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.0s - p l a n ep1Ð( jw- p )p20s - p l a n ep1p2z1qq12f1q121qrsjw12|jw - p |211jwsjw1fr2� � Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 321 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 − z1)(jω − z2) . . . (jω − zm−1)(jω − zm) . (1)(jω − p1)(jω − p2) . . . (jω − pn−1)(jω − pn) and if each of the vectors from the n system poles to a test point s = jω has a magnitude and an angle: |jω − pi| = = �σ2 + (ω − ωi)2 ,qii (s − pi) = θi = tan−1 �ω −−σωii � , and similarly for the m zeros |jω − zi| = ri = �σ2 + (ω − ωi)2 ,i � (s − zi) = φi = tan−1 �ω −−σωii � , the value of the frequency response at the point jω is r1 . . . rm |H(jω)| = K q1 . . . qn � H(jω) = (φ1 + . . . + φm) − (θ1 + . . . + θn) 1copyright cD.Rowell 2008 32–10s - p l a n ep1p2z1qq12f1q121qrjwswfr221.0.1 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 ωn−mω→∞ | | and that all of the angles of the vectors approach π/2, with the result π lim � H(jω) = −(n − m)2ω→∞ 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 0 we note the following → Magnitude Response: The magnitude response for the s-plane is r1 . . . rm |H(jω)| = Kq1 . . . qn If any of the ri 0, then H(jω) 0, and if any qi 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 at zero frequency. ωlim 0 |H(jω)| = 0 if there are zeros at the origin →lim H(jω) if there are poles at the origin ω→0 | | = ∞ r1 . . . rm ωlim 0 |H(jω)| = Kq1 . . . qn otherwise →32–20sjw1jwpp121qw i n c r e a s i n gi n t h e p r o x i m i t y o f p t h e l e n g t h q d e c r e a s e s t o am i n i m u m .11Phase Response: � H(jω) = (φ1 + . . . + φm) − (θ1 + . . . + θn) As ω 0: → All real-axis l.h.p. poles and zeros contribute 0 rad. to the phase response. • Each complex conjugate pole or zero pair contributes a total of 2π rad. to the phase • response (effectively adding 0 rad. to the total response). A pole at the origin (s = 0 + j0) contributes −π/2 rad. to the phase response. • A zero at the origin (s = 0 + j0) contributes +π/2 rad. to the phase response. • A r.h.p real zero contributes +π rad. to the phase response. • The low frequency phase response is therefore π lim H(jω) = −(N − M) + Lπ rad. ω 0 � 2→where N is the number of poles at the origin, M is the number of zeros at the origin, and L is the number of r.h.p. real zeros. 1.0.3 Behavior in the Proximity of Poles and Zeros Close to the Imaginary Axis Consider a second-order system with a damping ratio ζ � 1, so that the pair of complex conjugate poles are located close to the imaginary axis. K |H(jω)| = q1q2 � H(jω) = −(θ1 + θ2) In this case there are a pair of vectors connecting the two poles to the imaginary axis, and the following conclusions may b e drawn by noting how the lengths and angles of the vectors change as the test frequency moves up the imaginary axis: As the input frequency is increased and the test point on the imaginary axis approaches the pole, one of the vectors (associated with the pole in the second quadrant) decreases in length, and at some point reaches a minimum. 32–30sjw1jwjw2ÐH ( jw)ww|H ( jw)|00- 1 8 0w1wpp12( a )( b )Because q1 appears in the denominator of the magnitude function, over this range there is an increase in the value of H(jω) .| |• If a system has a pair of complex conjugate poles close to the imaginary axis, the magnitude of the frequency response has a “peak”, or resonance at frequencies in the proximity of the pole. If the pole pair lies directly upon the imaginary axis, the system exhibits an infinite gain at that frequency. • Similarly, if a system has a pair of complex conjugate zeros close to the imaginary axis, the frequency response. Over this range has a “dip” or “notch” in its magnitude function at frequencies in the vicinity of the zero. Should the pair of zeros lie directly upon the imaginary axis, the response is identically zero at the frequency of the zero, and the system does not respond at all to sinusoidal excitation at that frequency. Similarly in the proximity of the pole there is a rapid change of the angle θ1 associated with the pole p1. 2 Logarithmic (Bode) Plots In system dynamic analyses, frequency response characteristics are almost always plotted using logarithmic scales. In particular, the magnitude function H(jω) is plotted against | |frequency on a log-log scale, and the phase � H(jω) is plotted on a linear-log scale. For example, the frequency response functions of a typical first-order system τdy/dt + y = u(t) is plotted below on (a) linear axes, and (b) logarithmically scaled axes. 32–40 . 0 10 . 0 30 . 10 . 3131 0 3 01 0 00 . 0 10 . 0 3 0 . 1 0 . 3131 03 01 0 0- 9 0- 8 0- 7 0- 6 0- 5 0- 4 0- 3 0- 2 0- 1 001 0w tN o r m a l i z e d f r e q u e n c yN o r m a l i z e d f r …
View Full Document
Unlocking...