Chapter 5-Laplace Transform

Chapter 5 – Laplace Transform

M. Khusairi Osman Page 95

CHAPTER 5 – LAPLACE TRANSFORM

4.1 Introduction

The Laplace transform offers a simple alternative of solving linear

differential equation.

With Laplace transform, problem are solved in complex frequency domain

(function of complex variable, ) instead of the usual time domain.

By Laplace transform, differential equation is transformed into an algebraic

function in terms of complex variable, s where .

Then it can be manipulate to produce the solution in term of s.

Finally, the solution in time domain can be obtained by referring to the

Laplace transform pairs.

4.2 Bilateral and Unilateral Laplace Transform

The Laplace transform of a signal x(t) is given by:

If

If

The unilateral Laplace transform is actually the Laplace transform of a

causal signal.

Since signal of all system is causal, only the unilateral Laplace transform is

considered.

The lower limit can be written as as long s x(t) does not include a

singularity function at t = 0.

Example:

Find the Laplace transform for

i. x(t) = A,

ii. x(t) =

iii. iv.

where A is a constant for t ≥ 0



Solution:

Using Eular’s identity



4.3 Properties of Unilateral Laplace transform

4.3.1 Property of Homogeneity (Amplitude Scaling)

Example:

4.3.2 Property of Superposition

Example:



4.3.3 Property of Linearity

Example:

4.3.4 Shifting in the Time Domain

Example:

Find the Laplace transform for the following signals, given that for

and is a constant.

From Example (i)



4.3.5 Shifting in the s Domain

Example:

4.3.6 Time Scaling

Example:



4.3.7 Multiplication by t

Example:

4.3.8 Differentiation in Time-Domain

Example:

Given

Substitute



4.3.9 Integration in the time domain

4.3.10 Initial value theorem

4.3.11 Final value theorem

Example:

Find the initial and the final value of x(t) given X(s) as below:

Initial value Final value



4.4 Inverse Laplace Transform

The inverse Laplace transform equation is given by:

However this equation is not used to evaluate the inverse Laplace transform

due to the integration in the complex plane.

In most cases, the Laplace transform exists in the following form:

Where values of s that will set B(s) to zero such are

known as ZEROs of G(s) while values of s that will set A(s) to zero such

are known as POLEs of G(s).

The degree of polynomial B(s) is less than the degree of A(s) (m < n).

The inverse Laplace transform can be obtained by applying the partial

fraction.

Otherwise, long division must be applied first to reduce the expression such

that m is less than n.

4.4 Partial Fraction Expansion Method

A technique to reduce proper rational functions as in equation into a

sum of simple terms as follows:

Where has the following term:

Step in inverse Laplace transform:

1. Expand G(s) in terms of a sum of partial fractions

2. Express A(s) as a product of factors

3. Inverse Laplace transform is obtained by summing the inverse

Laplace transform f each of the partial fractions.



Case 1: G(s) contains only distinct poles:

If polynomial G(s) contains only linear non-repeated factors, it can be

rewritten as follows:

Values of can be obtained by multiplying both sides of the equation by

Refer to Table of Unilateral Laplace Transform

Solution:

Since degree of polynomial B(s) is higher than A(s), long division is performed.


M. Khusairi Osman 4

By partial fraction:

Method I:

Method II:

Compare coefficient:



Case 2: G(s) contains multiple poles:

If polynomial G(s) contains repeated factors, it can be rewritten as follows:

The multiple poles of of multiplicity r can be obtained in the following

way:

The other roots are assumed to be distinct, and should be treated as in Case 1

Example:

Solution:

Method I



Method II:

Or




Case 3: G(s) contains complex conjugate poles:

If polynomial G(s) contains non-repeated irreducible second degree

polynomial, it can be rewritten as follows:

Solution:


S = 0:



Side A Side B

Side A:

Side B:



4.5 Application of Laplace Transform

APPLICATION OF

LAPLACE TRANSFORM

SOLUTION OF

DIFFERENTIAL EQUATION

DETERMINATION OF

TRANSFER FUNCTION

STABILITY IN THE S-

DOMAIN

4.5.1 Solution of Differential Equation

Procedure:

1. Apply Laplace transform to both side of the differential equation and

substitute the appropriate initial condition, to obtain an algebraic

equation in terms of Y(s)

2. Solve the algebraic equation for Y(s) in sum of and 3. Apply inverse Laplace transform to obtain y(t)

Example:

A second order linear time invariant differential equation is given by:

Using Laplace transform method, determine:

i. The zero input solution

ii. The zero state solution

iii. The total solution

Given

Solution:

i. Zero input solution



Substitute

ii. Zero state solution:

For zero state solution, substitute , hence



iii. Total solution:

Example:

Using Laplace transform theorem, determine the initial and the final value for:

Then prove that the values are correct using inverse Laplace transform.

Solution:

Using the Initial Value Theorem

Using the Final Value Theorem



Using the inverse Laplace transform:


Inverse Laplace transform:

Initial value:

Final value:

Both methods produced the same answer.



4.5.2 Determination of transfer function:

Transfer function: a mathematical modeling to characterize the input-output

relationships of a system

Procedure to derive transfer function:

1. Obtain the differential equation that describes the dynamics of the

system.

2. Take the Laplace transform of the differential equation. Assume all

the initial conditions are zero.

3. Take the ratio of output function to the input function.

Example:

Determine the transfer function of the following differential equation:

Solution:

Assume all the initial conditions are zero or



Example:

Determine the transfer function for the circuit as shown in Figure 4.1

R

L

C)t(Vin )t(Vouti (t)

Figure 4.1

Solution:

Component Laplace transform

Resistor

Inductor

Capacitor



4.5.2 Stability in the s-domain

The poled and zeros can be represented on the complex plane or s-plane.

A small circle [ O ] is used to denote the location of a zero while a small cross

[ X ] is used to denote the location of a pole in the s-plane.

The complex plane consists of real axis and imaginary axis

. The complex plane can be divided into:

1. Left half plane

2. Right half plane

A linear system is stable if and only if all it poles of its transfer function are

located on the left half plane (all the roots of the characteristic polynomial

have negative real parts.

j

LHS

LHS

RHS

RHS

x

x

x

Example:

Determine whether the system with the following transfer function is stable.

Draw its pole-zero plot.

Solution:

Characteristic polynomial:



Pole-zero plot:

j

LHS

LHS

RHS

RHS

x

x

x

1

2

1

3

2 3-1

-2

-3

-1-2-3

Example:

Determine whether the system with the following transfer function is stable.

Draw its pole-zero plot.

Solution:

Characteristic polynomial:

j

LHS

LHS

RHS

RHS

x x1

2

1

3

2 3-1

-2

-3

-1-2-3x

Chapter 5-Laplace Transform

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