Questions: Frequency Response: Magnitude and Phase Relationships

Question 1

A feedback control system is designed to eliminate errors using negative feedback. At a particular frequency, the open-loop system has accumulated −180° of phase shift and its magnitude is 2 (greater than 1). What will happen to the closed-loop system at this frequency?

• A. The system will be stable — negative feedback always corrects errors regardless of phase
• B. The system will oscillate and become unstable — the −180° phase shift turns negative feedback into positive feedback, and gain > 1 amplifies the error
• C. The system will exhibit a steady-state tracking error proportional to the phase lag
• D. The system will attenuate signals at that frequency, making the closed loop inherently more stable

Answer: (B)

Negative feedback works by feeding back an inverted error signal to correct deviations. But if the system itself introduces −180° of phase shift, the fed-back signal arrives inverted again — turning intended subtraction into addition. At −180° with |G| > 1, the error is fed back amplified and in phase with the disturbance: the system now reinforces errors rather than correcting them. This is the mechanism of instability. Gain margin and phase margin measure how close a system is to this condition. The critical insight: negative feedback is only stabilizing if phase lag remains below 180° where gain exceeds 1.

Question 2

A system's transfer function gives |G(jω)| = 0.1 at a particular frequency. What does this mean for a sinusoidal input at that frequency?

• A. The output amplitude is ten times larger than the input — the system amplifies by a factor of 10
• B. The output amplitude is one-tenth of the input — the system attenuates that frequency by a factor of 10
• C. The output phase leads the input by 0.1 radians at that frequency
• D. The system is unstable at that frequency because the gain is non-unity

Answer: (B)

|G(jω)| is the ratio of output amplitude to input amplitude. If |G(jω)| = 0.1, a sinusoidal input of amplitude A produces an output of amplitude 0.1A — attenuation by a factor of 10. In decibels: 20·log₁₀(0.1) = −20 dB. Values greater than 1 mean amplification; values less than 1 mean attenuation; a value of exactly 1 (0 dB) means the amplitude passes through unchanged. Phase information is carried separately by ∠G(jω), not by the magnitude.

Question 3

A phase shift of −90° at frequency f represents a real time delay: the output arrives exactly one quarter-period (1/(4f) seconds) after the input.

• T. True
• F. False

Answer: True

Phase shift translates directly into time delay. One full cycle (360°) corresponds to a period of T = 1/f seconds. A −90° shift is 90/360 = 1/4 of a full cycle, corresponding to a time delay of T/4 = 1/(4f) seconds. This is why phase lag is not a cosmetic nuisance — it represents real latency between input and output. In a feedback loop, this delay means the corrective action always arrives late, and accumulated delays (across multiple poles) can push the total lag past −180°, causing instability.

Question 4

In a linear time-invariant (LTI) system, a sinusoidal input at frequency ω will generally produce an output containing energy at multiple frequencies — not just ω.

• T. True
• F. False

Answer: False

This is the defining property of linear time-invariant systems: a sinusoidal input at frequency ω always produces a sinusoidal output at the SAME frequency ω. The output may differ in amplitude (scaled by |G(jω)|) and phase (shifted by ∠G(jω)), but no new frequencies are introduced. This is why frequency response analysis works: you can characterize the system's behavior at each frequency independently. Non-linear systems (and linear time-VARIANT systems) can generate outputs at frequencies not present in the input — but that is precisely what 'linear' rules out.

Question 5

Explain why accumulated phase lag — rather than the magnitude of the gain — is the primary concern for stability in a negative feedback control system.

Model answer: Negative feedback relies on inverting the error signal (a built-in 180° inversion) and subtracting it from the input to reduce error. If the system accumulates additional phase lag in the loop, the fed-back signal arrives with a different phase relationship to the input. At exactly −180° of additional lag (total −360°, which is equivalent to 0°), the inversion disappears and the loop acts as positive feedback. If the gain at that frequency is greater than 1, any disturbance is amplified each time around the loop, causing oscillation or divergence. This is why the phase margin (how many degrees short of −180° the phase is at unity gain) is the critical stability measure — the magnitude can be reduced (gain margin), but phase lag from dynamics is harder to compensate.

This is why phase-lag compensators and lead compensators exist: a lead compensator adds positive phase shift at the crossover frequency to increase the phase margin. Pure gain changes cannot fix a phase problem — they change where unity gain occurs on the frequency axis but do not change the fundamental phase relationship at that point.