Mumbai University > Electronics Engineering > Sem 4 > Linear Control Systems

Topic: Stability Analysis in Time Domain

Difficulty : Medium

Marks : 5M

Rule 1 => The root locus is symmetrical about the real axis

Rule 2 => Total number of loci => Each branch of root locus originates from an open - loop pole corresponding to K=0 and terminates at either on an finite open loop zero corresponding to k=$\infty$. the number of branches of the root locus terminating on infinity is equal to p-z

Rule 3 => Real axis loci =>

segments of the real axis having an odd number of real axis open loop poles plus zeros to their rights are parts of the root locus.

Rule 4 => Angle of asymptotes =>

The (p-z) root locus branches that tends to infinity do so along straight line asymptotes making angle with the real axis given by,

$\phi_A$ = $\frac{180(2q+1)}{p-z}$

where q = 0,1,2,3----

Rule 5 => Centroid =>

The point of intersection of the asymptotes with the real axis is at S+$\sigma A$

where,

$\sigma A$ = $\frac{sum of poles - sum of zeros}{p-z}$

Rule 6 => Breakaway point =>

The breakingaway points of the root locus are determined from the roots of the equation $\frac{dk}{ds}$ = 0

If r number of branches of root locus meet at a point, than they break awy at an angle of $ \pm \frac{180^0}{r}$

Rule 7 => Angle of departure/Arrival =>

The angle of departure from a complex open loop poles is given by

$\phi_D=\phi_p = \pm 180(2q+1)-\phi$ where q = 0,1,2

OR $\phi_D=\phi_p = 180 - \phi$

and Angle of Arrival from a complex open loop zero is given by

$\phi_A=\phi_z = \pm 180(2q+1)+\phi$ where q = 0,1,2

OR $\phi_D=\phi_p = 180 + \phi$

Rule 8 => Intersection with imaginary axis =>

This rule gives us point on the imaginary axis, which the root locus will cut while moving to the R.H.S.

Rule 9 => The open loop gain K at any point

$S = S_a$ on the Root locus is given by,

K = [$\frac{Product of vector length from open loop poles to the point S}{Product of vector length from open loop zeros to the point Sa}$]