Question

The magnitude and phase plots shown in the figure match with the transfer-function ______.

• A. $\dfrac{10000}{s^2+2s+10000}$
• B. $\dfrac{10000}{s^2+2s+10000}\;e^{-0.05s}$
• C. $\dfrac{10000}{s^2+2s+10000}\;e^{-0.5\times10^{-12}s}$
• D. $\dfrac{100}{s^2+2s+100}$

Hint:

A time delay $e^{-s\tau}$ does {not} change magnitude but adds a linear phase lag $-\omega\tau$ (radians). Use this to distinguish between choices with identical magnitudes.

Answer 1

The Correct Option is B

The magnitude peak near $\omega_n\!\approx\!100$ rad/s indicates a lightly damped 2nd-order low-pass: $\dfrac{10000}{s^2+2s+10000}$ (since $2\zeta\omega_n=2\Rightarrow \zeta\approx0.01$ gives a tall resonance). Options (A) and (B) share this magnitude. The plotted phase, however, decreases far beyond $-180^\circ$ (approaching about $-700^\circ$ by 200 rad/s), which requires an additional frequency-proportional lag. A pure time delay $e^{-0.05s}$ contributes phase $-\omega(0.05)$ (radians), matching the observed extra linear drop. Hence (B).