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HW3soln

# HW3soln - PHYS3360/AEP3630 Electronic Circuits Spring 2011...

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PHYS3360/AEP3630 Electronic Circuits, Spring 2011, HW3 Solutions 1 AC Circuits Solutions 1. a. Choose any convenient value for 𝑡 0 and perform integration of one period. 𝑉 𝑖𝑛 , 𝑛 = 2 𝑗𝑛𝜋 (1 ( 1) 𝑛 ), 𝑛 > 0, 𝑉 𝑖𝑛 , 0 = 0, 𝑛 = 0. Units of 𝑉 𝑖𝑛 , 𝑛 are volts. Note: all even harmonics are 0. This is true for any half-wave (odd) symmetric shape, e.g. a shape for which each half-cycle is the mirror image of the next half-cycle. b. through d. The new Fourier coefficients corresponding to the output are given by 𝑉 𝑜𝑢𝑡 , 𝑛 = 𝑉 𝑖𝑛 , 𝑛 𝑗𝑛𝜔 / 𝜔 𝑐 1 + 𝑗𝑛𝜔 / 𝜔 𝑐 . Here 𝜔 = 2 𝜋 /1ms = 6.28 kHz is the fundamental angular frequency, and 𝜔 𝑐 = 1/0.1ms = 10 kHz is the corner frequency of the high-pass filter. The corresponding plots are shown above. It is seen that with 𝑛 𝑚𝑎𝑥 = 19 the most salient features of the shapes (both input and output) are being reproduced with truncated Fourier series. There are some artifacts, such as ringing and overshoot (undershoot) near abrupt steps. The latter is known as Gibbs phenomenon, which

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