Topic: Control Systems
The system with the open loop transfer function $G(s)H(s)=\frac{1}{s(s^{2}+s+1)}$ has a gain margin of
- A. -6 dB
- B. zero
- C. 3.5 dB
- D. 6 dB
Answer: B. zero
Explanation: To find the phase crossover frequency $\omega_{pc}$, equate the imaginary part of $G(j\omega)H(j\omega)$ to zero.
$G(j\omega)H(j\omega) = \frac{1}{j\omega(-\omega²+j\omega+1)} = \frac{1}{-\omega² + j\omega(1-\omega²)}$.
For the imaginary part to be zero, $1 - \omega² = 0 \Rightarrow \omega_{pc} = 1~\text{rad/sec}$.
Magnitude at $\omega = 1$ is $|G(j1)H(j1)| = \left|\frac{1}{-1^2 + j(1)(1-1^2)}\right| = \left|\frac{1}{-1}\right| = 1$.
Gain Margin = $-20\log_{10}(|G(j\omega_{pc})H(j\omega_{pc})|) = -20\log_{10}(1) = 0~\text{dB}$.