Topic: Control Systems
Consider a unity feedback system having an open loop transfer function
$G(j\omega)=\frac{k}{j\omega(j0.2\omega+1)(j0.05\omega+1)}$
Find open loop gain (k) with gain margin of 20 dB
- A. 5.2
- B. 2.5
- C. 0.1
- D. 2.25
Answer: B. 2.5
Explanation: $G(j\omega)=\frac{k}{j\omega(j0.2\omega+1)(j0.05\omega+1)}$
Gain margin $= 20 \text{ dB} = 10$
$\angle G(j\omega) = -90^{\circ} - \tan^{-1}(0.2\omega) - \tan^{-1}(0.05\omega)$
Equating it equal to $-180^{\circ}$ for phase cross over frequency.
$-180^{\circ} = -90^{\circ} - \tan^{-1}\left(\frac{0.2\omega_p+0.05\omega_p}{1-\frac{\omega_p^2}{100}}\right)$
$1-\frac{\omega_p^2}{100}=0 \Rightarrow \omega_p=10 \text{ rad/sec}$
$|G(j\omega)|_{\omega=\omega_p} = \frac{1}{\text{Gain Margin}}$
$\frac{1}{10} = \frac{K}{10\sqrt{2^2+1^2}\sqrt{(0.5)^2+1}}$
$K = \sqrt{5}\sqrt{1.25} = 2.5$