Laplace Transform (2)

Hany FerdinandoDept. of Electrical Eng.

Petra Christian University

Laplace Transform (2) - Hany Ferdinando 2

Overview

Unilateral Laplace Transform Two-sided Laplace Transform Application in electric circuit Application in differential equation Stability Frequency response analysis Laplace transform for periodic signal

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Unilateral Laplace Transform

This is applied for causal function only The general form is from 0 to ∞ for the

time variable It is only positive part of the whole

function

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Inverse (unilateral only)

Make the form of the function in s-domain as sum of rational function use partial fraction expansion

From the table, find the formula with the highest similarity

Use the properties to help you to find the result

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Inverse (two-sided)

This is applied for non causal function The RoCs are needed Make the form of the function in s-

domain as sum of rational function use partial fraction expansion

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Inverse (two-sided)

The location of the poles of the F(s) with respect to the RoC determines whether a given singularity refers to a positive or negative region Poles to the left of RoC give rise to a

positive time portion of f(t) Poles to the right of RoC give rise to a

negative time portion of f(t)

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Inverse (two-sided)

Pole ‘a’ lies to left of the RoC it give rise the positive time portion of f(t)

Pole ‘b’ lies to right of the RoC it give rise the negative time portion of f(t)

a b

a < Re(s) < b

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Inverse (two-sided)

For the positive part, use the unilateral approach

For negative part, use the following chart…

f(-t) F(-s)

f(t) F(s)

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Application in Electric Circuit

Transform all components to s-domain R R L sL C 1/(sC) Source use table

Use DC analysis to write the standard equation (you can use node, mesh or superposition)

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Application in Electric Circuit

Solve the equation in ‘s’ Use inverse Laplace transform to get

the result in time domain (do not forget to do this!!!)

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Application in Electric Circuit

V1

10 V 10 Hz 0Deg

R1

10 Ohm

L11.0H

Calculate the current which flows in the circuit!

Source: 20 cos (3t+1)

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Application in differential equation

Solving differential equation with ordinary way sometimes is difficult

We can use Laplace transform to simplify it

The differential equation is transformed to s-domain and then solve it

Do not forget to inverse the result…!!!

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Application in differential equation

)0(...)0()0()()( )1()1(21 nnnn

n

n

ffsfssFsdt

tfd

Use the following property…

Apply that property to solve this…

)()()()1( tuetayty t

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Stability

What is stability?

Is it important? Why?

B

A

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Stability

Simple poles of the form c/(s+a) Complex conjugates poles of the form

c/[(s+)2+2] Complex conjugates poles of the form

c/(s2+2)

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Frequency Response Analysis

It is evaluated along the jw axis Substitute ‘s’ with jw and solve it as

you do in the Fourier analysis

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Periodic Signal

If f(t) is periodic signal with period T, then the Laplace transform of f(t) is defined as

sT

Tst

e

dtetf

1

)(0

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Periodic Signal

Find Laplace transform for f(t) Then calculate the voltage across the

inductor

T/2 T

1

-1

f(t) V1

10 V 10 Hz 0Deg

R1

10 Ohm

L11.0H