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Home Laplace Transforms 5. Transform of Periodic Functions

5. Laplace Transform of a Periodic Function f(t)

If function f(t) is:

Periodic with period p > 0, so that f(t + p) = f(t), and

f1(t) is one period (i.e. one cycle) of the function, written using Unit Step functions,

then

NOTE: In English, the formula says:

The Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of onecycle of the function, divided by .

Examples

Find the Laplace transforms of the periodic functions shown below:

(a)

Answer

From the graph, we see that the first period is given by:

and that the period is .

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0 0

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Now

So

Hence, the Laplace transform of the periodic function, f(t) is given by:

(b) Saw-tooth waveform:

Answer

We can see from the graph that

and that the period is .

So we have

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\0 10^ \0 10 ^

0 10 0 10 10

\0 10^ \0 10 ^

\0 10^ \0 10 10 ^

\0 10^ \0 10 ^ \10 ^

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0 0

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(We next subtract, then add a " " term in the middle, to achieve the required form.)

(We now find the Laplace Transform of the individual pieces.)

So the Laplace Transform of the periodic function is given by

(c) Full-wave rectification of sin t:

Answer

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\0 10 0 10 ^

\0 10 0 10 ^

\0 10 0 10 10 ^

< \0 10^ \0 10 ^

\ 10 ^>

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