The Laplace Transform Method 16.1
The relation between the response yðtÞ and excitation xðtÞ in RLC circuits is a linear diﬀerential equation of the form an yðnÞ þ þ aj yð jÞ þ þ a1 yð1Þ þ a0 y ¼ bm xðmÞ þ þ bi xðiÞ þ þ b1 xð1Þ þ b0 x ð jÞ
where y and x are the jth and ith time derivatives of yðtÞ and xðtÞ, respectively. If the values of the circuit elements are constant, the corresponding coeﬃcients aj and bi of the diﬀerential equation will also be constants. In Chapters 7 and 8 we solved the diﬀerential equation by ﬁnding the natural and forced responses. We employed the complex exponential function xðtÞ ¼ Xest to extend the solution to the complex frequency s-domain. The Laplace transform method described in this chapter may be viewed as generalizing the concept of the s-domain to a mathematical formulation which would include not only exponential excitations but also excitations of many other forms. Through the Laplace transform we represent a large class of excitations as an inﬁnite collection of complex exponentials and use superposition to derive the total response.
THE LAPLACE TRANSFORM
Let f ðtÞ be a time function which is zero for t 0 and which is (subject to some mild conditions) arbitrarily deﬁned for t > 0. Then the direct Laplace transform of f ðtÞ, denoted l½ f ðtÞ, is deﬁned by ð1 l½ f ðtÞ ¼ FðsÞ ¼ f ðtÞest dt ð2Þ 0þ
Published on May 10, 2013