Frequency response signature – constant magnitude, linear phase lag A dynamic element exhibits unit (constant) magnitude at all frequencies and a negative phase that increases linearly in absolute value with frequency. Which transfer function matches this behavior?

Introduction / Context:
Frequency-domain fingerprints help identify elementary dynamic elements without time-domain tests. A pure time delay (dead time) has a distinctive Bode plot: flat magnitude and linearly increasing phase lag with frequency. This question asks you to map that signature to the correct transfer function.

Given Data / Assumptions:

• Magnitude ratio is constant (unity) over all frequencies.
• Phase is negative and proportional to ω (linear in frequency).
• No amplitude roll-off or resonant features observed.

Concept / Approach:
A pure time delay of amount τ has transfer function G(s) = e^{−τ s}. In the frequency domain, G(jω) has magnitude |G| = |e^{−j ω τ}| = 1 for all ω and phase ∠G = −ω τ (radians), a straight line through the origin with slope −τ. By contrast, first-order lag 1/(τ s + 1) shows decreasing magnitude and asymptotic −90° phase; second-order terms add curvature and possible resonance; an integrator K/s has magnitude decreasing 20 dB/decade and −90° phase, not constant magnitude.

Step-by-Step Solution:

Verification / Alternative check:
Plot Bode: delay line shows 0 dB across, with a straight-line phase lag increasing with ω; others do not.

Why Other Options Are Wrong:

• 1/(τ s + 1): Low-pass magnitude and saturating phase.
• 1/(τ^2 s^2 + 1): Second-order behavior, not constant magnitude.
• K/s: Magnitude falls with frequency; constant −90° phase, not linear unbounded lag.

Common Pitfalls:
Approximating delay by extra poles; only e^{−τ s} preserves unit magnitude.

Final Answer:
e^{-τ s}