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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec34

# KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec34

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1 Chapter 13 Frequency Response Analysis Sinusoidal Forcing of a First-Order Process For a first-order transfer function with gain K and time constant , the response to a general sinusoidal input, is: τ ( 29 sinω = x t A t ( 29 ( 29 2 2 ωτ ωτcosω sinω (5-25) ω τ 1 - = - + + t KA y t e t t Note that y(t) and x(t) are in deviation form. The long-time response, y l (t), can be written as: ( 29 ( 29 2 2 sinω φ for (13-1) ω τ 1 = + + l KA y t t t ( 29 1 φ tan ωτ - = - where:

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2 Chapter 13 Figure 13.1 Attenuation and time shift between input and output sine waves ( K = 1). The phase angle of the output signal is given by , where is the (period) shift and P is the period of oscillation. φ φ Time shift / 360 = - o P t
3 Chapter 13 1. The output signal is a sinusoid that has the same frequency, ϖ, as the input.signal, x(t) = A sin ϖ t . 2. The amplitude of the output signal, , is a function of the frequency ϖ and the input amplitude, A : ˆ A 2 2 ˆ (13-2) ω τ 1 = + KA A Frequency Response Characteristics of a First-Order Process

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KING FAHD UNIVERSITY CHEMICAL ENGINEERING COURSE NOTES (Process Control)-Lec34

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