Laplace Transform

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Laplace Transform
The Laplace Transform is one of the mathematical
tools for solving ordinary linear differential
equation. - The homogeneous equation and the
particular Integral are solved in one
operation. - The Laplace transform converts the
ODE into an algebraic eq. in s (? j? plane)
domain. It is then possible to manipulate the
algebraic eq. by simple algebraic rules (-x?) to
obtain the solution in the s- domain. The final
solution is obtained by taking the inverse
Laplace transform.
Definition of Laplace Transform
The Laplace transform of f(t) is defined as
s is referred to as the Laplace operator, which
is a complex variable s (? j? plane)
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Laplace Transform
Ex Let f(t) be a unit step function that is
defined to have a value of unity for t gt 0 and a
zero value for t lt 0.
Known f(t) u(t)
Find Laplace Transform of f(t)
Solution
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Laplace Transform
Ex Consider the exponential function
Where a is a constant
Known f(t) e-at
Find Laplace Transform of f(t)
Solution
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Inverse Laplace Transform
Definition of Inverse Laplace Transform
The Inverse Laplace transform of F(s) is defined
as
c is a real constant that is greater than the
real parts of all singularities of F(s)
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Laplace Transform Table
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Laplace Transform Table
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Important Theorems of the Laplace Transform
1) Multiplication by a constant
Where F(s) is the Laplace transform of f(t)
2) Sum and Difference
Where F1(s) and F2(s) is the Laplace transform of
f1(t) and f2(t)
3) Differential
In general, for higher-order derivatives,
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Laplace Transform Table
4) Integration
In general, for n th-order integration,
5) Shift in Time
6) Initial-Value Theorem
7) Final-Value Theorem
If sF(s) is analytical on the imaginary axis and
in the right half of the s-plane then,
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Laplace Transform
Ex Consider the following function, and find
steady state value of f(t)
Known F(s)
Find f(t) as t -gt ??
Solution
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Inverse Laplace Transform by Partial-Fraction
Expansion
When the Laplace transform solution of a
differential equation is a rational function in
s, it can be written as
P(s) and Q(s) is a polynomial of s, it is assumed
that the order of Q(s) in s is greater than that
of P(s). The polynomial Q(s) may be written
Where a1, , an are real coefficient. The
solution of Q(s) 0 or poles of X(s) (the points
in which X(s) is not analytic) are either real or
in complex-conjugate pairs, in simple or multiple
order.
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Partial-Fraction Expansion when all the
poles of X(s) are simple
Where -s1, -s2, , -sn are real or imaginary
numbers. Applying the partial fraction expansion
technique,
The coefficient Ks1, Ks2, , Ksn (i 1,2,,n)
are determined by multiplying both side of Eqs.
By the factor (s si) and then setting s equal
to si.
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Laplace Transform
Ex Consider the function
Known X(s)
Find partial fraction coefficient
Solution
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Partial-Fraction Expansion when some poles
of X(s) are multiple
If r of the n poles of X(s) are identical or the
pole of s -si is of multiplicity r,
The X(s) can be expanded as
(n - r) terms of simple poles
r terms of repeated poles
The coefficient Ks1, Ks2, , Ksn can be determined
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Partial-Fraction Expansion when some poles
of X(s) are multiple
The coefficient for multiple-order poles can be
determined
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Laplace Transform
Ex Consider the function
Known X(s)
Find partial fraction coefficient
Solution
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Laplace Transform
Ex Consider the differential equation
The initial condition
and
Solution
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Laplace Transform
Ex Consider the differential equation
The initial condition
and
Solution
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Transfer function
Transfer function of a linear time-invariant
system is defined as the Laplace transform of the
impulse response, with all the initial conditions
set to zero.
Let G(s) denote the transfer function of a system
with input x(t) and output y(t). Then, the
transfer function G(s) is defined as
The transfer function G(s) is related to the
Laplace transform of the input and output through
With all initial conditions set to zero, where
Transfer function defines the mathematical
operation that the measurement system performs on
input to yield the time response of the system
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Transfer function
nth Order ordinary linear differential equation
with constant coefficient
To obtain the transfer function of the linear
system, we simply take the Laplace transform and
assume zero initial conditions.
The transfer function between x(t) and y(t) is
given by
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Transfer function
For the first-order system
Laplace Transform
Where Y(s) and X(s) Laplace Transform of y(t)
and x(t)
This can be rewritten
Zero input response
Zero state response
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Transfer function
The frequency response of a system can be derived
from the transfer function by substituted s with
i?
For the second-order system
Transfer function
Frequency response
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Coupled system
When a measurement system consists of more than
one instrument, measurement system behavior can
become more complicated.
y1(t)
Measurement system 1 G1(s)
Measurement system 2 G2(s)
x(t)
y(t)
Y1(s)
G1(s)
G2(s)
X(s)
Y(s)
Equivalent system
The overall transfer function of the combined
system is the product of the transfer function of
each system