Topic: Control Systems
The magnitude of frequency response of an under-sampled second order system is 5 at 0 rad/sec and peaks to $10/\sqrt{3}$ at $5\sqrt{2}~\text{rad/sec}$. The transfer function of the system is
- A. $\frac{500}{(s^{2}+10s+100)}$
- B. $\frac{375}{(s^{2}+5s+75)}$
- C. $\frac{720}{(s^{2}+12s+144)}$
- D. $\frac{1125}{(s^{2}+25s+225)}$
Answer: A. $\frac{500}{(s^{2}+10s+100)}$
Explanation: Checking option (a): $T(s) = \frac{500}{s²+10s+100}$.
At $\omega=0$, $T(0) = \frac{500}{100} = 5$.
At $\omega = 5\sqrt{2}~\text{rad/sec}$, magnitude $|T(j\omega)| = \frac{500}{\sqrt{(100 - (5\sqrt{2})^2)^2 + (10 \times 5\sqrt{2})^2}}$.
$|T(j5\sqrt{2})| = \frac{500}{\sqrt{(100-50)^2 + (50\sqrt{2})^2}} = \frac{500}{\sqrt{2500 + 5000}} = \frac{500}{\sqrt{7500}} = \frac{500}{50\sqrt{3}} = \frac{10}{\sqrt{3}}$.
This perfectly matches the given conditions.