09_1_Frequency_response_analysis_E2009

# 09_1_Frequency_response_analysis_E2009 - 1 28150...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 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 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 ( 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 First-order process: response to sinusoidal input 2 November 2009 28150 First-order 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 • 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...
View Full Document

## This note was uploaded on 11/20/2009 for the course CHME DTU-abroad taught by Professor Rafiqulgani during the Fall '09 term at Rensselaer Polytechnic Institute.

### Page1 / 9

09_1_Frequency_response_analysis_E2009 - 1 28150...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online