**Root locus**

Root** **locus is
defined as the locus of closed loop poles when system gain is varied from to
zero to infinity. It determine the relative stability of system.

**Steps to
design a root locus**

**Step 1:-** The root
locus is symmetrical about real axis

**Step 2:-** Let

P =
number of open loop poles

Z = number of open loop zeroes

And P > Z then

No. of branches of root locus = P

Number of branches terminating at zeroes is equal to Z

Number branches terminating at infinity P - Z

**Step 3:-** A point on
real axis is said to be on root locus if to right side of this point the sum of
open loop poles and zeroes is odd. Root locus is outward from poles and inwards
to zeroes

**Step 4:- Angle of asymptotes**

The p-z branches terminating at infinity will go along
certain straight lines known as asymptotes of root locus.

The no of asymptotes = P – Z

Angle of asymptotes ^{ɵ}^{ =}

**Step 5:-** **Centroid**

It is the intersection of asymptotes on the real axis. It may
or may not be part of root locus.

^{Centroid =}

**Step 6:-** **Breakaway
points **_{}

They are those points where multiple roots of the
characteristics equation occur.

1 + G(s).H(s) = 0

Write in terms of K

dk/ds = 0 will
give breakaway point

Whenever there are two adjacently placed poles on the real axis with the section of real axis between them as a part of root locus then there exist a breakaway point between the adjacently placed poles.

Whenever
there are two adjacently placed zeroes on the real axis with the section of
real axis between them as a part of root locus then there exist a breakinn
point between the adjacently placed zeroes.

**Step 7:-**

**Intersection of root locus with imaginary axis**

**Step 8:-**

**Angle of departure and angle of arrival**

Ø The angle of arrival is obtained at complex zeroes

^{0}+ ɸ

^{0}- ɸ

_{Z}- Ʃɸ

_{P}= ɸ

**Angle and magnitude condition.**

^{0}– tan

^{-1}(0/1) ≈ 180

^{0}

^{0}

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