Questions: Bode Plot Construction and Interpretation

Question 1

A system's open-loop Bode phase plot shows −165° at the gain crossover frequency (where the magnitude equals 0 dB). What is the phase margin?

• A. −165°
• B. 165°
• C. 15°
• D. −15°

Answer: (C)

Phase margin is defined as 180° plus the phase angle at the gain crossover frequency: PM = 180° + ∠H(jω_gc). With a phase of −165°, PM = 180° + (−165°) = 15°. This is a positive but small phase margin — the system is stable but with limited robustness, since only 15° of additional phase lag would push it to the boundary of instability. A common error is confusing 180° minus the phase (which gives 345°) or taking the absolute value of the phase (165°). The formula exists because −180° phase shift plus unity gain is precisely the condition for sustained oscillation.

Question 2

Adding a zero (a factor of the form (1 + s/ω₀)) to a transfer function changes the high-frequency Bode magnitude slope by:

• A. −20 dB/decade — zeros attenuate high-frequency signals
• B. +20 dB/decade — zeros add a positive slope contribution above the corner frequency
• C. −40 dB/decade — a zero introduces a two-decade rolloff
• D. 0 dB/decade — zeros only affect phase, not magnitude

Answer: (B)

A first-order zero of the form (1 + jω/ω₀) contributes a magnitude of |1 + jω/ω₀|. For ω >> ω₀, this approximates |jω/ω₀| = ω/ω₀, which increases at +20 dB/decade. A first-order pole contributes −20 dB/decade above its corner frequency. Option D is the key misconception to reject: every magnitude change is accompanied by a phase change (they are related by the Hilbert transform for minimum-phase systems), but the converse is false — zeros absolutely affect magnitude, not just phase.

Question 3

Using a logarithmic frequency axis on a Bode plot transforms products of transfer function factors into sums, enabling graphical construction by adding individual factor contributions.

• T. True
• F. False

Answer: True

True. This is the mathematical key to the Bode plot's utility. The magnitude of a product H(jω) = H₁(jω) · H₂(jω) · ... in dB becomes 20log|H₁| + 20log|H₂| + ... — a sum. On a log-frequency axis, each factor's asymptotic magnitude contribution is a straight line, and straight lines are trivially added graphically. This decomposition is impossible on a linear scale, where the multiplicative structure of the transfer function provides no such convenience. The same logic applies to phase: the phase of a product is the sum of the phases of the factors.

Question 4

The bandwidth of a system can be read from a Bode plot as the frequency where the phase response crosses −90°.

• T. True
• F. False

Answer: False

False. Bandwidth is conventionally defined as the frequency at which the *magnitude* drops 3 dB below its passband (low-frequency) value — it is read from the magnitude plot, not the phase plot. The −90° phase frequency is a meaningful quantity (it relates to gain and phase margins) but is not the definition of bandwidth. Conflating the two leads to incorrect bandwidth estimates, which matters for filter design and control system specifications. On a Bode magnitude plot, bandwidth is simply the frequency where the curve crosses the −3 dB line.

Question 5

Why does expressing gain in dB on a logarithmic frequency scale make Bode plot construction practical, when a linear-scale plot of the same system would be unreadable?

Model answer: Two reasons. First, the frequency range of interest typically spans many orders of magnitude (e.g., 1 Hz to 1 MHz) — on a linear scale, most of this range is compressed into a thin sliver near the origin. A logarithmic axis spreads decades evenly, making all frequency ranges visually accessible. Second, expressing magnitude in dB (20log₁₀|H|) converts multiplication into addition: the combined magnitude of multiple factors is the sum of their individual dB contributions. This means each factor contributes a simple straight-line asymptote, and the total plot is built by adding these lines graphically — a construction that has no analogue on a linear scale.

The Bode plot exists specifically because these two transformations (log frequency, dB magnitude) together make complex frequency response tractable for hand analysis. Understanding *why* these scales are chosen — not just how to use them — reveals the mathematical structure that enables asymptotic approximation and the piece-by-piece construction technique.