Laplace Transform Electrical Engineering (EE) Notes | EduRev

Electrical Engineering (EE) : Laplace Transform Electrical Engineering (EE) Notes | EduRev

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 Laplace Transform
Laplace Transform enables one to get a very simple and elegant method of solving linear differential equation by transforming them into algebraic equations. It is well known that chemical processes are mathematically represented through a set of differential equations involving derivatives of process states. Analytical solution of such mathematical models in time domain is not only difficult but sometimes impossible without taking the help of numerical techniques. Laplace Transform comes as a good aid in this situation. For this reason, Laplace Transform has been included in the text of this “Process Control” course material though it is purely a mathematical subject. 

 Definition of Laplace Transform
Consider a function f(t). The Laplace transform of the function is represented by f(s) and defined by the following expression:

Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                    1

Hence, the Laplace Transform is a transformation of a function from the -domain (time domain) to -domain (Laplace domain) where both and are independent variables.

 Properties of Laplace Transform
•  The variable is defined in the complex plane as S= a + jb where j = √-1.
•  Laplace Transform of a function exists if the integral  Laplace Transform Electrical Engineering (EE) Notes | EduRev  has a finite value, i.e. , it remains bounded; eg . if f(t) = eat, then f(s) exists only for s>a, as the integral becomes unbounded for s<a.
•  Laplace Transform is a linear operation. 

Laplace Transform Electrical Engineering (EE) Notes | EduRev

 Laplace transform of a few basic functions

The Fig.1 shows a few basic functions which are frequently used in process control applications
Laplace Transform Electrical Engineering (EE) Notes | EduRev Laplace Transform Electrical Engineering (EE) Notes | EduRev

Laplace Transform Electrical Engineering (EE) Notes | EduRev Laplace Transform Electrical Engineering (EE) Notes | EduRev

Fig.1: Few basic functions which are frequently used in process control applications

Step function : See Fig. 1(a) for the schematic of a step function  Laplace Transform Electrical Engineering (EE) Notes | EduRev

Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                      2

hence  Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                3

Ramp function : See Fig. 1(b) for the schematic of a ramp function f(t)=at for t>0 where a is a constant
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                           4

hence
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                                 5

Exponential function : Laplace Transform Electrical Engineering (EE) Notes | EduRev  for t>0 where a is a constant

Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                          6

hence
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                        7

Sinusoidal function Laplace Transform Electrical Engineering (EE) Notes | EduRev

Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                         8

hence
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                    9

Delayed function :f(t - td) , i.e .f(t) is delayed by td seconds

Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                 10

Now, let us take  Laplace Transform Electrical Engineering (EE) Notes | EduRev , hence  Laplace Transform Electrical Engineering (EE) Notes | EduRev . At  Laplace Transform Electrical Engineering (EE) Notes | EduRev and at  Laplace Transform Electrical Engineering (EE) Notes | EduRev . Thus,

Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                        11

hence
Laplace Transform Electrical Engineering (EE) Notes | EduRev

Pulse function : See Fig. 1(c) for the schematic of an unit pulse function. The area under the pulse is 1. The duration of pulse is and hence it achieves maximum intensity of 1/T. Thus the 
Function is defined by  Laplace Transform Electrical Engineering (EE) Notes | EduRev

It can also be defined as the “addition” of two step functions which are equal but with opposite intensity, however, the second function is delayed by 
Laplace Transform Electrical Engineering (EE) Notes | EduRev

Hence, it is evident that f2(t) is equal to -f1(t) in intensity however it is delayed by time T.

Thus,  Laplace Transform Electrical Engineering (EE) Notes | EduRev . Since f1(t) is a step function of intensity 1/T, the following expression will hold.

Laplace Transform Electrical Engineering (EE) Notes | EduRev

Hence

Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                  14

Impulse function : See Fig. 1(d) for the schematic of an unit impulse function. This is analogous to a pulse function whose duration is shrinked to zero without losing the strength. Hence the area under the impulse remains 1. The function can be expressed as the following:
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                              15

As the duration of the impulse tends to zero, its maximum intensity ideally tends to ∞. Mathematically it is termed as Dirac Delta function and is represented as  Laplace Transform Electrical Engineering (EE) Notes | EduRev . The following relation holds for unit impulse:
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                              16

Thus the Laplace transform of the impulse function can be derived as the following: 
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                             17

L'Hospital's rule has been applied in the above derivation. Hence, 
Laplace Transform Electrical Engineering (EE) Notes | EduRev                                                                                                                                                                                                 18

The following table presents the Laplace transforms of various functions.
Table 1: Laplace transforms of various functions (t>0) 

Laplace Transform Electrical Engineering (EE) Notes | EduRev

     Laplace Transform Electrical Engineering (EE) Notes | EduRev

           Laplace Transform Electrical Engineering (EE) Notes | EduRev

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