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