 I am Mr. Sarvajna Gandhi working as assistant professor in Department of Mechanical Engineering from Walshan Institute of Technology, Soolapur. So today we are going to have a discussion on topic named as Root Locusts. At the end of the sessions students will be able to understand the concept of Root Locusts and rules to construct it. So introduction to Root Locusts. Root Locusts is a graphical method introduced by W.R. Evans in 1948. It indicates movement of poles in S plane by changing specific system parameters, those are called as gains from 0 to infinity. In designing a linear control system, Root Locusts method proves to be quite useful. Root Locusts methods specify Root Locusts plot clearly showing the contribution of each open loop pole and 0 to the location of closed loop poles. So if we change the value of k that is called as gain from 0 to plus infinity, it is called as direct root locusts that are generally used. And if we vary the value of k that is called as gain from minus infinity to 0, it is called as inverse root locusts. Thus we can say that Root Locusts of a closed loop pole obtain when a system gain that is k varied from minus infinity to plus infinity is called as Root Locusts. So transfer function and characteristic equations. So as you can see in figure one which represents block diagram for closed loop control system, we can see the block k and G dash of S are transfer functions in forward path whereas H of S is a block with transfer function in backward direction or we can call it as feedback. And this closed loop system is negative feedback system. So if we try to write the closed loop transfer function for this block, so we can write it as k into G dash of S divided by 1 plus k into G dash of S into H of S. So we can substitute the value of k into G dash of S as G of S. So the equation comes to be G of S divided by 1 plus G of S into H of S. Then if we want to write the characteristic equation, so we can write it as 1 plus G of S into H of S. So if you observe the closed loop transfer function, so whatever is present at the denominator that represents characteristic equation of closed loop control system. So if we take the roots of this characteristic equation then those roots are called as closed loop poles. And if we want to take the open loop transfer function for this block, we can write it as G of S into H of S. And if we take the roots of this G of S into H of S, then those are called as open loop poles or open loop zeros. So generalized representation of open loop transfer function can be written as G of S into H of S is equal to k into S plus Z0 S plus Z1 dot dot dot up to S plus Zn divided by S plus P0 into S plus P1 dot dot dot up to S plus Pn. So we can have this as numerator upon denominator. So whatever is present at the numerator those are called as roots and whatever is present at the denominator are called as poles. So at the numerator you can have a polynomial equation or you can have the simplified version of it. The same case applicable for denominator also you can have a polynomial equation or you can have the simplified version of it. So if you see this open loop transfer function carefully, so you can see at the top that represents numerator. So the roots lie at S is equal to minus Z, S is equal to minus 1, S is equal to minus 2 up to minus Zm. So if we see the denominator part, so we can find that there are poles existing there and the roots are at S is equal to minus P0 minus P1 up to minus Pn. So system stability, so whenever a control system is designed, the inherent capability should be stable. The system should be stable. So for that stability is the desired property of any design control system. A system is said to be stable if the system eventually comes back to the equilibrium state when the system is subjected to initial conditions. A system is unstable if the output deviates without bound from its equilibrium state then we say that the system to be unstable. And a system is said to be marginally stable if the system tends to oscillate in its equilibrium state when subjected to initial conditions. So these are some concepts which tell us regarding the system stability whether it is stable, unstable or marginally stable. But in the control system we are going to define a system to be stable or unstable depending upon the roots where they lie. So it is advantageous for us to know the movement of closed loop poles in the S plane. So which is found by using closed loop characteristic equation. So in this slide we are going to study regarding movable and immoveable roots. The roots which are obtained from open loop transfer function. So those represent open loop poles or open loop zeros and these roots are called as immoveable roots whereas the roots which are obtained from closed loop characteristic equation. So those roots are called as movable roots and which are obtained by varying the value of k correct. So the zeros whatever we are going to get are represented by a circle symbol whereas the pole that we get are represented by cross symbol in S plane. So what does S plane means? So in this we are going to have x and y axis on x axis we are going to represent the real part and on y axis we are going to represent the imaginary part. So if you consider the example of open loop transfer function which is given to us G of S into H of S we have at numerator k into S plus 2 divided by S plus 3. So whatever is present at numerator that represent zeros and whatever is present at the denominator those represent poles. So if we try to plot this on the S plane we get roots at minus 2 and minus 3. So as you can see on the figure 2 which represents S plane so it is there on the left hand side which is shown in green color. So the roots which are lying on the left hand side of the S plane represents that the system is stable and if the roots are lying on the right hand side of the S plane then we say that the system is unstable and if the roots are lying on the imaginary axis. So we have two cases we can say that the system is marginally stable or system is unstable. So it depends upon the roots fine. So now we are going to discuss regarding angle and magnitude condition for a given closed loop characteristic equation that is represented as 1 plus G of S into H of S equal to 0. So we are going to rearrange the terms so we can write it as G of S into H of S is equal to minus 1 plus J into 0. So we are writing it in complex form because we are going to represent this in S plane. So for any point to be on S plane so we need to see whether it lies on root locus. So for that it needs to satisfy the angle and magnitude condition. So first condition is called as angle condition where we are going to have angle signs on both the sides that is on right hand side on left hand side. So if we try to equate that then whatever the angle we are going to get should be odd multiple of plus or minus 180. So it can be plus or minus 180 plus or minus 540 or it can be plus or minus 900. So it can be equated to plus or minus 2Q plus 1 into 180. So if this angle condition is satisfied then we can say that the point lies on root locus. If the angle condition is satisfied then we go for checking of the magnitude condition. So we are going to put mod on both the sides of the equations and we are going to find what is the value of K marginal from the magnitude condition. So while checking the magnitude and angle condition we have to note that magnitude condition can only be used when a point lies in S plane is confirmed by the existence of angle condition. So one question for you, what will happen to root and root locus by changing the value of K? These are my references. Thank you.