The Laplace Transform
I introduced the Laplace Transform which takes a function f(t) in the time domain
and transforms it to another function, F(s), in the Laplace domain. The variable
"s" used in the Laplace domain will be referred to as the "Laplace Variable."
The power of the Laplace Transform, for our purposes, is that it will turn differential
equations into algebraic equations. We won't need to guess at the form of
the solution anymore either. The Laplace Transform also obviates the need
to find initial conditions at t=0+, you only need the conditions at t=0-.
I showed the definition of the Laplace Transform, and derived the transforms
of a few functions. I have a
posted on the web as well (ignore the Z-Transform entries for now). Note particularly
that the Laplace Transform of an impulse is unity - this will make working with
the impulse response very easy.
Properties of the Laplace Transform
Several properties of the Laplace Transform were developed., these and others
are listed in a
table of properties. There are several properties that will be the
most immediately useful to us, and that you should easily understand how to apply:
Linearity - We will use this constantly (probably for every problem we do),
and mostly without referring explicitly to it.
Differentiation - Multiplication by s. This property allows us to transform
a differential equation into an algebraic equation. This will make solution
of differential equations much easier. Another advantage is that initial
conditions are taken at t=0- (instead of 0+) so you don't
need to figure out what they are, for example, after a switch is thrown.
Convolution - Convolution in the time domain is powerful, but performing
the integration can be hard and/or tedious. The Laplace Transform makes
convolution very easy.
Other theorems you should feel comfortable with (though we
will use them less) are:
- Initial Value Theorem - useful for finding the value of a function at t=0+,
(e.g., right after the switch is thrown).
- Final Value Theorem - useful for finding the steady state value of a time
domain function from its Laplace domain equivalent.
- Integration - division by "s", the inverse operation to differentiation.
This is useful in analog computers and numerical simulations -- you will use
it in a lab in the future.
- Time Shift - at this point we will use this mostly for developing
complex functions from simple ones.