Explanation Given data: Refer to Figure 17.71 in the textbook. In the circuit, the voltage source v s is same as the function f 2 ( t ) shown in Figure 17.65(b). The inductance L is 1 H . The capacitor C is 1 F . Formula used: Write the general expression to calculate trigonometric Fourier series of v s ( t ) . v s ( t ) = a 0 + ∑ n = 1 ∞ ( a n cos n ω 0 t + b n sin n ω 0 t ) (1) Here, a 0 is the dc component of x ( t ) , a n and b n are the Fourier coefficients, n is an integer, and ω 0 is the angular frequency. Write the expression to calculate the fundamental angular frequency. ω 0 = 2 π T (2) Here, T is the period of the function. Write the expression to calculate the dc component of the function x ( t ) . a 0 = 2 T ∫ 0 T 2 f 2 ( t ) d t (3) Write the expression to calculate Fourier coefficients. a n = 4 T ∫ 0 T 2 f 2 ( t ) cos n ω 0 t d t (4) Calculation: The Figure 17.65(b) is redrawn as shown in Figure 1. The time period of the function in Figure 1 is T = 3 and the signal is even. Therefore, b n = 0 . From Figure 1, the Fourier series function of the waveform is defined as, f 2 ( t ) = { 1 , 0 < t < 1 2 , 1 < t < 1.5 (5) Substitute 3 for T in equation (2) to find ω 0 . ω 0 = 2 π 3 Applying equation (5) in equation (3) to find a 0 . a 0 = 2 3 [ ∫ 0 1 1 d t + ∫ 1 1.5 2 d t ] = 2 3 [ ( t ) 0 1 + 2 ( t ) 1 1.5 ] = 2 3 [ ( 1 − 0 ) + 2 ( 1.5 − 1 ) ] = 2 3 [ 1 + 1 ] Simplify the above equation to find a 0 . a 0 = 4 3 (6) Applying equation (5) in equation (4) to find a n . a n = 4 3 [ ∫ 0 1 ( 1 ) cos ( 2 n π t 3 ) d t + ∫ 1 1.5 ( 2 ) cos ( 2 n π t 3 ) d t ] { ∵ ω 0 = 2 π 3 } = 4 3 [ ( sin ( 2 n π t 3 ) ( 2 n π 3 ) ) 0 1 + ( ( 2 ) sin ( 2 n π t 3 ) ( 2 n π 3 ) ) 1 1.5 ] = 4 3 [ ( ( 3 2 n π ) sin ( 2 n π t 3 ) ) 0 1 + ( ( 6 2 n π ) sin ( 2 n π t 3 ) ) 1 1.5 ] = 4 3 [ [ ( 3 2 n π ) sin ( 2 n π ( 1 ) 3 ) − ( 3 2 n π ) sin ( 2 n π ( 0 ) 3 ) ] + [ ( 6 2 n π ) sin ( 2 n π ( 1.5 ) 3 ) − ( 6 2 n π ) sin ( 2 n π ( 1 ) 3 ) ] 1 1.5 ] Simplify the above equation as follows, a n = 4 3 [ [ ( 3 2 n π ) sin ( 2 n π 3 ) − ( 0 ) ] + [ ( 6 2 n π ) sin ( 3 n π 3 ) − ( 6 2 n π ) sin ( 2 n π 3 ) ] ] = 4 3 [ ( 3 2 n π ) sin ( 2 n π 3 ) + ( 3 n π ) sin ( n π ) − ( 3 n π ) sin ( 2 n π 3 ) ] = 4 3 [ ( 3 2 n π ) sin ( 2 n π 3 ) + 0 − ( 3 n π ) sin ( 2 n π 3 ) ] { ∵ sin ( n π ) = 0 } = 4 3 ( 3 n π ) [ ( 1 2 ) sin ( 2 n π 3 ) − sin ( 2 n π 3 ) ] Simplify the above equation as follows, a n = ( 4 n π ) sin ( 2 n π 3 ) [ 1 2 − 1 ] = ( 4 n π ) sin ( 2 n π 3 ) ( 1 − 2 2 ) = ( − 1 2 ) ( 4 n π ) sin ( 2 n π 3 ) a n = − 2 n π sin ( 2 n π 3 ) (7) Substitute equation (6) and equation (7) in equation (1) to find v s ( t )

EE 98: Fundamentals of Electrical Circuits - With Connect Access

6th Edition

ISBN: 9781259981807

Author: Alexander

Publisher: MCG

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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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