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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 ( 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 PoleZero Plot (continued) We showed that if H ( j ) = K ( j z 1 )( j z 2 ) ... ( j z m 1 )( j z m ) . (1) ( j p 1 )( j p 2 ) ... ( j p n 1 )( j p n ) and if each of the vectors from the n system poles to a test point s = j has a magnitude and an angle:  j p i  = = 2 + ( i ) 2 , q i i ( s p i ) = i = tan 1 i i , and similarly for the m zeros  j z i  = r i = 2 + ( i ) 2 , i ( s z i ) = i = tan 1 i i , the value of the frequency response at the point j is r 1 ...r m  H ( j )  = K q 1 ...q n H ( j ) = ( 1 + ... + m ) ( 1 + ... + n ) 1 copyright c D.Rowell 2008 321 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 ( j ) = K n m   and that all of the angles of the vectors approach / 2, with the result lim H ( j ) = ( n m ) 2 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 we note the following Magnitude Response : The magnitude response for the splane is r 1 ...r m  H ( j )  = K q 1 ...q n If any of the r i 0, then H ( j ) 0, and if any q i 0, then H ( j )     If a system has one or more zeros at the origin of the splane (corresponding to a pure differentiation), then the system will have zero gain at = 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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 Spring '08
 DerekRowell
 Mechanical Engineering

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