EET 309 CHAPTER3 - 1 Time Time Response. Response. CHAPTER...

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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) Pole-zero 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 e-at 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....
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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.

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EET 309 CHAPTER3 - 1 Time Time Response. Response. CHAPTER...

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