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 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 βA_OL = −1. (That is, the magnitude of βA_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 f₁₈₀, 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 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 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 βA_OL reaches −180°, denoted here as frequency f₁₈₀. Using this frequency, the Bode magnitude plot finds the magnitude of βA_OL. If |βA_OL|₁₈₀ = 1, the amplifier is unstable, as mentioned. If |βA_OL|₁₈₀ < 1, instability does not occur, and the separation in dB of the magnitude of |βA_OL|₁₈₀ from |βA_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 log₁₀( |βA_OL|₁₈₀) = 20 log₁₀( |A_OL|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 |βA_OL| reaches unity, denoted here as frequency f_0dB. Using this frequency, the Bode phase plot finds the phase of βA_OL. If the phase of βA_OL( f_0dB) > −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 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 < 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.