Week 12-Handout

# Week 12-Handout - Notes on Nyquist Plots Patricio A Vela...

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Unformatted text preview: Notes on Nyquist Plots Patricio A. Vela, [email protected] April 9, 2009 Nyquist plots are usually rough the first time around, so let’s go through the steps carefully so that you may see the process involved. That will be followed by a few extra examples. In general we have the following procedure. 1. Find the zeroes and poles of the transfer function G ( s ) and plot them on the graph showing the contour to follow. 2. Indent around any singularities in the imaginary axis. The resulting contour is the one we want to evaluate ourselves around. 3. Begin by determining the contour at ± j ∞ by using z = Re iθ in the transfer function and letting R → ∞ . We want to keep track of the phase, so don’t drop the exponent term. Determine how the interval [ π 2 ,- π 2 ] maps under G (you will get a new interval). Also figure out the phase change,i.e., Δ ∠ G ( ± j ∞ ) by subtracting the two endpoints of the resulting interval. If it is more than 2 π , then there will be encirclements according to how the interval tells you to traverse. If not, then just connect the points accordingly. Note that if they map to them same point, then this change in argument is not essential. (This interval is what closes the contour and connects ± j ∞ , plus it tells you from where the curves will leave from or head towards infinity). Of course, it does give hints as to how the curve comes in or leaves from the points, so it helps if you compute it. 4. Do the same for the point at the origin. If there is a pole at zero, then examine it the same as you did for ± j ∞ by setting z = re iθ and letting r → 0. (Here the phase is for the indentation as you go around it from below- π/ 2 around to above it π/ 2 (Note that it is the opposite direction as for the point at infinity). If not, you can simply evaluate it at zero to get a real number. Of course you will lose angle information, but you may be able to compensate for that with your intuition. 5. Now, let’s work out the indentations. The analysis is the same as for a singularity at zero, but you evaluate z = p + re iθ , where p is the location of the imaginary axis pole to analyze, and take the limit r → 0. Note that you will have two imaginary axis singularities due to complex cunjugates. The analysis will be similar for both, the substantive difference being in the phase. Don’t forget to mark them on the complex plane. 6. Lastly determine any axis crossings (especially on the real axis). This can be done by solving for the real and imaginary parts of the transfer function (not always an easy task). Mark them on the resulting plot for reference as to how the contour maps. The negative real axis crossings are critical, they will determine the stability margins....
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Week 12-Handout - Notes on Nyquist Plots Patricio A Vela...

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