Question

3. For the Nichols chart (shown below) for a unity feedback system with loop gain L(jw) = Gew)G(jw), using the following frequency table find: a. Peak resonance point, Mor of the closed-loop system b. Resonant frequency, w, of the closed-loop system c. 3dB bandwith. Wsw of the closed-loop system d. Phase Margin of the system 1 rad/ s 3 rad/s 6 rad/s 10 rad/s -0.10 dB 0.25 dB 0.5 dB G G ) -0.25 dB 5 1.0dB -0.5 dB 10 -10 dB Loop gain GG (decibels) 2 dB 3 dB 5dB 4 dB - 6 dB 9 dB 12 dB - 3 dB -4 dB -5 dB -6 dB -12 -210 -180 -150 -120 -90 -60

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Answer 1

a) Peak Resonant Point Mpr Of the closed-loop system from the Nichols Chart is the point at which the M Contour is tangent to the Loop Gain  $L\left ( j\omega \right )$  curve. It is clearly visible that from the chart the 6dB M contour is tangent to the loop gain curve. Therefore the Resonant peak value Mpr is 6dB.

b) The Resonant Frequency $\omega _{r}$ of the closed-loop system is the frequency of the loop gain curve at the tangency point. From the Nichols chart, it is observable that the frequency at the tangency point is $\omega _{2}$ , which is given in the table as 3 rad/s.

c) 3dB bandwidth $\omega _{BW}$ , of the Nichols chart, is the frequency at which the plot crosses the -3dB m contour. From the given chart the frequency at which -3dB crossed is $\omega _{4}$ , which from the table 10rad/s.

d) Phase Margin from the Nichols chart is the horizontal distance from the critical point which is the center of M and N circles. From the given chart it is observable that the horizontal distance between the critical point to the curve is 30o. So the Phase Margin is 30o.

Please see the image to get a clear idea of information for each question.

-0.10 dB 0.25 dB 0.5 dB GjwGwi -0.25 dB $ 1.0dB -0.5 dB -10 dB Loop gain G G (decibels) 3 dB 5 dB 4 dB -2 dB - 3 dB -4dB 9 dBV 12 dB -5 dB -6 dB Horizontal distance from - the critical point Critical Point M Contour is tanget to the curve -3dB Crossing point 12 - 210 -180 -150 -120 -90 -60