The introduction of a feedback to a system causes some instability , therefore an unstable system can not perform the control task requires of it.

while in the analysis of a given system, the very first investigation that needs to be made is whether the system is stable or not?

However, the determination of stability of a system is necessary but not sufficient.

A stable system with low damping is also unwanted.

a design problem in which the designer is required to achieve the desired performance for a system by adjusting the location of its close loop poles in the S-plane by varying one (or) more system parameters.

The Routh’s criterion obviously does not help much in such problems.

for determining the location of closed-loop poles one may resort to the classical techniques of factoring the characteristic equation and determining it’s roots.

when the degree is higher (or) repeated calculations are required as a system parameter is varied for adjustments.

a simple technique, known as the root locus technique, for finding the roots of the ch.eqn introduced by W.R.Evans.

This technique provides a graphical method of plotting the locus of the roots in the S-plane as a given system parameter is varied from complete range of values (may be from zero to infinity).

The roots corresponding to a particular value of the system parameter can then be located on the locus (or) value of the parameter for a desired root location can be determined from the locus.

Root Locus:-

- In the analysis and design for stable systems and gives information about transient response of control systems.
- It gives information about absolute stability and relative stability of a system.
- It clearly shows the ranges of stability and instability.
- used for higher order differential equations.
- value of k for a particular root location can be determined.
- and the roots for a particular k can be determined using Root Locus.

ch. equation is

let

To find the whether the roots are on the Root locus (or) not

They have to satisfy ‘2’ criteria known as

- Magnitude Criterion.
- Angle Criterion.

Magnitude criterion:-

the magnitude criterion states that will be a point on root locus, if for that value of s

i.e,

Angle criterion:-

where q=0,1,2……….

if is odd multiple of , a point s on the root locus, if is odd multiple of at of , then that point is on the root locus.

Root Locus definition:-

The locus of roots of the Ch. eqn in the S-plane by the variation of system parameters (generally gain k) from to is known as Root locus.

It is a graphical method

to Inverse Root Locus

to Direct Root Locus

generally Root Locus means Direct Root Locus.

m= no .of zeros

n= no.of poles

from magnitude criterion

The open loop gain k corresponding to a point on Root Locus can be calculated

product of length of vectors from open loop poles to the point /product of length of vectors from open loop zeros to the point .

from the Angle criterion,

i.e,( sum of angles of vectors from Open Loop zeros to point )-(sum of angles of vectors from Open Loop poles to point)

where q=0,1,2………