Problems

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Problems

Section 17.2 Trigonometric Fourier Series

17.1 Evaluate each of the following functions and see if it is periodic. If periodic, find its period.

(a)

$$f(t) = \cos \pi t + 2 \cos 3\pi t + 3 \cos 5\pi t$$

\n(b) $y(t) = \sin t + 4 \cos 2 \pi t$
\n(c) $g(t) = \sin 3t \cos 4t$
\n(d) $h(t) = \cos^2 t$
\n(e) $z(t) = 4.2 \sin(0.4\pi t + 10^{\circ}) + 0.8 \sin(0.6\pi t + 50^{\circ})$
\n(f) $p(t) = 10$

$$(g) q(t) = e^{-\pi t}$$

17.2 Using MATLAB, synthesize the periodic waveform for which the Fourier trigonometric Fourier series is

$$f(t) = \frac{1}{2} - \frac{4}{\pi^2} \left( \cos t + \frac{1}{9} \cos 3t + \frac{1}{25} \cos 5t + \cdots \right)$$

17.3 Give the Fourier coefficients a0, an, and bn of the waveform in Fig. 17.47. Plot the amplitude and phase spectra.

For Prob. 17.3.

17.4 Find the Fourier series expansion of the backward sawtooth waveform of Fig. 17.48. Obtain the amplitude and phase spectra.

For Probs. 17.4 and 17.66.

17.5 Obtain the Fourier series expansion for the waveform shown in Fig. 17.49.

Figure 17.49

For Prob. 17.5.

17.6 Find the trigonometric Fourier series for

$$f(t) = \begin{cases} 7.5 & 0 < t < \pi \\ 15 & \pi < t < 2\pi \end{cases} \quad \text{and} \quad f(t + 2\pi) = f(t).$$

17.7 Determine the Fourier series of the periodic function in Fig. 17.50.

Figure 17.50

For Prob. 17.7.

17.8 Using Fig. 17.51, design a problem to help other students better understand how to determine the exponential Fourier series from a periodic wave shape.

* An asterisk indicates a challenging problem.

17.9 Determine the Fourier coefficients an and bn of the first three harmonic terms of the rectified cosine wave in Fig. 17.52.

17.10 Find the exponential Fourier series for the waveform in Fig. 17.53.

Figure 17.53

For Prob. 17.10.

17.11 Obtain the exponential Fourier series for the signal in Fig. 17.54.

*17.12 A voltage source has a periodic waveform defined over its period as

$$v(t) = 120t(2\pi - t) \text{ V}, \qquad 0 < t < 2\pi$$

Find the Fourier series for this voltage.

17.13 Design a problem to help other students better understand obtaining the Fourier series from a periodic function.

17.14 Find the quadrature (cosine and sine) form of the Fourier series

$$f(t) = 7.5 + \sum_{n=1}^{\infty} \frac{37.5}{n^3 + 1} \cos\left(2nt + \frac{n\pi}{4}\right)$$

17.15 Express the Fourier series

$$f(t) = 10 + \sum_{n=1}^{\infty} \frac{4}{n^2 + 1} \cos 10nt + \frac{1}{n^3} \sin 10nt$$

(a) in a cosine and angle form,

(b) in a sine and angle form.

17.16 The waveform in Fig. 17.55(a) has the following Fourier series:

$$v_1(t) = \frac{1}{2} - \frac{4}{\pi^2} \left( \cos \pi t + \frac{1}{9} \cos 3\pi t + \frac{1}{25} \cos 5\pi t + \cdots \right) \text{V}$$

Obtain the Fourier series of v2(t) in Fig. 17.55(b).

Figure 17.55

For Probs. 17.16 and 17.69.

Section 17.3 Symmetry Considerations

17.17 Determine if these functions are even, odd, or neither.

(a) 1 + t (b) t ² − 1 (c) cos nπt sin nπt (d) sin2 πt (e) e^−t

17.18 Determine the fundamental frequency and specify the type of symmetry present in the functions in Fig. 17.56.

(c)

Figure 17.56 For Probs. 17.18 and 17.63.

17.19 Obtain the Fourier series for the periodic waveform in Fig. 17.57.

17.20 Find the Fourier series for the signal in Fig. 17.58. Evaluate f(t) at t = 2 using the first three nonzero harmonics.

17.21 Determine the trigonometric Fourier series of the signal in Fig. 17.59.

Figure 17.59 For Prob. 17.21.

17.22 Calculate the Fourier coefficients for the function in Fig. 17.60.

Figure 17.60 For Prob. 17.22.

17.23 Using Fig. 17.61, design a problem to help other students better understand finding the Fourier series of a periodic wave shape.

Figure 17.61 For Prob. 17.23.

• (a) find the trigonometric Fourier series coefficients a2 and b2,
• (b) calculate the magnitude and phase of the component of f(t) that has ωn = 10 rad/s,
• (c) use the first four nonzero terms to estimate f(π∕2),
• (d) show that

$$\frac{\pi}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \frac{1}{11} + \cdots$$

Figure 17.62 For Probs. 17.24 and 17.60.

17.25 Determine the Fourier series representation of the function in Fig. 17.63.

17.26 Find the Fourier series representation of the signal shown in Fig. 17.64.

For Prob. 17.26.

17.27 For the waveform shown in Fig. 17.65 below,

• (a) specify the type of symmetry it has,
• (b) calculate a3 and b3,
• (c) find the rms value using the first five nonzero harmonics.

Figure 17.65 For Prob. 17.27.

For Prob. 17.28.

17.29 Determine the Fourier series expansion of the sawtooth function in Fig. 17.67.

Figure 17.67

For Prob. 17.29.

17.30 (a) If f(t) is an even function, show that

$$c_n = \frac{2}{T} \int_0^{T/2} f(t) \cos n\omega_o t \, dt$$

(b) If f(t) is an odd function, show that

$$c_n = -\frac{j2}{T} \int_0^{T/2} f(t) \sin n\omega_o t \, dt$$

17.31 Let an and bn be the Fourier series coefficients of f(t) and let ωo be its fundamental frequency. Suppose f(t) is time-scaled to give h(t) = f(t). Express the aʹ ⁿ and bʹ ⁿ, and ωʹ ^o, of h(t) in terms of an, bn, and ωo of f(t).

Section 17.4 Circuit Applications

17.32 Find i(t) in the circuit of Fig. 17.68 given that

$$i_s(t) = 3.5 + \sum_{n=1}^{\infty} \frac{4}{n^2} \cos 3nt$$

A

Figure 17.68 For Prob. 17.32.

17.33 In the circuit shown in Fig. 17.69, the Fourier series expansion of vs(t) is

$$v_s(t) = 10 + \frac{5}{\pi} \sum_{n=1}^{\infty} \frac{1}{n} \sin(n\pi t)$$

Find vo(t).

Figure 17.69

For Prob. 17.33.

17.34 Using Fig. 17.70, design a problem to help other students better understand circuit responses to a Fourier series.

Figure 17.70 For Prob. 17.34.

17.35 If vs in the circuit of Fig. 17.71 is the same as function f2(t) in Fig. 17.56(b), determine the dc component and the first three nonzero harmonics of vo(t).

Figure 17.71 For Prob. 17.35.

17.36 Find the response io for the circuit in Fig. 17.72(a), where vs(t) is shown in Fig. 17.72(b).

Problems 805

17.37 If the periodic current waveform in Fig. 17.73(a) is applied to the circuit in Fig. 17.73(b), find vo.

17.38 If the square wave shown in Fig. 17.74(a) is applied to the circuit in Fig. 17.74(b), find the Fourier series for vo(t).

Figure 17.74 For Prob. 17.38.

17.39 If the periodic voltage in Fig. 17.75(a) is applied to the circuit in Fig. 17.75(b), find io(t).

Figure 17.75 For Prob. 17.39.

*17.40 The signal in Fig. 17.76(a) is applied to the circuit in Fig. 17.76(b). Find vo(t).

For Prob. 17.40.

17.41 The full-wave rectified sinusoidal voltage in Fig. 17.77(a) is applied to the low-pass filter in Fig. 17.77(b). Obtain the output voltage vo(t) of the filter.

Figure 17.77

For Prob. 17.41.

17.42 The square wave in Fig. 17.78(a) is applied to the circuit in Fig. 17.78(b). Find the Fourier series of vo(t).

Section 17.5 Average Power and RMS Values

17.43 The voltage across the terminals of a circuit is

$$v(t) = [30 + 20 \cos(60\pi t + 45^{\circ}) + 10 \cos(120\pi t - 45^{\circ})] \text{ V}$$

If the current entering the terminal at higher potential is

$$i(t) = 6 + 4\cos(60\pi t + 10^{\circ})$$ $$- 2\cos(120\pi t - 60^{\circ})$$

A

find:

(a) the rms value of the voltage,

(b) the rms value of the current,

(c) the average power absorbed by the circuit.

*17.44 Design a problem to help other students better

understand how to find the rms voltage across and the rms current through an electrical element given a Fourier series for both the current and the voltage. In addition, have them calculate the average power delivered to the element and the power spectrum.

17.45 A series RLC circuit has R = 10 Ω, L = 2 mH, and C = 40 μF. Determine the effective current and average power absorbed when the applied voltage is

v(t) = 100 cos 1000t + 50 cos 2000t + 25 cos 3000t V

17.46 Use MATLAB to plot the following sinusoids for 0 < t < 5:

(a) 5 cos 3t − 2 cos(3t − π∕3) (b) 8 sin(πt + π∕4) + 10 cos(πt − π∕8)

17.47 The periodic current waveform in Fig. 17.79 is applied across a 2-kΩ resistor. Find the percentage of the total average power dissipation caused by the dc component.

Figure 17.79 For Prob. 17.47.

17.48 For the circuit in Fig. 17.80,

$$i(t) = 20 + 16 \cos(10t + 45^{\circ})$$

• 12 \cos(20t - 60^{\circ}) mA

(a) find v(t), and

(b) calculate the average power dissipated in the resistor.

Problems 807

Figure 17.80

• For Prob. 17.48.
• 17.49 (a) For the periodic waveform in Prob. 17.5, find the rms value.
  • (b) Use the first five harmonic terms of the Fourier series in Prob. 17.5 to determine the effective value of the signal.
  • (c) Calculate the percentage error in the estimated rms value of z(t) if

s value of

$$z(t)$$

if
\n% error = $\left(\frac{\text{estimated value}}{\text{exact value}} - 1\right) \times 100$

Section 17.6 Exponential Fourier Series

• 17.50 Obtain the exponential Fourier series for f(t) = t, −1 < t < 1, with f(t + 2n) = f(t) for all integer values of n.
• 17.51 Design a problem to help other students better understand how to find the exponential Fourier series of a given periodic function.
• 17.52 Calculate the complex Fourier series for f(t) = et , −π< t < π, with f(t + 2πn) = f(t) for all integer values of n.
• 17.53 Find the complex Fourier series for f(t) = e^−t , 0 < t < 1, with f(t + n) = f(t) for all integer values of n.
• 17.54 Find the exponential Fourier series for the function in Fig. 17.81.

Figure 17.81 For Prob. 17.54.

17.55 Obtain the exponential Fourier series expansion of the half-wave rectified sinusoidal current of Fig. 17.82.

Figure 17.82

• For Prob. 17.55.
• 17.56 The Fourier series trigonometric representation of a periodic function is

$$f(t) = 10 + \sum_{n=1}^{\infty} \left( \frac{1}{n^2 + 1} \cos n\pi t + \frac{n}{n^2 + 1} \sin n\pi t \right)$$

Find the exponential Fourier series representation of f(t).

17.57 The coefficients of the trigonometric Fourier series representation of a function are:

$$b_n = 0,$$

$a_n = \frac{6}{n^3 - 2},$ $n = 0, 1, 2, ...$

If ωn = 50n, find the exponential Fourier series for the function.

17.58 Find the exponential Fourier series of a function that has the following trigonometric Fourier series coefficients:

$$a_0 = \frac{\pi}{4}$$

, $b_n = \frac{(-1)^n}{n}$ , $a_n = \frac{(-1)^n - 1}{\pi n^2}$

Take T = 2π.

17.59 The complex Fourier series of the function in Fig. 17.83(a) is

$$f(t) = \frac{1}{2} - \sum_{n=-\infty}^{\infty} \frac{je^{-j(2n+1)t}}{(2n+1)\pi}$$

Find the complex Fourier series of the function h(t) in Fig. 17.83(b).

Figure 17.83 For Prob.17.59.

• 17.60 Obtain the complex Fourier coefficients of the signal in Fig. 17.62.
• 17.61 The spectra of the Fourier series of a function are shown in Fig. 17.84. (a) Obtain the trigonometric Fourier series. (b) Calculate the rms value of the function.

For Prob. 17.61.

• 17.62 The amplitude and phase spectra of a truncated Fourier series are shown in Fig. 17.85.
  • (a) Find an expression for the periodic voltage using the amplitude-phase form. See Eq. (17.10).
  • (b) Is the voltage an odd or even function of t?

17.63 Plot the amplitude spectrum for the signal f2(t) in Fig. 17.56(b). Consider the first five terms.

17.64 Design a problem to help other students better understand the amplitude and phase spectra of a given Fourier series.

17.65 Given that

$$f(t) = \sum_{\substack{n=1 \ n \equiv odd}}^{\infty} \left( \frac{20}{n^2 \pi^2} \cos 2nt - \frac{3}{n\pi} \sin 2nt \right)$$

plot the first five terms of the amplitude and phase spectra for the function.

Section 17.7 Fourier Analysis with PSpice

• 17.66 Determine the Fourier coefficients for the waveform in Fig. 17.48 using PSpice or MultiSim.
• 17.67 Calculate the Fourier coefficients of the signal in Fig. 17.58 using PSpice or MultiSim.
• 17.68 Use PSpice or MultiSim to find the Fourier components of the signal in Prob. 17.7.
• 17.69 Use PSpice or MultiSim to obtain the Fourier coefficients of the waveform in Fig. 17.55(a).
• 17.70 Design a problem to help other students better
• understand how to use PSpice or MultiSim to solve circuit problems with periodic inputs.
• 17.71 Use PSpice or MultiSim to solve Prob. 17.40.

Section 17.8 Applications

17.72 The signal displayed by a medical device can be approximated by the waveform shown in Fig. 17.86. Find the Fourier series representation of the signal.

(a)

Figure 17.86

For Prob. 17.72.

• 17.73 A spectrum analyzer indicates that a signal is made up of three components only: 640 kHz at 2 V, 644 kHz at 1 V, 636 kHz at 1 V. If the signal is applied across a 10-Ω resistor, what is the average power absorbed by the resistor?
• 17.74 A certain band-limited periodic current has only three frequencies in its Fourier series representation: dc, 50 Hz, and 100 Hz. The current may be represented as

$$i(t) = 4 + 6 \sin 100\pi t + 8 \cos 100\pi t$$

• 3 sin 200 $\pi t$ – 4 cos 200 $\pi t$ A

(a) Express i(t) in amplitude-phase form.

• (b) If i(t) flows through a 2-Ω resistor, how many watts of average power will be dissipated?
• 17.75 Design a low-pass RC filter with a resistance R = 2 kΩ. The input to the filter is a periodic rectangular pulse train (see Table 17.3) with A = 1 V, T = 10 ms, and τ = 1 ms. Select C such that the dc component of the output is 50 times greater than the fundamental component of the output.
• 17.76 A periodic signal given by vs(t) = 10 V for 0 < t < 1 and 0 V for 1 < t < 2 is applied to the high-pass filter in Fig. 17.87. Determine the value of R such that the output signal vo(t) has an average power of at least 70 percent of the average power of the input signal.

Figure 17.87 For Prob. 17.76.