Questions: Bode Plot Phase Response: Calculation and Interpretation

Question 1

A transfer function has 3 poles and no zeros. What is the total phase of the system as frequency approaches infinity?

• A. 0°, because the poles cancel each other at high frequency
• B. -90°, because only the nearest pole contributes at high frequency
• C. -270°, because each pole contributes -90° asymptotically
• D. +270°, because poles contribute positive phase at high frequencies

Answer: (C)

Each first-order pole contributes -90° of phase asymptotically (well above its corner frequency). With 3 poles and no zeros, the total asymptotic phase is 3 × (-90°) = -270°. This accumulation of phase lag is why high-order systems are harder to stabilize — the increasing phase lag at gain crossover erodes the phase margin.

Question 2

At the gain crossover frequency, a system's phase angle is measured at -140°. What is the phase margin, and is the system stable in closed loop?

• A. Phase margin = -140°; the system is unstable
• B. Phase margin = 40°; the system is likely stable with acceptable performance
• C. Phase margin = 140°; the system has excess stability
• D. Phase margin = -40°; the system is marginally stable

Answer: (B)

Phase margin is defined as 180° + ∠G(jω) at the gain crossover frequency (where |G(jω)| = 0 dB). Here PM = 180° + (-140°) = 40°. Typical design targets are 45°–60°, so 40° indicates a stable system with reasonable transient behavior — slightly oscillatory but not dangerously so. A negative phase margin would indicate instability.

Question 3

A first-order pole contributes exactly -45° of phase at its corner frequency.

• T. True
• F. False

Answer: True

The phase contribution of a first-order pole at s = -p transitions from 0° (far below the corner frequency p) to -90° (far above), passing through exactly -45° at ω = p. This is a fundamental result of the arctan function that governs phase: ∠(jω/p + 1)^{-1} = -arctan(ω/p), which equals -45° when ω = p. The magnitude asymptote approximation breaks at this same frequency.

Question 4

Adding zeros to a transfer function usually increases phase lag at high frequencies.

• T. True
• F. False

Answer: False

Zeros contribute positive phase (phase lead), not phase lag. A first-order zero at s = -z transitions from 0° to +90°. Adding zeros therefore reduces total phase lag at high frequencies and can improve phase margin. This is exactly how lead compensators work: a lead compensator adds a zero closer to the origin than its pole, contributing net positive phase near the gain crossover frequency to improve stability margins.

Question 5

Why does the phase at the gain crossover frequency specifically (rather than phase at some other frequency) determine the phase margin and predict closed-loop stability?

Model answer: Gain crossover is where the loop gain equals 1 (0 dB), meaning the feedback loop can sustain oscillations at that frequency if the phase shift is -180°. The phase margin measures how far the system is from -180° at exactly this critical frequency. At other frequencies the loop gain is either too small to sustain oscillation (above crossover) or the loop attenuates disturbances before they close (below crossover). Phase elsewhere is irrelevant to the stability boundary.

The Nyquist stability criterion reduces, for most practical systems, to examining the loop gain at the -180° phase crossing (gain margin) and the phase at the 0 dB gain crossing (phase margin). The gain crossover frequency is the frequency that 'matters' because it is where the feedback loop has unit gain — small phase errors there directly determine whether the closed-loop response rings, oscillates, or diverges.