Frequency response of a first-order system under sinusoidal forcing:\nFor a first-order system subjected to a sinusoidal input, what is the ratio of output amplitude to input amplitude (the sinusoidal steady-state gain)?

Introduction / Context:
First-order systems are ubiquitous (thermal lags, level dynamics, simple RC networks). In frequency response, their magnitude declines with frequency. This question checks your qualitative understanding of sinusoidal steady-state behaviour without resorting to detailed calculations.

Given Data / Assumptions:

• Standard first-order transfer: G(jω) = K / (1 + jωτ).
• Sinusoidal forcing with angular frequency ω.
• We ask about amplitude ratio |G(jω)|.

Concept / Approach:
The magnitude of a first-order transfer is |G(jω)| = K / sqrt(1 + (ωτ)^2). For unity static gain (K = 1), this is always less than or equal to 1, with equality only at ω → 0. At nonzero frequencies, the denominator exceeds 1, so the amplitude ratio is strictly less than 1. Even for K ≠ 1, the frequency-dependent factor is 1 / sqrt(1 + (ωτ)^2) < 1, so the dynamic portion reduces amplitude relative to the static gain.

Step-by-Step Solution:

Verification / Alternative check:
Bode magnitude plots for first-order lags show 0 dB at low frequency rolling off to −20 dB/decade past the corner frequency ωc = 1/τ, confirming amplitude reduction.

Why Other Options Are Wrong:

• 1: Occurs only at zero frequency (DC), not for a sinusoid with finite ω.
• > 1: First-order lag cannot amplify sinusoidal amplitude.
• None of these / Exactly 0: Contradicted by standard frequency response.

Common Pitfalls:
Confusing static gain with dynamic gain; forgetting that phase lag and magnitude roll-off start as soon as ω > 0.

Final Answer:
< 1