A control engineer finds her system has a gain margin of 20 dB but a phase margin of only 8°. What should she conclude?
AThe system is robustly stable because the gain margin far exceeds the 6 dB rule of thumb
BShe should increase loop gain to push the gain crossover frequency higher and improve phase margin
CThe system is poorly conditioned — a small phase margin means it is nearly unstable despite the large gain margin, and both margins must be adequate simultaneously
DShe should switch to Nyquist analysis because Bode plots cannot assess systems with very small phase margins
Both gain margin AND phase margin must be adequate simultaneously. A phase margin of 8° means the system is operating very close to -180° phase at the gain crossover — a small perturbation could cause instability. Standard design targets require GM > 6 dB AND PM > 45°. A large gain margin with tiny phase margin still produces a poorly damped, nearly unstable system.
Question 2 Multiple Choice
In a negative feedback loop, why does having open-loop gain = 1 (0 dB) and phase = -180° at the same frequency cause instability?
AThese conditions cause the Laplace transform poles to become undefined
BThe controller loses all authority over the plant at this frequency
CThe feedback signal becomes a same-phase, full-strength copy of the input: the negative feedback summing junction adds rather than subtracts, reinforcing the signal without bound
DThe system's bandwidth collapses to zero, preventing any response
Negative feedback subtracts the feedback signal from the input. If the loop shifts the phase by -180°, the fed-back signal is inverted — which means the subtraction in the summing junction becomes addition. If the magnitude is also 1 (0 dB), this full-strength, same-phase signal is added to the input, causing unbounded self-reinforcement. Bode stability analysis checks how close the system comes to this critical condition at any frequency.
Question 3 True / False
The Bode plot used for stability margin analysis is the open-loop transfer function G(jω)H(jω), not the closed-loop frequency response.
TTrue
FFalse
Answer: True
Stability margins (gain margin and phase margin) are defined in terms of the open-loop frequency response G(jω)H(jω). The gain crossover and phase crossover frequencies are properties of the open loop. Reading margins from a closed-loop Bode plot would give the wrong answer — the closed-loop plot already incorporates the feedback and does not directly expose the critical 0 dB / -180° crossing relationships.
Question 4 True / False
A minimum-phase system with gain margin GM = 12 dB is very likely to be stable regardless of its phase margin, since it can tolerate a fourfold increase in gain.
TTrue
FFalse
Answer: False
Both gain margin and phase margin must be adequate simultaneously. A large GM with a small PM (e.g., 5°) still results in a poorly damped, nearly unstable closed-loop system. The two margins measure different dimensions of stability robustness — GM measures how much extra gain can be added, PM measures how much extra phase lag can be tolerated — and a deficiency in either is dangerous.
Question 5 Short Answer
Why does Bode's stability criterion (reading gain and phase margins from the open-loop Bode plot) apply only to minimum-phase systems, and what must be used instead for systems with right-half-plane zeros or time delays?
Think about your answer, then reveal below.
Model answer: Minimum-phase systems have a unique relationship between magnitude and phase: the phase response is completely determined by the magnitude response (via Hilbert transform relations). This is what makes reading margins off the Bode plot valid. Non-minimum-phase systems (with RHP zeros or time delays) have more phase lag than their magnitude would predict, so the standard margin-reading rule gives incorrect stability conclusions. The Nyquist criterion, which tracks encirclements of the -1 point in the complex plane, handles arbitrary loop transfer functions correctly.
The minimum-phase assumption is often glossed over but is essential for Bode stability analysis to be valid. Time delays introduce phase lag of -ωT radians that grows without bound with frequency, fundamentally changing the stability picture in ways the magnitude plot cannot capture.