# 2. Definition of the Laplace Transform

The Laplace transform provides a useful method of solving certain types of differential equations when certain initial conditions are given, especially when the initial values are zero.

The Laplace transform is also very useful in the area of circuit analysis (which we see later in the Applications section) . It is often easier to analyse the circuit in its Laplace form, than to form differential equations.

The techniques of Laplace transform are not only used in circuit analysis, but also in

- Proportional-Integral-Derivative (PID) controllers
- DC motor speed control systems
- DC motor position control systems
- Second order systems of differential equations (underdamped, overdamped and critically damped)

## Definition of Laplace Transform of *f*(t)

The Laplace transform ℒ, of a function *f*(*t*) for *t* > 0 is defined by the following integral over `0` to `oo`:

ℒ`{f(t)}=int_0^[oo]e^[-st] f(t) dt`

The resulting expression is a function of *s*, which we write as *F*(*s*). In words we say

"The Laplace Transform of

f(t) equals functionFofs".

and write:

ℒ`{f(t)}=F(s)`

Similarly, the Laplace transform of a function *g*(*t*) would be written:

ℒ`{g(t)}=G(s)`

## The Good News

In practice, we do not need to actually find this infinite integral for each function *f*(*t*) in order to find the Laplace Transform. There is a **table** of Laplace Transforms which we can use.

Go to the Table of Laplace Transformations.

## Scope of this Chapter

In this chapter, we deal only with the Laplace transform *f*(*t*) to *F*(*s*) (and the reverse process).

Also, we restrict ourselves to functions like

Unit step

functions: `f(t)=u(t)`, and

Ramp functions: `f(t)=t`.

We do not deal with impulse functions: `f(t) = δ(t)`, since it is beyond the scope of this introduction to Laplace Transform.

## Reminder: Unit, Ramp and Impulse Functions

Unit step function: *f*(*t*) = *u*(*t*):

Graph of `f(t)=u(t)`, the unit step function.

Ramp function: *f*(*t*) = (*t*):

Graph of `f(t)=t · u(t)`, the unit ramp function.

Impulse function `f(t)=δ(t)`:

Graph of `f(t)=δ(t)`, the impulse function.

`f(t)=δ(t)` represents an infinite impulse at *t* = 0 and has value 0 otherwise.

We do not cover the Laplace Transform of `δ(t)` in this chapter.