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)

闭环系统的过冲和相位裕度关系的分

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)

闭环系统的过冲和相位裕度关系的分

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

(3)

闭环系统的过冲和相位裕度关系的分

And,

(4)

闭环系统的过冲和相位裕度关系的分

闭环系统的过冲和相位裕度关系的分

(5)

闭环系统的过冲和相位裕度关系的分

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.

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