l26 - 16.06 Lecture 26 Frequency Response Analysis November...

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16.06 Lecture 26 Frequency Response Analysis November 5, 2003 Today’s Topics: 1. Steady state system responses to sinusoidal inputs 2. Second order system example
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Consider the following experiment in which a stable linear system is driven by a sinusoidal input The Laplace Transform of this input is- If G ( s ) is the system transfer function then the Laplace Transform of the output is 2
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which can be written as- Where N ( s ) and D ( s ) are the numerator and denominator polynomials of G ( s ) . Then Where ± z 1 , ± z 2, .... are the system zeros and ± p 1 , ± p 2 , .... are the system poles A partial fraction expansion of C ( s ) obtains 3
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The inverse Laplace Transform would then be Since the system is stable all of the system poles lie in the left half plan. Hence all of the p ' s have positive real parts so each of the ± p i t terms of the form e will decay to zero as time increases. In particular, if we wait for a long time after startup then This is the steady state response to the sinusoidal input at the frequency Z .
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