17.1 Introduction

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17.1 Introduction

We have spent a considerable amount of time on the analysis of circuits with sinusoidal sources. This chapter is concerned with a means of ana lyzing circuits with periodic, nonsinusoidal e xcitations. The notion of periodic functions w as introduced in Chapter 9; it w as mentioned there that the sinusoid is the most simple and useful periodic function. This chapter introduces the F ourier series, a technique for e xpressing a periodic function in terms of sinusoids. Once the source function is expressed in terms of sinusoids, we can apply the phasor method to analyze circuits.

The F ourier series is named after Jean Baptiste Joseph F ourier (1768–1830). In 1822, F ourier’s genius came up with the insight that any practical periodic function can be represented as a sum of sinusoids. Such a representation, along with the superposition theorem, allo ws us to find the response of circuits to arbitrary periodic inputs using phasor techniques.

We begin with the trigonometric F ourier series. Later we consider the exponential Fourier series. We then apply F ourier series in circuit analysis. Finally, practical applications of Fourier series in spectrum analyzers and filters are demonstrated.

17.2 Trigonometric Fourier Series

While studying heat flow, Fourier discovered that a nonsinusoidal periodic function can be expressed as an infinite sum of sinusoidal functions. Recall that a periodic function is one that repeats every T seconds. In other words, a periodic function f (t) satisfies

$$f(t) = f(t + nT)$$

(17.1)

where n is an integer and T is the period of the function.

According to the Fourier theorem, any practical periodic function of angular frequenc y ω0 can be e xpressed as an infinite sum of sine or cosine functions that are inte gral multiples of ω0. Thus, f(t) can be expressed as

$$f(t) = a_0 + a_1 \cos \omega_0 t + b_1 \sin \omega_0 t + a_2 \cos 2\omega_0 t + b_2 \sin 2\omega_0 t + a_3 \cos 3\omega_0 t + b_3 \sin 3\omega_0 t + \cdots$$

(17.2)

or

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

(17.3)

where ω0 = 2π∕T is called the fundamental angular frequency in radians per second. The sinusoid sin nω0t or cos nω0t is called the nth harmonic of f(t); it is an odd harmonic if n is odd and an even harmonic if n is even. Equation 17.3 is called the trigonometric Fourier series of f(t). The constants an and bn are the Fourier coefficients. The coefficient a0 is the dc component or the average value of f(t). (Recall that sinusoids have zero average values.) The coefficients an and bn (for n ≠ 0) are the amplitudes of the sinusoids in the ac component. Thus,

The Fourier series of a periodic function f (t) is a representation that resolves f (t) into a dc component and an ac component comprising an infinite series of harmonic sinusoids.

A function that can be represented by a Fourier series as in Eq. (17.3) must meet certain requirements, because the infinite series in Eq. (17.3) may or may not converge. These conditions on f(t) to yield a convergent Fourier series are as follows:

1. f(t) is single-valued everywhere.

• 2. f(t) has a finite number of finite discontinuities in any one period.
• 3. f(t) has a finite number of maxima and minima in any one period.
• 4. The integral ∫ t0 t0+T ∣ f(t)∣ dt < ∞ for any t0.

The harmonic angular frequency ωⁿ is an integer multiple of the fundamental angular frequency ω0, i.e., ωn = nω0.

Historical note: Although Fourier published his theorem in 1822, it was P. G. L. Dirichlet (1805–1859) who later supplied an acceptable proof of the theorem.

A software package like Mathcad or Maple can be used to evaluate the Fourier coefficients.

These conditions are called Dirichlet conditions. Although they are not necessary conditions, the y are sufficient conditions for a Fourier series to exist.

A major task in Fourier series is the determination of the Fourier coefficients a0, an, and bn. The process of determining the coefficients is called Fourier analysis. The following trigonometric inte grals are very helpful in Fourier analysis. For any integers m and n,

$$\int_0^T \sin n\omega_0 t \, dt = 0 \tag{17.4a}$$ $$\int_0^T \cos n\omega_0 t \, dt = 0 \tag{17.4b}$$