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Root Locus

The document explains the Root Locus technique, which illustrates the movement of closed-loop system poles as a function of open loop gain (k) in the s-plane. It outlines the general method for constructing Root Locus, including symmetry, the number of loci, real axis loci, angle of asymptotes, and the centroid of asymptotes. An example is provided to calculate the angles of asymptotes and centroid for a specific system transfer function.

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Muhammad Qumar Nazeer
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28 views8 pages

Root Locus

The document explains the Root Locus technique, which illustrates the movement of closed-loop system poles as a function of open loop gain (k) in the s-plane. It outlines the general method for constructing Root Locus, including symmetry, the number of loci, real axis loci, angle of asymptotes, and the centroid of asymptotes. An example is provided to calculate the angles of asymptotes and centroid for a specific system transfer function.

Uploaded by

Muhammad Qumar Nazeer
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Root Locus

 If some parameters of the close-loop system are


varied, there is movement of its poles.
 The cognizance of such movement of poles with
small variation of the parameters is remarkably
useful in the design of closed-loop system.
 Root locus technique provides knowledge of such
movement of poles by graphical method in the s-
plane. In Root locus method parameter (k) of the
system is varied from 0 to infinity.
 Root locus is defined as locus of the plot of closed
loop poles as a function of open loop gain (k)
when k is varied from –ve infinity to + infinity.
 If k is varied from zero to infinity it is known
as direct root locus.
Example: Find the root locus of unity feed fack
system having G(s)=k/s+1
General Method for Construction of
Root Locus
1. Symmetry: Root locus is always symmetrical
about the real axis. The roots are either real or
complex conjugates or a combination of both. So
the locus is symmetrical about the real axis s-
plane.
2. Number of Loci: let the number of OLTF and
OLTF zeros be n and m respectively. If n>m i.e
the number of OLTF poles is greater than the
number of OLTF zeros, the of loci is n.here (n-m)
loci will end at infinity and total loci is n.
3. Real axis loci: some of the loci will lie on the
real axis. A point on the real axis will lie on
the root locus if and only if the sum of the
OLTF poles and zeros to the right of the
point is odd.
G(s)H(s)= k(s+2)/s(s+1)(s+4)
4. Angle of asymptotes: Since the number of
poles is greater than number of zeros i.e
n>m n-m branches will move to infinity and
these branches move along the asymptote.
Asymptote is defined as a line on which root
the root locus touches infinity
The number of asymptotes=n-m
The angle of asymptote is given by
Angle=[2q+1/n-m]x180
5. Centre of asymptotes: Since only the
angles of asymptotes are not sufficient, the
location of asymptotes in s-plane has equal
importance. The point where the asymptotes
touch the real axis is called centroid
(sigmac).
Sigmac=[sum of real parts of poles of
G(s)H(s)-sum of real parts of zeros of
G(s)H(s)]/n-m
Calculate the angles of asymptotes and the
centroid for system having.
G(s)H(s)= k(s+2)/s(s+1)(s+4)

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