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Solved Problems-17

In the figure, Vi(t) = 10 sgn(t) volt. Using the Fourier transform method, find Vc(t) and sketch Vc(t) versus time, t. Given: R = 5 ohm, C = 1F.
vi(t) = 10 sgn(t)
Description: 4429.png Description: 4435.png

Transfer function of the circuit
Description: 4453.png
where, Vc(jï) is the Fourier transform of Vc(t)
∴ Description: 4459.png
Description: 4465.png

Taking inverse Laplace transform,
Description: 4471.png
To plot this curve, we follow the following steps:
  • From –  < t < 0, vi(t) = –10V, vC(t) = –10V;
  • At t = 0, vi(t) jumps from –10V to 10V and thus, vC(t) approaches its final value of 10V exponentially with time-constant of 5 second.
Description: 1966.png

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