Sinusoidal response magnitude check: For which element is the amplitude ratio |Output/Input| strictly less than 1 for all nonzero frequencies (assuming unity static gain)?

Introduction / Context:
Frequency-response intuition allows quick predictions without solving time-domain equations. This question asks which listed element guarantees an amplitude ratio less than 1 at all nonzero frequencies when normalised to unity DC gain. Knowing these signatures helps in controller design and filter selection.

Given Data / Assumptions:

• Steady-state sinusoidal input at frequency ω.
• Unity static (DC) gain for fair comparison.
• Standard textbook forms for each element.

Concept / Approach:
A unity-gain first-order lag G(jω) = 1 / (1 + jωτ) has magnitude |G(jω)| = 1 / sqrt(1 + (ωτ)^2) < 1 for any ω > 0. A pure time delay e^(−jωL) has magnitude exactly 1 at all frequencies (it only adds phase). A unity-gain second-order system can exhibit resonance with |G(jω)| > 1 when damping ζ < 1/√2, so it is not guaranteed to stay below 1. Therefore, only the first-order lag always satisfies the “< 1 for all ω > 0” condition under the stated assumptions.

Step-by-Step Solution:

Verification / Alternative check:
Bode magnitude plots confirm a constant 0 dB for delay, a monotonic roll-off for first-order lag, and a possible resonant peak for underdamped second-order systems.

Why Other Options Are Wrong:

Common Pitfalls:
Forgetting that delay only shifts phase; it does not attenuate amplitude.

Final Answer:
First-order system