Laplcae Transform of Electrical Network

       In the use of Laplace in electrical systems, it is always easy to redraw the system by finding Laplace transform of the given network. Electrical network mostly consists of the parameters R, L and C. the various expressions related to these parameters in time domain and Laplace domain are given in the table below.
       From the table it can be seen that after taking Laplace transform of time domain equations, neglecting the initial conditions, the resistance R behaves as R, the inductance behaves as sL, while the capacitance behaves as 1/(sC) and all time domain functions get converted to Laplace domain like i(t) to I(s), v(t) to V(s) and so on. 
       By using these transformations, the parameters can be replaced by their Laplace transform to get Laplace transform of the entire network. Once this is obtained, simple algebraic equations relating Laplace of various voltages and currents can be directly obtained. This eliminates the step of writing the integrodifferential equations and taking Laplace of them.
       e.g. Consider a network shown below,
       The Laplace of the above network can be obtained by following replacements.
       The other variables then can be introduced which will be directly Laplace variables to obtain the Laplace domain equations directly. Such Laplace of network is shown in the Fig.2.


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