![]() Compare the coefficients of all powers of s in the numerator of both sides which will give simultaneous equations interms of K 0, K 1, K 2,… Solving these equations we can obtain the coefficients K 0, K 1, K 2,…įor ease of solving simultaneous equations, we can find out the coefficient K 0 by the same method as discussed for simple roots. ![]() ![]() of the entire right hand side and express numerator interms of K 0, K 1, K 2,… The numerator N(s) on left hand side is known. Where N′(s)/D′(s) represents remaining terms of the expansion of F(s). The method of writing the partial fraction expansion for such multiple roots is, Here there is multiple root of the order ‘n’ existing at s = a. Hence once F(s) is expressed in terms partial fractions, with coefficients K 1, K 2 … K n, the inverse Laplace transform can be easily obtained. The values of K 1, K 2, K 3, … can be obtained as, Where K 1, K 2, K 3, … are called partial fraction coefficients. The degree of N(s) should be always less than D(s). Where a, b, c … are the simple and real roots of D(s). Hence the function F(s) can be expressed as, Let us discuss these three cases of roots of D(s). The method of finding partial fractions for each type is different. The roots of denominator polynomial D(s) play an important role in expanding the given F(s) into partial fractions. Once F(s) is expanded interms of partial fractions, inverse Laplace transform can be easily obtained by adjusting the terms and referring to the table of Standard Laplace transform pairs. Now in the remainder, degree of N′(s) is less than D'(s) and hence F 1(s) can be expressed in the partial fraction form. Q = Quotient obtained by dividing N(s) by D(s) Hence if degree of N(s) is equal or higher than D(s) then mathematically divide N(s) by D(s) to express F(s) in quotient and remainder form as, The given function F(s) can be expressed in partial fraction form only when degree of N(s) is less than D(s). The F(s), in partial fraction method, is written in the form as,
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