E12

Partial Fraction Expansion
(The Inverse Laplace Transform)

What follows is a brief description of partial fraction expansion.  I have a much more detailed version available.  The resulsts of the examples done in class are here.  The Matlab examples are available as a script, and as a web page.


The inverse Laplace Transform (going from the Laplace domain (s) to the time domain (t)) can be done analytically.  However, this is very difficult and requires mathematical tools that you don't have at this time -- we will never do the inverse transform analytically.  The method we will use is to split up the Laplace domain function into a sum of simpler functions that we can find in a Laplace Transform table.  Then we will simply use the table to get the inverse transform.  The method we will use is called "partial fraction expansion".

We will consider 4 cases of partial fraction expansion.  Even though we only did three and a half of them in class today, I will list all 4 here for convenience (we will finish on Monday).

Case 1-3 are used when the order of the numerator is less than the order of the denominator.  Case 4 is used when the order of the numerator is equal to the order of the denominator.  We will never have any cases when the order of the numerator is greater than the order of the denominator.

Case 1 - Distinct Real Roots

This is the simplest case.  I did a few examples in class, which I won't repeat here. 

Case 2 - Repeated Real Roots

This case is somewhat more complex.  Whenever you have an nth order pole in the denominator of the function being expanded, you need to include a term with an nth order pole, and (n-1)st order pole, all the way down to a first order pole.

Case 3 - Complex Conjugate Poles

As you recall from E11, complex conjugate poles result in decaying oscillations.  We will use two methods for doing partial fraction expansion with complex conjugate poles.  I only got to the first of the two methods in class Friday - we'll do the other one Monday.

Case 4 - Order of numerator=Order of denominator.

If the order of the numerator polynomial is equal to the order of the denominator polynomial, you must do a long division to pull a constant term out, and have a remainder that is a ratio of polynomials that can be solved according to cases 1-3.

Case 5 - An Exponential in the Numerator

I know I said there were only 4 cases, but sometimes you'll see an exponential in the numerator of the function being expanded.  The exponential does not get expanded.  To deal with it, do the partial fraction expansion as if the exponential weren't there (using the 4 cases listed above), then when doing the inverse transform the exponential simply becomes a time delay (see the time delay in Laplace Transform Properties Table).