# Bode Plot - Gain Margin and Phase Margin

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 AFB is the gain of the amplifier with feedback (the closed-loop gain), β is the feedback factor and AOL is the gain without feedback (the open-loop gain). The gain AOL 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 βAOL = −1. (That is, the magnitude of βAOL 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 f180, is the frequency where the open-loop gain flips sign. The second, labeled here f0dB, is the frequency where the magnitude of the product | β AOL | = 1 (in dB, magnitude 1 is 0 dB). That is, frequency f180 is determined by the condition:

where vertical bars denote the magnitude of a complex number (for example, | a + j b | = 1/2 ), and frequency f0dB 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 βAOL reaches −180°, denoted here as frequency f180. Using this frequency, the Bode magnitude plot finds the magnitude of βAOL. If |βAOL|180 = 1, the amplifier is unstable, as mentioned. If |βAOL|180 < 1, instability does not occur, and the separation in dB of the magnitude of |βAOL|180 from |βAOL| = 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 log10( |βAOL|180) = 20 log10( |AOL|180) − 20 log10( 1 / β ).

Another equivalent measure of proximity to instability is the phase margin. The Bode magnitude plot locates the frequency where the magnitude of |βAOL| reaches unity, denoted here as frequency f0dB. Using this frequency, the Bode phase plot finds the phase of βAOL. If the phase of βAOL( f0dB) > −180°, the instability condition cannot be met at any frequency (because its magnitude is going to be < 1 when f = f180), and the distance of the phase at f0dB 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 f0dB < f180. 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.

### 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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