Laplace Transform Partial Fraction:

Most transform methods depend on the partial fraction of a given transform function. Given any solution of the form N(s) = P(s)/Q(s), the inverse Laplace transform can be determined by expanding it into partial fractions. The Laplace Transform Partial Fraction depend on the type of factor. It is to be assumed that P(s) and Q(s) have real coefficients and contain no common factors. The degree of P(s) is lower than that of Q(s).

Case 1:

When roots are real and simple

In this case

N(s) = P(s)/Q(s)

where

Q(s) = (s – a)(s – b)(s – c)

Expanding N(s) into partial fractions, we get

To obtain the constant A, multiplying Eq. 13.2 with (s — a) and putting s = a, we get

Similarly, we can get the other constants

Case 2:

When roots are real and multiple

In this case

N(s) = P(s)/ Q(s)

where

Q(s) = (s – a)ⁿ Q₁(s)

The partial fraction expansion of N(s) is

where P₁(s)/Q₁(s) = R(s) represents the remainder twins of expansion. To obtain the constant A₀,A₁,…, A_n-1, let us multiply both sides of Eq. 13.3 by (s – a)ⁿ

Thus

where R(s) indicates the remainder terms.

Putting s = a, we get

Differentiating  Eq. 13.4 with respect to s, and putting s = a, we get

Similarly,

In general,

Case 3:

When roots are complex

Consider a function

The partial fraction expansion of N(s) is

where P₁(s)/Q₁(s) is the remainder term.

Multiplying Eq. 13.5 by (s – α – jβ) and putting s = α + jβ, we get

Similarly,

In general, B = A* where A* is complex conjugate of A.

If we denote the inverse transform of the complex conjugate terms as f(t)

where A and A* are conjugate terms.

If we denote A = C + jD, then