First-order element with unity gain: when subjected to a sinusoidal input of frequency ω = 1/τ, what is the amplitude ratio (magnitude of frequency response)?

Introduction / Context:
Frequency response of a first-order system is foundational in process control. The magnitude at a particular excitation frequency shows how much the output is attenuated relative to the input. The most commonly referenced point is ωτ = 1, because it sits at the “corner” of the Bode magnitude plot.

Given Data / Assumptions:

• First-order transfer function with unity DC gain: G(s) = 1 / (1 + τs).
• Sinusoidal input at angular frequency ω = 1/τ.
• Amplitude ratio AR = |G(jω)|.

Concept / Approach:
Compute AR using the standard magnitude formula for a first-order lag: |G(jω)| = 1 / sqrt(1 + (ωτ)^2). Substituting ωτ = 1 yields AR = 1 / sqrt(1 + 1) = 1 / √2, which is approximately 0.707. This is the familiar −3 dB point on the Bode magnitude plot for a single-pole system.

Step-by-Step Solution:
Write G(jω) = 1 / (1 + jωτ).Compute magnitude: |G(jω)| = 1 / sqrt(1 + (ωτ)^2).Set ω = 1/τ → ωτ = 1 → AR = 1 / √2.

Verification / Alternative check:
In decibels, 20 log10(1/√2) ≈ −3.01 dB, the well-known half-power point for a single pole.

Why Other Options Are Wrong:
1: Would imply no attenuation; incorrect at the corner frequency.0.5 or 0.25: Excessive attenuation for a single pole at ωτ = 1.2/√5 ≈ 0.894: Not a recognized first-order corner magnitude.

Common Pitfalls:
Confusing amplitude ratio with amplitude ratio squared; mixing up time constant τ and corner frequency 1/τ.

Final Answer:
1/√2