--- title: "Relative stability from Bode plot" subject: "Control Systems" chapter: 6 key: "semester_3/control_systems:bode-relative-stability" source: https://www.bctengineer.com/q/semester_3/control_systems/bode-relative-stability site: BCT Engineering --- # Relative stability from Bode plot Control Systems · Chapter 6: Frequency Response Techniques - Asked in: 2070 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/bode-relative-stability - Whole chapter: https://www.bctengineer.com/md/semester_3/control_systems/chapter-6 ## Question Discuss how Bode plot is used for determining relative stability. ## Short answer Relative stability is measured by gain margin (GM) and phase margin (PM) from the open-loop Bode plot. GM = −20 log|GH| at phase crossover (∠GH=−180°); PM = 180° + ∠GH at gain crossover (0 dB). Positive margins indicate stability; typical design values are GM ≥ 6 dB and PM 30°–60°. PM ≈ 100ζ for damping estimation. ## Answer **Relative stability** tells how far a stable system is from becoming unstable. On a Bode plot it is measured by the **gain margin (GM)** and **phase margin (PM)**, read directly from the magnitude and phase curves of the open-loop transfer function $G(j\omega)H(j\omega)$. #### Procedure 1. Draw the magnitude plot (dB) and phase plot of $G(j\omega)H(j\omega)$ on the same frequency axis. 2. **Gain crossover frequency $\omega_{gc}$:** where the magnitude curve crosses 0 dB. 3. **Phase crossover frequency $\omega_{pc}$:** where the phase curve crosses $-180^\circ$. 4. Gain margin: $$ GM = -20\log|G(j\omega_{pc})H(j\omega_{pc})|\ \text{dB} $$ i.e. the distance of the magnitude curve **below** 0 dB at $\omega_{pc}$. 5. Phase margin: $$ PM = 180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc}) $$ i.e. the distance of the phase curve **above** $-180^\circ$ at $\omega_{gc}$. ```text dB |\ 0 |-\----------+-------- w | \ wgc | ^ GM | \ | v deg | | -180 |----+-------x-------- w | ^ PM wpc ``` #### Interpretation (minimum-phase systems) | Condition | Closed-loop system | |---|---| | GM > 0 dB and PM > 0°, i.e. $\omega_{gc}<\omega_{pc}$ | Stable | | GM = 0 dB, PM = 0°, $\omega_{gc}=\omega_{pc}$ | Marginally stable (sustained oscillation) | | GM < 0 dB or PM < 0°, i.e. $\omega_{gc}>\omega_{pc}$ | Unstable | - **Larger margins mean more relative stability.** GM tells by how much the gain can be increased before instability; PM tells how much extra phase lag (e.g. a time delay) can be added. - Good design values: GM ≥ 6 dB and PM of about 30°–60°. - PM is linked to damping: approximately $\zeta \approx PM/100$ for PM below about 70°. Small PM means a large overshoot. **Example:** for a system with $\omega_{gc}=1$ rad/s, phase $-150^\circ$ there, and magnitude $-10$ dB at $\omega_{pc}$: PM = 30°, GM = 10 dB, so the system is stable with reasonable margin. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. On a Bode plot, how is the gain margin read? A. The magnitude below 0 dB at the phase crossover frequency B. The phase above -180 degrees at the gain crossover frequency C. The magnitude at the gain crossover frequency D. The frequency where the phase is 0 degrees Answer: A. GM is the distance of the magnitude curve below 0 dB at w_pc, where the phase is -180 degrees. (Key point: Gain margin definition) 2. How is the phase margin obtained? A. PM = -20 log|GH| at w_pc B. PM = 180 degrees + phase of GH at w_gc C. PM = phase of GH at w_pc D. PM = 90 degrees + phase at w_gc Answer: B. PM is 180 degrees plus the phase at the gain crossover frequency, i.e. the distance above -180 degrees. (Key point: Phase margin definition) 3. For a minimum-phase system, which condition shows a stable closed loop? A. GM < 0 dB and PM < 0 B. w_gc > w_pc C. GM > 0 dB and PM > 0, i.e. w_gc < w_pc D. w_gc = w_pc Answer: C. Both margins positive (gain crossover below phase crossover) means stable; equal frequencies is marginal. (Key point: Stability condition) 4. What does a small phase margin imply? A. Very small overshoot B. A highly oscillatory response with large overshoot C. Zero steady-state error D. Faster pole at the origin Answer: B. PM is linked to damping (zeta about PM/100), so a small PM means low damping and a large overshoot. 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