# Introduction to Root Locus

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.

$\frac{C(s)}{R(s)}=\frac{G(s)}{1+G(s)H(s)}$

ch. equation is $1+G(s)H(s)=0$

let $G(s)H(s)=D(s)$

$1+D(s)=0$

$D(s)=-1$

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

They have to satisfy ‘2’ criteria known as

1. Magnitude Criterion.
2. Angle Criterion.

Magnitude criterion:-

$\left&space;|&space;D(s)&space;\right&space;|=1$

$\left&space;|&space;G(s)H(s)&space;\right&space;|=1$

the magnitude criterion states that  $s=s_{a}$  will be a point on root locus, if for that value of s

i.e, $\left&space;|&space;G(s)H(s)&space;\right&space;|=1$

Angle criterion:-

$\angle&space;D(s)&space;=&space;\angle&space;G(s)H(s)=\pm&space;180^{o}(2q+1)$

where q=0,1,2……….

if $\angle&space;D(s)&space;=&space;\pm&space;180^{o}(2q+1)$ is odd multiple of $180^{o}$, a point s on the root locus, if $\angle&space;D(s)$ is odd multiple of at $s=s_{a}$ of $180^{o}$, 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 $0$ to $\infty$ is known as Root locus.

It is a graphical method

$-\infty$ to $0$       $\rightarrow$     Inverse Root Locus

$0$  to $\infty$    $\rightarrow$  Direct Root Locus

generally Root Locus means Direct Root Locus.

$D(s)&space;=&space;G(s)H(s)&space;=&space;k&space;\frac{(s+z_{1})(s+z_{2})(s+z_{3})....}{(s+p_{1})(s+p_{2})(s+p_{3})....}$

$\left&space;|&space;D(s)&space;\right&space;|&space;=&space;k&space;\frac{\left&space;|&space;s+z_{1}&space;\right&space;|\left&space;|&space;s+z_{2}&space;\right&space;|\left&space;|&space;s+z_{3}&space;\right&space;|....}{\left&space;|&space;s+p_{1}&space;\right&space;|\left&space;|&space;s+p_{2}&space;\right&space;|\left&space;|&space;s+p_{3}&space;\right&space;|....}$

$\left&space;|&space;D(s)&space;\right&space;|&space;=&space;k&space;\frac{\prod_{i=1}^{m}\left&space;|&space;s+z_{i}&space;\right&space;|}{\prod_{i=1}^{n}\left&space;|&space;s+p_{i}&space;\right&space;|}$

m= no .of zeros

n= no.of poles

from magnitude criterion $\left&space;|&space;D(s)&space;\right&space;|&space;=&space;1$

$k&space;=\frac{\prod_{i=1}^{n}\left&space;|&space;s+p_{i}&space;\right&space;|}{\prod_{i=1}^{m}\left&space;|&space;s+z_{i}&space;\right&space;|}$

The open loop gain k corresponding to a point $s=s_{a}$ on Root Locus can be calculated

$k=$ product of length of vectors from open loop poles to the point $s=s_{a}$/product of length of vectors from open loop zeros to the point $s=s_{a}$.

from the Angle criterion,

$\angle&space;D(s)&space;=&space;\angle&space;(s+z_{1})+\angle&space;(s+z_{2})+\angle&space;(s+z_{3}).....&space;-\angle&space;(s+p_{1})+\angle&space;(s+p_{2})+\angle&space;(s+p_{3}).....$

$\angle&space;D(s)&space;=&space;\sum_{i=1}^{m}\angle&space;(s+z_{i})&space;-\sum_{i=1}^{n}\angle&space;(s+p_{i})$

$\sum_{i=1}^{m}\angle&space;(s+z_{i})&space;-\sum_{i=1}^{n}\angle&space;(s+p_{i})=\pm&space;180^{o}(2q+1)$

i.e,( sum of angles of vectors from Open Loop zeros to point $s=s_{a}$)-(sum of angles of vectors from Open Loop poles to point$s=s_{a}$$=\pm&space;180^{o}(2q+1)$

where q=0,1,2………

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