Please show me step-by-step. Nicholes chart question Ql. Given the following Nic
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Question
Please show me step-by-step. Nicholes chart question
Ql. Given the following Nicholes chart where a,-1 rad/s, a-3 rad/s, as-6 rad/s and a4-10 rad/s, is this system stable? Explain why. If the system is stable, then estimate 1. the phase margin PM 2. the peak resonance Mp in dB 3. the resonant frequency a, 4. the 3-dB bandwidth 30 -0.10 dB 0.25 dB 24 0.5 dBGe(j)Gja 0.25 dB 18 1.0 dEB -0.5 dB 12 dB 3 dB dB 4 dB bD -30 -2 dB -3 dB -4 dB 5 dB 6 dB 6 dB 9 dB 0 12 dB -12 -210 -180 -150 -120 90 60 Loop phase (degrees)Explanation / Answer
Nichols combined the advantages of the Nyquist plot and the Bode plot pair. The frequency is the parameter along the curve from right to left.The chart consisting constant-magnitude loci and constant phase-angle loci (curves of constant closed-loop magnitude and phase angle) in the log-magnitude versus phase diagram is called Nichols chart.
for ex. -1+j0 as the point 0 dB, 180degree.
We can easily determine the phase margin, gain margin, resonant peak magnitude, resonant peak frequency, and bandwidth of the closed loops system from the plot. To check stability we check gain and phase margins which shows the closeness with border of stability and instability(gap between system and instability point). both should be positive for satbility.
As we can see that from graph so system is stable
Gain margin = - magnitude at (-180 degree) = -(-9) db = 9 db
1) Phase margin = 180 + angle at 0 db = 180-150 = 30 degree (approximate)
2) for peak resonance we check the plot point which is tangantial to m circles here, at w2 plot is tangantial to 6 db line hence Mpw = 6 db
3) Wr = W2 = 3 rad/sec (from point 2)
4) bandwidth is frequency where plot crosses -3 db line so from graph it is 10 rad/sec