Laplace TransformLaplace Transform
Submitted By:Md. Al Imran
BhuyanID: 143-19-1616Department of
ETEDaffodil
International University
Submitted to:Engr. Md. Zahirul Islam
Senior LecturerDepartment of ETE
Daffodil International
University
The French NewtonPierre-Simon Laplace
Developed mathematics in astronomy, physics, and statistics
Began work in calculus which led to the Laplace Transform
Focused later on celestial mechanics
One of the first scientists to suggest the existence of black holes
History of the Transform
Euler began looking at integrals as solutions to differential equations in the mid 1700’s:
Lagrange took this a step further while working on probability density functions and looked at forms of the following equation:
Finally, in 1785, Laplace began using a transformation to solve equations of finite differences which eventually lead to the current transform
Definition
The Laplace transform is a linear operator that switched a function f(t) to F(s).
Specifically: Go from time argument with real input to a
complex angular frequency input which is complex.
Laplace Transform Theory Laplace Transform Theory
•General Theory
•Example
THE SYSTEM FUNCTION
We know that the output y(t) of a continuous-time LTI system equals theconvolution of the input x ( t ) with the impulse response h(t); that is, y(t)=x(t)*h(t)For Laplace, Y(s)=X(s)H(s)
H(s)=Y(s)/X(s)
Properties of ROC of Laplace TransformThe range variation of σ for which the Laplace transform converges is called region of convergence.ROC doesn’t contains any polesIf x(t) is absolutely integral and it is of finite duration, then ROC is entire s-plane.If x(t) is a right sided sequence then ROC : Re{s} > σo.If x(t) is a left sided sequence then ROC : Re{s} < σo.If x(t) is a two sided sequence then ROC is the combination of two regions.
Real-Life ApplicationsReal-Life Applications
Semiconductor mobilitySemiconductor mobilityCall completion in Call completion in wireless networkswireless networksVehicle vibrations on Vehicle vibrations on compressed railscompressed railsBehavior of magnetic Behavior of magnetic and electric fields and electric fields above the atmosphereabove the atmosphere
Ex. Semiconductor MobilityEx. Semiconductor Mobility
MotivationMotivation semiconductors are commonly made semiconductors are commonly made
with super lattices having layers of with super lattices having layers of differing compositionsdiffering compositions
need to determine properties of need to determine properties of carriers in each layer carriers in each layer
concentration of electrons and holesconcentration of electrons and holesmobility of electrons and holes mobility of electrons and holes
conductivity tensor can be related to conductivity tensor can be related to Laplace transform of electron and hole Laplace transform of electron and hole densitiesdensities
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