---
title: "Short note: Gain margin and phase margin"
subject: "Control Systems"
chapter: 6
key: "semester_3/control_systems:short-gm-pm"
source: https://www.bctengineer.com/q/semester_3/control_systems/short-gm-pm
site: BCT Engineering
---

# Short note: Gain margin and phase margin

Control Systems · Chapter 6: Frequency Response Techniques

- Asked in: 2068 Chaitra, 4 marks
- Video: [Gain and Phase Margins Explained!](https://www.youtube.com/watch?v=ThoA4amCAX4) — Brian Douglas
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/control_systems/short-gm-pm
- Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-6

## Question

Write a short note on gain margin and phase margin.

## Short answer

Gain margin (GM) is the factor by which gain can increase before instability: GM = −20 log|GH| at ωpc where ∠GH=−180°. Phase margin (PM) = 180° + ∠GH at ωgc where |GH|=1. Positive GM and PM imply stability; typical design values: GM ≥ 6 dB, PM 30°–60°. PM ≈ 100ζ relates to damping.

## Answer

Gain margin and phase margin are measures of **relative stability**: they show how close the closed-loop system is to instability, using the open-loop frequency response $G(j\omega)H(j\omega)$.

#### Gain margin (GM)

The factor by which the open-loop gain can be increased before the closed-loop system becomes marginally stable.

$$
GM = \frac{1}{|G(j\omega_{pc})H(j\omega_{pc})|},\qquad GM_{dB} = -20\log|G(j\omega_{pc})H(j\omega_{pc})|
$$

- $\omega_{pc}$ = **phase crossover frequency**, where $\angle GH = -180^\circ$.
- On the polar plot: if the curve cuts the negative real axis at $-a$, then $GM = 1/a$.

#### Phase margin (PM)

The additional phase lag that can be added at the gain crossover frequency before the system becomes marginally stable.

$$
PM = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc})
$$

- $\omega_{gc}$ = **gain crossover frequency**, where $|GH|=1$ (0 dB).

```text
        Im
         |   unit circle
     ----+----
   /     |     \
 -1 o----+------+--- Re
   \ PM /|
     \_/ |   GM = 1/a
```

#### Stability from margins (minimum-phase systems)

| GM, PM | System |
|---|---|
| both positive | stable |
| both zero | marginally stable |
| either negative | unstable |

- Typical good values: GM ≥ 6 dB, PM 30°–60°.
- A larger PM means less overshoot (approximately PM ≈ 100ζ).

**Example:** if $|GH|=0.25$ at $\omega_{pc}$ then GM $=4=12$ dB; if $\angle GH=-140^\circ$ at $\omega_{gc}$ then PM $=40^\circ$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What does the gain margin represent?
   A. The extra phase lag that can be added at the gain crossover
   B. The factor by which gain can be increased before the closed loop becomes marginally stable
   C. The magnitude at the gain crossover frequency
   D. The frequency at which the magnitude is 1
   Answer: B. GM = 1/|GH| at the phase crossover frequency, the gain increase factor that brings the system to marginal stability. (Key point: Gain margin meaning)

2. Which expression gives the phase margin?
   A. PM = 180 degrees - angle GH at w_gc
   B. PM = 180 degrees + angle GH at w_gc
   C. PM = 180 degrees + angle GH at w_pc
   D. PM = -20 log|GH| at w_gc
   Answer: B. PM is 180 degrees plus the angle of GH evaluated at the gain crossover frequency, where |GH| = 1. (Key point: Phase margin formula)

3. If |GH| = 0.25 at the phase crossover frequency, what is the GM?
   A. 0.25 or -12 dB
   B. 4, i.e. 12 dB
   C. 4, i.e. 6 dB
   D. 2, i.e. 6 dB
   Answer: B. GM = 1/0.25 = 4, and 20 log 4 = 12 dB. (Key point: Numerical GM)

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