Short Notes: The Root Locus Technique | Control Systems - Electrical Engineering (EE) PDF Download

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Ro o t Lo cus T ec hnique: Short Notes
The Ro ot Lo cus T ec hnique is a graphical metho d used in con trol systems to analyze
ho w the ro ots of the c haracteristic equation (p oles of the closed-lo op system) v ary with
a system parameter, t ypically t he gain K . It helps in assessing system stabilit y and
p erformance.
1. System Configuration
Consider a unit y feedbac k system with op en-lo op transfer functionG(s)H(s) . The closed-
lo op transfer function is:
T(s) =
G(s)
1+G(s)H(s)
The c haracteristic equation, whose ro ots are the closed-lo op p oles, is:
1+G(s)H(s) = 0 or G(s)H(s) =-1
In p ol ar form, this b ecomes:
|G(s)H(s)| = 1 and ?G(s)H(s) = (2k +1)p, k = 0,±1,±2,...
2. Definition
The ro ot lo cus is the plot of the closed-lo op p oles in the s -plane as the gain K v aries
from 0 to8 . It starts at the op en-lo op p oles (K = 0 ) and ends at the op en-lo op zeros
or infinit y ( K?8 ).
3. R ules for Constructing Ro ot Lo cus
1. Num b er of Lo ci: The n um b er of ro ot lo cus branc hes equals the n um b er of op en-lo op
p oles.
2. Starting and Ending P oin ts:
• Lo ci start at op en-lo op p ol es (K = 0 ).
• Lo ci end at op en-lo op zeros or infinit y ( K?8 ).
3. Real-Axis Segmen ts: A p oin t on the real axis is on the lo cus if the sum of op en-lo op
p oles and zeros t o its righ t is o dd.
4. Asymptotes: F or largeK , lo ci approac h asymptotes. The angles of asymptotes are:
? =
(2k +1)p
n-m
, k = 0,1,...,|n-m|-1
where n is the n um b er of p oles and m is the n um b er of zeros. The asymptotes
in tersect t he real axis at:
s =
?
p oles-
?
zeros
n-m
1
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Ro o t Lo cus T ec hnique: Short Notes
The Ro ot Lo cus T ec hnique is a graphical metho d used in con trol systems to analyze
ho w the ro ots of the c haracteristic equation (p oles of the closed-lo op system) v ary with
a system parameter, t ypically t he gain K . It helps in assessing system stabilit y and
p erformance.
1. System Configuration
Consider a unit y feedbac k system with op en-lo op transfer functionG(s)H(s) . The closed-
lo op transfer function is:
T(s) =
G(s)
1+G(s)H(s)
The c haracteristic equation, whose ro ots are the closed-lo op p oles, is:
1+G(s)H(s) = 0 or G(s)H(s) =-1
In p ol ar form, this b ecomes:
|G(s)H(s)| = 1 and ?G(s)H(s) = (2k +1)p, k = 0,±1,±2,...
2. Definition
The ro ot lo cus is the plot of the closed-lo op p oles in the s -plane as the gain K v aries
from 0 to8 . It starts at the op en-lo op p oles (K = 0 ) and ends at the op en-lo op zeros
or infinit y ( K?8 ).
3. R ules for Constructing Ro ot Lo cus
1. Num b er of Lo ci: The n um b er of ro ot lo cus branc hes equals the n um b er of op en-lo op
p oles.
2. Starting and Ending P oin ts:
• Lo ci start at op en-lo op p ol es (K = 0 ).
• Lo ci end at op en-lo op zeros or infinit y ( K?8 ).
3. Real-Axis Segmen ts: A p oin t on the real axis is on the lo cus if the sum of op en-lo op
p oles and zeros t o its righ t is o dd.
4. Asymptotes: F or largeK , lo ci approac h asymptotes. The angles of asymptotes are:
? =
(2k +1)p
n-m
, k = 0,1,...,|n-m|-1
where n is the n um b er of p oles and m is the n um b er of zeros. The asymptotes
in tersect t he real axis at:
s =
?
p oles-
?
zeros
n-m
1
5. Breaka w a y/Break-in P oin ts: P oin ts where lo ci depart or arriv e on the real axis.
F ound b y solving :
d
ds
[G(s)H(s)] = 0
6. Angle of Departure/Arriv al: F or complex p oles/zeros, the angle of departure (from
a p ole) o r arriv al (at a zero) is calculated using the angle criterion:
?G(s)H(s) =
?
?( zeros)-
?
?( p oles) = (2k +1)p
7. In tersection with Imaginary Axis: F ound b y applying Routh-Hurwitz criterion to
the c haracteristic equation to determine the gain K at whic h p oles cross the imag-
inary axis.
4. Gain Calculation
A t an y p oin t s
1
on the ro ot lo cus, the gain K is:
K =
1
|G(s
1
)H(s
1
)|
=
?
| p ol e distances|
?
| zero distances|
5. Applications
• Determine closed-lo op p ole lo cations for v arying K .
• Assess system stabilit y (p oles in left-half plane indicate stabilit y).
• Design con trollers to ac hiev e desired transien t resp onse a nd stabilit y .
6. Key P oin ts
• The ro ot lo cus is symmetric ab out the real axi s.
• It pro vides insigh t in to ho w gain affects system dynamics.
• Soft w are to ols (e.g., MA TLAB) can plot ro ot lo ci acc urately .
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