7. Laplace Transform – Network Analysis and Synthesis

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 eσt, 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+ , 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 = + ).

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) = eat.

Solution: By definition, we can get the following:

Substituting f(t) = eat, 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 tn.

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.

Table 7.1 Laplace Transform of Certain Functions

f(t) L[f(t)]
1. L[1]
2.L[t]
3. L[tn]
4. L[eat]
5. L[eat]
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 − 5t3]

Solution:

Example 7.10 Find the Laplace transform of sin2 3t.

Solution:

Therefore,

Example 7.11 Evaluate L[cos2 5t]

Solution:

Example 7.12 Find the Laplace transform of sinh2 3t.

Solution:

Example 7.13 Find the Laplace transform of cosh2 4t.

Solution:

Example 7.14 Evaluate L[sin3 2t]

Solution:

Example 7.15 Evaluate L[cos3 4t]

Solution:

Example 7.16 Evaluate L[cosh3 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[e2tt3]

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 tn

Example 7.22 Evaluate L[t cos 3t]

Solution: We have

Therefore,

Example 7.23 Evaluate L [t2e−3t sin2t]

Solution: First, we will find

Then, we will use multiplication by tn 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:

Example 7.24 Evaluate

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,

Example 7.27 Evaluate

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

Property 4:

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 e4t cos 2t.

Solution: Given f(t) = e4t 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:

Solution: Given

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:

Example 7.40 Evaluate

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 s2, 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:

Example 7.45 Evaluate

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:

Example 7.47 Evaluate

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 f1(t) = eαt and f2(t) = eβt, then evaluate the convolution oftwo functions.

Solution:Given

Convolution of f1(t) and f2(t) = f1(t) * f2(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:

and

REVIEW QUESTIONS

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. et − e −2t]

4.Define Laplace transform of a time function. Determine Laplace transforms for

(i)The impulse function

(ii)The unit step function

(iii)tn eat

[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 f1(t) and f2(t) when f1 (t) = eat and f2(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 + 5et − 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]

MULTIPLE CHOICE QUESTIONS

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)Eeat

(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 e2t

(d)NOT

10.Laplace transform equation tn eat is

(a)

(b)

(c)

(d)

11.Inverse Laplace transform for is

(a)1 − e−3t

(b)1 + e−3t

(c)1 + e3t

(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 + 6e2t + 4e−4t

(d)−2e−t + 6e−2t + 4e4t

14.The value of function at t = 0 is

(a)10

(b)4

(c)0

(d)

15.Laplace transforms of tn 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 + e250t)

(b)40(1 − e250t)

(c)40(1 − e−250t)

(d)-40(1 + e250t)

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 f1(t) = e −at and f2(t) = t, then convolution of f1(t) and f2 (t0) is

(a)

(b)

(c)

(d)

21.Laplace transform of cos2t 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 + 4e4t

(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 + 5et + 6e−2t

ANSWERS

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