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Depok, October, 2009 Laplace Transform Electric Circuit Circuit Applications of Laplace Transform Electric Power & Energy Studies (EPES) Department of Electrical Engineering University of Indonesia http://www.ee.ui.ac.id/epes Chairul Hudaya, ST, M.Sc Depok, October, 2009 Electric Circuit
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Depok, October, 2009 Laplace Transform Electric Circuit Circuit applications 1.Transfer functions 2.Convolution integrals 3.RLC circuit with initial conditions
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Depok, October, 2009 Laplace Transform Electric Circuit Transfer function h(t)h(t) y(t)y(t)x(t)x(t) In s-domain, In time domain, Network System
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Depok, October, 2009 Laplace Transform Electric Circuit Example 1 For the following circuit, find H(s)=V o (s)/V i (s). Assume zero initial conditions.
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Depok, October, 2009 Laplace Transform Electric Circuit Solution Transform the circuit into s-domain with zero i.c.:
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Depok, October, 2009 Laplace Transform Electric Circuit Using voltage divider
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Depok, October, 2009 Laplace Transform Electric Circuit Example 2 Obtain the transfer function H(s)=V o (s)/V i (s), for the following circuit.
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Depok, October, 2009 Laplace Transform Electric Circuit Solution Transform the circuit into s-domain (We can assume zero i.c. unless stated in the question)
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Depok, October, 2009 Laplace Transform Electric Circuit We found that
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Depok, October, 2009 Laplace Transform Electric Circuit Example 3 Use convolution to find v o (t) in the circuit of Fig.(a) when the excitation (input) is the signal shown in Fig.(b).
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Depok, October, 2009 Laplace Transform Electric Circuit Solution Step 1: Transform the circuit into s-domain (assume zero i.c.) Step 2: Find the TF
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Depok, October, 2009 Laplace Transform Electric Circuit Step 3: Find v o (t) For t < 0 For t > 0
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Depok, October, 2009 Laplace Transform Electric Circuit Circuit element models Apart from the transformations we must model the s-domain equivalents of the circuit elements when there is involving initial condition (i.c.) Unlike resistor, both inductor and capacitor are able to store energy
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Depok, October, 2009 Laplace Transform Electric Circuit Therefore, it is important to consider the initial current of an inductor and the initial voltage of a capacitor For an inductor : –Taking the Laplace transform on both sides of eqn gives or
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Depok, October, 2009 Laplace Transform Electric Circuit
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For a capacitor Taking the Laplace transform on both sides of eqn gives or
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Depok, October, 2009 Laplace Transform Electric Circuit
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Example 4 Consider the parallel RLC circuit of the following. Find v(t) and i(t) given that v(0) = 5 V and i(0) = −2 A.
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Depok, October, 2009 Laplace Transform Electric Circuit Solution Transform the circuit into s-domain (use the given i.c. to get the equivalents of L and C)
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Depok, October, 2009 Laplace Transform Electric Circuit Then, using nodal analysis
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Depok, October, 2009 Laplace Transform Electric Circuit Since the denominator cannot be factorized, we may write it as a completion of square: Finding i(t),
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Depok, October, 2009 Laplace Transform Electric Circuit Using partial fractions, It can be shown that Hence,
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Depok, October, 2009 Laplace Transform Electric Circuit Example 5 The switch in the following circuit moves from position a to position b at t = 0 second. Compute i o (t) for t > 0.
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Depok, October, 2009 Laplace Transform Electric Circuit Solution The i.c. are not given directly. Hence, at first we need to find the i.c. by analyzing the circuit when t ≤ 0 :
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Depok, October, 2009 Laplace Transform Electric Circuit Then, we can analyze the circuit for t > 0 by considering the i.c. Let
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Depok, October, 2009 Laplace Transform Electric Circuit Using current divider rule, we find that Using partial fraction we have
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