Assertion: Gain margin is the factor by which the system gain can be decreased to drive it to the verge of instability.

Reason: Gain margin is the reciprocal of the gain at a frequency at which the phase angle becomes 180°

Gain margin (GM):

The gain margin of the system defines by how much the system gain can be increased so that the system moves on the edge of stability

∴ Assertion Statement is incorrect.

It is determined from the gain at the phase cross over frequency, i.e.

\(GM = \frac{1}{{{{\left| {G\left( {j\omega } \right)H\left( {j\omega } \right)} \right|}_{\omega = {\omega _{pc}}}}}}\)

Phase crossover frequency (ωpc) is the frequency at which phase angle of G(s) H(s) is -180°, i.e.

\(\angle G\left( {j\omega } \right)H\left( {j\omega } \right){|_{\omega = {\omega _{pc}}}} = - 180^\circ \)

The Reason is therefore correct.

NOTES:

Phase margin (PM):

The phase margin of the system defines by how much the phase of the system can increase to make the system unstable.

\(PM = 180^\circ + \angle G\left( {j\omega } \right)H\left( {j\omega } \right){|_{\omega = {\omega _{gc}}}} = - 180^\circ \)

It is determined from the phase at the gain cross over frequency.

Gain crossover frequency (ωgc): It is the frequency at which the magnitude of G(s) H(s) is unity.

\({\left| {G\left( {j\omega } \right)H\left( {j\omega } \right)} \right|_{\omega = {\omega _{gc}}}} = 1\)

• If both GM and PM are positive, the system is stable (ωgc < ωpc)
• If both GM and PM are negative, the system is unstable (ωgc > ωpc)
• If both GM and PM are zero, the system is just stable (ωgc = ωpc)