback. With multiple crossings or unstable open-loop poles, simple read-off rules are insufficient; assess Nyquist stability and model assumptions.

How do I check the result and units for Bode plot: gain and phase margins?+

Write the quantity and unit from the variable list next to every value: Frequency in rad/s; phase in degrees; GM dimensionless or dB = 20 log₁₀(GM). Substitute only after converting to a consistent unit system, and state sign or reference direction explicitly. The example For L(s)=20/((s+1)(s+2)(s+5)), the 0-dB crossing is ωgc≈1.286 rad/s with phase −99.30°, giving PM≈80.70°. The −180° crossing is ωpc=√17≈4.123 rad/s; |L|=20/126=10/63, so GM=6.30≈15.99 dB. Check: the three pole phases sum to −180° at ωpc. is a calculation check: result, unit and order of magnitude must fit the described situation. Round after intermediate calculations so small contributions are not lost prematurely. Assume a specified open-loop transfer function with consistent unity feedback. With multiple crossings or unstable open-loop poles, simple read-off rules are insufficient; assess Nyquist stability and model assumptions.

Which model limit matters for Bode plot: gain and phase margins?+

The background states this starting condition: For the usual unity-feedback loop, phase margin at each 0-dB crossing is 180° plus open-loop phase; gain margin. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Assume a specified open-loop transfer function with consistent unity feedback. With multiple crossings or unstable open-loop poles, simple read-off rules are insufficient; assess Nyquist stability and model assumptions.

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Bode plot: gain and phase margins

Phase and gain margins measure, on a Bode plot, the distance of the open loop from critical feedback conditions.

AdvancedUniversity/collegeBachelor foundationsSubject specialization

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Variables & units

Symbol

Meaning

Unit

L(jω)

Open-loop transfer function at angular frequency ω

dimensionless transfer ratio

ωgc

Gain-crossover frequency where |L|=1 (0 dB)

rad/s

ωpc

Phase-crossover frequency where phase is −180°

rad/s

PM

Phase margin at ωgc

GM

Gain factor to the critical magnitude at ωpc

dimensionless or dB

Derivation & background

Exam blueprint

Validity range

Continuous linear unity-feedback loop; inspect every relevant 0-dB and −180° crossing. With unstable open-loop poles or multiple crossings, use a full Nyquist analysis.

Derivation steps

A margin measures the additional phase or gain distance to the critical feedback condition.

1At |L|=1, determine phase φ: PM=180°+φ.
2At φ=−180°, determine magnitude: GM=1/|L|.
3Express GM in dB as 20log₁₀(GM); document convention and all crossings.

How do you calculate with Bode plot: gain and phase margins?

Here is how to work through a typical Bode plot: gain and phase margins (PM = 180° + φ bei |L|=1; GM = 1/|L| bei φ=−180°) task step by step:

Task

For L(s)=20/((s+1)(s+2)(s+5)), calculate margins given ωgc≈1.286 rad/s with φ≈−99.30° and ωpc=√17 with |L|=10/63.

Solution path

PM=180°−99.30°=80.70°.
GM=63/10=6.30; GMdB=20log₁₀(6.30)≈15.99 dB.

Check: 10^(15.99/20)≈6.30.

Frequently asked questions about Bode plot: gain and phase margins

Where is phase margin read?+

How is gain margin obtained?+

What do positive margins indicate?+

How do I check the result and units for Bode plot: gain and phase margins?+

Which model limit matters for Bode plot: gain and phase margins?+

Retain Bode plot: gain and phase margins for exams

Create a curated FSRS exam set for PM = 180° + φ bei |L|=1; GM = 1/|L| bei φ=−180°: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential