Laplace

Chapter #3 TRANSIENT ANALYSIS USING THE LAPLACE TRANSFORM TECHNIQUES

3.1 INTRODUCTION In the introductory courses of circuit analysis the transient response is usually examined for relatively simple circuits of one or two energy storage elements. This analysis is based on general (or classical) techniques, involves writing the differential equations for the network, and proceeds to use them to obtain the ff differential equation in terms of one variable. Then the complete solution, ff including the natural and forced responses, has to be obtained. The tedium and complexity of using this technique is in determining the initial conditions of the unknown variables and their derivatives and then evaluating the arbitrary constants by utilizing those initial conditions. This procedure usually requires a great amount of work, which increases with the complexity of the network. Therefore, we now focus our attention on more effective methods of transient ff analysis. A simplification of solving different problems can be achieved by using ff mathematical transformation. We are already familiar with one kind of mathematical transformation: the phasor transform technique, which allows simplifying the solution of the circuit steady-state response to sinusoidal sources. As we have seen, this very useful technique transforms the trigonometrical equations describing a circuit in the time domain into the algebraic equations in the frequency domain. Then the solution for the desirable variable (being actually manipulated by complex numbers) is transformed back to the time domain. In this chapter a very powerful tool for the transient analysis of circuits, i.e., the L aplace transform techniques, will be introduced. This method enables us to convert the set of integro-differential equations describing a circuit in its tranff sient behavior in the time domain to the set of linear algebraic equations in the complex frequency domain. Then using an algebraic operation, one may solve them for the