0195136047.pdf

122 TIME-DEPENDENT CIRCUIT ANALYSIS

whereω= 2 πf= 2 π/Tis thefundamentalangular frequency,a 0 is theaverageordinate or the

dccomponent of the wave, (a 1 cosωt+b 1 sinωt)isthefundamental component, and (ancosnωt

+bnsinnωt), forn≥2, is thenthharmoniccomponent of the function.

TheFourier coefficientsare evaluated as follows:

a 0 =

1

T

∫T

0

f(t)dt (3.1.46)

an=

2

T

∫T

0

f(t)cosnωt dt, n≥ 1 (3.1.47)

bn=

2

T

∫T

0

f(t)sinnωt dt, n≥ 1 (3.1.48)

Theexponential formof the Fourier series can be shown to be given by

f(t)=

∑∞

n=−∞

C ̄nejnωt (3.1.49)

where

C ̄n=^1

T

∫T

0

f(t)e−jnωtdt (3.1.50)

Even though the exact representation of the nonsinusoidal, periodic wave requires an infinite

number of terms in the Fourier series, a good approximation for engineering purposes can be

obtained with comparatively few first terms, since the amplitude of the harmonics decreases

progressively as the order of the harmonic increases.

The system response is determined by the principle of superposition, and the phasor technique

yields the steady-state response. Each frequency component of the response is produced by

the corresponding harmonic of the excitation. The sum of these responses becomes the Fourier

series of the system response. Note that while the phasor method is employed to determine each

frequency component, the individual time functions must be used in forming the series for the

system response. Such a method of analysis is applicable to all linear systems.

EXAMPLE 3.1.10

(a) Find the Fourier series for the square wave shown in Figure E3.1.10(a).

(b) Let a voltage source having the waveform of part (a) with a peak value of 100 V and a

frequency of 10 Hz be applied to anRCseries network withR= 20 andC= 0 .1F.

Determine the first five nonzero terms of the Fourier series ofvC(t).

Solution

(a) From Equation (3.1.46),