Chapter 7 ME 332 Lecture Notes
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Transcript of Chapter 7 ME 332 Lecture Notes
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Chapter 7
ROOT LOCUS METHOD
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ROOT LOCUS
Root Loci: Gives graphical picture of the effect of selectedparameters on system poles and suggest what values
should be chosen to meet system design specificationon time constant and damping ratio to improve speedof the response
The root locus method is two stages process
1) Construction of the root loci which is the loci of closedloop poles position in s-plane as K or any other designparameter changesRoot loci are constructed from angle condition alone asthe loci of all points s for which the sum of vectorangles I from all open loop zeros to s minus the sumof the vector angles j from all open loop poles to sequal to odd number multiple of 180
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ROOT LOCUS
2) The magnitude condition shows that the
value of k for which closed loop poles willbe located at a given points s along alocus equal the product of vector length B
jfrom all open loop poles to s divided by theproduct of vector length Ai from all open
loop zeros to s
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n=[1];
den=[1 6 11 6];
%ors1=[1 1];s2=[1 2];s3=[1 3];d=conv(s1,conv(s2,s3))
sys=tf(n,d);
sym2=tf(n,den)
rlocus(sys)
figure
rlocus(sym2)
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ROOT LOCUS
EXAMPLES
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Example 1
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Example Solution
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Example Solution Contd
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Example Solution Contd
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Example Solution Contd
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Example Solution Contd
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Example Solution Contd
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Example Solution Contd
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Example Solution Contd
A line of constant damping is found by finding the angle anddrawing a line that will cross the root locus at at least one point as
shown in the previous root locus plot
Trial point is picked and the angle criteria is applied
Hence the point lie on the root locus
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Example Solution Contd
Value of open Loop gain constant K: Applying the magnitude criterion to
the above point gives
Closed loop poles for K=11.35: Since the closed loop system is a third order
system, there are three closed loop poles. Two of them are found at the point
(complex conjugate poles). The third lies on the real locus that extends from -5 to - . The value is calculated using the magnitude criterion as shownBelow
Solving for x gives x=0.73 which implies that s1= -5.73. hence the closed
loop poles at K=11.35 are
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Example Solution Contd
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Example 2
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Example 2 Solution
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Example 2 Solution Contd
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Example 2 Solution Contd
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Example 2 Solution Contd
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Example 2 Solution Contd
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Example 2 Solution Contd
Plot of constant damping ratio on the root locus plot and test
trial points a long it using angle criterion as shown in theprevious figure at s=-0.8+j2.9 gives
Hence the point lie on the root locus