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Laplace Transform

  • The idea of Laplace transform is to solve an equation containing differential and
    integral terms by transforming the equation into the S-space
  • This is done by applying the Laplace transforming function as follows:-

clip_image001[14]{ f(t)} = clip_image002[6]

  • The resulting expression is a function of s which we say F(s)
  • Normally, people do not have to solve the entire formula as we can use standard
    Laplace transformations. Below is the table :-

 

Time Function f(t) 
  f(t) = clip_image001[15]-1{F(s)}

Laplace Transform of f(t)
F(s) = clip_image001[16]{ f(t)}

F1

clip_image003[10]      s > 0

t (unit-ramp function)

clip_image004[6]     s > 0

tn (n, a positive integer)

clip_image005[6]    s > 0

eat

clip_image006[5]   s > a

sin ωt

clip_image007[5]    s > 0

cos ωt

clip_image008[5]   s > 0

tng(t), for n = 1, 2, ...

clip_image009[5]

t sin ωt

clip_image010[5]      s > |ω|

t cos ωt

clip_image011[5]    s > |ω|

g(at)

clip_image012[5]       Scale
property

eatg(t)

G(sa)       Shift property

eattn, for n = 1, 2, ...

clip_image013[5]     s > a

te-t

clip_image014[5]     s > -1

1 − e-t/T

clip_image015[5]    s > -1/T

eatsin ωt

clip_image016[5]    s > a

eatcos ωt

clip_image017[5]    s > a

u(t)

clip_image003[11]     s > 0

u(ta)

clip_image018[5]    s > 0

u(ta)g(ta)

e-asG(s)
Time-displacement theorem

g'(t)

sG(s) − g(0)

g''(t)

s2 • G(s) − s • g(0) − g'(0)

g(n)(t)

sn • G(s) sn-1 • g(0) sn-2 • g'(0) − ... − g(n-1)(0)

clip_image019[5]

clip_image020[5]

clip_image021[5]


  • The last 5 on the table shows the true power of Laplace transform. A complex
    integral calculation then would be impossible to be performed in the time
    domain, when transformed into the S domain can be easily represented
  • Laplace transforms have several properties which are:-

·        Property1 - Constant Multiple

·        If
a is a constant and f(t) is a function of t, then L{a f(t)} = a L{f(t)}

·        Example
{7 sin t} = 7{sin t}

·        Property2 - Linearity Property

·        If
a and b are constants while f(t) and g(t) are functions of t, then

L{a f(t) + b g(t)} = a L{f(t)}
+ b L{g(t)}

·        Example
L{3t + 6t2 } = 3 L{t} + 6 L{t2}

·        Property3. Change of Scale Property

·        If
L{f(t)} = F(s) then L{f(at)} = 1/a F(s/a)

·        Example
L{F(5t) = (1/5)F(s/5)

·        Property4. Shifting Property (Shift Theorem)

·        L{exp(at)f(t)}
= F(s − a)

·        Example
L{exp(3t)f(t)} = F(s − 3)

·        Property5. Differential transformation

·        L{tf(t)}
= -F’(s)

·        Property6.

·        The Laplace transforms of the real (or imaginary) part of a complex function is
equal to the real (or imaginary) part of the transform of the complex function.

·        Let Re denote the real part of a complex function C(t) and Im denote the imaginary
part of C(t), then L{Re[C(t)]} = Re L{C(t)} and L{Im[C(t)]} = Im L{C(t)}


 

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