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Return to Blog# Bode Plot, Phase Margin, Crossover Frequency, and Stability

Since the inception of electronic communications, frequency response has been the common measurement unit for most instruments. Even after the introduction of the feedback amplifier, frequency response continued to be the basis for determining stability. Bode plots have been a great resource to visualize the frequency response of circuit to determine if a system is unstable.

Because the system roots are provided in factor form, it is in this condition that we must inspect the denominator to observe whether the real parts are positive or negative. This, however, can only be done if we know the closed-loop transfer function. Usually though, the closed-loop transfer function is not known.

In cases where the closed-loop transfer function is not known, we can determine stability by assessing the open-loop transfer function KG(j\omega ) and testing it. Here, we do not have to factor the closed-loop transfer function.

## Stability Margins

The majority of control system designs behave similarly with regard to stability. Most often, if the gain exceeds a certain critical point, the system loses stability. Gain margin and phase margin both measure a system’s stability margin. These two quantities relate directly to the following stability condition equation:

\left | KG (j\omega ) \right |<1 at \angle G(j\omega )=-180^{\circ}

### Gain Margin

The gain margin (GM) is the factor by which the gain is less than the neutral stability value. We can usually read the gain margin directly from the bode plot. This is done by calculating the vertical distance between the \left | KG (j\omega ) \right | curve and the \left | KG (j\omega ) \right |=1 at the frequency where \angle G(j\omega )=180^{\circ}.

### Phase Margin

Another quantity related to determining stability margin is the phase margin (PM). This is a different way to measure how well stability conditions are met in a given system. Phase margin is determined by how much the phase of G(j\omega ) exceeds -180^{\circ} when \left | KG (j\omega ) \right |=1.

The above figure shows that, for a system to be stable, a positive PM is required. From the figures, we can also see that the GM indicates the amount that the gain can increase before a system becomes unstable. The PM is calculated by measuring the difference between the G(j\omega ) and 180^{\circ} when KG(j\omega ) crosses the circle \left | KG(s) \right |=1. The stable case receives the phase margin’s positive value.

### Crossover Frequency

A gain of factor 1 (equivalent to 0 dB) where both input and output are at the same voltage level and impedance is known as unity gain. When the gain is at this frequency, it is often referred to as crossover frequency.

Frequency-response design is practical because we can easily evaluate how gain changes affect certain aspects of systems. With frequency-response design, we can determine the phase margin for any value of K without needing to redraw the magnitude or phase information. All we have to do is indicate where \left | KG (j\omega ) \right |=1 for certain trial values of K

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