ME451 S07ME451 S0722Transfer Function: Laplace Transform of Unit Impulse Transfer Function: Laplace Transform of Unit Impulse Response of the SystemResponse of the SystemInput signal:Input signal:Output signal:Output signal:Take Take Laplace transform ofLaplace transform ofthe impulse responsethe impulse responsedef: def: Transfer FunctionTransfer FunctionME451 S0723Frequency ResponseFrequency ResponseInputInputWe knowWe knowComplex numbers Complex numbers Phase shiftPhase shiftMagnitudeMagnitudeME451 S07ME451 S0724Frequency ResponseFrequency ResponseME451 S0725The Laplace Transform (Appendix B)The Laplace Transform (Appendix B)Laplace transform converts a calculus problem (the linear Laplace transform converts a calculus problem (the linear differential equation) to an algebra problemdifferential equation) to an algebra problemHow to Use it: How to Use it: Take the Take the Laplace transformLaplace transformof a linear differential equationof a linear differential equationSolve the algebra problemSolve the algebra problemTake the Take the Inverse Laplace transform Inverse Laplace transform to obtain the solution to the to obtain the solution to the original differential equationoriginal differential equationdef: Laplace transformdef: Laplace transformdef: Inverse Laplace transformdef: Inverse Laplace transformME451 S07ME451 S0726The Laplace Transform (Appendix B)The Laplace Transform (Appendix B)Laplace Transform of a function Laplace Transform of a function f(tf(t))Convolution integralConvolution integralME451 S0727Properties of Laplace Transforms (page 641Properties of Laplace Transforms (page 641--643)643)LinearityLinearityTime DelayTime DelayNonNon--rational functionrational functionME451 S07ME451 S0728Properties of Laplace TransformsProperties of Laplace TransformsShift in FrequencyShift in FrequencyDifferentiationDifferentiationME451 S0729Properties of Laplace TransformsProperties of Laplace TransformsDifferentiation ( in time domain Differentiation ( in time domain ⇔⇔s in Laplace domain)s in Laplace domain)Integration (Integration (in time domain in time domain ⇔⇔1/s in Laplace domain)1/s in Laplace domain)ME451 S07ME451 S0730Laplace Transform of Impulse and Unit StepLaplace Transform of Impulse and Unit StepImpulseImpulseUnit StepUnit StepME451 S0731Unit RampUnit RampME451 S07ME451 S0732Exponential FunctionExponential FunctionME451 S0733Sinusoidal FunctionsSinusoidal FunctionsME451 S07ME451 S0734PartialPartial--fraction Expansion (Text, page 637fraction Expansion (Text, page 637--641)641)F(sF(s) is rational, ) is rational, realizable condition (realizable condition (d/dtd/dtis not is not
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