There are two more HWs for the course. The final Exam will emphasis Nyquist and Bode. The rest of the course will cover Nyquist and Bode. Nyquist is considered the hardest part of the course. For motivation, we are moving to frequency domain now. So far we have not worked much in frequency domain. Now it will be the main emphasis. We will develop tools in the frequency domain. Some of the tools we did was Routh table for stability and second order system analysis. We did cover some frequency domain when we did noise attenuation and using low pass filter.
Nyquist method is a view of stability in the frequency domain. Why we care about this. We already have stability tools? Such as Routh table and Root locus also does tell us something about stability. Because Nyquist tells us more. Nyquist gives new information about stability and shows new ways a system can become unstable.
We have Routh table to check for stability. But Routh table assumes the model is perfect. But what if the model is not certain? What is there is amplifier that has frequency drift? This is called model imperfection.
The main measures of stability are gain and phase margins. This tells us how tolerant the model to imperfections. Nyquist is a graphical description of stability, while Routh table is algebraic. Engineers use Nyquist since it is considered CAS based method.
We will generate contour in complex plane, look at the contour and say right away if the system is stable or not. We will generate \(\Gamma _{GH}\) contour. \(\Gamma _{GH}\) is directed and closed curve. We use \(GH\) which is the open loop transfer function to generate \(\Gamma _{GH}\,.\) Then by looking at \(\Gamma _{GH}\) we will say if the closed loop is stable or not.
Suppose we have generated \(\Gamma _{GH}\) that looks like the following (we will later learn how to generate \(\Gamma _{GH}\)).
Now we apply the Nyquist criteria: The closed loop transfer function (we assume we have unity feedback), is stable, iff the number of net clockwise encirclements of the critical point \(-1\) on the real axis is equal to the number of unstable RHP poles of the open loop transfer function (strictly in the RHP). So we need to learn how to count encirclements. For this, we draw straight line from the point \(-1\) on the real line, outwards. It does not matter which direction we draw the line. Any one will end up giving the same result. This is due to how we will do the counting of the encirclements around \(-1\). To show this, the above graph is redrawn below with 3 lines on it. (but we only need one, any one will do).
In the above we have randomly drawn 3 lines (or rays). Now we count the number of times the straight line cross the \(\Gamma _{GH}\) graph. Each time it crosses \(\Gamma _{GH}\) we count the direction of \(\Gamma _{GH}\) at that point, if it is clock wise, or anticlock wise. We add one when it is clock wise, and subtract one when it is counter clock wise. For example, for ray \(1\) in the above diagram, we see that there are 2 counter clock wise crossings, and one clock wise crossing. There for the final result is one counter clock wise crossing. This is the same as saying the result is \(-1\) clock wise crossing. So think of counter clock wise as negative numbers, and clock wise as positive numbers and simply add them. If the final number that results from the above, is the same as the number open open loop poles in the RHP, then the system is stable.
Reader: The other rays (lines 2 and 3) will give the same final result of one counter clock net encirclements. Show this.
Example, suppose we have \(GH\) which has two unstable poles in the RHP such as one with denominator \(\left ( s-1\right ) \left ( s^{2}+s+2\right ) \left ( s-6\right ) \), then when we draw \(\Gamma _{GH}\) and count the encirclements, and find that the net number of clock wise encirclements is also \(2\), then we know the closed loop is stable.
Now that we know how to interpret \(\Gamma _{GH}\) and how to use it to find if the closed loop is stable or not, we need to learn how to generate \(\Gamma _{GH}\). The plan is to become an expert in generating \(\Gamma _{GH}\) and also learn how to use it to obtain other stability information from it. To generate \(\Gamma _{GH}\), we first generate \(\Gamma \), which is called the Nyquist contour. Then we map \(\Gamma \) to \(\Gamma _{GH}\). But first we need to generate \(\Gamma \). The first step is to mark all the open loop poles on the complex plane (we do not need to mark the open loop zeros, but we can do that as well if we want). The we draw a counter clock wise curve around that incloses all these poles in the RHP. If there are any open loop poles on the imaginary axis, we draw small circle around them to bypass them. Here are two examples
Now we show an example of \(\Gamma \) for the case when the open loop transfer function \(GH\) has poles on the imaginary axis.
