Root Locus

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Chapter 5

• Root Locus

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For sketching Root Locus
8 rules to follow
Rule 1 When K 0
Rule 2 When K 8
Rule 3 Real axis segment
Rule 4 Angle of asymptote
Rule 5 Centroid
Rule 6 Break away break in point
Rule 7 R-L crosses the j?-axis
Rule 8 Angle of Departure / Arrival
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Rule 8 Angle of departure (arrival) (if exist)
Finding angles of departure and arrival from
complex poles zero
Root locus start from open loop poles
? Angle of departure
Root locus terminates at open loop zeros
? Angle of arrival
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Angle of Departure
Angle of Arrival
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Example 1
Step 1 Determine poles zeros
Np3 Nz0
poles at
Step 2 Maps poles zeros on the s-plane
4
-3
0
-4
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Step 3 Calculate angle of asymptote
7
Step 4 Determine centroid
poles
Step 5 Sketch asymptotes line with dotted line
4
-3
0
-2
-4
8
Step 6 Determine the break-away or break in
points
Complex numbers
NO break away or break in points ? only exist on
real-axis of s-plane
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Step 7 Determine j?-crossing
From Characteristic equation
9
10
Construct the Routh array
10
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Force row s1 to be zero, thus 150-K0 K 150
Substitute K 150 into row s2
Since s j?
Thus
11
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Step 8 Since complex poles exist, calculate the
angle of departure
4
due to s0
due to its complex conjugate
-3
0
-4
How to determine ???
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To determine ?
4
-3
0
Thus
-4
Angle of departure
-36.87o
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Step 9 Sketch the root locus
-36.87o
4
Rule 7
-3
0
-2
-4
36.87o
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Final Sketch
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-36.87o
4
-3
0
-2
10 points
-4
36.87o
-5
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Try it
Sketch the root locus for
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How to use Matlab to sketch root locus
In Matlab
num 1 7 12
den 1 3 2
rlocus(num,den)
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Figure 810 page 438
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Any Qs ???