3.5.42 · D5Guidance, Navigation & Control (GNC)
Question bank — Gain margin, phase margin — stability margins
Before the traps, we fix the vocabulary so no symbol on this page is a mystery.
Recall The critical point in one sentence
A feedback loop self-destructs when the signal comes back exactly upside-down (phase ) and just as strong (magnitude 1) — that is the point on the complex plane. All margins are just distances to , measured two different ways.
Two annotated pictures anchor everything below. First the Bode view — where the two margins are read as vertical gaps:
Then the Nyquist view — the same two margins as distances from the curve to the point :
Keep both pictures in mind as you work the traps.
True or false — justify
True or false: A system with a large gain margin is always stable.
False. You need both GM dB and PM ; a loop can have plenty of gain room but still cross with gain above unity, giving negative phase margin.
True or false: Phase margin equals the phase of .
False. PM is the gap to : . The raw phase (e.g. ) is not the margin — the margin is the of room left.
True or false: Increasing the loop gain improves both stability margins.
False. Raising lifts the whole magnitude curve, so hits 1 at a higher frequency where phase has drooped further — this shrinks GM and usually PM too. Gain helps speed, hurts margins.
True or false: If the Nyquist curve never touches , the closed loop is stable.
False in general. Merely missing is not enough; the full Nyquist stability criterion counts encirclements of relative to open-loop right-half-plane poles. Missing the point but encircling it is unstable.
True or false: A pure time delay changes the gain but not the phase of the loop.
False. A delay leaves magnitude untouched () but adds phase lag that grows with frequency — it eats phase margin. See Time delay and Padé approximation.
True or false: Gain margin and phase margin are measured at the same frequency.
False. GM lives at the phase crossover (); PM lives at the gain crossover (). Each margin is measured where the other quantity has already hit its critical value.
True or false: A phase margin of is better than in every way.
False. Huge PM means very heavy damping — sluggish, over-cautious response. The sweet spot is roughly –; too much margin is as much a design smell as too little. See Damping ratio and overshoot.
True or false: Negative gain margin (in dB) means the loop is already unstable.
True. Negative dB means at the phase, so the curve already sits beyond along the negative-real axis — the loop feeds itself and grows.
True or false: Every stable loop has exactly one gain crossover and one phase crossover.
False. Curves can cross dB or multiple times (conditionally stable systems). Then a single margin number is ambiguous and you must inspect the whole Nyquist plot.
True or false: The rule works for any feedback loop.
False. It is only an approximation valid for a second-order, dominant-pole loop and only for PM up to roughly . Outside that (higher-order, extra poles/zeros, big PM) it can be quite wrong — treat it as a sanity-check, not a law. See Damping ratio and overshoot.
Spot the error
"." — what's wrong?
It should be the reciprocal: . GM is the factor you can multiply gain by before reaches 1, so if you have room for a factor of 6, not .
"." — what's wrong?
The sign is flipped; it is . With this gives (correct), whereas the wrong form gives .
"Since GM is in dB and PM is in degrees, you can't compare them, so just make GM big." — the flaw?
The flaw is treating them as interchangeable priorities. They measure different failure directions (extra gain vs extra phase lag). A design must satisfy both simultaneously; you cannot trade one away to inflate the other.
"The critical point is because that's where is largest." — the error?
The critical point is , where (denominator vanishes, poles blow up). makes the denominator — perfectly benign, not dangerous.
"A minimum-phase system with PM and GM dB could still be secretly unstable." — right or wrong?
Wrong for a minimum-phase, single-crossover, open-loop-stable loop — there the positive margins do guarantee stability. The caution applies to non-minimum-phase, conditionally-stable, or open-loop-unstable systems, not this one.
"To add phase margin, just increase the gain to push the crossover higher." — the error?
Increasing gain moves the gain crossover to a higher frequency where the phase has dropped more, so PM typically decreases. To add phase margin you shape the phase (e.g. a lead compensator), not raise raw gain — see Loop shaping.
" exactly, so PM means ." — the error?
The relation is an approximation for dominant second-order loops with PM up to ~; extrapolating to is outside its valid range and gives a misleading number. The true damping depends on the full pole–zero layout.
Why questions
Why is the critical point and not or ?
Because closed-loop poles are the roots of , i.e. . On , the value means the returning signal is unit-strength and perfectly inverted — a sinusoid that reinforces itself each cycle.
Why do we take the reciprocal of for gain margin instead of itself?
Because we want the multiplier that scales up to exactly 1. If current magnitude is , that multiplier is — the headroom before instability.
Why does phase margin relate to damping and overshoot?
For a dominant second-order loop, low PM means the crossover sits near , so the closed loop has lightly-damped poles → ringing and overshoot. The approximate rule (valid only up to ~) captures this trend; more PM ≈ more damping. See Damping ratio and overshoot.
Why does a time delay make even a "well-designed" loop unstable if you wait long enough?
Delay adds phase lag proportional to . Once exceeds the phase margin (in radians), the phase at gain crossover passes and PM goes negative. Max safe delay .
Why do we measure GM along the negative-real axis specifically?
At the phase crossover is a negative real number , lying on that axis. So the only way to reach is to slide along the real axis by scaling magnitude — GM measures exactly that scaling distance.
Why can two loops with identical gain margins behave very differently in the time domain?
GM only samples one point (the phase crossover) along one direction. Two loops can match there yet differ wildly in phase margin, bandwidth, and pole locations — so transient behavior (overshoot, settling) diverges. Margins are rulers, not full portraits.
Why do we prefer both Bode and Nyquist views of the same margins?
Bode reads GM and PM directly off two stacked curves (fast for design tweaks via Bode plots); Nyquist shows the global geometry and encirclements, catching conditionally-stable, multi-crossing, and open-loop-unstable cases the Bode margins can misreport.
Edge cases
What is the gain margin if the phase never reaches at any finite frequency?
GM is effectively infinite — there is no phase crossover, so no amount of extra gain lets the curve reach along that axis (e.g. a first-order loop whose phase asymptotes at ).
What is the phase margin if is below 1 for all ?
There is no gain crossover, so PM is conventionally taken as infinite (not literally "no value"): the curve never reaches the unit circle, hence can never hit from the phase direction, so the loop is robustly stable in gain terms. Say "infinite," not "undefined," to avoid implying instability.
What does GM (i.e. dB) mean physically?
The loop sits exactly on the boundary: at the phase crossover , so precisely. It is marginally unstable — a sustained oscillation that neither grows nor decays.
What does PM mean physically?
At the gain crossover the phase is exactly , so : the same marginal-oscillation boundary, reached this time from the phase direction rather than the gain direction.
How do gain and phase margins work when the open-loop plant is already unstable (has right-half-plane poles)?
The margins are still computed the same way, but their interpretation flips: a stable closed loop now requires the Nyquist curve to encircle the correct number of times (once per RHP open-loop pole). Positive GM/PM alone no longer guarantee stability, and such loops are often conditionally stable — you must use the full Nyquist stability criterion.
For an open-loop-unstable loop, why can too little gain be as dangerous as too much?
With RHP poles the loop needs a minimum gain just to produce the encirclement that stabilizes it. Drop the gain and the curve shrinks below , losing the required encirclement — so there is a lower stability boundary as well as an upper one. Root locus shows both boundaries clearly.
For a conditionally-stable system, why can decreasing the gain make it unstable?
Its Nyquist curve encircles safely only within a band of gains. Lowering can shrink the curve so a loop of the plot slips onto the wrong side of , flipping the encirclement count — trust Root locus / Nyquist over a single margin here.
For a non-minimum-phase loop (right-half-plane zero), why can healthy-looking margins still mislead?
RHP zeros add extra phase lag and impose fundamental bandwidth limits; the local GM/PM at one crossover may look fine while the true robustness is poor. The full Nyquist stability criterion and careful Loop shaping are required.
If PM is measured in degrees, why must you convert to radians before computing max time delay?
Because divides an angle by an angular frequency (rad/s); the units only cancel to seconds when the angle is in radians. Using degrees gives a nonsense number off by .