**Gain Margin and Phase Margin**

Bode plots are used to assess the stability of negative feedback amplifiers by finding the gain and phase margins of an amplifier. The notion of gain and phase margin is based upon the gain expression for a negative feedback amplifier given by

where A_{FB} is the gain of the amplifier with feedback (the **closed-loop gain**), β is the **feedback factor** and *A*_{OL} is the gain without feedback (the **open-loop gain**). The gain *A*_{OL} is a complex function of frequency, with both magnitude and phase. Examination of this relation shows the possibility of infinite gain (interpreted as instability) if the product β*A*_{OL} = −1. (That is, the magnitude of β*A*_{OL} is unity and its phase is −180°, the so-called **Barkhausen stability criterion**). Bode plots are used to determine just how close an amplifier comes to satisfying this condition.

Key to this determination are two frequencies. The first, labeled here as *f*_{180}, is the frequency where the open-loop gain flips sign. The second, labeled here *f*_{0dB}, is the frequency where the magnitude of the product | β *A*_{OL} | = 1 (in dB, magnitude 1 is 0 dB). That is, frequency *f*_{180} is determined by the condition:

where vertical bars denote the magnitude of a complex number (for example, | *a* + *j* *b* | = 1/2 ), and frequency *f*_{0dB} is determined by the condition:

One measure of proximity to instability is the **gain margin**. The Bode phase plot locates the frequency where the phase of β*A*_{OL} reaches −180°, denoted here as frequency *f*_{180}. Using this frequency, the Bode magnitude plot finds the magnitude of β*A*_{OL}. If |β*A*_{OL}|_{180} = 1, the amplifier is unstable, as mentioned. If |β*A*_{OL}|_{180} < 1, instability does not occur, and the separation in dB of the magnitude of |β*A*_{OL}|_{180} from |β*A*_{OL}| = 1 is called the *gain margin*. Because a magnitude of one is 0 dB, the gain margin is simply one of the equivalent forms: 20 log_{10}( |β*A*_{OL}|_{180}) = 20 log_{10}( |*A*_{OL}|_{180}) − 20 log_{10}( 1 / β ).

Another equivalent measure of proximity to instability is the **phase margin**. The Bode magnitude plot locates the frequency where the magnitude of |β*A*_{OL}| reaches unity, denoted here as frequency *f*_{0dB}. Using this frequency, the Bode phase plot finds the phase of β*A*_{OL}. If the phase of β*A*_{OL}( *f*_{0dB}) > −180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when *f = f*_{180}), and the distance of the phase at *f*_{0dB} in degrees above −180° is called the *phase margin*.

If a simple *yes* or *no* on the stability issue is all that is needed, the amplifier is stable if *f*_{0dB} < *f*_{180}. This criterion is sufficient to predict stability only for amplifiers satisfying some restrictions on their pole and zero positions (minimum phase systems). Although these restrictions usually are met, if they are not another method must be used, such as the Nyquist plot. Optimal gain and phase margins may be computed using Nevanlinna–Pick interpolation theory.

Read more about this topic: Bode Plot

### Other articles related to "gain margin and phase margin, gain":

**Gain Margin and Phase Margin**- Examples Using Bode Plots

... Figures 6 and 7 illustrate the

**gain**behavior and terminology ... Figure 6 compares the Bode plot for the

**gain**without feedback (the open-loop

**gain**) AOL with the

**gain**with feedback AFB (the closed-loop

**gain**) ... Because the open-loop

**gain**AOL is plotted and not the product β AOL, the condition AOL = 1 / β decides f0dB ...

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