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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_6

# KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_6

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1 More General Transfer Function Models Chapter 6 Poles and Zeros: The dynamic behavior of a transfer function model can be characterized by the numerical value of its poles and zeros. General Representation of ATF: There are two equivalent representations: ( 29 0 0 (4-40) m i i i n i i i b s G s a s = = =

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2 Chapter 6 where { z i } are the “zeros” and { p i } are the “poles”. ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 2 (6-7) m m n n b s z s z s z G s a s p s p s p - - - = - - - K K We will assume that there are no “pole-zero” calculations. That is, that no pole has the same numerical value as a zero. Review: in order to have a physically realizable system. n m i
3 Chapter 6 Example 6.2 For the case of a single zero in an overdamped second-order transfer function, ( 29 ( 29 ( 29 ( 29 1 2 τ 1 (6-14) τ 1 τ 1 a K s G s s s + = + + calculate the response to the step input of magnitude M and plot the results qualitatively. Solution The response of this system to a step change in input is ( 29 1 2 τ τ τ τ / τ 1 2 1 (6-15) τ τ τ τ 1 2 2 1 t t a a y t KM e e - - - - = + + - -

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4 Chapter 6 Note that as expected; hence, the effect of including the single zero does not change the final value nor does it change the number or location of the response modes. But the zero does affect how the response modes (exponential terms) are weighted in the solution, Eq. 6-15. ( 29 y t KM i = A certain amount of mathematical analysis (see Exercises 6.4, 6.5, and 6.6) will show that there are three types of responses involved here: Case a: Case b: Case c: 1 τ τ a 1 τ a < τ 0 a <
5 Chapter 6

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6 Chapter 6 Summary: Effects of Pole and Zero Locations 1. Poles Pole in “right half plane (RHP)” : results in unstable system (i.e., unstable step responses) ( 29 1 p a bj j = + = - x x x Real axis Imaginary axis x = unstable pole Complex pole : results in oscillatory responses Real axis Imaginary axis x x x = complex poles
7 Chapter 6 1. Zeros Pole at the origin (1/s term in TF model): results in an “integrating process” Note: Zeros have no effect on system stability. Zero in RHP: results in an inverse response to a step change in the input Zero in left half plane: may result in “overshoot” during a step response (see Fig. 6.3). x 1 y 0 t inverse response Real axis Imaginary axis

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8 Chapter 6 Inverse Response Due to Two Competing Effects An inverse response occurs if: 2 2 1 1 τ (6-22) τ K K -
9 Chapter 6 Time Delays Time delays occur due to: 1. Fluid flow in a pipe 2. Transport of solid material (e.g., conveyor belt) 3. Chemical analysis - Sampling line delay - Time required to do the analysis (e.g., on-line gas chromatograph) Mathematical description: A time delay, , between an input u and an output y results in the following expression: θ ( 29 ( 29 0 forθ (6-27) θ forθ t y t u t t < = -

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10 Chapter 6 Example: Turbulent flow in a pipe Let, fluid property (e.g., temperature or composition) at point 1 fluid property at point 2
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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Simulation)-Chapter_6

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