{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

problem_11_18

# problem_11_18 - Problem 11.18 of the Palm textbook Plant TF...

This preview shows pages 1–2. Sign up to view the full content.

Problem 11.18 of the Palm textbook Plant TF: G P s ( ) = s + 10 s 2 + 5 s + 6 Sketch the root locus. Determine the smallest damping ratio possible for positive values of proportional gain K. Determine the value of proportional gain K that is required to minimize the dominant time constant with ζ = 0.707, and determine this time constant. SOLUTION Closed-loop characteristic equation: 0 = 1 + K s + 10 s 2 + 5 s + 6 ! " # \$ % & = 1 + K N s ( ) D s ( ) ! " # \$ % & OR D CL s ( ) = D s ( ) + KN s ( ) = s 2 + 5 s + 6 + K s + 10 ( ) = s 2 + 5 + K ( ) s + 6 + 10 K 1. Number of branches : max N P , N Z { } = max 2,1 { } = TWO branches 2. Start/end points : OL zeros : z 1 = ! 10 OL poles : p 1 = ! 2, p 2 = ! 3, Therefore, the RL START at poles p 1 , p 2 . One RL branch ENDS at p 1 , and one RL branch ends at ! . 4. Since the RL can exist on the real axis only to the left of an ODD number of real OL zeros/poles, the following regions on the real axis can contain the RL: p 2 < s < p 1 and s < z 1 . 5. Here we have one branch of the RL going to ! ; we need to determine the asymptotes for these branches. We could use Rule 5 to determine this asymptote. Alternately, we can

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

problem_11_18 - Problem 11.18 of the Palm textbook Plant TF...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online