Frequency response statements: Select the correct statement about pure capacity (integrator) and pure time delay elements.

Introduction / Context:Frequency-domain intuition helps predict closed-loop behavior. Two canonical elements are the integrator (pure capacity) and pure time delay. Each has distinctive magnitude and phase characteristics.

Given Data / Assumptions:

• Pure capacity (integrator): G(s) = 1/s.
• Pure time delay: G(s) = exp(-Ts), T > 0.
• Frequency response evaluated at s = jω.

Concept / Approach:For G(jω) = 1/(jω), the magnitude |G| = 1/ω and phase = −90 degrees. For pure delay, |G| = 1 for all ω, while phase = −ωT (in radians), i.e., magnitude is constant but phase lag grows linearly with frequency.

Step-by-Step Solution:Compute integrator magnitude → |1/(jω)| = 1/ω → inversely proportional to frequency.Compute delay magnitude → |exp(-jωT)| = 1 (independent of ω).Compute delay phase → ∠ = −ωT, whose magnitude increases with frequency (more negative).Therefore, only the statement about the integrator magnitude varying as 1/ω is correct.

Verification / Alternative check:Standard Bode plots show a −20 dB/decade slope for an integrator and 0 dB flat magnitude for a pure delay with a steadily increasing phase lag.

Why Other Options Are Wrong:

• (a) “Unbounded for all frequencies” is false; only as ω → 0 does |G| → ∞.
• (b) Delay phase lag magnitude increases with ω; it does not decrease.
• (d) Delay amplitude does not increase; it remains 1.
• (e) Delay has nonzero phase lag except at ω = 0.

Common Pitfalls:Mixing up amplitude and phase effects of delays; equating an integrator with a first-order lag which has finite low-frequency gain.

Final Answer:The amplitude ratio of a pure capacity (integrator) process is inversely proportional to frequency