Questions: Gain and Phase Margins as Stability Measures

Question 1

A control system has gain margin = 20 dB and phase margin = 8°. A new sensor is added that introduces a 10 ms transport delay. What is the most likely effect on stability?

• A. The system becomes more stable because the sensor's filtering action reduces high-frequency noise in the loop
• B. The gain margin decreases because transport delays uniformly reduce loop gain at all frequencies
• C. The system may become unstable because the delay adds phase lag that could reduce the already-small phase margin below 0°
• D. Neither margin changes because transport delays only affect frequencies far above the system's bandwidth

Answer: (C)

Transport delay adds phase lag that increases with frequency: τ·ω radians at frequency ω. At the gain crossover frequency, even a 10 ms delay can subtract a significant number of degrees of phase margin. With PM = 8° to begin with, any meaningful phase lag is catastrophic — it can bring PM to zero or negative, causing oscillation or instability. The 20 dB gain margin offers no protection here; it only measures robustness to gain increases, not phase lag. This scenario illustrates exactly why both margins must be checked and why a large GM does not compensate for a small PM.

Question 2

At the gain crossover frequency of a control loop, the loop gain is 0 dB and the measured phase is −155°. What is the phase margin?

• A. 155°, because the phase has not yet reached −180° and has 155° of distance to travel
• B. 25°, because the phase must lag an additional 25° beyond −155° before reaching the instability condition at −180°
• C. −155°, because phase margin equals the phase at the gain crossover frequency
• D. −25°, indicating the system is already unstable

Answer: (B)

Phase margin = phase at gain crossover − (−180°) = −155° − (−180°) = 25°. It represents the additional phase lag the system can tolerate at unity gain before reaching the instability condition. A PM of 25° is below the recommended 45° threshold — it is functional but has limited robustness. Option A makes the common error of treating the phase angle magnitude as the margin; option C confuses the phase reading with the margin value.

Question 3

A control system with gain margin = 30 dB can still be fragile if its phase margin is small, even though its gain could triple without causing instability.

• T. True
• F. False

Answer: True

GM and PM measure robustness along two independent axes. A large GM means the gain could increase substantially (a factor of ~31× for 30 dB) before the instability condition is met at the phase crossover frequency. But this says nothing about what happens at the gain crossover frequency, where PM is measured. If PM is small, a small amount of added phase lag — from a transport delay, an unmodeled resonance, or a temperature-dependent component — can drive the loop to instability regardless of how large the GM is.

Question 4

Phase margin and gain margin measure the same aspect of stability robustness, so a system with a large gain margin is very likely to also have an adequate phase margin.

• T. True
• F. False

Answer: False

GM and PM are independent measures, evaluated at different frequencies. GM is measured at the phase crossover frequency (where phase = −180°); PM is measured at the gain crossover frequency (where gain = 0 dB). In many practical systems — especially those with transport delays, complex resonance structures, or lightly damped modes — these frequencies are well separated. A system can have GM = 25 dB (safe against gain variations) but PM = 5° (nearly unstable from phase lag), or vice versa. Both margins must be checked.

Question 5

Explain why a control engineer must check both gain margin and phase margin. Give an example of how a system could have an adequate margin in one dimension but be dangerously close to instability in the other.

Model answer: The instability condition is met when the loop gain equals 1 (0 dB) at the same frequency where phase = −180°. GM measures how far the gain is from 1 at the phase crossover frequency; PM measures how far the phase is from −180° at the gain crossover frequency. These two frequencies are typically different, so the margins are independent. A system with high PM but low GM is robust to phase variations (cable delays, temperature drift in sensors) but vulnerable to gain changes (component aging, operating point shifts). Conversely, a system with high GM but low PM will survive gain changes but can be destabilized by adding a small transport delay — even a few milliseconds of sensor latency can subtract the entire phase margin. A concrete example: a system designed with PM = 10° and GM = 20 dB looks safe on the gain axis but would be destabilized by a modest signal processing delay. Engineers typically target PM > 45° and GM > 6 dB as a minimum, but must evaluate both independently.