Questions: Feedback Control Systems and Stability Analysis

Question 1

A feedback amplifier's Bode plot shows |T(jω)| = 3 (about 10 dB) at the frequency where the loop phase equals −180°. What will this circuit do?

• A. Operate normally with slightly reduced gain
• B. Oscillate, because positive feedback with loop gain ≥ 1 at the −180° crossing is unstable
• C. Saturate once, then settle to a stable DC output
• D. Become stable at higher frequencies where gain rolls off

Answer: (B)

When the phase shift around the loop reaches −180°, the feedback that was negative (subtracting from the input) has become positive (adding to it). If the loop gain magnitude |T| is ≥ 1 at this frequency, the self-reinforcing loop satisfies the Barkhausen criterion for oscillation. A loop gain of 3 at the −180° crossing means the circuit has a large negative gain margin — it is far past the stability boundary and will oscillate. Option C (settling to DC) might occur in a nonlinear circuit, but from a linear stability analysis, the circuit is unstable.

Question 2

Increasing the feedback fraction β in a negative feedback amplifier reduces the closed-loop gain but makes the circuit more stable and predictable.

• T. True
• F. False

Answer: True

This is the fundamental trade of negative feedback. Increasing β increases the loop gain T = βA, which drives the closed-loop gain toward 1/β — a value that depends on the feedback network (often made from stable passive components) rather than on the amplifier's exact gain A. The circuit becomes less sensitive to variations in A (due to temperature, aging, or manufacturing variation) and has better-defined frequency response. The cost is that raw gain falls, but stability and predictability improve — the essence of why negative feedback is so widely used in amplifier design.

Question 3

A circuit with 60° of phase margin will oscillate if the loop gain magnitude exceeds 1 at some frequency below the unity-gain crossover.

• T. True
• F. False

Answer: False

Phase margin is measured at the specific frequency where |T(jω)| = 1 (0 dB). What matters for stability is not whether |T| > 1 somewhere, but whether it is ≥ 1 at the frequency where the phase crosses −180°. A circuit with 60° of phase margin has its phase at −120° (not −180°) when the gain crosses unity. The loop gain magnitude must be below 1 by the time the phase reaches −180° for stability. Having |T| > 1 at low frequencies (where phase is well above −180°) is completely normal and does not cause oscillation.

Question 4

Negative feedback can become positive feedback at high frequencies due to accumulated phase shift in real amplifiers.

• T. True
• F. False

Answer: True

Real amplifiers introduce phase shift that increases with frequency due to parasitic capacitances, finite transistor transit times, and pole-zero pairs in the gain function. At low frequencies, the feedback path subtracts from the input (true negative feedback, ~0° phase shift). As frequency rises, the accumulated phase shift grows. If total loop phase reaches −180°, the subtracted signal has been inverted again, making it additive — converting negative feedback into positive feedback. This is the fundamental instability mechanism in feedback amplifiers, and it is why stability analysis focuses on the high-frequency behavior of T(jω).

Question 5

Why is phase margin measured at the unity-gain (0 dB) frequency rather than at the −180° phase frequency?

Model answer: Phase margin is measured at the unity-gain frequency because that is the frequency at which instability would actually occur if phase were the binding constraint. If |T| = 1 and phase = −180°, the Barkhausen criterion for oscillation is exactly met. By measuring how many degrees the phase is above −180° at the unity-gain crossing, we directly quantify how much additional phase shift the system could tolerate before oscillating. Measuring at the −180° frequency would tell us the gain margin (how much gain increase would cause instability) — a complementary measure — but phase margin answers: 'how close to −180° are we right now, when the gain is at the critical threshold of 1?'

The two stability margins are complementary: phase margin asks 'how much phase can we afford to lose before oscillating?' (measured at 0 dB), while gain margin asks 'how much gain can we add before oscillating?' (measured at −180°). Both should be checked in a robust design because a system can fail either way — if gain increases unexpectedly (gain margin insufficient) or if additional poles add phase (phase margin insufficient).