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