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- The Routh's criterion gives a satisfactory answer to the question of stability but its adoption to determine the relative stability is not satisfactory and requires trial and error procedure even in the analysis problem.
- A simple technique, known as the root locus technique, for finding the roots of the characteristic equation, introduced by W.R.Evens, is extensively used in control engineering practice.
- This technique provides a graphical method of plotting the locus of the roots in the s-plane as a given system parameter is varied over the 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 the value of the parameter for a desired root location can be determined from the locus.
- Root locus is drawn with the help of spirule

**Advantages**

- The roots locus is a powerful technique as it brings into focus the complete dynamic response of the system and further, being a graphical technique, an approximate root locus sketch can be made quickly and the designer can easily visualize the effects of varying various system parameters on root locations.
- The root locus also provides a measure of sensitivity of roots to the variation in the parameter being considered.
- The root locus also provides a measure of sensitivity of roots to the variation in the parameter being considered.
- It may further be pointed out here that the root locus technique is applicable for single as well as multiple-loop system.
- In short it is defined as the locus of the roots of the characteristic equation as the gain parameter 'K' varies from 0 to ∞

**ANGLE & MAGNITUDE CONDITIONS****Angle condition**

- The angle condition is used for checking whether particular points are lying on root locus or not

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

G(s)H(s) = -1

G(s)H(s) = -1 +10

∠G(s)H(s) = 180^{0}

∠G(s)H(s) = +_{-} (2q + 1)180^{0}

- The angle condition may be stated as for a point to lie on root locus, the angle evaluated at that point must be an odd multiple of ±180º . Magnitude Condition
- This condition is used for finding the value of system gain K at that point on root locus.

G(s)H(s) =1

**RULES OF DRAWING THE ROOT LOCUS**

- Root locus starts from open loop poles with K= 0 (although practically it never happens as practically we have number of poles greater then number of zeros); and ends on open loop zeros with K = ∞
- Root locus is always symmetrical about real axis.
- A point on real axis lies on the root locus if number of poles + zeros to the right of the point are odd.

**Steps of Drawing the Root Locus**

Let, Number of poles = n (open loop poles)

Number of open loop zeros = m

- Number of root loci ending on infinite = n - m, n > m
- Root locus on real axis

- Here the root locus on real axis confirms above mentioned rule.
- Root locus moves always away from open loop poles and towards zero or infinity.
- Number of asymptotes = (n – m)
- Asymptotes are the paths along which root locus moves towards ∞ .
- Angle of asymptotes

r = Number of incoming branch of root locus

q = 0, 1, 2, ......, n – m – 1

(e) Centroid

(f) Determination of Breakaway or breaking point put and find out the value of 's'.

(g) Angle of departure or Angle of arrival

- Angle made by root locus with real axis when it departs from a complex open loop pole is called angle of departure.

∠GH' = angle of the function excluding the concerned poles at the poles itself

- Just calculate for one (s
_{1}, or s_{2}) and you can write for the other by putting negative sign. - Crossover at imaginary axis.
- The roots of the auxiliary equation in Routh array at K = K
_{mar}determines the intersection of rootlocus with imaginary axis. - Determination of 'K' from root-locus:

i.e for the following root locus

**Table: **Open-loop pole-zero configurations and the corresponding Root loci.

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