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Unformatted text preview: 1 28150. Introduction to process control 9. Frequency response analysis (1) Krist V. Gernaey 2 November 2009 28150 Learning objectives • At the end of this lesson you should be able to: – Derive and plot the frequency response of a transfer function using Bode diagrams or Nyquist diagram 2 November 2009 28150 Outline • Firstorder process with sinusoidal input • n thorder process with sinusoidal input • Bode diagrams • Frequency response characteristics of feedback controllers • Nyquist diagrams 2 November 2009 28150 Outline • Firstorder process with sinusoidal input • n thorder process with sinusoidal input • Bode diagrams • Frequency response characteristics of feedback controllers • Nyquist diagrams 2 November 2009 28150 From Chapter 5: Firstorder process: sinusoidal response • Input: • Time domain response (deviation variables): – With: ( ) t A t U ω sin sin = Time A P ( ) ( )( ) 2 2 1 ω τ ω + + ⋅ ⋅ ⋅ = s s A K s Y ( ) + + + ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ = 2 2 2 2 2 2 2 1 1 ω ω ω τ ω τ τ ω τ ω s s s s A K s Y ( ) ( ) t t e A K t Y t ω ω τ ω τ ω τ ω τ sin cos 1 / 2 2 + ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ = ( ) ( ) φ ω τ ω τ ω τ ω τ + + ⋅ ⋅ + ⋅ + ⋅ ⋅ ⋅ ⋅ = t A K e A K t Y t sin 1 1 2 2 / 2 2 ( ) τ ω φ ⋅ = 1 tan 2 November 2009 28150 Trigonometric identity ( ) ( ) ( ) tan tan 1 1 2 2 < ± = ≥ = + + = + a if a / b a if a / b x sin b a x cos b x sin a π ϕ ϕ ϕ ( ) ( ) t t e A K t Y t ω ω τ ω τ ω τ ω τ sin cos 1 / 2 2 + ⋅ ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ = ( ) ( ) φ ω τ ω τ ω τ ω τ + + ⋅ ⋅ + ⋅ + ⋅ ⋅ ⋅ ⋅ = t A K e A K t Y t sin 1 1 2 2 / 2 2 ( ) τ ω φ ⋅ = 1 tan τ ω ⋅ = = b a 1 2 2 November 2009 28150 Firstorder process: sinusoidal input • After sufficiently long time, the exponential term becomes negligible ( long term response ): • The output has the same frequency as the input ( ω ), but its phase is shifted relative to the input sine wave by the phase angle (or phase shift ) φ ( φ depends on ω !) . • The output amplitude also depends on ω , and is given by: • The amplitude ratio is given by: • And the normalized amplitude ratio (often used in frequency response analysis) is: ( ) ( ) φ ω τ ω + + ⋅ ⋅ = t A K t Y sin 1 2 2 1 ˆ 2 2 + ⋅ ⋅ = τ ω A K A 1 ˆ 2 2 + ⋅ = = τ ω K A A AR 1 1 2 2 + ⋅ = = τ ω K AR AR N 2 November 2009 28150 Firstorder process: response to sinusoidal input 2 November 2009 28150 Firstorder process: effect of ω ( ) ( ) φ ω τ ω + + ⋅ ⋅ = t A K t Y sin 1 2 2 Note: different scaling of X axis! ( ) τ ω φ ⋅ = 1 tan 2 November 2009 28150 Outline • Firstorder process with sinusoidal input • n thorder process with sinusoidal input • Bode diagrams • Frequency response characteristics of feedback controllers • Nyquist diagrams 2 November 2009 28150 n thorder process with sine input • A general TF G(s) multiplied with the Laplace transform of a sine wave input...
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This note was uploaded on 11/20/2009 for the course CHME DTUabroad taught by Professor Rafiqulgani during the Fall '09 term at Rensselaer Polytechnic Institute.
 Fall '09
 RafiqulGani

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