sawtooth laplace
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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 .
\"0^
\ 0^
"
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0 0
"
,
\ 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
\0 ^
\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^ g
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0 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
\ 0^
"
0
\0 10 0 10 ^
\0 10 0 10 ^
\0 10 0 10 10 ^
< \0 10^ \0 10 ^
\ 10 ^>
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