Page 247 - Electromagnetics

4.7.2 The phasor fields and Maxwell’s equations
Sinusoidal steady-state computations using the forward and inverse transform formulas
are unnecessarily cumbersome. A much more efficient approach is to use the phasor
concept. If we define the complex function
ˇ
ψ(r) = ψ 0 (r)e jξ(r)
˜
as the phasor form of the monochromatic field ψ(r,ω), then the inverse Fourier transform
ˇ
is easily computed by multiplying ψ(r) by e j ˇωt and taking the real part. That is,
ˇ
j ˇωt
ψ(r, t) = Re ψ(r)e = ψ 0 (r) cos[ ˇωt + ξ(r)]. (4.126)
Using the phasor representation of the fields, we can obtain a set of Maxwell equations
relating the phasor components. Let
3 3
E
ˇ
ˆ ˇ
ˆ jξ (r)
E(r) = i i E i (r) = i i |E i (r)|e i
i=1 i=1
represent the phasor monochromatic electric field, with similar formulas for the other
fields. Then
3
ˇ
j ˇωt
E
E(r, t) = Re E(r)e = ˆ i i |E i (r)| cos[ ˇωt + ξ (r)].
i
i=1
Substituting these expressions into Ampere’s law (2.2), we have
∂
ˇ
ˇ
ˇ
j ˇωt j ˇωt j ˇωt
∇× Re H(r)e = Re D(r)e + Re J(r)e .
∂t
Since the real part of a sum of complex variables equals the sum of the real parts, we
can write
" ∂ #
ˇ
ˇ
ˇ
Re ∇× H(r)e j ˇωt − D(r) e j ˇωt − J(r)e j ˇωt = 0. (4.127)
∂t
If we examine for an arbitrary complex function F = F r + jF i the quantity
j ˇωt
Re (F r + jF i )e = Re {(F r cos ˇωt − F i sin ˇωt) + j(F r sin ˇωt + F i cos ˇωt)} ,
we see that both F r and F i must be zero for the expression to vanish for all t.Thus
(4.127) requires that
ˇ
ˇ
ˇ
∇× H(r) = j ˇωD(r) + J(r), (4.128)
which is the phasor Ampere’s law. Similarly we have
ˇ
ˇ
∇× E(r) =− j ˇωB(r), (4.129)
ˇ
∇· D(r) = ˇρ(r), (4.130)
ˇ
∇· B(r) = 0, (4.131)
and
ˇ
∇· J(r) =− j ˇω ˇρ(r). (4.132)
The constitutive relations may be easily incorporated into the phasor concept. If we
use
E

ˇ
ˇ
D i (r) = ˜ (r, ˇω)E i (r) =|˜ (r, ˇω)|e jξ (r, ˇω) |E i (r)|e jξ (r) ,
i
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