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Laplace Transform Notes - Electrical Engineering (EE)

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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 Notes - Electrical Engineering (EE)                                                                                                                                                                                    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 Notes - Electrical Engineering (EE)  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 Notes - Electrical Engineering (EE)

 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 Notes - Electrical Engineering (EE) Laplace Transform Notes - Electrical Engineering (EE)

Laplace Transform Notes - Electrical Engineering (EE) Laplace Transform Notes - Electrical Engineering (EE)

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 Notes - Electrical Engineering (EE)

Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                                                      2

hence  Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                                                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 Notes - Electrical Engineering (EE)                                                                                                                                           4

hence
Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                                                                                                 5

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

Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                          6

hence
Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                                                                                        7

Sinusoidal function Laplace Transform Notes - Electrical Engineering (EE)

Laplace Transform Notes - Electrical Engineering (EE)                                                                                                         8

hence
Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                                                                                    9

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

Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                 10

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

Laplace Transform Notes - Electrical Engineering (EE)                                                                                                        11

hence
Laplace Transform Notes - Electrical Engineering (EE)

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 Notes - Electrical Engineering (EE)

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 Notes - Electrical Engineering (EE)

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

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

Laplace Transform Notes - Electrical Engineering (EE)

Hence

Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                                                                                  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 Notes - Electrical Engineering (EE)                                                                                                                                                                                              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 Notes - Electrical Engineering (EE) . The following relation holds for unit impulse:
Laplace Transform Notes - Electrical Engineering (EE)                                                                                                                                                                                                              16

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

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

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

Laplace Transform Notes - Electrical Engineering (EE)

     Laplace Transform Notes - Electrical Engineering (EE)

           Laplace Transform Notes - Electrical Engineering (EE)

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