# Why isn't Nyquist taught like this?

When I learnt how to do a Nyquist plot I was taught a really long-winded method that I don't understand to this day. I realised by myself that if you are given a system like below where $$\s=j\omega\$$ $$\frac{s+1}{s+10}$$ you can just replace $$\s\$$ with $$\j\omega\$$ and put in several values of $$\\omega\$$ and plot these outputs on the real-imaginary axis.

Is there a problem doing it this way because I cannot understand why it would not be taught like this if it is this easy?

• I was once challenged by an old-timer to not just accept how Nyquist can be used to come up with answers, but why. A lot of approaches are presented assuming that one day you may be asked to prove it. – user65586 Nov 2 '18 at 18:18

1. There is no problem with the method you propose. It's the one I've used for years designing real systems.
2. You were probably taught a graphical method (I'm curious as to what it is -- do you have a link?). The graphical methods were invented before digital computers, or even calculators, were ubiquitous, and were designed to make it easy for an engineer with pencil, paper, a ruler, and a slide rule to make the graphs needed to get the job done.
• Thanks for your response! This is the best thing I could find to answer you question about what method I was taught (the image I added in the edit). This is a lecture slide that explains how to choose C1 which is a contour that encircles the right-hand half of the plane. Is the graphical method still taught in most universities or are students expected just to use Matlab or something similar? – CoderEH Nov 2 '18 at 15:16
• So, the reason that they're obsessing on a contour is because the mathematical basis of the Nyquist plot is the Method of the Argument, which tells you how many more poles than zeros are within an area on the complex plane by counting how many times the graph revolves around zero. You actually need to do that to be absolutely 100% correct -- but you also need to know exactly how many unstable zeros you have in the system. I find it easier in practice to start with a known-stable system and investigate what changes I can make that will improve it's performance while keeping it stable. – TimWescott Nov 2 '18 at 15:28

As far as I know, the usual way to introduce the Nyquist plots are the one you just described. This is how it was defined in my university studies and this is what I have seen in multiple books. The slide what you attached is not an introduction or definition for Nyquist diagrams, but for some advanced methods using it.