How do you calculate the gain margin from a Bode plot?

A Bode plot is a graph that represents the frequency response of a system, showing both the magnitude (gain) and phase shift of the system as a function of frequency. To calculate the gain margin from a Bode plot, follow these steps:

Find the frequency at which the phase shift is -180 degrees. This is often called the "phase crossover frequency" or "gain crossover frequency."

Locate the corresponding magnitude (gain) value at the phase crossover frequency on the Bode plot.

The gain margin is the amount of gain (in dB) that the system can handle before the magnitude reaches 0 dB (unity gain). In other words, it's the difference between the 0 dB point and the magnitude value at the phase crossover frequency.

Mathematically, the gain margin (GM) is calculated as:

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GM = 0 dB - Magnitude at Phase Crossover Frequency

It's important to note that a system is considered stable if the gain margin is positive. If the gain margin is negative or zero, the system is at risk of instability.

Some Bode plots might not explicitly show the 0 dB reference line, in which case, you can determine the gain margin by calculating the difference between the magnitude values at the phase crossover frequency and the frequency at which the magnitude is 0 dB.