09_1_Frequency_response_analysis_E2009

09_1_Frequency_response_analysis_E2009 - 1 28150....

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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 First-order process with sinusoidal input n th-order process with sinusoidal input Bode diagrams Frequency response characteristics of feedback controllers Nyquist diagrams 2 November 2009 28150 Outline First-order process with sinusoidal input n th-order process with sinusoidal input Bode diagrams Frequency response characteristics of feedback controllers Nyquist diagrams 2 November 2009 28150 From Chapter 5: First-order process: sinusoidal response Input: Time domain response (deviation variables): With: ( ) t A t U sin sin = Time A P ( ) ( )( ) 2 2 1 + + = s s A K s Y ( ) + + + - + + = 2 2 2 2 2 2 2 1 1 s s s s A K s Y ( ) ( ) t t e A K t Y t sin cos 1 / 2 2 + - + =- ( ) ( ) + + + + =- t A K e A K t Y t sin 1 1 2 2 / 2 2 ( ) - =- 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 ( ) ( ) t t e A K t Y t sin cos 1 / 2 2 + - + =- ( ) ( ) + + + + =- t A K e A K t Y t sin 1 1 2 2 / 2 2 ( ) - =- 1 tan - = = b a 1 2 2 November 2009 28150 First-order process: sinusoidal input After sufficiently long time, the exponential term becomes negligible ( long- term response ): The output has the same frequency as the input ( ), but its phase is shifted relative to the input sine wave by the phase angle (or phase shift ) ( depends on !) . The output amplitude also depends on , and is given by: The amplitude ratio is given by: And the normalized amplitude ratio (often used in frequency response analysis) is: ( ) ( ) + + = t A K t Y sin 1 2 2 1 2 2 + = A K A 1 2 2 + = = K A A AR 1 1 2 2 + = = K AR AR N 2 November 2009 28150 First-order process: response to sinusoidal input 2 November 2009 28150 First-order process: effect of ( ) ( ) + + = t A K t Y sin 1 2 2 Note: different scaling of X axis! ( ) - =- 1 tan 2 November 2009 28150 Outline First-order process with sinusoidal input n th-order process with sinusoidal input Bode diagrams Frequency response characteristics of feedback controllers Nyquist diagrams 2 November 2009 28150 n th-order process with sine input A general TF G(s) multiplied with the Laplace transform of a sine wave input...
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09_1_Frequency_response_analysis_E2009 - 1 28150....

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