Skip to main content
Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

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

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

`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))`.

Examples

Find the Laplace transforms of the periodic functions shown below:

(a)

Graph of periodic unit ramp function.

Answer

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{f_1(t)}`

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

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

Now

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

So

`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)}`

`=1/s^2-e^(-s)/s^2-e^(-s)/s`

`=(1-e^(-s)-se^(-s))/(s^2)`

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))`

`=(1-e^(-s)-se^(-s))/(s^2(1-e^(-2s))`

(b) Saw-tooth waveform:

Graph of saw-tooth waveform.

Answer

We can see from the graph that

`f_1(t)=a/bt*[u(t)-u(t-b)]`

and that the period is `p = b`.

So we have

`Lap{f_1(t)}`

`= 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)}{:]`

`=a/b(1/s^2-(e^(-bs))/s^2-(be^(-bs))/s)`

`=(a(1-e^(-bs)-bse^(-bs)))/(bs^2)`

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`.

Answer

Here,

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

and the period, `p=pi`.

`Lap{f_1(t)}`

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

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

`=1/(s^2+1)+(e^(-pis))/(s^2+1)`

`=(1+e^(-pis))/(s^2+1)`

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

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

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