This preview shows pages 1–10. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 1 Time Time Response. Response. CHAPTER 3 CHAPTER 3 2 3.2 Poles and Zeros and System Response. s s R s s s G s R s G s C s K s G 1 ) ( 5 2 ) ( where ) ( ) ( ) ( ) ( = + + = = + = 5 5 / 3 5 / 2 5 ) 5 ( ) 2 ( ) ( ) ( ) ( + + = + + = + + = + = s s s B s A s s s s C s s K s C t t e K K t c e t c  + = + = 2 1 5 ) ( 5 3 5 2 ) ( Figure 3.1: (a) System showing input and output; (b) Polezero plot of the system; (c) Evolution of a system response. 3 The transfer function, For a unit step of; 1/s, Its response, A, B and B are constant. For K=1 and =1/a then, Time constant (1/a), is defined as the time for eat to decay to 37%of its initial value or the time it takes for step response to reach 63% of its final value. 3.3 First Order System. 1 ) ( ) ( ) ( + = = s K s R s C s G 1 1 1 ) ( + + = + = s B s A s K s s C t e B A t c + = ' ) ( at e t c = 1 ) ( 4 Step response , at e K K t c = 2 1 ) ( ) ( ) ( a s s K s C + = Contd Figure 3.2: (a) First Order Response to a Unit Step. 5 The pole of the transfer function is at a , the farther the pole from the imaginary axis, the faster the transient response. Rise time (T r ), the time the response to go from the 0.1 to 0.9 of its final value. T r =2.2/a . Settling time (T s ) , time range when the response to reach and stay within 2% of its final value. Let c(t) = 0.98 then the T s =4/a. Contd 6 3.4 Second Order System. The transfer function, For Impulse response, Where, Standard Form, Where K is the dc gain, is the damping ratio, n is the undammped natural frequency. 1 2 ) ( ) ( b s b s a s R s C + + = 2 2 1 1 ) ( + + + = s s s C 2 2 2 2 ) ( ) ( n n n s s K s R s C + + = 2 2 2 = + + n n s s 1 2 2 , 1  = n n s Where Where 7 Example of 2 nd order system responses. Figure 3.3: Second Order System, pole plots and Step Response. Contd 8 General 2 nd Order System. Natural Frequency ( n ), Damping Ratio ( ) , Example 3.1 : Find the Natural Frequency ( n ) and Damping Ratio ( ), Solution: ) ( ) ( ) ( ) ( ) ( 2 b as s s b s C s G s R s C + + = = n n n n n n n a b a b s s s G 2 2 2 ) ( 2 2 2 2 = = = = + + = Contd ) 36 2 . 4 ( 36 ) ( 2 + + = s s s G 9 From previous page, Figure 3.4: Left; Plot for an underdamped 2 nd Order System. Right; Step Response for 2 nd Order System Damping Cases....
View
Full
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.
 Spring '11
 MariahAhmad

Click to edit the document details