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ME451 S07 22
Transfer Function: Laplace Transform of Unit ImpulseTransfer Function: Laplace Transform of Unit Impulse Response of the SystemResponse of the System
Input signal:Input signal: Output signal:Output signal:
TakeTake Laplace transform ofLaplace transform of the impulse responsethe impulse response
def:def: Transfer FunctionTransfer Function
Frequency ResponseFrequency Response
InputInput We knowWe know
Complex numbersComplex numbers
Phase shift Phase shift
MagnitudeMagnitude
ME451 S07 24
Frequency ResponseFrequency Response
The Laplace Transform (Appendix B)The Laplace Transform (Appendix B)
Laplace transform converts a calculus problem (the linearLaplace transform converts a calculus problem (the linear differential equation) to an algebra problemdifferential equation) to an algebra problem How to Use it:How to Use it: ^ Take theTake the Laplace transformLaplace transform of a linear differential equationof a linear differential equation ^ Solve the algebra problemSolve the algebra problem Take theTake the Inverse Laplace transformInverse Laplace transform to obtain the solution to theto obtain the solution to the original differential equationoriginal differential equation
def: Laplace transform def: Laplace transform
def: Inverse Laplace transformdef: Inverse Laplace transform
ME451 S07 28
Properties of Laplace TransformsProperties of Laplace Transforms
Shift in FrequencyShift in Frequency
DifferentiationDifferentiation
Properties of Laplace TransformsProperties of Laplace Transforms
Differentiation (Differentiation ( in time domainin time domain ⇔⇔ s in Laplace domain)s in Laplace domain)
Integration (Integration ( in time domainin time domain ⇔⇔ 1/s in Laplace domain)1/s in Laplace domain)
ME451 S07 30
Laplace Transform of Impulse and Unit StepLaplace Transform of Impulse and Unit Step
ImpulseImpulse
Unit StepUnit Step
Unit RampUnit Ramp
PartialPartial--fraction Expansion (Text, page 637fraction Expansion (Text, page 637--641)641)
F(s) is rational,F(s) is rational, realizable condition (realizable condition (d/dtd/dt is notis not realizable)realizable)
zeros zeros polespoles
