Frequency response of a first-order system (τ = 1 min) A first-order process with time constant τ = 1 minute is tested at input frequency ω = 1 rad/min. What is the steady-state phase shift (in degrees) between output and sinusoidal input?

Introduction / Context: First-order dynamics are ubiquitous in process units (tanks, heat exchangers). Their frequency response is simple yet powerful: knowing the phase shift and attenuation at a given frequency helps in controller tuning and estimating closed-loop behaviour. This question focuses on computing the phase at a specific frequency using the well-known formula for a first-order lag.

Given Data / Assumptions:

• Transfer function form: G(jω) = K / (1 + jωτ).
• Time constant τ = 1 min; test frequency ω = 1 rad/min.
• We assume standard steady-state sinusoidal response.

Concept / Approach: For G(jω) = K / (1 + jωτ), the phase is φ(ω) = −tan^−1(ωτ). This is independent of K. Substituting τ = 1 and ω = 1 gives φ = −tan^−1(1) = −45°. The negative sign indicates the output lags the input, characteristic of a causal lag element.

Step-by-Step Solution:

Verification / Alternative check: The magnitude at this frequency is |G(j1)| = K / √(1 + 1^2) = K/√2, confirming a −3 dB point when ω = 1/τ. The corner frequency 1/τ = 1 rad/min also typically corresponds to a −45° phase, consistent with the result.

Why Other Options Are Wrong:

Common Pitfalls: Forgetting the negative sign (lag), or confusing the break frequency with the high-frequency asymptote.

Final Answer: −45°