What is Root locus analysis?

Root locus analysis is described as a graphical technique to determine the change in roots of a system with variations in different system parameters. In this technique, an open-loop transfer function is used to determine the stability of the closed-loop control systems.

The root locus diagram in the following section identifies the path of the closed-loop poles.

Now, let's look at some rules and procedures for plotting root locus.

Rules and Procedures for plotting root locus

Following are the general rules for plotting the root locus:

Rule 1: Determine all the open-loop poles and zeroes of the open-loop transfer function and plot them in an s-plane.

The representation of poles and zeroes in the root locus plot is shown as:

Here, poles are represented by the symbol (X), and zeroes are represented by the symbol (O).

Rule 2: Starting and ending point.

The root locus starts from the open-loop poles at $K=0$ and ends on open-loop zeroes at $K=\infty$

Rule 3: Number of branches of root locus.

The number of branches of root locus N equals either the number of finite open-loop poles P or the number of finite open-loop zeroes Z, whichever one is greater.

If $P>Z$ , then the number of branches of root locus equals the number of poles, and if  $Z>P$, then the number of branches of root locus equals the number of zeroes.

Example: If the number of poles in the system is 2 and the number of zeroes in the system is 5, determine the number of branches.

Solution: In the case given, $Z>P$ , the number of branches will equal the number of zeroes, i.e., $N=Z$

 since the root locus starts from the pole and terminates at zero. 3 branches will start from infinity, and 2 branches will start from open-loop poles and all 5 branches will terminate at 5 finite locations of zeroes.

Rule 4: Determination of the direction of root loci on the real axis.

A branch of root locus lies on the real axis if the total number of open-loop poles and zeroes to the right side of the point is odd.

Example: The root locus plot of a control system is shown below containing two poles and one zero. Determine the direction of pole.

$S=-2$

Solution: Take two points $P_1$ and $P_2$on the right and left sides of the pole, respectively. Now, count the number of poles and zeroes to the right of the point.

For point $P_1$, there is one zero and one pole to the right side, which means it has an even number of poles and zeroes.

For point $P_2$, there are two poles and one zero to the right side, which means it has an odd number of poles and zeroes.

Now, it is identified that the direction of the pole in the real axis is towards point or left or negative direction.

Rule 5: Breakaway point.

When two branches move toward each other on the real axis, the coincident point is termed as the breakaway point. It occurs among two adjacent open-loop poles on the real axis.

The diagrammatical representation for breakaway point in the root locus plot is shown as:

The breakaway point is calculated by differentiating the open-loop transfer function 

$G\left(s\right)$ with respect to s and then equating it to zero.

The formula for breakaway point is given as, 

$\frac{dG\left(s\right)}{ds}=0$

By simplifying the above equation, we get an appropriate value of s which is called as breakaway point.

Rule 6: On increasing the value of K, the root loci move away from the poles and zeroes.

For higher values of K, the root locus branches are approximated by Asymptotes.

Step (1): Determination of intersection point of asymptotes with the real axis.

The number of Asymptotes will be equal to the number of poles of the system.

The formula for the intersection point of asymptotes with the real axis is given as,

$x=\frac{\Sigma P-\Sigma Z}{P-Z}$

Here, $\Sigma P$ is the summation of all poles of the system or open-loop transfer function, 

$\Sigma Z$  is the summation of all zeroes of the system, P is the total number of poles, and Z is the total number of zeroes.

Step (2): Determination of angle of asymptotes.

Asymptotes are drawn at point x, making an angle with the real axis.

The formula for the angle of asymptotes is given as, 

$\theta =\frac{\left(2m+1\right)}{P-Z}\times 180°$

Here,  $m=1,2,3,..........\left(P-Z\right)-1$, P is the number of poles, and Z is the number of zeroes of the system.

Rule 7: To determine the points where the root loci branches intersect the imaginary axis.

The root locus is basically a locus of points or roots of the characteristic equation.

Following are the two ways or methods to identify intersection points:

Method 1: Put 

$s=j\omega$  in the characteristic equation and make real part and imaginary part equal to the zero and simplify the obtained equations for $\omega$  and K.

Method 2: By using Routh's stability criterion.

Rule 8: To determine the angles of departure and arrival in the case of complex poles.

The formula for the departure angle from complex pole is given as,

${\varphi }_{departure}=180°-\left({\varphi }_p-{\varphi }_z\right)$

Here,  ${\varphi }_p$  is the sum of all angles subtended by all other poles 

$\left({\varphi }_{p1}+{\varphi }_{p2}+{\varphi }_{p3}+..........{\varphi }_{pn}\right)$

and

${\varphi }_z$  is the sum of all angles subtended by all other zeroes 

$\left({\varphi }_{z1}+{\varphi }_{z2}+{\varphi }_{z3}+........{\varphi }_{zn}\right)$

The formula for the angle of arrival from complex zero is given as,

${\varphi }_{arrival}=-180°-\left({\varphi }_p-{\varphi }_z\right)$

Rule 9: Determining the value of K on root locus

The value of K is the ratio of the product of the length of vectors drawn from the poles of the open-loop transfer function to point P, to the product of lengths of vectors drawn from the zeroes of the open-loop transfer function to point P.

The formula for the value of K is given as,

$K=\frac{Product of length of vectors drawn from the poles of OLTF to point P}{Product of length of vectors drawn from the zeroes of OLTF to point P}$

Here, K is the gain of the system and P is the point laying on the root locus.

Following are the various possible conditions for K:

• When $K=0$, there will be no roots on the imaginary axis. 
• If $K<0$, the closed loop system is unstable.
• If $K>0$, the closed loop system is stable.

Common Mistakes

Following are the common mistakes performed by students:

• Students may get confused between the representation of poles and zeroes in the root locus plot of control systems.
• They may also assume that the number of branches of root locus is always equal to the number of zeroes.
• Sometimes, students get confused that the breakaway point is calculated by differentiating either an open or closed-loop transfer function.
• Students may also forget the concept for different cases of K values in closed-loop control systems.

Context and Applications

The topic of procedure for plotting root locus is significant in various courses and professional exams at the undergraduate, graduate, postgraduate, and doctorate levels. For example:

• Bachelor of technology in electrical engineering
• Bachelor of technology in electronics engineering
• Bachelor of technology in electrical and electronics engineering
• Master of technology in control systems
• Doctor of philosophy in electronics

Related Concepts

• Introduction to control systems
• Control systems - Stability
• Time response analysis
• Control systems - Characteristic equation

Practice Problems

Q1 Which of the following identifies the path of the closed-loop poles?

1. Root locus plot or diagram
2. Value of K on root locus
3. Asymptotes
4. None of these

Correct option: (a)

Explanation: The root locus diagram identifies the path of the closed-loop poles.

Q2. A branch of root locus lies on the real axis if the total number of open-loop poles and zeroes to the right side of the point is ____.

1. Zero
2. Even
3. Odd
4. One

Correct option: (c)

Explanation: A branch of root locus lies on the real axis if the total number of open-loop poles and zeroes to the right side of the point is odd.

Q3. The breakaway point is calculated by differentiating the _____ and then equating it to zero.

1. Closed-loop transfer function
2. Open-loop transfer function
3. Both open and closed-loop transfer functions
4. None of these

Correct option: (b)

Explanation: The breakaway point is calculated by differentiating the open-loop transfer function 

$G\left(s\right)$  with respect to s and then equating it to zero.

Q4. Which of the following is a correct formula for the departure angle from the complex pole?

1. ${\varphi }_{departure}=-180°-\left({\varphi }_p-{\varphi }_z\right)$
2. ${\varphi }_{departure}=-180°+\left({\varphi }_p-{\varphi }_z\right)$
3. ${\varphi }_{departure}=180°+\left({\varphi }_p-{\varphi }_z\right)$
4. ${\varphi }_{departure}=180°-\left({\varphi }_p-{\varphi }_z\right)$

Correct option: (d)

Explanation: The formula for the departure angle from complex pole is 

${\varphi }_{departure}=180°-\left({\varphi }_p-{\varphi }_z\right)$

Q5. Which of the following conditions represents the unstable closed-loop system?

1. $K=0$
2. $K>0$
3. $K<0$
4. None of these

Correct option: (c)

Explanation: When $K=0$ , there will be no roots on the imaginary axis. Similarly, if $K<0$ , the closed-loop system is unstable and if $K>0$ , the closed-loop system is stable.