- The Laplace transform is a linear operator that transforms a function of time (f(t)) into a function of complex frequency (F(s)). It was developed from the work of mathematicians like Euler, Lagrange, and Laplace. - The Laplace transform has many applications in fields like semiconductor mobility, wireless network call completion, vehicle vibration analysis, and modeling electric and magnetic fields. It allows transforming differential equations into algebraic equations that are easier to solve. - For example, in semiconductors with varying material layers, the Laplace transform can relate the conductivity tensor to the Laplace transforms of electron and hole densities, enabling the determination of key properties like carrier concentration and mobility in each layer.

1. Laplace TransformLaplace Transform
Submitted By:
Md. Al Imran Bhuyan
ID: 143-19-1616
Department of ETE
Daffodil International
University
Submitted to:
Engr. Md. Zahirul Islam
Senior Lecturer
Department of ETE
Daffodil International
University

2. The French Newton
Pierre-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

3. 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

4. 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.

5. Laplace Transform TheoryLaplace Transform Theory
•General Theory
•Example

6. THE SYSTEM FUNCTION
We know that the output y(t) of a continuous-time
LTI system equals the
convolution 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)

7. Properties of ROC of Laplace Transform
The range variation of for which the Laplaceσ
transform converges is called region of convergence.
ROC doesn’t contains any poles
If 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

8. Real-Life ApplicationsReal-Life Applications
Semiconductor mobilitySemiconductor mobility
Call completion inCall completion in
wireless networkswireless networks
Vehicle vibrations onVehicle vibrations on
compressed railscompressed rails
Behavior of magneticBehavior of magnetic
and electric fieldsand electric fields
above the atmosphereabove the atmosphere

9. Ex. Semiconductor MobilityEx. Semiconductor Mobility
MotivationMotivation

semiconductors are commonly madesemiconductors are commonly made
with super lattices having layers ofwith super lattices having layers of
differing compositionsdiffering compositions

need to determine properties ofneed to determine properties of
carriers in each layercarriers in each layer
concentration of electrons and holesconcentration of electrons and holes
mobility of electrons and holesmobility of electrons and holes

conductivity tensor can be related toconductivity tensor can be related to
Laplace transform of electron and holeLaplace transform of electron and hole
densitiesdensities