The Root Locus method

The root locus method is used to study the change in the pole-location of a closed-loop system in the s-plane with respect to the change in the control-gain. A simple closed loop system is described as follows:

Figure 23.5: A simple closed loop system

Following the block-diagram, the error E(s) is

Again

Hence,

Equation (24.11) is known as the closed-loop transfer function for a negative feed-back system. The characteristic equation corresponding to a unity feedback (H(s) =1) for the above system could be written as

The above equation could be expressed in terms of magnitude and phase as follows:

(23.13)

Using the phase relationship of the above equation, one can plot the locus of the roots of the characteristic equation as K varies from 0 to infinity.

The following are the basic rules used for plotting the root-locus corresponding to a transfer function having n open-loop poles and m open-loop zeros.

1. The n branches of root-locus start at the n open-loop poles and out of them m branches end at the m open-loop zeros, while the n-m branches end at infinity.
2. On the real axis, the root-loci exist on the left side of odd number of poles and zeros.
3. The root locus is symmetric about the real axis.
4. The asymptotes of root-locus at infinity have the following real-axis intercept σ and angle θ:

where, P and Zs are finite poles and zeros already defined by Eqn.23.5.

5. The angle of arrival and departure from the poles and zeros are calculated by choosing a point very close to such desired poles and zeros. Now, using a simple graphical technique, vectors are drawn to the desired pole/zero from all the other poles and zeros. Adding the angles of the zeros and subtracting that of the poles the net angle is obtained at the point as θ_net . Subtracting this angle from 180⁰ would yield the angle of departure/arrival of the root-locus.
6. The root-locus crosses the imaginary axis at the points where the angle of G(s)H(s) becomes equal to (2k+1)180⁰ .