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.

• One method for doing this involves taking derivatives with respect to "s".

• Another method that I demonstrated involves cross multiplication by the denominator of the function being expanded.  This yields two equal polynomials and the unknown coefficients can be determined by equating like powers.  This method is usually (but not always) easier than the method involving differentiation.

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.

• The first method is the simpler one conceptually, but the more complicated one computationally.  In this method you proceed as in case 1, and you treat each complex conjugate pole as a simple first order pole.  This yields complex coefficients for the partial fraction expansion.  In class I showed that the coefficients corresponding to complex conjugate pairs are themselves complex conjugates, and how to simplify down to a decaying cosine function.  This is hard to do because you have to deal with complex numbers, however it is the method we will use when using MatLab, because MatLab easily handles complex numbers.

• Another method for dealing with complex conjugate pairs is a method called completing the square.  If the denominator has a polynomial s2+as+b that has complex conjugate roots, we can rewrite it as
s2+as+b=s2+as+0.25a2+b-0.25a2=(s+0.5a)2+b-0.25a2
which can be equated to
(s+α)22
where
α=0.5a,  and ω2=b-0.25a2
Note that this term occurs in the denominator of many of the forms in the Laplace Transform Table.  If you use this method you must be sure to make the numerator of the term a first order polynomial, for example Cs+D.  The coefficients C and D are most easily found by the cross multiplication method.

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).