EET 309 CHAPTER4

# EET 309 CHAPTER4 - 4.0ROOTLOCUS OBJECTIVE Determination of...

This preview shows pages 1–8. Sign up to view the full content.

OBJECTIVE Determination  of  root  from  the  characteristic  equation  by  using  graphical  solution. Rules on sketching the root locus. Analysis of closed-loop using root locus 4.0 ROOT LOCUS

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

View Full Document
Root locus concept Root locus concept Consider Open-loop transfer function And Where If And KG ( s ) H ( s ) R ( s ) + E ( s ) Y ( s ) B ( s ) ) ( ) ( s H s KG ) ( ) ( ) ( 1 1 s P s Z s G = = + = + + + = m i i m ) z s ( ) z s ).... ( z s )( z s ( ) s ( Z 1 2 1 1 = + = + + + = n j j n p s p s p s p s s P 1 2 1 1 ) ( ) ).... ( )( ( ) (
Closed-loop transfer function ) ( ) ( ) ( ) ( ) ( 1 1 1 s KZ s P s KZ s R s Y + = ~ Number and position of zeros for open-loop and closed-loop are the same ~ Position of poles for the closed-loop depend on the position of poles, zeros and K . Characteristic equation is 0 ) ( ) ( 1 1 = + s KZ s P . If 1 ) ( s H Let ) ( ) ( ) ( 1 1 s P s Z s G = and ) ( ) ( ) ( 2 2 s P s Z s H = Its open-loop transfer function ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 s P s P s Z s Z K s H s KG = ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 s Z s KZ s P s P s P s KZ s R s Y + = and its closed-loop as ~ Position of poles for the closed-loop depend on the position of poles, zeros and K of the open-loop. transfer function. Root locus concept Root locus concept Where H(s) =1

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

View Full Document
As K varies the closed-loop poles follow similarly and form a locus. For that we define Root locus is a locus of characteristic equation as K varies from 0 to  . Referring to a characteristic equation 0 ) ( ) ( 1 = + s H s KG Let say = = + + = n j j m i i p s z s K s H s KG 1 1 ) ( ) ( ) ( ) ( 1 ) ( ) ( - = s H s KG Re-arrange -1 is a complex number π jr e 1 1 = - where .... , , , r 5 3 1 ± ± ± = Root locus concept Root locus concept
Magnitude condition Magnitude condition 1 ) ( ) ( = s H s KG Re-arrange m n m i i n j j z s z s z s p s p s p s z s p s K + + + + + + = + + = = = ...... ...... 2 1 2 1 1 1 where n p s p s p s + + + ...... 2 1 is the magnitude from a test point to open-loop poles n p p p - - - ...... , 2 1 and m z s z s z s + + + ...... 2 1 the magnitude from a test point to open-loop zeros n z z z - - - ...... , 2 1 From the magnitude condition, we can determine K . Let s be the test point, the magnitude are n n n ,..... 1 m m m × × ..... 1 and from open-loop zeros and poles respectively. Hence the magnitude condition is 1 ..... ..... 1 1 = × × × × m n m m n n

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

View Full Document
ϖ j σ s-plane n 1 n n m m m 1 s Magnitude condition Magnitude condition
Angle condition Angle condition     Revisiting the complex number of -1 π jr e 1 1 = - Its angle , 5 , 3 , 1 , ) ( ) ( ) ( ) ( 1 1 ± ± ± = = + - + = = = r r p s z s s H s KG n j j m i i Expand [ ] [ ] ..... , 5 , 3 , 1 ) ( ..... ) ( ) ( ( ..... ) ( ) ( 2 1 2 1 ± ± ± = = + + + + + + - + + + + + + r r p s p s p s z s z s z s n m where ( 29 ( 29 ( 29

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.

## This note was uploaded on 03/04/2011 for the course EET 309 taught by Professor Mariahahmad during the Spring '11 term at University of Malaya.

### Page1 / 43

EET 309 CHAPTER4 - 4.0ROOTLOCUS OBJECTIVE Determination of...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online