Electrical Engineering ⇒ Topic : Partial fraction expansion

PARTIAL FRACTION EXPANSION

The Laplace transform of any function may contain the ratio of two polynomials in s-domain, i.e.

                           ........... (1)

where Q(s) is of higher degree than P(s).

Therefore, the expansion of quotient into the sum of several fractions is required to obtain

inverse transforms. The partial fraction expansion method and another method popularly known as

Heaviside expansion theorem are introduced in this section.

Equation (1) can be written as

                                               ............ (1)

(a) General Case

Let us consider an example

                                    ................. (2)

Factorizing the denominator, we have

                                 ............... (3)

                                                  a1 + a2 = 1                                         .................. (4)

                                               3a2 - 2a1 = 2                                         .................... (5)

Solving Eqs. (4) and (5), we get

Putting the values of al and a2 in Eq. (3), we can show that 

                                               ............... (6)

The inverse Laplace transform of Eq. (6) gives the result as follows:

                                  ................ (7)