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8. Determine the sin/ cos Fourier series representation of f(t), where A = 10V. (1) A A/2- 0 -A/2- 1 2 3 4 1 (s)

8. Determine the sin/ cos Fourier series representation of f(t), where A = 10V. (1) A A/2- 0 -A/2- 1 2 3 4 1 (s)

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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**Problem 8: Determine the sin/cos Fourier series representation of \( f(t) \), where \( A = 10 \, \text{V}.** **Description of the Graph:** The graph represents a periodic waveform \( f(t) \) over the interval from \(-2\) to \(4\) seconds. It depicts a triangular waveform that repeats every 2 seconds. - **Time Axis (t-axis):** The horizontal axis is labeled \( t \) (s) indicating time in seconds. - **Amplitude Axis (f(t) axis):** The vertical axis is labeled \( f(t) \) and measures the amplitude in volts (V). The axis ranges from \(-A\) to \(A\), with specific points \(A/2\) and \(-A/2\) marked along the vertical. **Waveform Structure:** 1. **Negative Slope:** From \( t = -2 \) to \( t = -1 \), the waveform has a negative slope, moving from \( 0 \) towards \(-A\). 2. **Positive Slope:** Between \( t = -1 \) and \( t = 0 \), the waveform increases linearly from \(-A\) to \(A\). 3. **Negative Slope:** From \( t = 0 \) to \( t = 1 \), it decreases linearly back to \(0\). 4. **Repetition:** This pattern repeats for subsequent intervals \( t = 1 \) to \( t = 3 \) and beyond, with linear segments alternating between negative and positive slopes. **Characteristics:** - **Periodicity:** The waveform repeats every 2 seconds. - **Amplitude Values:** The maximum amplitude is \( A = 10 \, \text{V} \) and the minimum is \(-A = -10 \, \text{V} \). - **Symmetry:** The waveform shows odd symmetry, implying that only sine terms will appear in its Fourier series due to the antisymmetric nature about the vertical axis. This waveform is a typical example used in Fourier analysis to break down complex periodic signals into simpler sinusoidal components.
Transcribed Image Text:**Problem 8: Determine the sin/cos Fourier series representation of \( f(t) \), where \( A = 10 \, \text{V}.** **Description of the Graph:** The graph represents a periodic waveform \( f(t) \) over the interval from \(-2\) to \(4\) seconds. It depicts a triangular waveform that repeats every 2 seconds. - **Time Axis (t-axis):** The horizontal axis is labeled \( t \) (s) indicating time in seconds. - **Amplitude Axis (f(t) axis):** The vertical axis is labeled \( f(t) \) and measures the amplitude in volts (V). The axis ranges from \(-A\) to \(A\), with specific points \(A/2\) and \(-A/2\) marked along the vertical. **Waveform Structure:** 1. **Negative Slope:** From \( t = -2 \) to \( t = -1 \), the waveform has a negative slope, moving from \( 0 \) towards \(-A\). 2. **Positive Slope:** Between \( t = -1 \) and \( t = 0 \), the waveform increases linearly from \(-A\) to \(A\). 3. **Negative Slope:** From \( t = 0 \) to \( t = 1 \), it decreases linearly back to \(0\). 4. **Repetition:** This pattern repeats for subsequent intervals \( t = 1 \) to \( t = 3 \) and beyond, with linear segments alternating between negative and positive slopes. **Characteristics:** - **Periodicity:** The waveform repeats every 2 seconds. - **Amplitude Values:** The maximum amplitude is \( A = 10 \, \text{V} \) and the minimum is \(-A = -10 \, \text{V} \). - **Symmetry:** The waveform shows odd symmetry, implying that only sine terms will appear in its Fourier series due to the antisymmetric nature about the vertical axis. This waveform is a typical example used in Fourier analysis to break down complex periodic signals into simpler sinusoidal components.
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