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Circuit Applications of Laplace Transform

Circuit Applications of Laplace Transform. Chairul Hudaya, ST, M.Sc. Electric Power & Energy Studies (EPES) Department of Electrical Engineering University of Indonesia http://www.ee.ui.ac.id/epes.

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Circuit Applications of Laplace Transform

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  1. Circuit Applications of Laplace Transform Chairul Hudaya, ST, M.Sc Electric Power & Energy Studies (EPES) Department of Electrical Engineering University of Indonesia http://www.ee.ui.ac.id/epes Depok, October, 2009 Electric Circuit

  2. Circuit applications 1. Transfer functions 2. Convolution integrals 3. RLC circuit with initial conditions

  3. Transfer function h(t) x(t) y(t) Network System In time domain, In s-domain,

  4. Example 1 For the following circuit, find H(s)=Vo(s)/Vi(s). Assume zero initial conditions.

  5. Solution Transform the circuit into s-domain with zero i.c.:

  6. Using voltage divider

  7. Example 2 Obtain the transfer function H(s)=Vo(s)/Vi(s), for the following circuit.

  8. Solution Transform the circuit into s-domain (We can assume zero i.c. unless stated in the question)

  9. We found that

  10. Example 3 Use convolution to find vo(t) in the circuit of Fig.(a) when the excitation (input) is the signal shown in Fig.(b).

  11. Solution Step 1: Transform the circuit into s-domain (assume zero i.c.) Step 2: Find the TF

  12. Step 3: Find vo(t) For t < 0 For t > 0

  13. 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

  14. 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

  15. For a capacitor • Taking the Laplace transform on both sides of eqn gives or

  16. 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.

  17. Solution Transform the circuit into s-domain (use the given i.c. to get the equivalents of L and C)

  18. Then, using nodal analysis

  19. Since the denominator cannot be factorized, we may write it as a completion of square: Finding i(t),

  20. Using partial fractions, It can be shown that Hence,

  21. Example 5 The switch in the following circuit moves from position a to position b at t = 0 second. Compute io(t) for t > 0.

  22. 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:

  23. Then, we can analyze the circuit for t > 0 by considering the i.c. Let

  24. Using current divider rule, we find that Using partial fraction we have

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