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A circuit model
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Figure 61 A circuit model
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reality, the circuitry in a hi- stereo system is far more complex than the circuits that will be discussed in this chapter and in the homework problems However, from the standpoint of intuition and everyday experience, the audio analogy provides a useful example; it allows you to build a quick feeling for the idea of frequency response Practically everyone has an intuitive idea of bass, mid range, and treble as coarsely de ned frequency regions in the audio spectrum The material presented in the next few sections should give you a more rigorous understanding of these concepts
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Circuits
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According to this de nition, frequency response could be de ned in a variety of ways For example, we might be interested in determining how the load voltage varies as a function of the source voltage Then, analysis of the circuit of Figure 61 might proceed as follows To express the frequency response of a circuit in terms of variation in output voltage as a function of source voltage, we use the general formula HV (j ) = VL (j ) VS (j ) (61)
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One method that allows for representation of the load voltage as a function of the source voltage (this is, in effect, what the frequency response of a circuit implies) is to describe the source and attached circuit by means of the Th venin equivalent e circuit (This is not the only useful technique; the node voltage or mesh current equations for the circuit could also be employed) Figure 62 depicts the original circuit of Figure 61 with the load removed, ready for the computation of the Th venin equivalent e
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ZS VS
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Z1 Z2
+ VL VT = VS
+ _
ZT ZT = (ZS + Z1) || Z2
Z2 ZS + Z1 + Z2
Figure 62 Th venin equivalent source circuit e
Fourier Analysis In this brief introduction to Fourier theory, we shall explain in an intuitive manner how it is possible to represent many signals by means of the superposition of various sinusoidal signals of different amplitude, phase, and frequency Any periodic nite-energy signal may be expressed by means of an in nite sum of sinusoids, as illustrated in the following paragraphs Consider a periodic waveform, x(t) Its Fourier series representation is de ned below by the in nite summation of sinusoids at the frequencies n 0 (integer multiples of the fundamental frequency, 0 ), with amplitudes An and phases n x(t) = x(t + T0 ) x(t) =
One could also write the term 2 n/T0 as n 0 , where 0 = 2 = 2 f0 T0 (64)
T0 = period 2 nt + n T0
(62)
An cos
(63)
is the fundamental (radian) frequency and the frequencies 2 0 , 3 0 , 4 0 , and so on, are called its harmonics The notion that a signal may be represented by sinusoidal components is particularly useful, and not only in the study of electrical circuits in the sense that we need only understand the response of a circuit to an arbitrary sinusoidal excitation in order to be able to infer the circuit s response to more complex signals In fact, the frequently employed sinusoidal frequency response discussed in this chapter is a function that enables us to explain how a circuit would respond to a signal made up of a superposition (continued)
6
Frequency Response and System Concepts
of sinusoidal components at various frequencies These sinusoidal components form the spectrum of the signal, that is, its frequency composition; the amplitude and phase of each of the sinusoids contribute to the overall character of the signal, in the same sense as the timbre of a musical instrument is made up of the different harmonics that are generated when a note is played (the timbre is what differentiates, for example, a viola from a cello or a violin) An example of the amplitude spectrum of a square-wave signal is shown in Figure 63 In order to further illustrate how the superposition of sinusoids can give rise to a signal that at rst might appear substantially different from a sinusoid, the evolution of a sine wave
into a square wave is displayed in Figure 64, as more Fourier components are added The rst picture represents the fundamental component, that is, the sinusoid that has the same frequency as the square wave Then one harmonic at a time is added, up to the fth nonzero component (the ninth frequency component; see Figure 63), illustrating how, little by little, the rounded peaks of the sinusoid transform into the at top of the square wave! Although this book will not deal with the mathematical aspects of Fourier series, it is important to recognize that this analysis tool provides excellent motivation for the study of sinusoidal signals, and of the sinusoidal frequency response of electric circuits
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