topic-laplace transformation presented by harsh patel 130460111012
TRANSCRIPT
Topic-laplace transformationPresented by Harsh PATEL
130460111012
•Transforms -- a mathematical conversion from one way of thinking to another to make a problem easier to solve
linear differential equation
timedomainsolution
Laplacetransformed
equation
Laplacesolution
Laplace domain orcomplex frequency domain
algebra
Laplace transform
inverse Laplace transform
Convert time-domain functions and operations into frequency-domain f(t) F(s) (tR, sC Linear differential equations (LDE) algebraic
expression in Complex plane Graphical solution for key LDE characteristics Discrete systems use the analogous z-
transform
0)()()]([ dtetfsFtf stL
)(lim)(lim
)(lim)0(
)()()
)(1)(
)(
)0()()(
)()()]()([
0
0
2121
0
2121
ssFtf-
ssFf-
sFsFdτ(ττ)f(tf
dttfss
sFdttfL
fssFtfdt
dL
sbFsaFtbftafL
st
s
t
t
theorem valueFinal
theorem valueInitial
nConvolutio
nIntegratio
ationDifferenti
calingAddition/S
SIMPLE TRANSFORMATIONS
Impulse -- (to)
F(s) =
0
e-st (to) dt
= e-sto
f(t)
t
(to)
Step -- u (to)
F(s) =
0
e-st u (to) dt
= e-sto/s
e-at
F(s) =
0
e-st e-at dt
= 1/(s+a)
f1(t) f2(t)
a f(t)
eat f(t)
f(t - T)
f(t/a)
F1(s) ± F2(s)
a F(s)
F(s-a)
eTs F(as)
a F(as)
Linearity
Constant multiplication
Complex shift
Real shift
Scaling
Most mathematical handbooks have tables of Laplace transforms
PARTIAL FRACTION EXPANSION
Definition -- Partial fractions are several fractions whose sum equals a given fraction
Purpose -- Working with transforms requires breaking complex fractions into simpler fractions to allow use of tables of transforms
32)3()2(
1
s
B
s
A
ss
s Expand into a term for each factor in the denominator.
Recombine RHS
Equate terms in s and constant terms. Solve.
Each term is in a form so that inverse Laplace transforms can be applied.
)3()2(
2)3(
)3()2(
1
ss
sBsA
ss
s
3
2
2
1
)3()2(
1
ssss
s
1BA 123 BA
0)0(')0(2862
2
yyydt
dy
dt
yd • ODE w/initial conditions
• Apply Laplace transform to each term
• Solve for Y(s)
• Apply partial fraction expansion
• Apply inverse Laplace transform to each term
ssYsYssYs /2)(8)(6)(2
)4()2(
2)(
ssssY
)4(4
1
)2(2
1
4
1)(
sss
sY
424
1)(
42 tt eety
When the factors of the denominator are of the first degree but some are repeated, assume unknown numerators for each factor If a term is present twice, make the
fractions the corresponding term and its second power
If a term is present three times, make the fractions the term and its second and third powers
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