The critical value of gain for a system is 40. The system is operating at a gain of 20. The gain margin of the system is:

Concept:

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.

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

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

\(GM = \frac{Critical \ Gain}{Given \ Gain}\)  ----(1)

Calculation:

Given:

Critical Gain = 40

Operating Gain = 20

From equation 1:

\(GM = \frac{40}{20}=2\)

GM in dB = 20 log₁₀ (2)

GM in dB = 6 dB