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Unformatted text preview: 1 Chapter 4 Transfer Functions Convenient representation of a linear , dynamic model. A transfer function (TF) relates one input and one output: ( 29 ( 29 ( 29 ( 29 system x t y t X s Y s i The following terminology is used: x input forcing function cause y output response effect 2 Chapter 4 Definition of the transfer function: Let G ( s ) denote the transfer function between an input, x , and an output, y . Then, by definition where: ( 29 ( 29 ( 29 Y s G s X s = ( 29 ( 29 ( 29 ( 29 Y s y t X s x t = L = L Development of Transfer Functions Example: Stirred Tank Heating System 3 Chapter 4 Figure 2.3 Stirredtank heating process with constant holdup, V . 4 Chapter 4 Recall the previous dynamic model, assuming constant liquid holdup and flow rates: ( 29 (1) i dT V C wC T T Q dt = + ( 29 ( 29 ( 29 ( 29 , , 2 i i T T T T Q Q = = = Suppose the process is initially at steady state: where steadystate value of T, etc. For steadystate conditions: T = ( 29 (3) i wC T T Q = + Subtract (3) from (1): ( 29 ( 29 ( 29 (4) i i dT V C wC T T T T Q Q dt = + 5 Chapter 4 But, ( 29 because is a constant (5) d T T dT T dt dt = Thus we can substitute into (42) to get, ( 29 (6) i dT V C wC T T Q dt = + where we have introduced the following deviation variables , also called perturbation variables: , , (7) i i i T T T T T T Q Q Q i = = = ( 29 ( 29 ( 29 ( 29 ( 29 (8) i V C sT s T t wC T s T s Q s  = = Take L of (6): 6 Chapter 4 Evaluate ( 29 0 . T t i = By definition, Thus at time, t = 0, . T T T i = ( 29 ( 29 (9) T T T i = But since our assumed initial condition was that the process was initially at steady state, i.e., it follows from (9) that Note : The advantage of using deviation variables is that the initial condition term becomes zero. This simplifies the later analysis. ( 29 T T = ( 29 0. T i = Rearrange (8) to solve for ( 29 ( 29 ( 29 1 (10) 1 1 i K T s Q s T s s s @ = + + + ( 29 : T s i 7 Chapter 4 where two new symbols are defined: ( 29 1 and 11 V K wC w = = Transfer Function Between and Q i T i Suppose is constant at the steadystate value. Then, Then we can substitute into (10) and rearrange to get the desired TF: i T ( 29 ( 29 ( 29 0....
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 Spring '09
 Dr.MuhammadAlArfaj

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