FrequencyResponseLecture

# FrequencyResponseLecture - ME 279 Automatic Control of...

This preview shows pages 1–10. Sign up to view the full content.

Frequency Response ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers

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

View Full Document
Introduction 2 For a linear, time–invariant system, if the input is a sinusoid with a frequency ϖ , the steady–state output will be a sinusoid with a frequency ϖ . However, The output amplitude and phase will, in general, be different than those of the input. The frequency response of a dynamic system can be represented by a phasor Frequency Response Dr. Robert G. Landers input frequency response ( 29 ( 29 ( 29 ( 29 cos j M t M Me φ ϖ φ ϖ + = = output frequency response ( 29 ( 29 i i M ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 o o i i M M M = + ( 29 ( 29 M ( 29 ( 29 i i M ( 29 ( 29 o o M
Introduction 3 Frequency Response Dr. Robert G. Landers magnitude frequency response G(s) U(s) Y(s) ( 29 M ϖ phase frequency response frequency response ( 29 φ ϖ ( 29 ( 29 M A dynamic system is given above. The input is ( 29 ( 29 ( 29 ( 29 2 2 1 cos sin cos tan cos i i u t A t B t B A B t M t A φ - = + = + - = - ÷ ÷

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

View Full Document
Introduction 4 Frequency Response Dr. Robert G. Landers The input in the Laplace domain is ( 29 2 2 As B U s s ϖ + = + The output is ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 1 P.F.E. of G s 2 2 i G i G j j i G i G As B Y s G s s j s j M M e M M e s j s j φ - + + + = + - = + + + - where ( 29 ( 29 G G M G j G j = = ∠ If G(s) is stable, its terms approach zero in the steady state
Introduction 5 The steady state output in the Laplace domain is Frequency Response Dr. Robert G. Landers ( 29 ( 29 ( 29 1 2 i G i G j j ss i G e e Y s M M s j s j φ ϖ - + + = + + - The steady state output in the time domain is ( 29 ( 29 [ ] [ ] cos ss i G i G i i G G y t M M t M M = + + = A system’s frequency response is G(j ϖ )

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

View Full Document
Bode Plots 6 The magnitude Bode plot is the magnitude of G(j ϖ ) as a function of ϖ . The phase Bode plot is the phase (angle) of G(j ϖ ) as a function of ϖ . Frequency Response Dr. Robert G. Landers ( 29 ( 29 ( 29 ( 29 ( 29 1 1 k m n K s z s z G s s s p s p + + = + + L L The system transfer function is The magnitude, in dB, of G(j ϖ ) is ( 29 ( 29 10 10 10 1 10 10 10 1 10 20log 20log 20log 20log 20log 20log 20log k m n G j K j z j z j j p j p ϖ = + + + + + - - + - - + L L
Bode Plots 7 The magnitude is typically given in decibels (dB) and the phase is typically given in degrees. Frequency Response Dr. Robert G. Landers The phase of G(j ϖ ) is ( 29 G j K ϖ = ∠ ( 29 ( 29 ( 29 ( 29 ( 29 0 1 1 k m n j z j z j j p j p + ∠ + + + ∠ + - -∠ + - -∠ + L L

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

View Full Document
Bode Plots of G(s) = s+a 8 Frequency Response Dr. Robert G. Landers ( 29 G j j a a j ϖ = + = + Evaluating G(s) at s = j ϖ ( 29 ( 29 2 2 10 20log G j a = + ( 29 1 tan G j a - = ÷ Magnitude of G(j ϖ ) Phase of G(j ϖ )
Bode Plots of G(s) = s+a 9 Frequency Response Dr. Robert G. Landers ( 29 ( 29 ( 29 10 10 10 low 20log 0 20log 2 45 high 20log 90 a a a a aj a j ϖ ° = + ° ° Transfer Frequency Magnitude Phase Function

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.

## This note was uploaded on 02/01/2012 for the course MECH ENG 279 taught by Professor Landers during the Spring '11 term at Missouri S&T.

### Page1 / 54

FrequencyResponseLecture - ME 279 Automatic Control of...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online