Explanation Given data: Refer to Figure 17.14 in the textbook. Formula used: Write the expression to calculate the fundamental angular frequency. ω 0 = 2 π T (1) Here, T is the period of the function. Write the general expression to calculate trigonometric Fourier series of f ( t ) . f ( t ) = a 0 + ∑ n = 1 ∞ ( a n cos n ω 0 t + b n sin n ω 0 t ) (2) Here, a 0 is the dc component of f ( t ) , a n and b n are the Fourier coefficients, n is an integer, and ω 0 is the angular frequency. Calculation: The given waveform is drawn as Figure 1. The Figure 17.11(b) in the textbook is the odd periodic function. The Figure 17.11(b) in the textbook and Figure 1 is similar waveform function. Therefore, the function f ( t ) in Figure 1 is also an odd function. Write the expressions for Fourier coefficients for an odd function. a 0 = 0 a n = 0 b n = 4 T ∫ 0 T 2 f ( t ) sin n ω 0 t d t (3) Refer to the Figure 1. The Fourier series function of the waveform is defined as, f ( t ) = { 20 , − π < t < 0 − 20 , 0 < t < π The time period of the function in Figure 1 is, T = 2 π Substitute 2 π for T in equation (1) to find ω 0 . ω 0 = 2 π 2 π = 1 Substitute 1 for ω 0 , 0 for a 0 and 0 for a n in equation (2) to find f ( t ) . f ( t ) = 0 + ∑ n = 1 ∞ ( ( 0 ) cos n ( 1 ) t + b n sin n ( 1 ) t ) = 0 + ∑ n = 1 ∞ ( 0 + b n sin n t ) f ( t ) = ∑ n = 1 ∞ b n sin n t (4) Substitute 1 for ω 0 and 2 π for T in equation (3) to find b n . b n = 4 2 π ∫ 0 2 π 2 f ( t ) sin n ( 1 ) t d t = 2 π ∫ 0 π f ( t ) sin n t d t Substitute − 20 for f ( t ) with an interval 0 < t < π in above equation to find b n

EE 98: Fundamentals of Electrical Circuits - With Connect Access

6th Edition

ISBN: 9781259981807

Author: Alexander

Publisher: MCG

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Fourier Transform

In mathematics, the Fourier transformation is a mathematical transformation that rotates responsibilities by using region or time into tasks depending on the local or temporal frequency, such as the rendering of a musical track in step with existing volu…

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Chapter 17.3, Problem 3PP

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Find the Fourier series of the given function f(t) in Figure 17.14.

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Chapter 17.2, Problem 2PP

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Chapter 17 Solutions

EE 98: Fundamentals of Electrical Circuits - With Connect Access

Ch. 17.2 - Find the Fourier series of the square wave in Fig....Ch. 17.2 - Determine the Fourier series of the sawtooth...Ch. 17.3 - Prob. 3PP Ch. 17.3 - Find the Fourier series expansion of the function...Ch. 17.3 - Prob. 5PP Ch. 17.4 - Prob. 6PP Ch. 17.4 - If the input voltage in the circuit of Fig. 17.24...Ch. 17.5 - The voltage and current at the terminals of a...Ch. 17.5 - Find the rms value of the periodic current i(t) =...Ch. 17.6 - Obtain the complex Fourier series of the function...

Ch. 17.6 - Obtain the complex Fourier series expansion of...Ch. 17.7 - Prob. 12PP Ch. 17.8 - Rework Example 17.14 if the low-pass filter is...Ch. 17 - Which of the following cannot be a Fourier series?...Ch. 17 - If ft=t,0t,ft+n=ft, the value of 0 is (a) 1 (b) 2...Ch. 17 - Which of the following are even functions? (a) t +...Ch. 17 - Prob. 4RQ Ch. 17 - Prob. 5RQ Ch. 17 - If f(t) = 10 + 8 cos t + 4 cos 3t + 2 cos 5t + ,...Ch. 17 - Prob. 7RQ Ch. 17 - The plot of |cn| versus n0 is called: (a) complex...Ch. 17 - Prob. 9RQ Ch. 17 - Prob. 10RQ Ch. 17 - Evaluate each of the following functions and see...Ch. 17 - Using MATLAB, synthesize the periodic waveform for...Ch. 17 - Given that Fourier coefficients a0, an, and bn of...Ch. 17 - Find the Fourier series expansion of the backward...Ch. 17 - Prob. 5P Ch. 17 - Find the trigonometric Fourier series for f t =...Ch. 17 - Determine the Fourier series of the periodic...Ch. 17 - Using Fig. 17.51, design a problem to help other...Ch. 17 - Determine the Fourier coefficients an and bn of...Ch. 17 - Find the exponential Fourier series for the...Ch. 17 - Obtain the exponential Fourier series for the...Ch. 17 - Prob. 12P Ch. 17 - Prob. 13P Ch. 17 - Find the quadrature (cosine and sine) form of the...Ch. 17 - Express the Fourier series...Ch. 17 - The waveform in Fig. 17.55(a) has the following...Ch. 17 - Prob. 17P Ch. 17 - Prob. 18P Ch. 17 - Obtain the Fourier series for the periodic...Ch. 17 - Prob. 20P Ch. 17 - Prob. 21P Ch. 17 - Calculate the Fourier coefficients for the...Ch. 17 - Using Fig. 17.61, design a problem to help other...Ch. 17 - Prob. 24P Ch. 17 - Determine the Fourier series representation of the...Ch. 17 - Find the Fourier series representation of the...Ch. 17 - For the waveform shown in Fig. 17.65 below, (a)...Ch. 17 - Obtain the trigonometric Fourier series for the...Ch. 17 - Prob. 29P Ch. 17 - Prob. 30P Ch. 17 - Prob. 31P Ch. 17 - Prob. 32P Ch. 17 - Prob. 33P Ch. 17 - Prob. 34P Ch. 17 - Prob. 35P Ch. 17 - Prob. 36P Ch. 17 - If the periodic current waveform in Fig. 17.73(a)...Ch. 17 - Prob. 38P Ch. 17 - Prob. 39P Ch. 17 - The full-wave rectified sinusoidal voltage in Fig....Ch. 17 - Prob. 42P Ch. 17 - The voltage across the terminals of a circuit is...Ch. 17 - Prob. 44P Ch. 17 - A series RLC circuit has R = 10 , L = 2 mH, and C...Ch. 17 - Prob. 46P Ch. 17 - Prob. 47P Ch. 17 - Prob. 48P Ch. 17 - Prob. 49P Ch. 17 - Prob. 50P Ch. 17 - Prob. 51P Ch. 17 - Prob. 52P Ch. 17 - Prob. 53P Ch. 17 - Find the exponential Fourier series for the...Ch. 17 - Obtain the exponential Fourier series expansion of...Ch. 17 - The Fourier series trigonometric representation of...Ch. 17 - Prob. 57P Ch. 17 - Find the exponential Fourier series of a function...Ch. 17 - Prob. 59P Ch. 17 - Obtain the complex Fourier coefficients of the...Ch. 17 - The spectra of the Fourier series of a function...Ch. 17 - Prob. 62P Ch. 17 - Plot the amplitude spectrum for the signal f2(t)...Ch. 17 - Prob. 64P Ch. 17 - Prob. 65P Ch. 17 - Prob. 66P Ch. 17 - Prob. 67P Ch. 17 - Prob. 68P Ch. 17 - Prob. 69P Ch. 17 - Design a problem to help other students better...Ch. 17 - Prob. 71P Ch. 17 - Prob. 72P Ch. 17 - Prob. 73P Ch. 17 - Prob. 74P Ch. 17 - Prob. 75P Ch. 17 - Prob. 76P Ch. 17 - Prob. 77CP Ch. 17 - Prob. 78CP Ch. 17 - Consider the full-wave rectified sinusoidal...Ch. 17 - Prob. 82CP

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Chapter 17 Solutions