Sinusoidal response magnitude: for a pure transportation lag (dead time) element, identify the amplitude ratio (magnitude) of the frequency response.

Introduction / Context:
A transportation lag (dead time) element delays a signal by a fixed time L without changing its shape. In the frequency domain, dead time contributes phase lag that increases with frequency, but it does not attenuate amplitude in the ideal mathematical model.

Given Data / Assumptions:

• Ideal dead time with transfer function e^(−sL).
• Sinusoidal steady-state input of frequency ω.
• No additional dynamics in series.

Concept / Approach:
The frequency response of e^(−sL) evaluated at s = jω is e^(−jωL) whose magnitude is |e^(−jωL)| = 1 and whose phase is −ωL radians. Thus, the amplitude ratio (output amplitude / input amplitude) is unity for all frequencies, while the phase lag grows linearly with frequency.

Step-by-Step Solution:

Verification / Alternative check:
Bode plots for dead time show a flat 0 dB magnitude line (AR = 1) and a phase that decreases linearly with frequency.

Why Other Options Are Wrong:

Common Pitfalls:
Confusing practical dead time (with filters and measurement noise) with the ideal mathematical element; in practice, some attenuation can appear, but in theory AR = 1.

Final Answer:
Transportation lag