EE3530-3.1

EE3530-3.1 - Chapter 3 Dynamic Response Review of Laplace...

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hapter 3 Dynamic Response Chapter 3 Dynamic Response • Review of Laplace Transforms p • System Modeling Diagrams • Effects of Pole Locations • Time-Domain Specifications • Effects of Zeros and Additional Poles • Stability • Numerical Simulation • Obtaining Models from Experimental Data
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Review of Laplace Transforms Key Facts about Linear Time Invariant Systems: – Response obeys the principle of superposition – Response can be obtained by convolution of the input with the unit impulse response of the system. rinciple of superposition: Principle of superposition: Input u 1 (t) output y 1 (t) put u ) utput y ) Input u 2 (t) output y 2 (t) Then for input u= α u (t) + α u (t) output y= α y (t) + α y (t) 1 1 () 2 2 py 1 y 1 2 y 2
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Example et ( ) be the solution with ( ) i e yk y u t u t += & 11 1 Let () be the solution with (), i.e. d ( ) be the solution with ( ) i e yt y u t ut & 22 2 and () be the solution with (), i.e. hen for any input ( ) ( ) ( ) we can verify at y u t u t + & 2 2 Then for any input () () () we can verify that αα =+ ( ) ( ) ( ) y t satisfies the equation: - - ( )-( ) y u y y k y y u u + + + && & 1 1 2 2 2 2 = ( - )( - )0 y k yu y k α + ++ =
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Signal Decomposition Superposition shows that if we can decompose a general signal into a set of elementary signals, then the response an be obtained by appropriately summing the can be obtained by appropriately summing the responses to all the elementary signals. () i p t ct c e δ + Typical Elementary Signals: impulse and exponential i i Hint: any signal represented by a Laplace Transform U(s), it can be expanded as and the inverse Laplace Transform is i i i c c s p + i p t i i c e +
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Impulse The impulse is represented by () t δ The impulse can be regarded as a signal with huge magnitude in an extremely short duration.
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EE3530-3.1 - Chapter 3 Dynamic Response Review of Laplace...

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