Skip to main content
Search IntMath

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,


`Lap{f(t)}= Lap{f_1(t)}xx 1/(1-e^(-sp))`

NOTE: In English, the formula says:

The Laplace Transform of the periodic function f(t) with period p, equals the Laplace Transform of one cycle of the function, divided by `(1-e^(-sp))`.


Find the Laplace transforms of the periodic functions shown below:


Graph of periodic unit ramp function.


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

`f_1(t)=t*[u(t)-u(t-1)]` and that the period is `p=2`.


`= Lap{t*[(u(t)-u(t-1)]}`

`= Lap{t*u(t)}- Lap{t*u(t-1)}`


`t*u(t-1)` `=(t-1)*u(t-1)+u(t-1)`


`Lap{t*u(t)}- Lap{t*u(t-1)}`

`= Lap{t*u(t)}-` ` Lap{(t-1)*u(t-1)+u(t-1)}`

`= Lap{t*u(t)}- ` `Lap{(t-1)*u(t-1)}-` ` Lap{u(t-1)}`



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

`Lap{f(t)}` `=((1-e^(-s)-se^(-s))/s^2)xx1/(1-e^(-2s))`


(b) Saw-tooth waveform:

Graph of saw-tooth waveform.


We can see from the graph that


and that the period is `p = b`.

So we have


`= Lap{a/bt*[u(t)-u(t-b)]}`

`=a/b Lap{t*u(t)-t*u(t-b)}`

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

`=a/b Lap{t*u(t)-` `{:(t-b+b)*u(t-b)}`

`=a/b Lap{t*u(t)-` `(t-b)*u(t-b) -` `{: b*u(t-b)}`

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

`=a/b[ Lap{t*u(t)}-` ` Lap{(t-b)*u(t-b)}-` `{: Lap{b*u(t-b)}{:]`



So the Laplace Transform of the periodic function is given by

`Lap{f(t)}` `=(a(1-e^(-bs)-bse^(-bs))) / (bs^2(1-e^(-bs))`

(c) Full-wave rectification of sin t:

Graph of `f(t)=sin t*{u(t)-u(t-pi)}`, with period `pi`.



`f_1(t)=sin t*{u(t)-u(t-pi)}`

and the period, `p=pi`.


`= Lap{sin\ t*(u(t)-u(t-pi))}`

`= Lap{sin t*u(t)}+` ` Lap{sin(t-pi)*u(t-pi)}`



So the Laplace Transform of the periodic function is given by:


Problem Solver

AI Math Calculator Reviews

This tool combines the power of mathematical computation engine that excels at solving mathematical formulas with the power of GPT large language models to parse and generate natural language. This creates math problem solver thats more accurate than ChatGPT, more flexible than a calculator, and faster answers than a human tutor. Learn More.

Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.