PROBLEM: Given a unity feedback system that has the forward transfer function
$G(s)=\frac{K(s+2)}{(s^{2}+4s+13 )}$
do the following:

a. Calculate the angle of G(s) at the point ( -3 + j0) by finding the algebraic sum of angles of the vectors drawn from the zeros and poles of G(s) to the given point.

b. Determine if the point specified in a is on the root locus.
c. If the point specified in a is on the root locus, find the gain, K, using the lengths of the vectors.

Question

PROBLEM: Given a unity feedback system that has the forward transfer function

[latex]G(s)=\frac{K(s+2)}{(s^{2}+4s+13 )}[/latex]

do the following:

a. Calculate the angle of G(s) at the point ( -3 + j0) by finding the algebraic sum of angles of the vectors drawn from the zeros and poles of G(s) to the given point.

b. Determine if the point specified in a is on the root locus.
c. If the point specified in a is on the root locus, find the gain, K, using the lengths of the vectors.

Accepted Answer

a. First draw the vectors.

From the diagram,

[latex]\sum angles=180°- an ^{-1} \left(\frac{-3}{-1} ight) - an ^{-1}\left(\frac{-3}{1} ight)=180°-108.43°+108.43°=180° [/latex].

b. Since the angle is 180°, the point is on the root locus.

c. [latex]K=\frac{\Pi pole lengths}{\Pi zero lengths} =\frac{\left(\sqrt{1^{2}+3^{2} } ight)\left(\sqrt{1^{2}+3^{2}} ight) }{1} =10 [/latex]

Question 8.SE.2: PROBLEM: Given a unity feedback system that has the forward ...

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