Electrical Engineering ⇒ Topic : Heaviside Expansion Theorem

Heaviside Expansion Theorem

          ............. (1)

(i) The roots of Q(s) are real and unequal

Consider the example of Eq. (1), i.e.

(ii) The roots of Q(s) are real and equal:

To determine the coefficients of fractions, where the roots are not equal, the method has been demonstrated above. For equal and repeated roots, the general formula is

where p denotes the number of times the roots are repeated and j = 1, 2, ..., p.

Let us consider an example

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

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

The inverse Laplace transform will be

                                                     ................. (4)

(iii) The roots of Q(s) are complex:

                                            ................. (5)

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

Taking the inverse Laplace transforms, we have

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

The above methods except that at (a) were originally discovered by Oliver Heaviside.These are, therefore, known by the name Heaviside expansion theorem.