Chapter 7
Mathematics
CHAPTER 7 Laplace Transform Introduction For continuous-time signals, Laplace Transform (LT) is a way of frequency domain representation of a continuous – time signal at a generalized frequency and is generalization of CTFT (Continuous Time Fourier Transform) for continuous – time signal.
Laplace Transform Definition Let f(t) be a function of time. It is assumed that the value of the function is zero for t < 0, and the function is continuous for all values of t from 0 to the Laplace transform of f(t) is given by , ( )-
( )
( )
∫
---------(A)
S is the complex frequency and is given by s = imaginary part ).
= kernel of the function
is Laplace transform operator
+ jω (Here
is the real part of s , ω is the
Equation (A) evaluates unilateral / one sided LT. Also bilateral /two-sided LT can be evaluated for two-sided functions as below, , ( )-
( )= ∫
e
f(t)dt
---------(B)
Inverse Laplace Transform Given F(s),
, ( )-
( ) can be found by one of the following,
a. Inspection b. Using properties of LT c. By partial fractions d.
( )
, ( )-=
∫
(s)e ds
(i.e using the concept of the analytic function)
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Chapter 7
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Important Properties of Laplace transform Given , [f(t)] = F(s) 1. Homogeneity [k f(t)] = k F(s) 2. Superposition [f1(t) ] = F1 (s) and
[f2(t)] = F2 (s)
[f1(t) + f2(t) ] = F1(s) + F2 (s) and L {a f1(t) + b f2(t)} = a F1(s) + bF2(s) ( 3.
th
ge eity
)
Frequency shift [e-at f(t) ] = F(s + a) [eat f(t) ] = F(s - a)
4. Time shift [f(t – to)] = e-st F(s) 5. Differentiation in Time domain If
[ f(t) ] = F(s), then
[
f(t) ] = s F(s) – f(0)
Differentiating in time domain is equivalent to multiplying by s in frequency domain. Similarly,
[
f(t) ] = s F(s) –s f(0) - f (0)
Where f (0) is the initial value of [
f(t) ] at t = 0
6. Integration in Time domain 0∫ f(t)dt1
( )
and
2∫
( )+
f(t)dt3
∫
f(t)dt
Integration in time domain is equivalent to division by s in frequency domain. 7. Differentiation in frequency domain If
[ f(t) ] = F(s) , then
[ t f(t) ] =
( )
and
*t f(t)+
(
)
F(s))
Differentiation in frequency domain is equal to multiplication by t in time domain. THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Chapter 7
Mathematics
8. Integration in frequency domain If
[f(t)] = F(s), then 0
()
1 = ∫
(s) ds
Integration in frequency domain is equal to division by t in time domain. 9. Initial Value Theorem If f(t) and its derivative are Laplace transformable, then i
f(t)
i
s (s)
Note This theorem does not apply to the rational function F(s) in which the order of numerator polynomial is equal to or greater than the order of denominator polynomial. 10. Final Value Theorem If f(t) and its derivative f (t) are Laplace transformable, then i
f(t)
i
s (s)
Note For applying final value theorem, it is required that all the poles of s- plane (strictly) i.e. poles at
( ) be in the left half of
axis also not allowed.
11. Convolution theorem Let f1(t) and f2(t) be functions of time whose Laplace transforms are F1(s) and F2 (s) respectively. Then, , ( ) , ( )
( )-
( )
( )-
( )
( )
{* denotes convolution}
( )
{* denotes convolution}
12. Laplace transform of the periodic function If f(t) is periodic function with period T, then (f(t)) = Where
(
(s) = ∫ e
)
.
(s)
f(t)dt
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Chapter 7
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Laplace Transform of standard function LT of some of the standard functions are summarized in table below. Table 1. Laplace Transform of standard function S. No
Functions
Laplace Transform
( ) 1
1 2 3 4 5
si
6
c s
7
si h
8
c sh
+ +
9 10
c s (
) +
(
) +
11 12 13
√
√
6. Applications 1. LT is generalization of CTFT for continuous-time signals and hence signal can be characterized at any generalized frequency. 2. LT is helpful to perform transient and steady state analysis of any LTI system for any arbitrary input and initial conditions. 7. Impulse Function S(t) 0
+
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Chapter 7
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2. Unit Function u(t) u(t)
1 0 L{u(t)} = Example L {a} = L {a-u(t)} = a L {u(t)} = Important Points
If we are finding laplace transform of any differentiation in time domain, always multiply by ‘s’
-
Example ( )
*
( )
( )
If we are fi di g ap ace tra sf r
-
f a y i tegrati
i ti e d
ai , a ways divide by ‘s’
Example ( )
( ) +
*∫
Example 1 Given L{f(t)} =
(
)
then f(t) is _______ (assumed that ∫ f(t)dt
(A)
+ si t
(C)
+ si t
(B)
+ c st
(D)
+ c st
Solution We know, .
(
.
)
/
/
)
si t
∫ ∫ ∫ si u du
∫ ∫ (– c s u) dt ∫ ∫ (– c s t + ) dt ∫ (t
si t) dt
∫ ∫ ( – c s u)du
0 + c s t1
+ c st
Hence D THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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Chapter 7
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Example 2 The Laplace transform of the triangular wave of period 2P for the following function f(t) = t ,
0
= 2p – t , p < t < 2P, is (A)
0
(B)
(C) .ta h . / /
(
(D)
)
+ ta h
Solution Laplace transform of a periodic function f(t) with period T is given by L(f(t)) =
( )
∫
=
0∫
=
0∫
( )
( ) 1
+∫
+∫ (
)
1
Integrating, =
(
) (
)
( (
)
)(
=
)
ta h . /
Hence (B) Example 3 Inverse Laplace transform of log (A)
2 log
(B)
.
is _________
u(t)
(C)
/
sinh . /u(t) .
(D)
/
Solution f(t)= L-1 log .
/
As we know that if L ( f(t) ) = F(s) then t f(t) = L-1. ∴ t f(t) = L-1.
( )/ /+L-1.
/
6
4
5+
4
57
= 2sinh . /u(t) f(t) = sinh . /u(t) Hence (C) THE GATE ACADEMY PVT.LTD. H.O.: #74, Keshava Krupa (third Floor), 30th Cross, 10th Main, Jayanagar 4th Block, Bangalore-11 : 080-65700750,
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