Difference between revisions of "Why counterclockwise encirclements of -1?"

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There are really two questions here:  why counterclockwise? and why -1?  First things first: there's nothing innately special about traversing the contour counterclockwise...that's basically a sign convention.  The fact that counterclockwise traversal is the tradition is only because the contour integration that is necessary to justify the Nyquist Criterion rigorously is defined to be positive for traversal in the direction that places the interior of the domain on the left, and for this contour, that direction is counterclockwise.
 
There are really two questions here:  why counterclockwise? and why -1?  First things first: there's nothing innately special about traversing the contour counterclockwise...that's basically a sign convention.  The fact that counterclockwise traversal is the tradition is only because the contour integration that is necessary to justify the Nyquist Criterion rigorously is defined to be positive for traversal in the direction that places the interior of the domain on the left, and for this contour, that direction is counterclockwise.
  
Secondly, why -1?  That's because we want the number of RHP zeros of <math>1+L(s)</math>.  If we wanted to find the number of RHP zeros of the plain loop transfer function <math>L(s)</math>, we would count the number of encirclements of the ''origin'', but this isn't what we're interested, so we have to shift the winding center over to -1 to make the answer more relevant.
+
Secondly, why -1?  That's because we want the number of RHP zeros of <math>1+L(s)</math>.  If we wanted to find the number of RHP zeros of the plain loop transfer function <math>L(s)</math>, we would count the number of encirclements of the ''origin'', but this isn't what we're interested in, so we have to shift the winding center over to -1 to make the answer more relevant.
  
 
Constructing the Nyquist plot by hand directly from the transfer function is a huge pain, and I wouldn't recommend ever actually trying it.  If you want to do this, just use Matlab's builtin nyquist() command.  However, it is sometimes convenient to sketch a Nyquist plot by hand from a Bode plot.  To do this, you read off the gain and the phase at a given frequency on the Bode plot, and use those as the polar coordinates of the corresponding point on the Nyquist plot.  Going in the opposite direction (Bode plot from Nyquist plot) isn't terribly useful, because both plots are found from the same transfer function, and it's generally easier to plot the Bode plot from the transfer function directly.
 
Constructing the Nyquist plot by hand directly from the transfer function is a huge pain, and I wouldn't recommend ever actually trying it.  If you want to do this, just use Matlab's builtin nyquist() command.  However, it is sometimes convenient to sketch a Nyquist plot by hand from a Bode plot.  To do this, you read off the gain and the phase at a given frequency on the Bode plot, and use those as the polar coordinates of the corresponding point on the Nyquist plot.  Going in the opposite direction (Bode plot from Nyquist plot) isn't terribly useful, because both plots are found from the same transfer function, and it's generally easier to plot the Bode plot from the transfer function directly.

Revision as of 01:53, 13 November 2007

There are really two questions here: why counterclockwise? and why -1? First things first: there's nothing innately special about traversing the contour counterclockwise...that's basically a sign convention. The fact that counterclockwise traversal is the tradition is only because the contour integration that is necessary to justify the Nyquist Criterion rigorously is defined to be positive for traversal in the direction that places the interior of the domain on the left, and for this contour, that direction is counterclockwise.

Secondly, why -1? That's because we want the number of RHP zeros of 1+L(s). If we wanted to find the number of RHP zeros of the plain loop transfer function L(s), we would count the number of encirclements of the origin, but this isn't what we're interested in, so we have to shift the winding center over to -1 to make the answer more relevant.

Constructing the Nyquist plot by hand directly from the transfer function is a huge pain, and I wouldn't recommend ever actually trying it. If you want to do this, just use Matlab's builtin nyquist() command. However, it is sometimes convenient to sketch a Nyquist plot by hand from a Bode plot. To do this, you read off the gain and the phase at a given frequency on the Bode plot, and use those as the polar coordinates of the corresponding point on the Nyquist plot. Going in the opposite direction (Bode plot from Nyquist plot) isn't terribly useful, because both plots are found from the same transfer function, and it's generally easier to plot the Bode plot from the transfer function directly.

--George Hines 17:43, 12 November 2007 (PST)