7 Determine the Fourier series to represent a half-wave rectifier output current, i amperes, defined by Asin wt 0
7 Determine the Fourier series to represent a half-wave rectifier output current, i amperes, defined by Asin wt 0
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Problem Statement
**Determine the Fourier series to represent a half-wave rectifier output current, \( i \) amperes, defined by**
\[
i = f(t) =
\begin{cases}
A \sin \omega t & \text{for } 0 < t < \frac{T}{2} \\
0 & \text{for } \frac{T}{2} < t < T
\end{cases}
\]
**Subject to the condition**
\[
f(t + T) = f(t).
\]
### Explanation
The problem is to find the Fourier series for a periodic function that models the output current of a half-wave rectifier. The function \( f(t) \) is piecewise-defined over one period \( T \):
- The function is \( A \sin \omega t \) from \( t = 0 \) to \( t = \frac{T}{2} \).
- The function is zero from \( t = \frac{T}{2} \) to \( t = T \).
The periodicity condition \( f(t + T) = f(t) \) ensures that this pattern repeats every period \( T \).
### Purpose
This exercise involves using Fourier series to approximate periodic but non-sinusoidal waveforms, which is essential in understanding and analyzing signals in electrical engineering. The half-wave rectifier is a common example where such analysis is used to model and understand the behavior of alternating current (AC) circuits converted to direct current (DC).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F54376728-bf4e-4be6-b7b2-036009a3f75f%2F23e29827-9c11-40c4-9a21-fe1cae0b8033%2F0eviu68_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
**Determine the Fourier series to represent a half-wave rectifier output current, \( i \) amperes, defined by**
\[
i = f(t) =
\begin{cases}
A \sin \omega t & \text{for } 0 < t < \frac{T}{2} \\
0 & \text{for } \frac{T}{2} < t < T
\end{cases}
\]
**Subject to the condition**
\[
f(t + T) = f(t).
\]
### Explanation
The problem is to find the Fourier series for a periodic function that models the output current of a half-wave rectifier. The function \( f(t) \) is piecewise-defined over one period \( T \):
- The function is \( A \sin \omega t \) from \( t = 0 \) to \( t = \frac{T}{2} \).
- The function is zero from \( t = \frac{T}{2} \) to \( t = T \).
The periodicity condition \( f(t + T) = f(t) \) ensures that this pattern repeats every period \( T \).
### Purpose
This exercise involves using Fourier series to approximate periodic but non-sinusoidal waveforms, which is essential in understanding and analyzing signals in electrical engineering. The half-wave rectifier is a common example where such analysis is used to model and understand the behavior of alternating current (AC) circuits converted to direct current (DC).
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