lecture_32

# lecture_32 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . s - p l a n e p 1 Ð ( j w- p ) p 2 s - p l a n e p 1 p 2 z 1 q q 1 2 f 1 q 1 2 1 q r s j w 1 2 | j w- p | 2 1 1 j w s j w 1 f r 2 Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 32 1 Reading: Nise: 10.1 • • Class Handout: Sinusoidal Frequency Response 1 Frequency Response and the Pole-Zero Plot (continued) We showed that if H ( jω ) = K ( jω − z 1 )( jω − z 2 ) ... ( jω − z m − 1 )( jω − z m ) . (1) ( jω − p 1 )( jω − p 2 ) ... ( jω − p n − 1 )( jω − p n ) and if each of the vectors from the n system poles to a test point s = jω has a magnitude and an angle: | jω − p i | = = σ 2 + ( ω − ω i ) 2 , q i i ( s − p i ) = θ i = tan − 1 ω − − σ ω i i , and similarly for the m zeros | jω − z i | = r i = σ 2 + ( ω − ω i ) 2 , i ( s − z i ) = φ i = tan − 1 ω − − σ ω i i , the value of the frequency response at the point jω is r 1 ...r m | H ( jω ) | = K q 1 ...q n H ( jω ) = ( φ 1 + ... + φ m ) − ( θ 1 + ... + θ n ) 1 copyright c D.Rowell 2008 32–1 s - p l a n e p 1 p 2 z 1 G G 1 2 B 1 q 1 2 1 q r j M I M B r 2 2 1.0.1 High Frequency Response In Lecture 31 we saw that at high frequencies all vectors have approximately the same length, that is and 1 lim H ( jω ) = K ω n − m ω →∞ | | and that all of the angles of the vectors approach π/ 2, with the result π lim H ( jω ) = − ( n − m ) 2 ω →∞ If a system has an excess of poles over the number of zeros ( n > m ) the magnitude of the frequency response tends to zero as the frequency becomes large. Similarly, if a system has an excess of zeros the gain increases without bound as the frequency of the input increases. If n = m the magnitude function tends to a constant K . 1.0.2 Low Frequency Response As ω we note the following → Magnitude Response : The magnitude response for the s-plane is r 1 ...r m | H ( jω ) | = K q 1 ...q n If any of the r i 0, then H ( jω ) 0, and if any q i 0, then H ( jω ) → | | → → | | → ∞ If a system has one or more zeros at the origin of the s-plane (corresponding to a pure differentiation), then the system will have zero gain at ω = 0. Similarly, if the system has one or more poles at the origin (corresponding to a pure integration term in the transfer function), the system has infinite gain...
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lecture_32 - MIT OpenCourseWare http/ocw.mit.edu 2.004...

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