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-3 -2 -1 f(t) 0 0 1 2 3 t(s) ∞ ) = ao + E (an cos nwot + bn sin nwot) f (t) = ao + E An cos(n&ot + Фn) n=1 "=1

-3 -2 -1 f(t) 0 0 1 2 3 t(s) ∞ ) = ao + E (an cos nwot + bn sin nwot) f (t) = ao + E An cos(n&ot + Фn) n=1 "=1

Introductory Circuit Analysis (13th Edition)
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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(a) Find a0:

(b) Find a1 and b1:

(d) Find A1 and Φ1:

**Question 1**: Represent the following waveform into a Fourier series in cosine/sine and magnitude/phase forms where \( A = 6 \). **Graph Description**: The graph shows a periodic waveform \( f(t) \) over time \( t \) (in seconds). The waveform exhibits a sawtooth pattern with sharp peaks and linear increases and decreases. It has the following features: - The waveform amplitude is labeled as \( A = 6 \). - The waveform starts at \( t = -3 \) and ends at \( t = 3 \), with repeating cycles. - At \( t = -3 \), \( f(t) = 0 \), and it increases linearly to \( f(t) = A \) at \( t = -2 \), then drops sharply to \( f(t) = 0 \) at \( t = -1 \). This pattern repeats through \( t = 0 \) to \( t = 3 \). **Formulas**: 1. **Cosine/Sine Form**: \[ f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos n\omega t + b_n \sin n\omega t \right) \] 2. **Magnitude/Phase Form**: \[ f(t) = a_0 + \sum_{n=1}^{\infty} A_n \cos(n\omega t + \phi_n) \] These equations represent the decomposition of the waveform into its Fourier series components. The first equation uses cosine and sine functions, while the second uses magnitude and phase angles.
Transcribed Image Text:**Question 1**: Represent the following waveform into a Fourier series in cosine/sine and magnitude/phase forms where \( A = 6 \). **Graph Description**: The graph shows a periodic waveform \( f(t) \) over time \( t \) (in seconds). The waveform exhibits a sawtooth pattern with sharp peaks and linear increases and decreases. It has the following features: - The waveform amplitude is labeled as \( A = 6 \). - The waveform starts at \( t = -3 \) and ends at \( t = 3 \), with repeating cycles. - At \( t = -3 \), \( f(t) = 0 \), and it increases linearly to \( f(t) = A \) at \( t = -2 \), then drops sharply to \( f(t) = 0 \) at \( t = -1 \). This pattern repeats through \( t = 0 \) to \( t = 3 \). **Formulas**: 1. **Cosine/Sine Form**: \[ f(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos n\omega t + b_n \sin n\omega t \right) \] 2. **Magnitude/Phase Form**: \[ f(t) = a_0 + \sum_{n=1}^{\infty} A_n \cos(n\omega t + \phi_n) \] These equations represent the decomposition of the waveform into its Fourier series components. The first equation uses cosine and sine functions, while the second uses magnitude and phase angles.
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