Questions: Gain and Phase Margins: Stability Robustness

Question 1

An engineer reads a Bode plot. At the phase crossover frequency, the open-loop magnitude is −3 dB. At the gain crossover frequency, the phase is −150°. What are the gain margin and phase margin?

• A. Gain margin = 3 dB, Phase margin = 30°
• B. Gain margin = −3 dB, Phase margin = −150°
• C. Gain margin = 3 dB, Phase margin = −150°
• D. Gain margin = −3 dB, Phase margin = 30°

Answer: (A)

Gain margin is measured at the phase crossover frequency (where phase = −180°): the magnitude is −3 dB there, so the gain could increase by 3 dB before crossing 0 dB — gain margin = +3 dB. Phase margin is measured at the gain crossover frequency (where magnitude = 0 dB): the phase is −150°, which is 30° away from −180° — phase margin = 30°. Both positive means the system is stable. Option D reverses the sign of gain margin; option C applies the phase measurement to the wrong quantity.

Question 2

A feedback system has a gain margin of 8 dB and a phase margin of 15°. An engineer concludes the system is adequately robust because the gain margin exceeds the 6 dB standard. Is this correct?

• A. Yes — gain margin above 6 dB satisfies the primary robustness requirement
• B. No — both margins must independently meet their requirements; a 15° phase margin indicates poor robustness to phase lag
• C. Yes — gain margin is the primary stability indicator; phase margin is secondary
• D. No — 8 dB gain margin is insufficient; the standard requires at least 12 dB

Answer: (B)

Both margins must independently meet their requirements. A 15° phase margin means only 15° of additional phase lag at the gain crossover frequency would push the system to the edge of instability — corresponding to a poorly damped closed-loop response (damping ratio around 0.15) with significant ringing. Small delays, flexible modes, or sensor resonances easily contribute 15° of extra lag. The conventional minimum is 30–45°. A sufficient gain margin does not compensate for an insufficient phase margin; they measure robustness against different types of model uncertainty.

Question 3

A phase margin of 0° means the closed-loop system is unstable and will produce oscillations that grow without bound.

• T. True
• F. False

Answer: False

A phase margin of 0° produces *marginal stability* — constant-amplitude, sustained oscillations rather than growing ones. For instability (growing oscillations), the phase margin must be *negative* — the phase already exceeds −180° at the gain crossover, meaning positive feedback is occurring with gain above unity. Marginally stable systems oscillate indefinitely; unstable systems diverge. Both are unacceptable in most control applications, but the distinction matters for analysis.

Question 4

Gain margin is measured at the gain crossover frequency — the frequency where the open-loop magnitude equals 0 dB.

• T. True
• F. False

Answer: False

Gain margin is measured at the *phase crossover frequency* — where the open-loop phase equals −180°. The gain margin tells you how much the gain could increase at that specific frequency before the system loses stability. Phase margin is what's measured at the gain crossover frequency (where magnitude = 0 dB). Confusing these two frequencies is the most common error when reading stability margins from Bode plots.

Question 5

Why does −180° of phase shift combined with 0 dB of loop gain cause a feedback system to become unstable? What does phase margin measure in relation to this threshold?

Model answer: A feedback system subtracts the output from the reference to form an error signal — negative feedback. If the loop introduces −180° of phase shift, the signal is inverted: what was supposed to subtract now adds, turning negative feedback into positive feedback. If the loop gain is simultaneously 1.0 (0 dB) at that frequency, the system has positive feedback with unity gain and will sustain growing oscillations. Phase margin is the angular distance from −180° at the gain crossover frequency: if the phase is −150° when gain = 0 dB, the phase margin is 30°, meaning 30° of additional lag would push the system to the instability threshold.

The physical intuition is: −180° + 0 dB is the critical point where the intended correction becomes an amplification of error. Phase margin quantifies how close the system is to that point in terms of phase lag. Real systems accumulate extra lag from unmodeled delays (computation, actuator dynamics, flexible modes), so a margin of 30–45° provides a buffer against these unavoidable sources of additional phase shift.