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*G1(s)/H1(s) where G1(s), H1(s) are polynomials. The characteristic equation can be rearranged and written as G1(s) +K*H1(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 s1 be the root of characteristic equation 1+K*G1(s)*H1(s) = 0 at s=s1, G1(s)*H1(s) = -1/K hence for s1 to be on RL K should be positive hence the magnitude of G1 (s1)*H1 (s1) should be real and the phase of G1(s)*H1(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 G1(s)*H1(s) will be infinite, so s must approach the poles of G1(s)*H1(s).
- Similarly the K=infinite points on the root loci are at the zeros of G(s)*H(s).Since at K=infinite G1(s)*H1(s) will be zero, so s must approach the zeros of G1(s)*H1(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*s2 + b*s + c = 0 with order 2 will have two roots, a cubic equation a*s3 + b*s2 + 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 θi = (2*i+1)*180/(|n-m|), where n is not equal to m and i = 0,1,2,……,|n-m|-1.
- 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.