21 Phasor form of Maxwell’s equations and damped
waves in conducting media
•
When the felds and the sources in Maxwell’s equations are all monochro
∇·
D
=
ρ
∇·
B
=0
∇×
E
=

∂
B
∂t
∇×
H
=
J
+
∂
D
∂t
.
matic Functions oF time expressed in terms oF their phasors, Maxwell’s
equations can be transFormed into the phasor domain.
–
In the phasor domain all
∂
∂t
→
jω
and all variables
D
,
ρ
, etc. are replaced by their phasors
˜
D
,
˜
ρ
,
and so on. With those changes Maxwell’s equations take the Form
shown in the margin.
∇·
˜
D
=˜
ρ
∇·
˜
B
=0
∇×
˜
E
=

jω
˜
B
∇×
˜
H
=
˜
J
+
jω
˜
D
–
Also in these equations it is implied that
˜
D
=
±
˜
E
˜
B
=
μ
˜
H
˜
J
=
σ
˜
E
where
±
,
μ
, and
σ
could be a Function oF Frequency
ω
(as, strictly
speaking, they all are as seen in Lecture 11).
–
We can derive From the phasor Form Maxwell’s equations shown
in the margin the TEM wave properties obtained earlier on using
the timedomain equations by assuming
˜
ρ
=
˜
J
=0
.
1
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˜
J
=0
with
˜
J
=
σ
˜
E
to examine how TEM waves propagate in conducting media.
•
With
˜
ρ
=
˜
J
=0
the phasor form Maxwell’s equation take their simpli
Fed forms shown in the margin.
∇·
˜
E
=0
∇·
˜
H
=0
∇×
˜
E
=

jωμ
˜
H
∇×
˜
H
=
jω±
˜
E
–
Using
∇×
[
∇×
˜
E
=

jωμ
˜
H
]
⇒ ∇
2
˜
E
=

jωμ
∇×
˜
H
which combines with the Ampere’s law to produce
∇
2
˜
E
+
ω
2
μ±
˜
E
=0
.
–
±or
x
polarized waves with phasors
˜
E
=ˆ
x
˜
E
x
(
z
)
,
the phasor wave equation above simpliFes as
∂
2
∂z
2
˜
E
x
+
ω
2
μ±
˜
E
x
=0
.
–
Try solutions of the form
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 Summer '10
 KUDEKI
 Wave propagation, e−αz cos, Damped wave snapshot, phasor wave solutions

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