Question

The gain cross over frequency is the frequency at which the \(|G(s)H(s)|\) is (Note: s should be \(j\omega\) for frequency response magnitude) \(|G(j\omega)H(j\omega)|\)

• A. 0
• B. --1
• C. 1
• D. \(\infty\)

Hint:

Gain Crossover Frequency (\(\omega_{gc}\)): Frequency where \(|G(j\omega)H(j\omega)| = 1\) (or 0 dB).
Phase Crossover Frequency (\(\omega_{pc}\)): Frequency where \(\angle G(j\omega)H(j\omega) = -180^\circ\).
Phase Margin (PM) is measured at \(\omega_{gc}\): PM = \(180^\circ + \angle G(j\omega_{gc})H(j\omega_{gc})\).
Gain Margin (GM) is measured at \(\omega_{pc}\): GM = \(1 / |G(j\omega_{pc})H(j\omega_{pc})|\) (or \(-20\log_{10}|G(j\omega_{pc})H(j\omega_{pc})|\) in dB).

Answer 1

The Correct Option is C

The gain crossover frequency (\(\omega_{gc}\)) is defined as the frequency at which the magnitude of the open-loop transfer function \(|G(j\omega)H(j\omega)|\) is equal to unity (or 0 dB). \[ |G(j\omega_{gc})H(j\omega_{gc})| = 1 \] This frequency is important in stability analysis using Bode plots, particularly for determining the phase margin. The phase crossover frequency (\(\omega_{pc}\)) is the frequency at which the phase of \(G(j\omega)H(j\omega)\) is \(-180^\circ\). So, at gain crossover frequency, \(|G(j\omega)H(j\omega)| = 1\). \[ \boxed{1} \]