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2012/12/24
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Introduction to Laplace Transforms•Introduction•Definition of
the Laplace Transform•Properties of the Laplace Transform•The
Inverse Laplace Transform•The Convolution Integral•Application to
Integrodifferential Equations•Summary
Introduction•A very powerful tool for analyzing
circuits•Differential equations are converted into
algebraic equations•Laplace transform method
–Transformation from time domain to frequencydomain
–Obtain the solution–Transformation from frequency domain to
time
domain by applying inverse Laplace transform
•It provides the total response (natural/forced)in one single
operation
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2012/12/24
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Definition of the Laplace Transform
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Convergenceregion(>c)
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2012/12/24
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Cont’d
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Determine L[e-atu(t)]
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2012/12/24
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Properties of the Laplace Transform
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6
Time Shift Property
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Time Differentiation Property
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Time Integration Property
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Frequency Differentiation Property
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10
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11
Final-Value Theorem
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Summary
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Summary
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2012/12/24
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The Inverse Laplace Transform
•Zeros: the roots of N(s) = 0•Poles: the roots of D(s) = 0•Steps
to find the inverse Laplace transform.
–Decompose F(s) into simple terms using partialfraction
expansion.
–Find the inverse of each term.
)()(
)(sDsN
sF
Case 1: Simple Poles
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21
21
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Simple Poles (Cont’d)
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|)()(ssSet
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:findingformethodResidue
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Repeated Poles (Cont’d)
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Example 1
)(782)(7
)1()3(129
)2(12
)()3(
8)1)(2(
124)3(
12)()2(
2)3)(2(
12)3)(2(
12)(
givesmethodresidueApplying32
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)(if)(Find
32
3
2
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2
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Example 2
)(2213141)(
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2012/12/24
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Example 3
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The Convolution Integral•Useful in finding the response y(t) of
a system to an
excitation x(t) when the system impulse response h(t)is
defined.
)()()(or
)()()(
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dtxHtxHty
thtHdtxtx
)()()()(
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system,linearaofpropertylinearityApply the
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18
dtxtx
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ii
ii
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(Cont’d)
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t)()(
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)()()(.or0for0)(then
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Properties of Convolution Integral
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t)()()()()(.7
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)()()()()()()(:veDistributi.2
)()()()(:eCommutativ.1
'''
00
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20
)()()()(
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Example
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2012/12/24
21
• Folding: Take the mirror image of h() about theordinate axis
to obtain h().
• Displacement: Shift or delay h() by t to obtainh(t).
• Multiplication: Find the product of h(t) and x().•
Integration: For a given time t, calculate the area
under the product h(t) x() for 0
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22
Example 1 (Cont’d)
0 < t < 1 1 < t < 2 2 < t < 3
3 < t < 4 t > 4
Step 3
)()( 21 txtx
Solving Linear, Time-Invariant,Differential Equations
•Steps–By taking the Laplace transform, the time-
domain differential equation is converted into ans-domain
algebraic equation.
–Obtain the solution of the s-domain algebraicequation.
–The time-domain solution is obtained byapplying inverse Laplace
transform.
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2012/12/24
23
Example
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