7. Laplace Transform
CHAPTER OBJECTIVES
After carefully studying this chapter, you should be able to do the following:
➢Provide a explanation of the concept of Laplace transform.
➢Distinguish between the functions of Laplace transform and inverse Laplace transform.
➢Determine the Laplace transform of standard functions.
➢Determine Laplace transform of certain functions using standard transformation formula.
➢Use the properties of Laplace transform to solve problems on determining Laplace transform of different functions.
➢State and explain initial value theorem.
➢State and explain final value theorem.
➢Solve problems using initial value and final value theorems.
➢Use standard formula to carryout inverse Laplace transform of different functions.
➢State and explain convolution theorem.
7.1 CONCEPT OF LAPLACE TRANSFORM
In this chapter, we introduce the Laplace transform which is used for providing the solution of network problems.
We have known that Fourier transform is used in finding solutions of large variety of engineering problems. Fourier transform enables understanding of system behaviour in frequency domain. This is done by expressing a signal f(t) as a continuous sum of complex exponentials. The Fourier transform is defined as follows:
There are some special functions where Fourier transform is not possible. The introductio of a convergence factor in the form of e^{−σt}, where σ is a real number, makes the integra convergent.
The introduction of a convergence factor into the Fourier transform creates a new transformation, which is called Laplace transform. Thus, the Laplace transform is defined as in the following:
It is to be noted that the lower limit of the integral has been taken as 0 instead of −∞ This is because with σ > 0, the convergence factor e^{−σt} will diverge rather than converge when t tends to −∞.
If we substitute s = σ+ jω, then
F(s) is called the Laplace transform of f(t).
Laplace transform is a mathematical tool to convert the function from time domain to frequency domain. Let f(t) be any time domain function, then its Laplace transform is defined as follows:
where s is a complex frequency (s = -σ + jω).
The integral 0 to ∞ does not take care of any information contained in f(t), when t tends to −∞.
Thus, the Laplace transform changes the time domain function f(t) to frequency domain function F(s).
Inverse Laplace transformation converts a frequency domain function into time domain function.
The Laplace transformation method of solving differential equations has number of advantages.
The solution follows a systematic procedure where the initial conditions are taken care automatically in the transform operation, and a complete solution is obtained in one operation.
7.2 LAPLACE TRANSFORM OF STANDARD FUNCTIONS
We will now find Laplace transform of some standard functions.
Example 7.1 Determine the Laplace transform of unity, that is, f(t) = 1.
Solution: By definition, we can write the following equation:
Therefore, the following is obtained:
Example 7.2 Find the Laplace transform of exponential function f(t) = e^{at}.
Solution: By definition, we can get the following:
Substituting f(t) = e^{at}, we get the following:
Example 7.3 Calculate the Laplace transform of t.
Solution: By definition, the equation can be written as follows:
Substituting f(t) = t, we get the following:
Integrating by parts, the equation can be written as follows:
Example 7.4 Calculate the Laplace transform of t^{n}.
Solution: By definition, the following form is obtained:
Integrating by parts, we get the equation as follows:
Again integrating by parts, we get the equation as follows:
Example 7.5 Calculate the Laplace transform of sin at.
Solution: By definition,
Now using the formula, we get the form as follows:
Then, we have the following form:
Example 7.6 Calculate the Laplace transform of cos at.
Solution: By definition, we get the following:
Therefore,
Hence,
Example 7.7 Calculate the Laplace transform of sinh at.
Solution: By definition, we get the following:
Example 7.8 Calculate the Laplace transform of cosh at.
Solution: By definition,
The summary of formulae for Laplace transform is given in a tabular form in Table 1.
f(t) | L[f(t)] |
1. L[1] | |
2.L[t] | |
3. L[t^{n}] | |
4. L[e^{at}] | |
5. L[e^{at}] | |
6. L[sin at] | |
7. L[cos at] | |
8. L[sinh at] | |
9. L[cosh at] |
7.3 LAPLACE TRANSFORM PROBLEMS BASED ON STANDARD FORMULA
Some more examples based on Laplace transform are provided as follows:
Example 7.9 Evaluate L[3e^{−5t} + 8cos3t + 2 sinh 2t − 5t^{3}]
Solution:
Example 7.10 Find the Laplace transform of sin^{2} 3t.
Solution:
Therefore,
Example 7.11 Evaluate L[cos^{2} 5t]
Solution:
Example 7.12 Find the Laplace transform of sinh^{2} 3t.
Solution:
Example 7.13 Find the Laplace transform of cosh^{2} 4t.
Solution:
Example 7.14 Evaluate L[sin^{3} 2t]
Solution:
Example 7.15 Evaluate L[cos^{3} 4t]
Solution:
Example 7.16 Evaluate L[cosh^{3} 2t].
Solution:
Example 7.17 Evaluate L[sin 2t cos 3t]
Solution:
Example 7.18 Evaluate L[cos 3t cos 2t]
Solution:
7.4 PROPERTIES OF LAPLACE TRANSFORM
In this section, we will state the various properties of Laplace transform and solve problems using these properties.
7.4.1 Property 1: First Shifting Property
Example 7.19 Evaluate L[e^{−3t} cos 2t].
Solution: We have
Now, by the first shifting property, we get the following:
Example 7.20 Evaluate L[e^{2t}t^{3}]
Solution: Now,
Therefore, by the shifting property,
Example 7.21 Evaluate L[sinh 2t cos 3t]
Solution: Now,
7.4.2 Property 2: Multiplication By t^{n}
Example 7.22 Evaluate L[t cos 3t]
Solution: We have
Therefore,
Example 7.23 Evaluate L [t^{2}e^{−3t} sin2t]
Solution: First, we will find
Then, we will use multiplication by t^{n} property.
At last, we will use the first shifting property and we get the equation as follows:
7.4.3 Property 3: Division By ‘t’
Property 3 states the following:
Solution: Firstly, we will find
Now, we will use division by ‘t’ property, and the equation can be written as follows:
Example 7.25 Evaluate
Solution: Firstly, we will find
Now, we will use the division by ‘t’ property to get the following:
At last, we will use the first shifting property and the following is obtained:
7.4.4 Property 4
Example 7.26 Evaluate
Solution: Now,
Solution: We have
7.5 SUMMARY OF USEFUL PROPERTIES OF LAPLACE TRANSFORM
Property 1: First shifting property
Property 2: Multiplication by f
Property 3: Division by t
7.6 INITIAL VALUE THEOREM
It states that or f(0), that is, the initial value of a function f(t) in any time domain is equal to . That is, it can be expressed as follows:
Example 7.28 Verify the initial value theorem for the function e^{4t} cos 2t.
Solution: Given f(t) = e^{4t} cos 2t
Now, the L.H.S of the initial value theorem can be given as follows:
The R.H.S of the initial value theorem is as follows:
From equations (7.1) and (7.2), it is clear that . Hence, the initial value theorem is verified.
Example 7.29 Find the initial value of 2e^{−3t} cos t using the initial value theorem.
Solution: Given f(t) = 2e^{−3t} cos t
Firstly, let us find L[f(t)] for which
Now, using the first shifting property, we get the following form:
and
Now, by the initial value theorem,
Initial value off (t) = Lt s F(s) Substituting the value of F (s)
Example 7.30 Find the initial value of the function, where the Laplace transform is given as follows:
Solution: Given
Now, by the initial value theorem, we get the following:
Initial value of
Substituting the given value of F(s), the following form is obtained:
7.7 FINAL VALUE THEOREM
Final value theorem states that or the final value of a function f(t) is expressed as in the following:
where F(s) = L[f(t)] of any time domain function f(t).
The numericals based on the final value theorem are given as follows.
Example 7.31 Verify the final value theorem for a function
Solution: Given f (t) = 4 + e^{-t} sin t
and
Using the first shifting property, the form is obtained as follows:
Now, the L.H.S of the final value theorem can be written as follows:
Further, the R.H.S of the final value theorem can be given as in the following:
From equations (7.3) and (7.4), it is clear that . Hence, the final value theorem is verified.
Example 7.32 Find the final value of f(t) = 8 (2 − e^{−4t}) using the final value theorem.
Solution: Given
Now,
Now, by the final value theorem, we obtain the final value of
Example 7.33 Find the value of a function whose Laplace transform is given as follows:
Example 7.34 Find the Laplace transform of unit step function.
Solution: The unit step function is defined as follows:
Now,
Example 7.35 Find the Laplace transform of unit ramp function.
Solution: The unit ramp function is defined as follows:
Therefore,
Integrating by parts, we get the following:
Example 7.36 Find the Laplace transform of impulse function.
Solution: We know the following:
Unit impulse function = time derivative of unit step function
7.8 INVERSE LAPLACE TRANSFORM
Some useful formulas of inverse transform are provided in a tabular form.
Example 7.37 Evaluate
Solution:
Example 7.38 Evaluate
Solution:
Example 7.39 Evaluate the inverse Laplace transform of
Solution:
(Making perfect square, by adding and subtracting the square of coefficient of s.)
Using the second shifting property, we get the following:
Solution:
Using the second shifting property, the following can be obtained:
Example 7.41 Evaluate
Solution:
Example 7.42 Evaluate
Solution:
Let us make partial fractions of the following form:
To find the values of A and B, we write the equation as follows:
Substitute s = −2 in equation (7.7), and we obtain the value of A.
In order to find B, we substitute s = −3 in equation (7.7), we get the value as follows:
Substituting the values of A and B in equation (7.6), the following form is obtained:
By substituting this value in equation (7.5), we get the equation as follows:
By taking the inverse Laplace transform, the equation can be written as follows:
Example 7.43 Evaluate
Solution:Now firstly, we will make partial fractions equation as in the following:
To find the values A, B and C
Substituting s = 1 in equation (7.9), we get the following:
Substituting s = −2 in equation (7.9), we obtain the value as follows:
To find the value of A, let us expand equation (7.9) as in the following:
Equating the co-efficient of s^{2}, s and constant terms on both sides, we get the following set of equations:
The value of A can be found from any of the above equations.
From equation (7.10), we get the following value:
Substituting the values of A, B and C in equation (7.5), the following form is obtained:
Taking the inverse Laplace transform of the equation, we get the following:
Example 7.44 Evaluate
Solution:Firstly, let us make partial fractions as in the following:
To find the values of A, B, C and D
As there is no simple fraction in this case, all the fractions are quadratic. Hence, we cannot find any value directly. We will have to expand equation (7.14) as follows:
Equalising the co-efficient on both sides, we get the following form:
From equations (7.16) and (7.18), we get the values of B and D as follows:
Now, solving equations (7.15) and (7.17), the values of A and C are obtained as follows:
Now substituting the values of A, B, C and D in equation (7.13), we get the equation as follows:
Now, taking the inverse Laplace transform on both sides, the following form can be obtained:
Solution:Let us make partial fractions of
Let us find the values of A, B and C.
In this case, there is one simple factor (s − 1), and therefore, one value can be obtained by substituting s = 1 in equation (7.20)
Substituting s = 1 in equation (7.20), we get the equation as follows:
As there is no other simple factor, and therefore, other values will be obtained by expanding equation (7.20).
Equating the co-efficient on both sides, we get the form as follows:
Substituting the value of in equation (7.21), we get
Further, substituting the value of B in equation (7.22), we get
Now, substituting the values of A, B and C in equation (7.20), we get the form as follows:
Now, taking the inverse Laplace transform on both sides, the equation can be written as in the following:
Example 7.46 Evaluate
Solution:Firstly, let us make partial fractions
Let us find the values of A, B, C and D.
Since in this case, there is no simple factor. Both the factors are quadratic. Therefore, all the values will be obtained by expanding equation (7.25)
Expanding equation (7.25), we get the form as follows:
Equating the coefficients on both sides, the following form is obtained:
From equation (7.26) and (7.28), we get A = C = 0.
By solving equations (7.27) and (7.29), we get, and
Substituting the values of A, B, C and D in equation (7.24), we get the following:
Taking the inverse Laplace transform, the equation is written as follows:
Using the second shifting property, we get the following form:
Solution:
Example 7.48 Evaluate
Solution:
7.9 CONVOLUTION THEOREM
The theorem states the following:
That is, convolution of f (t) and g(t) is as follows:
Example 7.49 Evaluate using the convolution theorem.
Solution:
Now, by the convolution theorem, we get the following form:
Example 7.50 Evaluate using the convolution theorem.
Solution:Let
Now, by the convolution theorem, the following equations are obtained:
Example 7.51 If f_{1}(t) = e^{αt} and f_{2}(t) = e^{−βt}, then evaluate the convolution oftwo functions.
Solution:Given
Convolution of f_{1}(t) and f_{2}(t) = f_{1}(t) * f_{2}(t)
Example 7.52 Evaluate
Solution:
Using the second shifting property, we get the following form:
Further,
Now, by the convolution theorem, the equation can be written as follows:
Short Answer Type
1.What is convolution in time domain? What is the Laplace transform of convolution of two time domain functions?
2.State the advantage of using Laplace transform in networks. Given S domain representations for resistance, inductance and capacitance.
3.State and prove convolution theorem.
Numerical Questions
1.Define Laplace transform of a function f(t). Find the Laplace transform for the function
[Ans. ]
2.Determine the Laplace transform of the function
[Ans. ]
3.Using convolution theorem, find the inverse laplace transform of
[Ans. e^{t} − e ^{−2t}]
4.Define Laplace transform of a time function. Determine Laplace transforms for
(i)The impulse function
(ii)The unit step function
(iii)t^{n} e^{at}
[Ans. ]
5.Find the inverse Laplace transform for
[Ans. (1 + 2 cos 2t)]
6.State the initial value theorems in the Laplace transform. What is the value of the function at t = 0, if
[Ans. Initial value = 4]
7.Find the Laplace transform of the function cosh ωt.
[Ans. ]
8.Find the Laplace transform of the function
[Ans. ]
9.Using the partial fraction method, obtain the inverse Laplace transform of
[Ans. −40(1−e^{−250t})]
10.State the final value theorem and using this theorem find the final value of the function where Laplace transform is
[Ans. Final value = 2]
11.Find the Laplace transform of the function
[Ans. ]
12.Find the convolution of f_{1}(t) and f_{2}(t) when f_{1} (t) = e^{−at} and f_{2}(t) = t
[Ans. ]
13.Find the Laplace transform of the functions
[Ans. ]
14.Find the inverse Laplace transform of
[Ans. 1 − 4e^{−2t} + 8e^{−t}]
15.Define the unit step, ramp and impulse function. Determine the Laplace transform for three functions.
16.Find the inverse Laplace transform of
[Ans. 1 + 5e^{−t} − 6e^{−t}]
17.Find the Laplace transform of
[Ans. ]
18.Find the final value of the function whose Laplace transform is
[Ans. Final value = 1.8]
1.Laplace transform of a unit impulse function is
(a)S
(b)0
(c)e^{-s}
(d)1
2.Laplace transforms of a damped sine wave e^{−αt} sin (θt) · u(t) is
(a)
(b)
(c)
(d)
3.The final value of f(t) for a given
(a)Zero
(b)1/15
(c)1/8
(d)1/6
4.Laplace transforms of the function e^{−2t} is
(a)
(b)(s + 2)
(c)
(d)2s
5.The Laplace transform of a function is . The function is
(a)E sin ωt
(b)Ee^{at}
(c)Eu(t − a)
(d)E cos ωt
6.If f(t) = r(t − α), f (s) =
(a)
(b)
(c)
(d)
7.The integral of a step function is
(a)A ramp function
(b)An impulse function
(c)Modified ramp function
(d)A sinusoidal function
8.Laplace transform of the function f (t) = (1 − e^{−αt}) sin αt, where α is a constant is
(a)
(b)
(c)
(d)NOT
9.Laplace inverse equation is
(a)e^{−t} − e^{−2t}
(b)e^{−} e^{−2t}
(c)e^{−} e^{2t}
(d)NOT
10.Laplace transform equation t^{n} e^{at} is
(a)
(b)
(c)
(d)
11.Inverse Laplace transform for is
(a)1 − e^{−3t}
(b)1 + e^{−3t}
(c)1 + e^{3t}
(d)1 − e ^{3t}
12.Inverse Laplace transform for is
(a)1 + cos 2t
(b)1 + 3cos 2t
(c)1 + 2cos 2t
(d)1 + 3cos t
13.Inverse Laplace transform for is
(a)-2e^{−t} + 6 e^{−2t} + 4e^{−4t}
(b)-2e^{−t} + 6 e^{−2t} − 4e^{−4t}
(c)−2e^{−t} + 6e^{2t} + 4e^{−4t}
(d)−2e^{−t} + 6e^{−2t} + 4e^{4t}
14.The value of function at t = 0 is
(a)10
(b)4
(c)0
(d)∞
15.Laplace transforms of t^{n} u(t) is
(a)
(b)
(c)
(d)
16.Laplace transform of cosh ωt u(t) is
(a)
(b)
(c)
(d)
17.Obtain the inverse Laplace transforms of
(a)40(1 + e^{250t})
(b)40(1 − e^{250t})
(c)40(1 − e^{−250t})
(d)-40(1 + e^{250t})
18.The final value of the function is
(a)1
(b)2
(c)3
(d)0
19.The Laplace transform of t cos 4t is
(a)
(b)
(c)
(d)NOT
20.If f^{1}(t) = e ^{−at} and f_{2}(t) = t, then convolution of f_{1}(t) and f_{2} (t0) is
(a)
(b)
(c)
(d)
21.Laplace transform of cos^{2}t is
(a)
(b)
(c)
(d)
22.Laplace transform of t sin 2t is
(a)
(b)
(c)
(d)
23.If , then V (t) = ?
(a)1 + 4e^{−2t} + 8e ^{−t}
(b)1 − 4e^{−2t} + 8e^{−t}
(c)1 − 4e^{−2t} − 8e^{−t}
(d)1 − 4e^{−2t} + 8e^{−t}
24.If , then i(t) is
(a)e^{−t} + 4e^{-4t}
(b)e^{−t} + 4e^{4t}
(c)−e^{−t} + 4e^{−4t}
(d)−e^{−t} − 4e^{−4t}
25.The inverse Laplace transforms for is
(a)1 + e ^{−t} − 6e^{−2t}
(b)1 + 5e^{−t} − 6e^{−2t}
(c)1 − 5e^{−t} − 6e^{−2t}
(d)1 + 5e^{t} + 6e^{−2t}
1.d
2.c
3.a
4.c
5.c
6.a
7.a
8.c
9.b
10.c
11.b
12.c
13.b
14.b
15.d
16.d
17.c
18.b
19.a
20.c
21.d
22.a
23.b
24.c
25.b