Question 10.8: Find the gain and phase margin for the system of Example 10....
Find the gain and phase margin for the system of Example 10.7 if K = 6.
To find the gain margin, first find the frequency where the Nyquist diagram crosses the negative real axis. Finding G(jω)H(jω), we have
G( jω) H( jω) = \frac{6}{(s² + 2s + 2) (s + 2)} \mid_{s \rightarrow jω } (10.47)
= \frac{6 [4(1 − ω²) − jω (6 − ω²)]}{16 (1 − ω²)² + ω² (6 − ω²)²}
The Nyquist diagram crosses the real axis at a frequency of \sqrt{6} rad/s. The real part is calculated to be −0.3. Thus, the gain can be increased by (1/0.3) = 3.33 before the real part becomes −1. Hence, the gain margin is
G_{M} = 20 log 3.33 = 10.45 dB (10.48)
To find the phase margin, find the frequency in Eq. (10.47) for which the magnitude is unity. As the problem stands, this calculation requires computational tools, such as a function solver or the program described in Appendix H.2. Later in the chapter, we will simplify the process using Bode plots. Equation (10.47) has unity gain at a frequency of 1.253 rad/s. At this frequency, the phase angle is −112.3°. The difference between this angle and −180° is 67.7°, which is the phase margin.
Students who are using MATLAB should now run ch10apB3 in Appendix B. You will learn how to use MATLAB to find gain margin, phase margin, zero dB frequency, and 180° frequency. This exercise solves Example 10.8 using MATLAB.
MATLAB’s Linear System Analyzer, with the Nyquist diagram selected, is another method that may be used to find gain margin, phase margin, zero dB frequency, and 180° frequency. You are encouraged to study Appendix E, which contains a tutorial on the Linear System Analyzer as well as some examples. Example E.2 solves Example 10.8 using the Linear System Analyzer.