## 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 0.360 0.622 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 feedback loop 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 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)
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 following expression for the closed loop gain.
 (8)
Equation 8 is the transfer function for a second order system. The general form for the response of a second order system, where system properties are described by its Q and resonant frequence wo, is shown in Equation 10.
 (9)
By comparing Equation 8 to Equation 9 we can get an expression for the resonant frequency and Q of the amplifier closed 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 frequency where the magnitude of the loop gain is unity.
 (13)
where is 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 the loop 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 is a 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 to create 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.