17.9 Summary

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17.9 Summary

• 1. A periodic function is one that repeats itself every T seconds; that is, f(t ± nT) = f(t), n = 1, 2, 3, ….
• 2. Any nonsinusoidal periodic function f(t) that we encounter in electrical engineering can be e xpressed in terms of sinusoids using Fourier series:

$$f(t) = \underbrace{a_0}_{\text{dc}} + \underbrace{\sum_{n=1}^{\infty} (a_n \cos n\omega_0 t + b_n \sin n\omega_0 t)}_{\text{ac}}$$

where ω0 = 2π∕T is the fundamental frequency. The Fourier series resolves the function into the dc component a0 and an ac component containing infinitely many harmonically related sinusoids. The Fourier coefficients are determined as

$$a_0 = \frac{1}{T} \int_0^T f(t) dt, \qquad a_n = \frac{2}{T} \int_0^T f(t) \cos n\omega_0 t dt$$ $$b_n = \frac{2}{T} \int_0^T f(t) \sin n\omega_0 t dt$$

If f(t) is an e ven function, bn = 0, and when f(t) is odd, a0 = 0 and an = 0. If f(t) is half-wave symmetric, a0 = an = bn = 0 for even values of n.

3. An alternative to the trigonometric (or sine-cosine) Fourier series is the amplitude-phase form

$$f(t) = a_0 + \sum_{n=1}^{\infty} A_n \cos(n\omega_0 t + \phi_n)$$

where

$$A_n = \sqrt{a_n^2 + b_n^2}$$

, $\phi_n = -\tan^{-1} \frac{b_n}{a_n}$

• 4. Fourier series representation allo ws us to apply the phasor method in analyzing circuits when the source function is a nonsinusoidal periodic function. We use phasor technique to determine the response of each harmonic in the series, transform the responses to the time domain, and add them up.
• 5. The average-power of periodic voltage and current is

$$P = V_{\rm dc} I_{\rm dc} + \frac{1}{2} \sum_{n=1}^{\infty} V_n I_n \cos(\theta_n - \phi_n)$$

In other w ords, the total average power is the sum of the a verage powers in each harmonically related voltage and current.

6. A periodic function can also be represented in terms of an exponential (or complex) Fourier series as

$$f(t) = \sum_{n = -\infty}^{\infty} c_n e^{jn\omega_0 t}$$

where

$$c_n = \frac{1}{T} \int_0^T f(t) e^{-jn\omega_0 t} dt$$

and ω0 = 2π∕T. The e xponential form describes the spectrum of f(t) in terms of the amplitude and phase of ac components at posi tive and negative harmonic frequencies. Thus, there are three basic forms of F ourier series representation: the trigonometric form, the amplitude-phase form, and the exponential form.

• 7. The frequency (or line) spectrum is the plot of An and n or ∣cn∣ and θⁿ versus frequency.
• 8. The rms value of a periodic function is given by

$$F_{\rm rms} = \sqrt{a_0^2 + \frac{1}{2} \sum_{n=1}^{\infty} A_n^2}$$

The power dissipated by a 1-Ω resistance is

$$P_{1\Omega} = F_{\text{rms}}^2 = a_0^2 + \frac{1}{2} \sum_{n=1}^{\infty} (a_n^2 + b_n^2) = \sum_{n=-\infty}^{\infty} |c_n|^2$$

This relationship is known as Parseval’s theorem.

• 9. Using PSpice, a F ourier analysis of a circuit can be performed in conjunction with the transient analysis.
• 10. Fourier series find application in spectrum analyzers and filters. The spectrum analyzer is an instrument that displays the discrete Fourier spectra of an input signal, so that an analyst can determine the fre quencies and relative energies of the signal’s components. Because the Fourier spectra are discrete spectra, filters can be designed for great effectiveness in blocking frequenc y components of a signal that are outside a desired range.

Review Questions

• 17.1 Which of the following cannot be a Fourier series?
  • (a) t − t 2 __ 2 ^+t 3 __ 3 − t 4 __ 4 ^+t 5 __ 5 (b) 5 sin t + 3 sin 2t − 2 sin 3t + sin 4t
  • (c) sin t − 2 cos 3t + 4 sin 4t + cos 4t
  • (d) sin t + 3 sin 2.7t − cos πt + 2 tan πt

(e)

$$1 + e^{-j\pi t} + \frac{e^{-j2\pi t}}{2} + \frac{e^{-j3\pi t}}{3}$$

17.2 If f(t) = t, 0 < t < π, f(t + nπ) = f(t), the value of ω0 is

(a) 1 (b) 2 (c)

$$\pi$$

(d) $2\pi$

17.3 Which of the following are even functions?

2 (a) t + t	2 (b) t cos t	(c) et2
2 4 (d) t + t	(e) sinh t

17.4 Which of the following are odd functions?

(a) sin t + cos t	(b) t sin t
(c) t ln t	3 (d) t cos t
(e) sinh t

17.5 If f(t) = 10 + 8 cos t + 4 cos 3t + 2 cos 5t + …, the magnitude of the dc component is:

(a) 10	(b) 8	(c) 4
(d) 2	(e) 0

17.6 If f(t) = 10 + 8 cos t + 4 cos 3t + 2 cos 5t + …, the angular frequency of the 6th harmonic is

(a) 12	(b) 11	(c) 9
(d) 6	(e) 1

• 17.7 The function in Fig. 17.14 is half-wave symmetric.
  • (a) True (b) False
• 17.8 The plot of ∣cn∣ versus nω0 is called:

(a) complex frequency spectrum (b) complex amplitude spectrum (c) complex phase spectrum