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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec30

# KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec30

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1 Chapter 11 Figure 11.25 Stability regions in the complex plane for roots of the charact- eristic equation.

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2 Chapter 11 Figure 11.26 Contributions of characteristic equation roots to closed-loop response.
3 Chapter 11 Direct Substitution Method The imaginary axis divides the complex plane into stable and unstable regions for the roots of characteristic equation, as indicated in Fig. 11.26. On the imaginary axis, the real part of s is zero, and thus we can write s=j ϖ . Substituting s=j ϖ into the characteristic equation allows us to find a stability limit such as the maximum value of K c . • As the gain K c is increased, the roots of the characteristic equation cross the imaginary axis when K c = K cm .

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4 Chapter 11 Example 11.12 Use the direct substitution method to determine K cm for the system with the characteristic equation given by Eq. 11-99. Solution Substitute and K c = K cm into Eq. 11-99: ω s j = 3 2 10ω 17ω 8 ω 1 0 cm j j K - - + + + = or (11-105) ( 29 ( 29 2 3 1 17ω 8ω 10ω 0 cm K j + - + - = 3 2 10 17 8 1 0 (11-99) c s s s K + + + + =
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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec30

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