**Root Locus introduction**

Root Locus is a frequency domain technique used in investigating the roots of characteristic equation when a certain parameter varies. In general it can be applied to any algebraic equation of the form

**F(x) =P(x) +K*Q(x) =0 **

with P(x) is a polynomial of order n and Q(x) is a polynomial of order m (n, m are integers) K as variable parameter and locus of all the values of x that satisfy this equation is root loci.

The root loci is categorized into two types based on the value of K

a) **RL**: it is the portion of root loci when K is positive; 0<K<infinity

b) **Complementary Root Loci**: the portion of root locus when K is negative;(-infinity)<K<0

Consider a closed loop system with transfer function G(s)/(1+G(s)*H(s)) where G(s) is forward transfer function and H(s) is transfer function of feedback loop. The characteristic equation is written as 1+G(s)*H(s) = 0. Suppose loop transfer function G(s)*H(s) has a variable parameter K as a multiplying factor, such that the rational function can be written as G(s)*H(s) = K*G_{1}(s)/H_{1}(s) where G_{1}(s), H_{1}(s) are polynomials. The characteristic equation can be rearranged and written as G_{1}(s) +K*H_{1}(s) = 0 for which we can apply root locus techniques to study the behavior of roots of characteristic equation as K varies.

**Properties of Root Loci**

- Let s
_{1}be the root of characteristic equation 1+K*G_{1}(s)*H_{1}(s) = 0 at s=s_{1},_{ }G_{1}(s)*H_{1}(s) = -1/K hence for s_{1}to be on RL K should be positive hence the magnitude of G_{1}(s_{1})*H_{1}(s_{1}) should be real and the**phase of G**._{1}(s)*H_{1}(s) should be odd multiples of 180 Degrees - The
**K=0 points on the root loci are at the poles of G(s)*H(s)**. Since at K=0 G_{1}(s)*H_{1}(s) will be infinite, so s must approach the poles of G_{1}(s)*H_{1}(s). - Similarly the
**K=infinite points on the root loci are at the zeros of G(s)*H(s)**.Since at K=infinite G_{1}(s)*H_{1}(s) will be zero, so s must approach the zeros of G_{1}(s)*H_{1}(s). - Each branch in root locus is the locus of one root of characteristic equation as K varies. Hence the
**number of branches in root locus is equal to the order of characteristic equation**, as the number of roots of characteristic equation is equal to the order of the equation. For example a quadratic equation of the form a*s^{2}+ b*s + c = 0 with order 2 will have two roots, a cubic equation a*s^{3}+ b*s^{2}+ c*s + d = 0 of order 3 will have 3 roots. - The
**root locus is symmetrical with respect to the real axis of s-plane**. This is due to the fact that the complex roots of an equation always occur in pairs. - The
**root loci are symmetrical with respect to axes of symmetry of pole-zero configuration of G(s)*H(s)**. The axes of symmetry of pole-zero configuration can be thought as a new complex plane obtained through a linear transformation. - When the order of polynomial P(s) n is not equal to the order of polynomial Q(s) m some of the loci will approach infinity in the s-plane. The properties of root loci near infinity in the s-plane are described by asymptotes which are the tangents to the curve at infinity.
**The number of asymptotes = 2*|n-m|,**where n is the number of finite poles and m is the number of finite zeros. - The root loci are asymptotic to asymptotes with angles given by
**θ**where n is not equal to m and i = 0,1,2,……,|n-m|-1._{i}= (2*i+1)*180/(|n-m|), - The intersect of the 2*|n-m| asymptotes of the root loci lies on the real axis of the s-plane and is given as
**σ1 = (Σ real parts of poles of G(s)*H(s) – Σ real parts of zeros of G(s)*H(s)) / (n-m)** - On a given section of real axis,
**RL are found in the section only if the total number of poles and zeros of G(s)*H(s) to the right of the section is odd**. - The angle of departure or arrival of a root locus at a pole or zero, respectively, of G(s)*H(s) denotes the angle of the tangent to the locus near the point.
- The points where the root loci meets the imaginary axis can be found by mans of Routh Hurwitz criteria.
**Breakaway points on the root loci of an equation correspond to multiple order roots of the equation**. The breakaway points must satisfy the equation dG1(s)*H1(s) / ds = 0 in addition to the necessary condition that the breakaway points should lie on root loci- n root loci arrive or depart a breakaway pint at 180/n degrees apart.