Answered: Obtain the Fourier Series Expression…

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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$**Title: Fourier Series Expression for Output Voltage Waveform** **Problem Statement:** Obtain the Fourier Series Expression for the output voltage waveform shown in Fig 2. Draw the harmonic profile curve for it. **Description of Fig 2:** - **Vertical Axis:** Represents the voltage with two labeled points, $ V_{dc} $ at both the positive and negative peaks of the waveform. - **Horizontal Axis:** Represents the time or angular position with several labeled points: - $ \alpha_1, \alpha_2, \alpha_3 $ are points on the positive side before $ \pi/2 $. - $ \pi/2 $ marks the halfway point of one positive pulse. - $ \pi $ marks the end of the positive voltage waveform and the start of the negative waveform. **Waveform Description:** The waveform is a periodic rectangular pulse that switches between $ +V_{dc} $ and $ -V_{dc} $. The positive pulses are shown on the left side until $ \alpha_3 $, with transitions at $ \alpha_1 $ and $ \alpha_2 $. The negative pulses reach $ -V_{dc} $ and are mirror images, occurring after $ \pi $. **Task:** 1. Derive the Fourier Series Expression, which represents this periodic waveform in terms of sine and cosine functions. 2. Illustrate the harmonic profile curve, showing the contribution of each harmonic frequency in the series to reconstruct the waveform. This task involves mathematical analysis and plotting, helping to visualize and understand the frequency components present in complex waveforms.$

Transcribed Image Text:**Title: Fourier Series Expression for Output Voltage Waveform** **Problem Statement:** Obtain the Fourier Series Expression for the output voltage waveform shown in Fig 2. Draw the harmonic profile curve for it. **Description of Fig 2:** - **Vertical Axis:** Represents the voltage with two labeled points, $ V_{dc} $ at both the positive and negative peaks of the waveform. - **Horizontal Axis:** Represents the time or angular position with several labeled points: - $ \alpha_1, \alpha_2, \alpha_3 $ are points on the positive side before $ \pi/2 $. - $ \pi/2 $ marks the halfway point of one positive pulse. - $ \pi $ marks the end of the positive voltage waveform and the start of the negative waveform. **Waveform Description:** The waveform is a periodic rectangular pulse that switches between $ +V_{dc} $ and $ -V_{dc} $. The positive pulses are shown on the left side until $ \alpha_3 $, with transitions at $ \alpha_1 $ and $ \alpha_2 $. The negative pulses reach $ -V_{dc} $ and are mirror images, occurring after $ \pi $. **Task:** 1. Derive the Fourier Series Expression, which represents this periodic waveform in terms of sine and cosine functions. 2. Illustrate the harmonic profile curve, showing the contribution of each harmonic frequency in the series to reconstruct the waveform. This task involves mathematical analysis and plotting, helping to visualize and understand the frequency components present in complex waveforms.

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