Why counterclockwise encirclements of -1?
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 . If we wanted to find the number of RHP zeros of the plain loop transfer function (for instance), we would count the number of encirclements of the origin, or if we wanted the number of RHP zeros of , we would count the number of encirclements of -2, but these aren't what we're interested in.
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)