ROOT LOCUS
-is a graphical presentation of the closed-loop poles as a system
parameter is varied, and is a powerful method of analysis and
design for stability and transient response
BASICS OF ROOT LOCUS
✓It plots the system’s dynamic characteristics
✓It explains the plot of the system’s roots (poles and Zeros) on
S-Plane
✓Explains the system’s response and region of stability on S-Plane
S-PLANE
Imaginary j𝝎
Poles with negative real Poles with positive real
values, the system is a values, the system is a
STABLE system UNSTABLE system
Real 𝝈
For Poles with zero real values, the system
is MARGINALLY STABLE system
RULES OF ROOT LOCUS
❑ Root locus plot emerges from poles and gets enclosed to the zeros or asymptotes.
❑ Root locus will start at the poles and terminates at the zeros.
❑ Root locus must be symmetric with respect to the real axis.
❑ There are a few standard steps but these steps are not compulsory, but based on requirements we
need to calculate steps.
❑ Step 1 : Identify the roots (poles and zeros) of the system.
❑ Step 2 : Number of Asymptotes (X) = total Poles (P) – total Zeros (Z)
2𝑛+1
❑ Step 3 : Angle of Asymptotes = (180˚) , where n = 0, 1, 2, … (X-1)
𝑃 −𝑍
σ 𝑅𝑒𝑎𝑙 𝑃𝑎𝑟𝑡 𝑜𝑓 𝑃𝑜𝑙𝑒𝑠−σ 𝑅𝑒𝑎𝑙 𝑃𝑎𝑟𝑡 𝑜𝑓 𝑍𝑒𝑟𝑜𝑠
❑ Step 4 : Centroid of Asymptotes 𝜎𝑐 =
𝑃 −𝑍
RULES OF ROOT LOCUS
❑ Step 5 : Find the break away points:
• Find characteristic equation [1+G(s)H(s) = 0]
• Compute K = Polynomial
𝑑𝐾
• Compute = 0 and find S = break away points
𝑑𝑆
❑ Step 6 : Find angle of Departure 𝜃𝑑 = 180° − σ 𝜃𝑝 − σ 𝜃𝑧 ; not compulsory – only if P and Z
are imaginary values
❑ Step 7 : Intersection to Imaginary Axis
• Find characteristic equation [1+G(s)H(s) = 0]
• Construct Routh array
• Find K for marginally stable system (from 1st column Routh array)
• Place K in auxiliary equation (for 2nd order equation in Routh array)
SOME IMPORTANT CONCEPTS
Asymptote – a line that a curve approaches as it moves toward infinity.
*” Root locus will start at the poles and terminates at the zeros.”*
Branches - the individual paths or curves that represent the location of closed-loop poles as the gain K of the
system varies from 0 to infinity. Each branch originates from a pole of the open-loop transfer function and
terminates at a zero (or goes to infinity)
If P > Z , number of branches = P
If Z < P, number of branches = Z.
Example: P = 2 and Z = 5 ; number of branches = 5
>> 2 branches starts from poles and the remaining 3 will start at infinity
>> all 5 branches will terminate at 5 location of zeros.
Example: P = 5 and Z = 3 ; number of branches = 5
>> all 5 branches will start at 5 location of poles
>> 3 branches will terminate at 3 location of zeros and the remaining 2 will terminate at infinity.
SOME IMPORTANT CONCEPTS
As the value of K increases, the root loci moves further away from the poles and zeros and in order
to plot it, the root locus branches are approximated by asymptotes.
Break away point happens when two branches moves toward each other.
On the real axis, coincident points are called break away points.
Example 1: Obtain Root-Locus Plot for the Unity feedback system with system gain
𝐾
G s = .
𝑠(𝑠+2)
Imaginary j𝝎
Step 1 : Obtain Loci
Poles: 𝑃1 = 0 , 𝑃2 = −2
Total Poles (P) = 2
Zeros: none
Total Zeros (Z) = 0 X X
-2 0 Real 𝝈
Number of Loci = Max (P , Z)
=2
Example 1: Obtain Root-Locus Plot for the Unity feedback system with system gain
𝐾
G s = .
𝑠(𝑠+2)
Imaginary j𝝎
Step 2 : Number of Asymptotes
X=P–Z=2–0=2
Step 3 : Angle of Asymptotes
2𝑛+1
𝜎𝑛 = (180°) ; n = 2-1 = 1 ∴ 𝑛 = 0, 1
𝑃 −𝑍
2(0) + 1 X X
𝜎0 = 180° = 90° -2 0 Real 𝝈
2 −0
2(1) + 1
𝜎1 = 180° = 270°
2 −0
Example 1: Obtain Root-Locus Plot for the Unity feedback system with system gain
𝐾
G s = .
𝑠(𝑠+2)
Imaginary j𝝎
Step 4 : Centroid of Asymptotes
90°
σ 𝑅𝑒𝑎𝑙 𝑃𝑎𝑟𝑡 𝑜𝑓 𝑃𝑜𝑙𝑒𝑠 − σ 𝑅𝑒𝑎𝑙 𝑃𝑎𝑟𝑡 𝑜𝑓 𝑍𝑒𝑟𝑜𝑠
𝜎𝑐 =
𝑃 −𝑍
−2 + 0 − 0
𝜎𝑐 = = −1
2−0 > >>
X> >> X
Step 5 : Break Away point
-2 -1 0 Real 𝝈
1 + G(s)H(s) = 0 ; H(s) = 1
𝐾
1+ 1 = 0 ; 𝐾 = −𝑠 𝑠 + 2
𝑠 𝑠+2
K = s2 – 2s 270°
Example 1: Obtain Root-Locus Plot for the Unity feedback system with system gain
𝐾
G s = .
𝑠(𝑠+2)
Imaginary j𝝎
K=– s2 – 2s
90°
𝑑𝐾
=0;
𝑑𝑆
0 = –2s – 2 ; s = –1 Break away point
> >>
> >>
X> >> X
-2 -1 0 Real 𝝈
> >>
270°
Example 2: Obtain Root-Locus Plot for the Unity feedback system with system gain
𝐾
G s = .
𝑠(𝑠+2)(𝑠+4)
Example 3: Obtain Root-Locus Plot for the Unity feedback system with system gain
𝐾(𝑠+9)
G s = 2 .
𝑠(𝑠 +4𝑠+11)
