Gp4_Chapter 3 - EE2010 Systems and Control (Chapter 3) 3-1...

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Unformatted text preview: EE2010 Systems and Control (Chapter 3) 3-1 System transfer function is the ratio of the Laplace transform of the output signal, y ( t ), over the Laplace transform of the input driving function, u ( t ), under the assumption that all initial conditions are zero Transfer function models the input-output relationship of a system and can be represented by a pictorial representation, called block diagram Transfer Function of LTI Systems ) ( ) ( ) ( , i.e. ) ( ) ( ) ( ) ( ) ( s G s U s Y s U s Y t u t y s G L L U ( s ) Y ( s ) ) ( ) ( ) ( s U s Y s G EE2010 Systems and Control (Chapter 3) 3-2 Suppose the LTI system is modeled by the following differential equation Transfer function may be derived from the Transform of derivative rule. Since all initial conditions are zero, Example: Find transfer function of M m m m m N n n n n dt t u d b dt t y d a ) ( ) ( ) ( ) ( s X s dt t x d n n n ) ( ) ( ) ( ) ( 1 2 t u b t y a t y a t y a 1 2 2 1 2 2 1 2 2 1 2 ) ( ) ( function, Transfer ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( a s a s a b s U s Y y y s U b s Y a s a s a s U b s Y a y s sY a y sy s Y s a t u b t y a t y a t y a L L EE2010 Systems and Control (Chapter 3) 3-3 For a general N th order system: Numerator polynomial in the numerator is formed using coefficients of the input function and its derivative(s) Denominator polynomial is formed using coefficients of the output signal and its derivative(s) System order is N , the largest power of s in the denominator of G ( s ) 1 1 1 1 ... ... ) ( ) ( ) ( ) ( a s a s a b s b s b s U s Y t u s b t y s a N N N N M M M M M m m m N n n n L L EE2010 Systems and Control (Chapter 3) 3-4 Example Example: Derive the transfer function of series RC-circuit and V c ( t ) Method 1 for deriving transfer function: The V c ( t ) to V ( t ) D.E. is ) ( ; ) ( ) ( ) ( ) ( ) ( : Inductor ) ( ; 1 ) ( ) ( ) ( ) ( 1 ) ( : Capacitor ) ( ) ( ) ( ) ( ) ( : Resistor L L L c c t c i sL s I s V s G dt t di L t V V Cs s I s V s G d i C t V R s I s V s G t Ri t V ) ( ) ( ) ( t V t V dt t dV RC c c EE2010 Systems and Control (Chapter 3) 3-5 Performing Laplace Transform, Assuming zero initial conditions, i.e., V c (0) = 0 and re-arranging the equation, the transfer function is Method 2 for deriving transfer function: Transfer function of capacitor is ) ( ) ( ) ( ) ( ) ( ) ( ) ( s V s V RCV s RCsV t V t V dt t dV RC c c c c c L L 1 1 ) ( ) ( ) ( sRC s V s V s G c ) ( ; 1 c V sC EE2010 Systems and Control (Chapter 3) 3-6 Performing voltage division,...
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Gp4_Chapter 3 - EE2010 Systems and Control (Chapter 3) 3-1...

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