FrequencyResponseLecture

FrequencyResponseLecture - ME 279 Automatic Control of...

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

View Full Document Right Arrow Icon
Frequency Response ME 279 Automatic Control of Dynamic Systems Dr. Robert G. Landers
Background image of page 1

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

View Full DocumentRight Arrow Icon
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
Background image of page 2
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 - = + = + - = - ÷ ÷
Background image of page 3

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

View Full DocumentRight Arrow Icon
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
Background image of page 4
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 )
Background image of page 5

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

View Full DocumentRight Arrow Icon
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
Background image of page 6
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
Background image of page 7

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

View Full DocumentRight Arrow Icon
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 )
Background image of page 8
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
Background image of page 9

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

View Full DocumentRight Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

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 Right Arrow Icon
Ask a homework question - tutors are online