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Solutions_hw7 - Homework 7 ECE147A The Solutions from the...

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Homework 7 ECE147A November 24, 2009 The Solutions from the Solution Manual follow at the end, first points and some remarks. Problem 1 For full points, you need to explain in a few words why your sketch looks the way it looks, e.g. “The magnitude starts at 0 dB while the phase is 0 , so the Nyquist Plot starts at (1 , 0 j ).” 3 points for each Nyquist Plot, 2 each for the statement whether we have a stable system for the K you picked in your sketch or not. Problem 2 Do not be confused by the last figure where it seems like you could move the point - 1 K out of the contour – you cannot, because the semi-circle has infinite radius, which is just a little hard to draw. 15, if you explained what you are drawing why. Problem 3 2 points for the Phase Margin, 5 points for finding all Gain Margins. 4 points for describing what happens for small and large gains and in between - stability changes 3 times! The sketch of the Bode Plot is a little unsatisfactory, here’s what I would suggest: The phase starts at -90 . 1 point The phase is -180 at ω 0 , ω L and ω H . Furthermore, ω 0 < ω L < ω H . 2 points Between ω 0 and ω L and after ω H , the phase is below -180 , between ω L and ω H it is above. 3 points 1
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the phase approaches -270 for ω → ∞ . 2 points The magnitude at ω * is 0 dB, at ω L it is β dB = 20 log( β ) and at ω H it is α dB . 2 points The phase at ω * is φ - 180 . 2 points An example Bode Plot is shown in Figure 1. The system I used is H ( s ) = 2000 s 2 + 6000 s + 4500 s 5 + 40 . 2 s 4 + 408 s 3 + 80 . 4 s 2 + 4 s . Don’t worry how I found that, it took me a while. The relevant part of the root locus is shown in Figure 2. The Nyquist Plot also resembles the one shown in the book somewhat, however, you need to zoom a lot to see that, so I did not include it here. 4 points for a root locus that has the porperty that the closed loop is stable, then unstable, then stable and finally unstable again for increasing k . Bode Diagram Frequency (rad/sec) 10 -2 10 -1 10 0 10 1 10 2 10 3 -270 -225 -180 -135 -90 Phase (deg) -150 -100 -50 0 50 100 150 Magnitude (dB) ϖ H ϖ * ϖ L ϖ 0 φ -180 α dB β dB Figure 1: Problem 3: An example Bode Plot Problem 4 For each of the ranges 3 points, for the number of unstable poles 2 points and 3 points for the sketch of the root locus.
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