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Unformatted text preview: Angles of departure and arrival • Looking at the open loop poles and zeros below, the root locus starts at the poles and ends at zeros but at what angles? j ϖ • Taking a point ( ε ) close to a complex pole. Assuming this point lies on the root locus, the angles to this point must be (2 k +1)180 o . Us ing the angles from the other poles and zeros to ( ε ) pole will give us angle close to (2 k + 1)180 o , the difference being the angle of de parture. 1 θ 1 + θ 2 + θ 3 θ 4 θ 5 + θ 6 = (2 k + 1)180 o θ 1 = θ 2 + θ 3 θ 4 θ 5 + θ 6 (2 k + 1)180 o θ j ϖ 6 5 4 3 2 1 θ θ θ θ θ • This is also the case with a point ε close to a complex zero for angles of arrival. 2 θ θ 3 θ 5 θ 6 θ 4 θ j ϖ 2 1 θ 1 + θ 2 + θ 3 θ 4 θ 5 + θ 6 = (2 k + 1)180 o θ 2 = θ 1 θ 3 + θ 4 + θ 5 θ 6 + (2 k + 1)180 o Example: Given a unity feedback system that has a forward transfer function G ( s ) = K ( s + 2) s 2 4 s + 13 a)Sketch the root locus 3 b)Find the imaginary axis crossing;...
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 Fall '10
 GU
 Root Locus, Complex number, real axis

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