Chapter 14

# 14.4 Gain and Phase Margins vs. Polar Plots

**14.4.1 Example Gain Margin from Polar Plot**

Let the crossover frequency be defined as [latex]\omega_{cg}[/latex], the frequency at which the phase plot crosses over the [latex]-180^ \circ[/latex] line. On the polar plot, this corresponds to the plot crossing the negative part of Real axis. Remember the definition of Gain Margin [latex]G_{m}[/latex] :

[latex]G_m = \frac{K_{crit}}{K}[/latex] [latex]G_m>1[/latex] system stable [latex]G_m<1[/latex] system unstable |
Equation 14-6 |

[latex]G_m = \frac{1}{ \mid A \mid}[/latex] |
Equation 14-7 |

On the polar plot, Gain Margin [latex]G_m[/latex] can be found as an inverse of the coordinate A of the polar plot crossover with the Real axis, as shown next. If the crossover is to the right of (-1, j0) point,[latex]\mid A \mid < 1[/latex] ,[latex]G_{m}>1[/latex] , and the system is stable. If the crossover is to the left of (-1, j0) point,[latex]\mid A \mid>1[/latex] ,[latex]G_m[/latex] [latex]<1[/latex] , and the system is unstable.

**14.4.2 Example Phase Margin from Polar Plot**

The crossover frequency defined as [latex]\omega_{cp}[/latex], is the frequency at which the polar plot crosses over the unit circle ([latex]0[/latex] dB = 1 Volt/Volt). Phase Margin [latex]\Phi_m[/latex] is defined as [latex]\Phi_m=180 ^ \circ + \angle GH( \omega_{cp} ).[/latex] Therefore, on the polar plot Phase Margin [latex]\Phi_m[/latex] can be found as the angle between the Real axis and the crossover of the polar plot with the unit circle, as shown in **Error! Reference source not found.** If this angle is above the Real axis, the system is unstable, if this angle is below the Real axis, the system is stable.

**14.4.3 Solved Example**

Consider a unit feedback closed loop system where the open loop transfer function [latex]G(s)[/latex] is known to be unstable and its transfer function [latex]G(s)[/latex] is known as [latex]G(s) = \frac{s+2}{s(s-2)} .[/latex] Such system can be stabilized by using an appropriately large value of the controller gain. We need to establish the critical gain [latex]K_{crit}[/latex] .

**Solution Part 1: **Let’s tackle this problem in s-domain. The system closed loop characteristic equation is:

[latex]1+KG(s)=0[/latex]

[latex]1+K \frac{s+2}{s(s-2)} =0[/latex]

[latex]s^2-2s+Ks+2K=0[/latex]

[latex]s^2+(K-2)s+2K=0[/latex]

For the 2^{nd} order system the necessary and sufficient condition of stability is that all coefficients of the characteristic polynomial are positive:

[latex]K-2>0[/latex]

[latex]K>0[/latex]

The critical value of the gain is [latex]K_{crit}=2[/latex] and the range of gains for stable system performance is:

[latex]2

The frequency of oscillations [latex]\omega_{osc}[/latex] at the critical gain is equal to [latex]2[/latex] rad/s:

[latex]s^2+(K-2)s+2K=0[/latex]

[latex]K_{crit}=2[/latex]

[latex]s^2+4=0[/latex]

[latex]s= \pm j2[/latex]

[latex]\omega_{osc} =2[/latex]

The upper limit of the gain range will be determined by practical issues such as saturation.

**Solution Part 2: **Now let’s try to apply the Gain Margin and Phase Margin definitions to this system. The open loop frequency response has to be simulated as the system is open-loop unstable and no measurements on the open loop are possible. From the plot shown in **Error! Reference source not found.** we read:

[latex]G_m=+6dB=2[/latex]

[latex]\omega_{cg}=2[/latex]

The positive Gain Margin [latex]G_m= 6 dB = 2[/latex] Volt/Volt is measured at the crossover frequency [latex]\omega_{cp}[/latex] = 2 rad/s. This would have to be interpreted as the system being stable for gains **lower **than 2, which as we know from the s-domain analysis is not correct. On the other hand, the Phase Margin [latex]\Phi_m[/latex] is negative, indicating the system is unstable for gains [latex]< 2[/latex]. This is an example of when the Gain and Phase Margin definitions cannot be applied consistently to determine the system stability limits. A new, more general frequency domain based stability criterion will now be defined.