Root Locus Analysis (1) Hany Ferdinando Dept. of Electrical Eng. Petra Christian University

General Overview This section discusses how to plot the Root Locus method This section discusses how to plot the Root Locus method Step by step procedure is used inline with an example Step by step procedure is used inline with an example Finally, some comments are given as the complement for this section Finally, some comments are given as the complement for this section

Why Root Locus Closed-loop poles location determine the stability of the system Closed-loop poles location determine the stability of the system Closed-loop poles location is influenced as the gain is varied Closed-loop poles location is influenced as the gain is varied Root locus plot gives designer information how the gain variation influences the stability of the system Root locus plot gives designer information how the gain variation influences the stability of the system

Plot Example

Important Notes: Poles are drawn as x while zeros are drawn as o Poles are drawn as x while zeros are drawn as o Gain at poles is zero, while gain at zeros is infinity Gain at poles is zero, while gain at zeros is infinity Pole is the starting point and it must finish at zero; therefore, for every pole there should be corresponding zero Pole is the starting point and it must finish at zero; therefore, for every pole there should be corresponding zero Root locus is plot on the s plane Root locus is plot on the s plane

Standardization G(s) H(s) + - R(s)C(s) Find Characteristic Equation!! 1 + G(s)H(s) = 0

How to make it? 1. Start from the characteristic equation 2. Locate the poles and zeros on the s plane 3. Determine the root loci on the real axis 4. Determine the asymptotes of the root loci 5. Find the breakaway and break-in points 6. Determines the angle of departure (angle of arrival) from complex poles (zeros) 7. Find the points where the root loci may cross the imaginary axis

Example (1) 1. Start from the characteristic equation

Example (2) 2. Locate the poles and zeros on the s plane

Example (3) 3. Determine the root loci on the real axis

Example (4a) 4. Determine the asymptotes of the root loci

Example (4b) 4. Determine the asymptotes of the root loci

Example (5) 5. Find the breakaway and break-in point From the characteristic equation, find then calculate… s = and s =

Example (6) 6. Determine the angle from complex pole/zero This example has no complex poles and zeros, therefore, this step can be skipped!!! pole = 0 – sum from pole + sum from zero pole = 0 – sum from pole + sum from zero zero = 0 – sum from zero + sum from pole zero = 0 – sum from zero + sum from pole

Example (7) 7. Find the points where the root loci may cross the imaginary axis Do this part by substituting j for all s in the characteristic equation = ±2, K = 6 or = 0, K = 0 = ±2, K = 6 or = 0, K = 0

Example (8)

Comments n th degree algebraic equation in s n th degree algebraic equation in s If n-m2 then a1 is negative sum of the roots of the equation and is independent of K If n-m2 then a1 is negative sum of the roots of the equation and is independent of K It means if some roots move on the locus toward the left as K increased then the other roots must move towards the right as K is increased It means if some roots move on the locus toward the left as K increased then the other roots must move towards the right as K is increased

Root Locus in Matlab Function rlocus(num,den) draws the Root Locus of a system. Another version in state space is rlocus(A,B,C,D) The characteristic equation Those functions are for negative feedback (normal transfer function)

Next… Topic for the next meeting is Root Locus in positive feedback