Lecture 7.1 bugs
Notes and bugs from Lecture 7.1.
UPDATE (12 Nov 2007 10pm): the arrows on the online copies of the handouts have been corrected to align with the convention in the book, of CCW traversal of the "D contour". So bugs 1 and 2 have been addressed in the online version of the lecture notes.
The Nyquist D contour is defined as being traversed in the counter-clockwise direction. On Page 5, if there were arrows on the "D" countour, they would show the tracing starting at the origin and moving downard toward \(-j\infty\), traversing around toward \(+j \infty\) at \(|s|=R\) where <amsmath>R \approx \infty</amsmath>, and then traversing back to the origin. Then the Nyquist mapping (the second figure on P. 5) shows the value of \(L(s)\) in the complex plane as \(s\) takes on all of the values of the "D" contour.
Bug 1: With that convention, which is the one used in the book, the Nyquist mapping would have arrows in the opposite direction as shown. They are different because that plot was generated in MATLAB, which uses a convention that the traverse is performed in the opposite direction. So you should mentally reverse the arrows in all Nyquist plots shown in this lecture.
With this convention established, then the Nyquist theorem works with N=#counter-clockwise encirclements of -1, as stated in the notes. As stated in lecture, amnyquist() from the book's web page will draw the nyquist diagrams with the arrows in the convention used in the course and by mathematicians.
Bug 2: On page 13, N is #CCW encirclements, not #CW encirclements (as above).
Bug 3: The Nyquist diagrams on P. 10 and 13 show a unit circle referring to unity gain magnitude. But they are not drawn to scale according to the axes on the figures -- the width is correct, but they should be squished vertically (along the imaginary axis) by a factor of 2.
--Fuller 16:33, 12 November 2007 (PST)