Page 245 - Electromagnetics

4.7 Monochromatic fields and the phasor domain
The Fourier transform is very efficient for representing the nearly sinusoidal signals
produced by electronic systems such as oscillators. However, we should realize that the
elemental term e jωt by itself cannot represent any physical quantity; only a continuous
superposition of such terms can have physical meaning, because no physical process can
be truly monochromatic. All events must have transient periods during which they are
established. Even “monochromatic” light appears in bundles called quanta, interpreted
as containing finite numbers of oscillations.
Arguments about whether “monochromatic” or “sinusoidal steady-state” fields can
actually exist may sound purely academic. After all, a microwave oscillator can create
a wave train of 10 10 oscillations within the first second after being turned on. Such a
waveform is surely as close to monochromatic as we would care to measure. But as with
all mathematical models of physical systems, we can get into trouble by making non-
physical assumptions, in this instance by assuming a physical system has always been
in the steady state. Sinusoidal steady-state solutions to Maxwell’s equations can lead to
troublesome infinities linked to the infinite energy content of each elemental component.
For example, an attempt to compute the energy stored within a lossless microwave cavity
under steady-state conditions gives an infinite result since the cavity has been building up
energy since t =−∞. We handle this by considering time-averaged quantities, but even
then must be careful when materials are dispersive (§ 4.5). Nevertheless, the steady-
state concept is valuable because of its simplicity and finds widespread application in
electromagnetics.
Since the elemental term is complex, we may use its real part, its imaginary part, or
some combination of both to represent a monochromatic (or time-harmonic) field. We
choose the representation

ψ(r, t) = ψ 0 (r) cos[ ˇωt + ξ(r)], (4.121)
where ξ is the temporal phase angle of the sinusoidal function. The Fourier transform is

∞

˜ − jωt
ψ(r,ω) = ψ 0 (r) cos[ ˇωt + ξ(r)]e dt. (4.122)
−∞
Here we run into an immediate problem: the transform in (4.122) does not exist in the
ordinary sense since cos( ˇωt +ξ) is not absolutely integrable on (−∞, ∞). We should not
be surprised by this: the cosine function cannot describe an actual physical process (it
extends in time to ±∞), so it lacks a classical Fourier transform. One way out of this
predicament is to extend the meaning of the Fourier transform as we do in § A.1. Then
the monochromatic field (4.121) is viewed as having the generalized transform

˜

ψ(r,ω) = ψ 0 (r)π e jξ(r) δ(ω − ˇω) + e − jξ(r) δ(ω + ˇω) . (4.123)
We can compute the inverse Fourier transform by substituting (123) into (2):
1 ∞ jξ(r) − jξ(r) jωt
ψ(r, t) = ψ 0 (r)π e δ(ω − ˇω) + e δ(ω + ˇω) e dω. (4.124)
2π
−∞
By our interpretation of the Dirac delta, we see that the decomposition of the cosine
function has only two discrete components, located at ω =± ˇω. So we have realized our

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