Laplace transform Lab

Please refer a text book for a very detailed study about Laplace transforms

Laplace Transform (LT) helps in converting differentiation, integration and many other complex functions into simple algebraic functions or expressions there by making the analysis of a system easier. LT are applied to systems which use continuous time signals and many a times on problems which involve absolutely non-integrable functions like impulse response of an unstable system. Also the convolution(shift and add repeatedly) also can be performed by pure multiplication after converting the functions into respective LT’s.

LT exist in 2 varieties namely Unilateral(one-side) and Bilateral(2-sided) transforms. Many properties are similar in nature hence studying one of them and extending it to the other form is felt easier(Changes to be mentioned where ever necessary).

Definition

LT of a function f(t) is defined as

It contains complex exponentials,e^-st and s= σ + jω

We will be using laplace transform to find the solutions of differential equations, Operations on elementary signals, checking the basic properties of functions etc.

Inverse LT can be achieved by complex contour integrals or by partial fraction method. We will be following the partial fraction method and then find the Inverse Laplace Transform

Let us start with some basic elementary functions (signals)

Unit step function

For an impulse function, sifting property holds good and this will be dealt later when we are taking the convolution of signals.

Properties of LT

Matlab programs

Some cautions before going in for the programs

1.Save the file name with no spaces

2.It must have *.m extension

3.Make sure you are committing any spelling mistakes for variables and functions

4.take care for right division and left division

5.Use “syms” for single character objects

6.Use “sym” for strings but make sure you put it in single quotes

7.Work out the problem by hand to make sure your results are correct.

8. If there an error displayed in a function then check for spelling mistakes. If it did not solve the problem then type the following lines in all your codes or type in command line
close all;
clear all;

Find laplace transform of e^-2t, 10e^-5t, te^-3t, 5d²y/dt²
/***********/
%Find the LT of e^-2t
syms t;
solution = laplace(exp(-2*t));
/***********/
%Find the LT of 10e^-5t
syms t;
solution = laplace(10*exp(-5*t));
/***********/
%Find the LT of te^-3t
syms t;
syms s; %can be omitted
solution = laplace(t*exp(-3*t));
/***********/
%Find the LT of 5d²y/dt²
sym ‘y(t)’;
solution = laplace(5*diff(‘y(t)’,2));
/***********/

Find the inverse LT of

1). Y(s) = 1/s – 2/(s+3) + 10/(s+1)
2). Y(s) = s(s+1)/((s+2)(s²+2s+1))

%Program for inverse laplace transform of Y(s) = 1/s – 2/(s+3) + 10/(s+1)
syms t; %can be omitted
syms s;
soln = ilaplace(1/s – 2/(s+3) + 10/(s+1));
/***********/
%Program for inverse laplace transform of Y(s) = s(s+1)/((s+2)(s²+2s+1))
syms s;
soln = ilaplace((s*(s+2))/((s+2)*(s^2+2*s+1));
/************/

Solve the DE by laplace transform
D²(y) + 12 D(y) +32y = 32u(t)

Initial and Final Value Theorem

Lt sX(s) = x(0⁺)
s->inf

Lt sX(s) = x(inf)
s->0