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 of a function f(t) for t > 0 is defined by the following integral defined over 0 to ∞:
{ f(t)} =
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 function F of s"
and write:
Similarly, the Laplace transform of a function g(t) would be written:
The Good News
In practice, we do not need to actually find this infinite integral for each function f(t) that we have to find the Laplace Transform for.
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)
Ramp function: f(t) = (t)
Impulse function: f(t) = δ(t)
δ(t) represents an impulse at t = 0 and has value 0 otherwise.
We do not cover the Laplace Transform of δ(t) in this chapter.
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