# Lec-37 Mapping of Control in the Complex-Plane

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### Lec-37 Mapping of Control in the Complex-Plane

We have been looking at the root locus method for getting some idea of the transient response of the feedback control system or even open loop control system and we saw that some of the ideas associated with the root locus can be used to determine the value of the gain beyond which the system may become unstable Now we look at another approach to the same problem which is based on what is called frequency response Historically, the root locus method was invented in the 1950’s whereas the frequency response approach to the feedback control system analysis and design was developed in the 1930’s and 40’s. As I told you earlier, people who have working with telephone network, first came across this problem of stabilizing of amplifier and Harold Black invented the idea of or rather applied the idea of feedback to stabilize amplifiers and because of this people who were studying amplifiers till that time started looking at feedback in amplifiers and this is how the frequency response method was developed. In the case of an amplifier, whether it is useful speech which was the main application transmission of voice signal over long distances, over a large telephone network. This was the first application, today of course one can talk about other communication networks wired or wireless Now in these applications, frequency responses are very important concept a signal being sinusoidal is a very meaningful signal for example, a pure tone if one looks at it on the oscilloscope looks like almost a perfect sinusoidal signal and parts of speech also look like periodic waveforms and therefore, we know from Fourier series approach that it can be a thought of has consisting of number of sinusoidal signals because of this, it is quite natural to think of frequency response when studying amplifier performance and so, the frequency response techniques were developed in the 1930’s and 40’s for not only studying the performance of amplifiers but also studying the effect of feedback on the behavior of the performance of amplifiers So we will take a look at it now, although it seems that the root locus method which was invented later does a better job for this purpose than the frequency response base analysis for method. Now, we will start with the very simple result which is as follows you have all studied functions of a complex variable in your mathematics courses. So, suppose z is a complex variable that is z stands for any complex number, what so ever then you might have a function of z which is also a complex number that is its complex value of function of a complex variable and typically in the maths courses, the symbol z and q are use for the 2 complex numbers. So w equal to f z represents what is called a mapping

or a transformation of the complex plane into the complex plane and in order to show this diagrammatically, one shows the 2 numbers z and w, start with the complex number z apply the function f to it, you get the value f of z that is another complex number w So these 2 numbers z and w are shown into 2 different planes that is you think of instead of 1 single complex plane, you think of 2 different complex planes. So one is call the z plane, other is call the w plane. So let us say here is the z plane, in other words in this plane or in this part of the diagram, I am going to represent the complex number z and here is the other plane. Let us say, we call it the w plane that is in this part of the diagram, I am going to represent the value of the function at a complex number z So for example if I choose a complex number z, here it may be some complex number like, let us say 2 plus j 3 g, right. It is a complex number in the first quadrant of course, I am not put the real and imaginary axes heading here. So if I do that this is the positive real axis, this is the positive imaginary axis for the z plane likewise, I will have the real and imaginary axis that is real and imaginary parts of w, in the w plane. So let us say there is some function f such that its value it 2 plus j 3 is let us say minus 1 plus j 3. So where is the point minus 1 plus j 3 all that will be a point 1 plus j 3 all that will be a point somewhere here, here is the point minus 1 plus j 3, the real part is minus1 the imaginary part is plus 3 and so, one says that this point 2 plus j 3 is mapped into the point minus 1 plus j 3 under the function or mapping f and this function or mapping f, we will do different things to different points in the complex plane, different points in the complex plane will be mapped in the complex plane, z plane will be mapped into or taken into different points in the w plane and one uses this terminology of mapping z is transformed into w under the action of f or f maps a point in the z plane into point in the w plane So this is the way in which one looks at such functions and what they do to complex numbers Now, let us consider a very simple complex function namely w, which is f z is simply given by let us say Z minus 1. So, if I start with the complex number z then what this function f does it, it simply subtracts from it the number 1 and I get a new complex number and that is the value of the function, it is a function as simple as that Now, let us look at the following. Here is the z plane, in the z plane I draw simple closed curve and a very special one, I will draw what is called the unit circle. So suppose this represents the unit circle in the z plane, what does this mean to say that this is the unit circle in the z plane, it means that it is a set of all points z such that modular sub z is equal to 1, it is in way is equation of the circle unit circle in the z plane, mod z equal to 1. So, here is a point which is on the unit circle so here is the complex number z and this is the corresponding vector drawn from the origin to the complex number z or this vector can also represent the complex number z Then, the modulus of this z which is the length of this vector must be 1 and so all the points which lie at distance one from the origin will satisfy this condition and conversely, any points satisfying this condition must lie at a distance of 1 from the origin. So this is a curve in the complex z plane, it is a simple closed curve and you should go back to your complex variables or complex functions course and find out, what is the simple closed curve. Now one goes a little further, one thinks of tracing the curve or moving along the curve in a particular direction For example, I will put the arrow here this way which means now, I am going to trace the curve or move along the curve in the counter clockwise direction Now, such a curve with an arrow on it which indicates the direction in which you want to trace the curve is called a contour. So, I have chosen a contour in the complex plane and it is conventional to use the Greek letter capital gamma to denote any such contour So in this particular case, I have a closed curve gamma in the z plane which is traveled along or travels in the counter clockwise direction. I have this curve but there is a direction given a or marked on it which is the direction in which I am going to travel