船长 水手:Overshoot as a Function of Phase Margin

J. C. Daly
Electrical and Computer Engineering
University of Rhode Island
4/19/03

Figure 1

Figure 2   Amplifier frequency response.
PM wt/weq Q %OS
55o 0.700 0.925 13.3%
60o 0.580 0.817 8.7%
65o 0.470 0.717 4.7%
70o 0.3600.622 1.4%
75o 0.270 0.527 0.008%
Table I
PM is the phase margin.
wt is the unity gain frequency (rad/sec).
weq is the frequency of the equivalent higher order pole (rad/sec).
Q is the system Quality factor.
OS is the Over Shoot.
When an amplifier with a gain A(s) is put in a feedbackloop as shown in Figure 1, the closed loop gain,
Vo/Vin = ACL
(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 overshootin response to a stepinput.
The phase margin is a measure of how close the phase of the loopgain is to -180 degrees, when themagnitude of the loop gain is one.The phase margin is the additional phase required to bring the phase of theloop gain to -180 degrees. Phase Margin = Phase of loop gain - (-180).
The loop gain has a dominant pole at.Higherorder poles can be represented by an equivalent pole at.The amplifier is approximated by a function with two poles as shownin Equation 2.

(2)
Since for frequencies of interest where the loop gain magnitude is close to unity,
(3)
And,
(4)
Also, it can be shown,
(5)
Defining,
(6)
For frequencies of interest (frequencies close to the unity gain frequency),the amplifier gain can be written,
(7)
Plugging Equation 7 into Equation 1 results in the followingexpression for the closed loop gain.
(8)
Equation 8 is the transfer function fora second order system. The general form for the response ofa second order system, where system properties aredescribed by its Q and resonantfrequence wo, is shown in Equation 10.
(9)
By comparing Equation 8 to Equation 9 we canget an expression for the resonant frequency and Q of the amplifierclosed loop gain. (Equate coefficients of like powers of s in the dominators.)
(10)

(11)
The loop gain is the feedback factor ß multiplied by the amplifier gain A(s).
(12)
The phase margin is a function of the phase of the loop gain at the frequencywhere the magnitude of the loop gain is unity.
(13)
whereis the loop gain unity gain frequency.It follows from Equations 12 and 13 that,
(14)
Also, solving for wta and dividing by weq,
(15)
It follows from Equations 11 and 15 that,
(16)
The phase of the loop gain (Equation 13) is. Phase of loop gain
(17)
The phase margin is the additional phase required to bring the phase of theloop gain to -180 degrees.
Phase Margin = Phase of loop gain - (-180). Phase Margin
(18)
A well known property of second order systems is that the percent overshoot isa function of the Q and is given by,
(19)
Both phase margin (Equation 18) and Q (Equation 16)are a function of wt / weq.This allows us to use Equation 19 tocreate tables and plots of percent overshoot as a function of phase margin.As shown in Figures 3 and 4,and in Table I.
Figure 3   Overshoot as a function of phase margin.
Plot generated using MATLAB code.
Figure 4   Q as a function of phase margin.
Plot generated using MATLAB code.