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 small table 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.