The Laplace Transform Lab

Module 4 – Laplace Transform

 

The Laplace transform provides a significant algebraic characterization of continuous-time systems: the ratio of the Laplace transform of the output to that of the input—or the transfer function of the system. The transfer function concept unifies the convolution integral and the ordinary differential equations system representations in the linear time invariant case. Certain characteristics of continuous-time systems can only be studied via the Laplace transform. Such is the case of stability, transient and steady-state responses.

Since the Laplace transform requires integration over an infinite domain, it is necessary to consider if and where this integration converges—or the “region of convergence” in the s-plane.

The one-sided Laplace transform is of significance given that most of the applications consider causal systems and causal signals. The one-sided Laplace transform can be used to find the two-sided Laplace transform of any signal or impulse response.

 

Problem 1 : In order to compute Laplace transform of an expression, consider the following example:   

Example 1:

Consider computing a Laplace transform of  

Method 1: Laplace transform is computed by integrating the function

 L

L

   

L      

 

Method 2:   L can be found either by integrating or by simply converting a calculus problem into algebraic problem by using Laplace Transform table.

Look up Laplace transform table to find the Laplace Transform of the function

L

SAMPLE LAPLACE TRANSFORM PAIR TABLE: Here is a sample table to find the Laplace transform pair.

(Time Domain)	(Frequency or ‘s’ domain)
1. 1
2.

 

Example 2:

Consider computing a Laplace transform of 

L [ ] can be found either by integrating or by simply converting a calculus    problem into algebraic problem by using Laplace Transform table.

Look up table 3.2 to find the Laplace Transform of the function

Note:     Please use Laplace Transform Pair Table to solve the Laplace Transform for a function (where applicable). Convert a Calculus problem into an Algebraic form to solve problems and save time.

L [ ] =

 

 

Problem 2: In order to compute the Laplace transform, consider the following example:  

For example:

Consider computing a Laplace transform of    

Laplace transform is computed by integrating the function

 L [ ] =

From the multiplication property of Laplace transforms, Table 3.1:

 L [ ] =               -------- > From Table 3.1 (Page 199)

L [ ] = )   =     =    

Note:     Please use Laplace Transform Pair Table to solve the Laplace Transform for a function (where applicable). Convert a Calculus problem into an Algebraic form to solve problems and save time.

 

 

Problem 3:

The Laplace transform of the convolution integral of a causal signal with Laplace transform , and a causal impulse response with Laplace transform , is given by Equation 3.23.

 L

Convolution in time domain is multiplication in the frequency domain.                      

· First, find the Laplace transform of (i.e. L  ) and then Laplace transform of    (i.e.  L )

· Then you just need to multiply them L  to get the answer.

 

For example: Consider =   and as  =  .

Laplace transform of  L =   L  

And Laplace transform of L  =  L =   

Therefore, the Laplace of convolution of is just multiplied by

 

                                L

Note:     Please use Laplace Transform Pair Table to solve the Laplace Transform for a function (where applicable). Convert a Calculus problem into an Algebraic form to solve problems and save time.

 

Problem 4:

Inverse Laplace Transform

Inverting the Laplace transform consists in finding a function (either a signal or an impulse response of a system) that has the given transform with the given region of convergence. We will consider three cases:

· inverse of one-sided Laplace transforms giving causal functions,

· inverse of Laplace transforms with exponentials,

· inverse of two-sided Laplace transforms giving anti-causal or non-causal functions.

Follow these steps in finding an Inverse Laplace Transform.

Step 1: Perform partial fraction expansion.

Step 2: Find all the coefficients of expansion.

Step 3: Plug in the values of all the coefficients in the expression .

Step 4: Find the inverse of each term in using Laplace Transform Pair.

Step 5: Finally, write the expression for which is the inverse of .

 

For Example:

Compute the inverse Laplace transform of

Step 1: Perform partial fraction expansion.

                        

Step 2: Find all the coefficients of expansion. (A, B, C in this case)

                                For A,    = -2   ;    ;     ;

For C,    = -1  ;   ;     ;

For B,    = 0  ;   ;     ;

Step 3: Plug in the values of all the coefficients in the expression .

                          

Step 4: Find the inverse of each term in using Laplace Transform Pair.

                                 L

  L  

 L  

Step 5: Finally, write the expression for which is the inverse of .

                                ]

Note:     Please use Laplace Transform Pair Table to solve the Laplace Transform for a function (where applicable). Convert a Calculus problem into an Algebraic form to solve problems and save time.

 

 

Problem 5:

Analysis of LTI Systems:

LTI Systems Represented by Ordinary Differential Equations

Two ways to characterize the response of a causal and stable LTI system are:

· zero-state and zero-input responses, which have to do with the effect of the input and the initial conditions of the system,

· transient and steady-state responses, which have to do with close and far away behavior of the response.

For Example:

Use Laplace transforms to compute the solution to the differential equation given below:

 

    Where    

 

The Laplace transform of the ordinary differential equation gives:

   

 

                   

 

So we have as:

  

 

 

 

After replacing  =     ; We find that B1 = 1/2, B2 = 1 and B3 = −1/2,

 

Note:     Please use Laplace Transform Pair Table to solve the Laplace Transform for a function (where applicable). Convert a Calculus problem into an Algebraic form to solve problems and save time.

So that the complete response is:           

]

  

 

Problem 6:

A right sided signal's initial value    and final value   (if finite) can be found from its Laplace transform   by the following theorems:

· Initial value theorem: 

· Final value theorem: 

 

In order to find the final values of an expression, you would need to find  

 

For Example:  

Determine the final value of

First find

 

            Then find;

Therefore, the final value of the expression    is   1.

Note:     Please use Laplace Transform Pair Table to solve the Laplace Transform for a function (where applicable). Convert a Calculus problem into an Algebraic form to solve problems and save time.