First-order element under sinusoidal forcing The steady-state sinusoidal response of an ideal first-order system exhibits what phase lag relative to the input?

Introduction / Context:
Understanding frequency response is essential for controller tuning and stability analysis. A first-order process (with transfer function 1/(τs + 1)) has characteristic gain and phase behaviors that determine how it reacts to oscillatory inputs and noise. The phase lag is especially important for phase margin and closed-loop robustness.

Given Data / Assumptions:

• Standard first-order transfer function: G(jω) = 1 / (1 + jωτ).
• We examine the steady-state sinusoidal response.
• Phase lag φ(ω) = arctan(ωτ) (expressed as a positive lag magnitude).

Concept / Approach:
The phase lag of a first-order system increases monotonically with frequency from 0° at ω → 0 to a limiting value as ω → ∞. The asymptotic maximum lag is 90°. At the corner frequency (ω = 1/τ), the phase lag equals 45°. Hence, the best single descriptive value for the maximum phase lag is 90° (noting this is the limit as frequency becomes very high).

Step-by-Step Solution:

Verification / Alternative check:
Bode plot of a first-order lag shows a straight-line phase characteristic approaching −90° at high frequency, confirming the result.

Why Other Options Are Wrong:

• 30°, 120°, 180°: Do not match first-order behavior; 180° is characteristic of sign reversal or second-order resonant conditions, not a simple first-order lag.

Common Pitfalls:
Interpreting 90° as an exact constant; for a first-order element it is an upper limit reached asymptotically with increasing frequency.

Final Answer:
90°