 I am Mr. Sarvatna Gandhi, working as assistant professor in Department of Mechanical Engineering from Valchand Institute of Technology, Solarpoor. Today in this session, we are going to study regarding rules to construct root locus. At the end of the session, students will be able to understand the rules to construct root locus and analyze the nature of root locus. Rules to construct root locus Rule 1 The root locus is always symmetric about real axis. The roots of the characteristic equation are either real or complex or combination of both. Therefore, their locus must be symmetric about the real axis of S plane. The roots which are real can be plus or minus 2. The roots which are complex can be minus 1 plus 2j and the roots which are complex conjugate can be minus 1 plus or minus 1j. Rule 2 So from this, we can know the number of branches which are going to approach infinity. From the given open loop transfer function that is G of S into H of S, we can find the number of zeros and number of poles. So whatever is present at the numerator that represents zeros and whatever is present at the denominator represents poles and we need to count that number. So Z represents number of open loop zeros whereas P represents number of open loop poles. Generally, the value of P is greater than Z. So from this, we are going to calculate the value of N that is P minus Z that is going to represent number of branches which are going to approach infinity. Whatever may be the case, the branch direction always remain from open loop poles where the value of K is zero towards open loop zeros where the K value tends to infinity. So if we try to draw the sketch of S plane which has real and imaginary part, so we can see that there are two roots out of which at minus one we have pole and at minus two we have zero and those are represented as cross and circles respectively. So now we can see that root locus moves from open loop pole to open loop zero where the value of K varies from K is equal to zero to K equal to infinity. So if we try to see the example of this pertaining to rule number two, so we can see in first example of open loop transfer function as K divided by S into S plus one into S plus two. Now in this case the simplified form of the polynomial equation is given to us. So from that we can see number of zeros present as zero whereas the number of poles present are three. So if we try to calculate the number of branches which are going to approach infinity, so that will be three minus zero. So that implies three branches are going to approach infinity. So if we take the second case where we have open loop transfer function as K into S plus two into S plus five divided by S into S plus one into S plus three into S plus four. So if we count the number of zeros, so it gives us two. If we count the number of poles, so we have four. So n represent number of branches which are approaching infinity, so that will be four minus two. So it implies that two branches are going to approach infinity. So rule number three, a point on a real axis lies on root locus if the sum of number of open loop poles and open loop zeros on the real axis to the right hand side of this point is odd. So whatever, so if we take a section in between the roots and if we try to see towards the right hand side of that section, if we are getting the count as odd, so we can say that in between those two roots, the root locus is going to exist. So for example, open loop transfer function is given to us as K into S plus one into S plus four divided by S plus three into S plus five. So if we try to draw the S plane for this, so we are going to have zeros which are at minus one and minus four which are represented by circle, whereas we are going to have the poles at minus three and minus five which are represented by cross. So we are going to see now where does root locus exist on this real axis. So if we try to take a section in between minus one and minus three at which the roots are existing and if we see the right hand side of that section, so we can find the count to be one which is odd. So we can say that root locus exist between minus one and minus three. Then we have the roots which are existing between minus three and minus four. So if we try to take a section in between that and see how many roots are there on its right hand side, the count goes to two. So it is even. So root locus will not exist between minus three and minus four. Next we have the roots at minus four and minus five. If we count the number of roots which are on its right hand side, we get the count to be three which is odd. So root locus is going to exist between minus four and minus five. Lastly, we have the root at minus five and if we take a section beyond minus five and observe how many roots are there on its right hand side. So we get the count to be four which is even. So root locus will not exist beyond point minus five. So the root locus is existing between minus one and minus three and minus four and minus five. So in these rule number three, we have one note while counting the number of roots which are on the right hand side of the section which is drawn by you. You are not supposed to count the complex conjugate roots. So for example, if the open loop transfer function is given to us as k into s plus two divided by s square into s square plus two s plus two into s plus three. So if you try to write the simplified version of the polynomial equation that is given to you in the denominator, then we can write it as k into s plus two divided by s square into s plus one plus j and s plus one minus j into s plus three. So if you try to count the number of zeros which are present, so it comes to be one and it is going to lie at minus two. Whereas the number of poles which are given in the open loop transfer function counts to five and the roots are going to lie at zero zero minus one minus j minus one plus j and minus three correct. So we are going to draw it on s plane fine. So we can see that at origin we have two roots whereas at minus two we have one and we have two roots which are complex conjugate at minus one plus or minus j. So if we try to take first section in between zero and minus one, so we have to count how many roots are there on the right hand side. So the count goes to be four by considering the complex conjugates. So if we don't consider that the count goes to be two. So two and four both are even. So there is no change in the nature by adding the complex conjugate correct. So root locus is not going to exist between zero and minus two. So if we take a section in between minus two and minus three without considering the complex conjugate the count goes to three which is odd and if we consider the complex conjugate the count goes to three plus two that is five that is also odd. So we are going to have the root locus in between minus two and minus three. So by counting the complex conjugate there is going to be no difference in the nature. It is going to remain even if the condition is even it is going to remain odd if the condition is odd. So if we try to take a section in between minus three and minus four the number of roots which are on the right hand side without considering the complex conjugate roots the count goes to four which is even. So root locus will not exist between minus three and minus four. In rule number four we are going to find the angle of asymptotes. So angle of asymptotes gives us the information that how many branches are approaching infinity. Generally the number of poles are more than the number of zeros. So P minus Z branches are going to approach infinity. But how are these branches going to approach infinity? So for those branches which are approaching infinity asymptotes will act as a guideline for them. So which is going to travel up to infinity. Angle of asymptote is represented as theta which is equal to two Q plus one into one eighty divided by P minus Z. So in this you need to vary the value of Q and for that the equation is P minus Z minus one. The value of Q ranges from zero one to up to infinity. Rule number five. So in this rule we are going to find the centroid. So that is the location of asymptotes on S plane. All the asymptotes intersect the real axis at a common point called as asymptotes. And that is represented by sigma. So the equation comes out to be summation of real part of pole of G of S H of S minus summation of real part of zeros of G of S into H of S divided by P minus Z. So in this you need to note centroid is always real. It may be located on negative real axis or it can be located on positive real axis. It may or may not be a part of root locus. So we will see one example for which we are going to apply all the five rules that we have studied. So open loop transfer function given to us is K divided by S plus one into S plus two plus two J S plus two minus two J. In step number one the number of zeros are zero. The number of poles are three which are at minus one minus two plus two J minus two minus two J and N number of branches which are approaching infinity are three minus zero. So that implies three branches are going to approach infinity. Then angle of asymptotes that is two Q plus one into 180 divided by P minus Z. Since the value of P minus Z is three we are going to select three values for Q that are zero one and two respectively. So we are going to get the angle of asymptotes as sixty one eighty and three hundred. In step number three we are going to find the centroid that is summation of real part of poles minus summation of real part of zeros divided by P minus Z. So it comes out to be minus five by three that is minus one point six seven. We can see the root locus exists between minus one up to infinity. The centroid is located as minus one point six seven and from that the asymptotes angle which are sixty one eighty and three hundred are going to pass. And we are going to measure the angle in anticlockwise positively. These are my references. Thank you.