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Signals and Spectra Chap. 2
where the phasor c = |c|e
jl c and Re{ # } denotes the real part of the complex quantity { # } .
We will refer to ce jv 0t
as the rotating phasor, as distinguished from the phasor c.
When c is given to represent a waveform, it is understood that the actual waveform that
appears in the circuit is a sinusoid as specified by Eq. (2–24). Because the phasor is a complex
number, it can be written in either Cartesian form or polar form; that is,
c ! x + jy = |c|e jw
(2–25)
where x and y are real numbers along the Cartesian coordinates and |c| and
lc = w = tan -1 (y/x) are the length and angle (real numbers) in the polar coordinate system.
Shorthand notation for the polar form on the right-hand side of Eq. (2–25) is |c| lw .
For example, 25 sin(2p500t + 45°) could be denoted by the phasor 25 l -45°,
since, from Appendix A, sin x = cos(x - 90°) and, consequently, 25 sin(v 0 t + 45°) =
25 cos(v 0 t - 45°) = Re{(25e -jp/4 ) e jv0t }, where v 0 = 2pf 0 and f 0 = 500 Hz. Similarly,
10 cos(v 0 t + 35°) could be denoted by 10 l35° .
Some other authors may use a slightly different definition for the phasor. For example,
w(t) may be expressed in terms of the imaginary part of a complex quantity instead of the real
part as defined in Eq. (2–24). In addition, the phasor may denote the RMS value of w(t), instead
of the peak value [Kaufman and Seidman, 1979]. In this case, 10 sin (v 0 t + 45°) should
be denoted by the phasor 7.07 l45° . Throughout this book, the definition as given by
Eq. (2–24) will be used. Phasors can represent only sinusoidal waveshapes.
2–2 FOURIER TRANSFORM AND SPECTRA
Definition
How does one find the frequencies that are present in a waveform? Moreover, what is the
definition of frequency? For waveforms of the sinusoidal type, we know that we can find the
frequency by evaluating f 0 = 1/T 0 , where T 0 is the period of the sinusoid. That is, frequency
is the rate of occurrence of the sinusoidal waveshape. All other nonsinusoidal waveshapes
have more than one frequency. †
In most practical applications, the waveform is not periodic, so there is no T 0 to use to calculate
the frequency. Consequently, we still need to answer this question: Is there a general
method for finding the frequencies of a waveform that will work for any type of waveshape? The
answer is yes. It is the Fourier transform (FT). It finds the sinusoidal-type components in w(t).
DEFINITION.
The Fourier transform (FT) of a waveform w(t) is
q
W1f2 = [w(t)] = [w(t)]e -j2pft dt
L
-q
(2–26)
† The constant-voltage or constant-current DC waveform has one frequency, f = 0. It is a special case of a
cosine wave (i.e. sinusoidal-type waveform) where T 0 : q and f 0 : 0. A periodic square wave has an infinite
number of odd-harmonic frequencies, as shown by Example 2–13.