EGR 261 – Laplace Transforms using MATLAB

Finding Laplace transforms using MATLAB

MATLAB has a powerful function

laplace( )

for finding Laplace transforms.

laplace(F)

- the Laplace transform of symbolic F with default independent variable t. The default return is a function of s. The Laplace transform is applied to a function of t and returns a function of s.

laplace(F, z)

- returns a function of z instead of a function of s -

laplace(F, s)

is the same as

laplace(F) laplace(F, w, z)

- finds the transform as a function of w and returns a function of z -

laplace(F, t, s)

is the same as

laplace(F) Note:

Since we are using one-sided Laplace transforms, there is an implied u(t) associated with functions. So to find

L(tu(t)),

use

laplace (t)

.

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Examples: See the following pages

EGR 261 – Laplace Transforms using MATLAB

Examples – using MATLAB

Verify the following relationships: L    1 s 2 L   2 ae 2t  be 3t  u(t)   2 s s  a 2  s b  3 L  10cos(2t)u (t)   s 2 10s  4 L  10u(t 4)   10e s 4s 2

EGR 261 – Laplace Transforms using MATLAB

Examples

Find Laplace transforms for the some of the following functions and use MATLAB to verify the results.

A) v(t) = 10sin(4t)u(t) 3 B) i(t) = 10te -2t u(t) C) v(t) = 10e -3t cos(4t)u(t) D) f(t) = 10t 2 e -3t u(t) E) i(t) = 4e -2(t - 3) u(t - 3)} F) v(t) = 10e -2(t - 4) sin(6[t - 4])u(t – 4) G) f(t) = 10tcos(3t)u(t)

EGR 261 – Laplace Transforms using MATLAB

Unit step functions

The

unit step function

, u(t), is sometimes called a

Heaviside function

.

Recall that u(t) is defined as follows:

u(t)

u(t)  0 1 t  0 t  0

1 t 0

Sometimes the unit step function is defined as: 0 t  0

u(t)

u(t)   0.5

1 t t   0 0

0.5

1 t 0

The

heaviside( )

function in MATLAB is defined like the second definition of u(t) above 4

EGR 261 – Laplace Transforms using MATLAB

Graphing unit step functions and piecewise-continuous functions in MATLAB

As noted earlier, the

heaviside( )

function can be used to represent a unit step function.

The waveform below (from an earlier example) could be formed using: 5

20*t.*(heaviside(t-2) – heaviside(t-6));

See the next slide for more details.

“window function” 20t[u(t-2) – u(t-6)] 120 40 0 2 6 t

EGR 261 – Laplace Transforms using MATLAB

MATLAB Example

u1: u(t) vs t u2: u(t-2) vs t u3: u(t)-u(t-2) vs t

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EGR 261 – Laplace Transforms using MATLAB

Finding derivatives in MATLAB

diff(f )

– finds the derivative of symbolic function f

diff(f, 2)

– finds the 2 nd derivative of f (same as diff(diff(f)) If f is a function of both x and y,

diff(f, x)

– partial derivative of f w.r.t. x

diff(f, y)

– partial derivative of f w.r.t. y

Examples

: (shown to the right) Before proceeding with more advanced derivatives and applications, it may be useful to discuss different ways of

representing functions in MATLAB

.

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EGR 261 – Laplace Transforms using MATLAB

Performing integration in MATLAB

int(S)

– finds the

indefinite integral

of a symbolic expression S with respect to its symbolic variable (or variable closest to x)

int(S, z)

– finds the

indefinite integral

of a symbolic expression S with respect to z

int(S, a, b)

– finds the

definite integral

of a symbolic expression S from a to b with respect to its symbolic variable (or variable closest to x)

int(S, z, a, b)

– finds the

definite integral

of a symbolic expression S from a to b with respect to z

double(int(S, z, a, b))

– finds a numeric result for the

definite integral

of a symbolic expression S from a to b with respect to z. This is useful in cases where MATLAB can’t find a symbolic solution.

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EGR 261 – Laplace Transforms using MATLAB

MATLAB Examples:

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