# 闭环系统的过冲和相位裕度关系的分

When an amplifier with a gain A(s) is put in a

feedback loop as shown in Figure 1, the closed loop gain, V o /V in = A CL

(1)

The system is unstable when the loop gain, ß A(s), equals -1. That is, ß A(s) has a magnitude of one and a phase of -180

degrees. An unstable system oscillates. A system close to being unstable has a large ringing overshoot in response to a step input.

The phase margin is a measure of how close the phase of the loop gain is to -180 degrees, when the magnitude of the loop gain is one. The phase margin is the additional phase required to bring the phase of the loop gain to -180 degrees. Phase Margin = Phase of loop gain - (-180). The loop gain has a dominant pole at

. Higher order poles can be

represented by an equivalent pole at . The amplifier is approximated by a function with two poles as shown in Equation 2.

(2)

Since for frequencies of interest where the loop gain magnitude is close to unity,

(3)

And,

(4)

(5)

Defining ,

Table I

∙ PM is the phase margin.

∙ w t is the unity gain frequency (rad/sec).

∙ w eq is the frequency of the equivalent higher order pole (rad/sec).

∙ Q is the system Quality factor.

∙ OS is the Over Shoot.