22 Phasor form of Maxwell’s equations and damped
waves in conducting media
• When the fields and the sources in Maxwell’s equations are all monochromatic functions of time expressed in terms of their phasors, Maxwell’s
equations can be transformed into the phasor domain.
– In the phasor domain all
∂
→ jω
∂t
∇·D = ρ
∇·B = 0
∂B
∂t
∂D
∇×H = J+
.
∂t
∇×E = −
and all variables D, ρ, etc. are replaced by their phasors D̃, ρ̃,
and so on. With those changes Maxwell’s equations take the form
shown in the margin.
– Also in these equations it is implied that
D̃ = $Ẽ
B̃ = µH̃
J̃ = σ Ẽ
where $, µ, and σ could be a function of frequency ω (as, strictly
speaking, they all are as seen in Lecture 11).
– We can derive from the phasor form Maxwell’s equations shown
in the margin the TEM wave properties obtained earlier on using
the time-domain equations by assuming ρ̃ = J̃ = 0.
1
∇ · D̃
∇ · B̃
∇ × Ẽ
∇ × H̃
=
=
=
=
ρ̃
0
−jω B̃
J̃ + jω D̃
We will do that, and and after that relax the requirement J̃ = 0 with
J̃ = σ Ẽ to examine how TEM waves propagate in conducting media.
• With ρ̃ = J̃ = 0 the phasor form Maxwell’s equation take their simplified forms shown in the margin.
– Using
∇ × [∇ × Ẽ = −jωµH̃] ⇒ −∇2Ẽ = −jωµ∇ × H̃
which combines with the Ampere’s law to produce
∇2Ẽ + ω 2µ$Ẽ = 0.
– For x-polarized waves with phasors
Ẽ = x̂Ẽx(z),
the phasor wave equation above simplifies as
∂2
2
Ẽ
+
ω
µ$Ẽx = 0.
x
∂z 2
– Try solutions of the form
Ẽx (z) = e−γz or eγz
where γ is to be determined.
2
∇ · Ẽ
∇ · H̃
∇ × Ẽ
∇ × H̃
=
=
=
=
0
0
−jωµH̃
jω$Ẽ
– Upon substitution into wave equation both of these lead to
(γ 2 + ω 2µ$)Ẽx = 0,
which yields
γ 2 + ω 2 µ$ = 0 ⇒ γ 2 = −ω 2µ$
from which one possibility is
√
with β ≡ ω µ$.
γ = jβ,
Thus viable phasor solutions are
Ẽx (z) = e∓jβz
as we already knew.
– Furthermore, using the phasor form Faraday’s law it is easy to
show that
!
µ
e∓jβz
with η =
.
H̃y = ±
η
ω
Note that we have recovered above the familiar properties of
plane TEM waves using phasor methods.
Next, the phasor method carries us to a new domain that cannot be
easily examined using time-domain methods.
3
• With ρ̃ = 0 but J̃ = σ Ẽ )= 0, implying non-zero conductivity σ, the
pertinent phasor form equations are as shown in the margin.
∇ · Ẽ
∇ · H̃
jω$ has been replaced by σ + jω$.
∇ × Ẽ
Thus, we can make use of phasor wave solutions above after ap- ∇ × H̃
plying the following modifications to γ and η:
– This is the same set as before, except that
=
=
=
=
=
0
0
−jωµH̃
σ Ẽ + jω$Ẽ
(σ + jω$)Ẽ
1.
"
⇒⇒
γ = −ω µ$ = (jωµ)(jω$)
γ = (jωµ)(σ + jω$)
σ )= 0
2
2.
2
η=
!
µ
=
$
#
jωµ
jω$
⇒⇒
η=
σ )= 0
#
jωµ
.
σ + jω$
Note that the modified γ and η satisfy
γη = jωµ and
γ
= σ + jω$
η
leading to useful relations shown in the margin (assuming real
valued σ and $).
4
µ =
γη
jω
γ
σ = Re{ }
η
1
γ
$ = Im{ }
ω
η
• In terms of γ and η above, we can express an x-polarized plane wave
propagating in z direction in terms of phasors
Ẽ = x̂Eoe∓γz and H̃ = ±ŷ
(a) Damped wave snapshot at t=0
together with exponential envelope
e−αz
Eo ∓γz
e
η
z
e−αz cos(ωt − βz)|t=0
where Eo is an arbitrary complex constant (complex wave amplitude).
• In expanded forms γ and η appear as:
"
γ = (jωµ)(σ + jω$) ≡ α+jβ, so that α = Re{γ} and β = Im{γ},
and
η=
#
jωµ
≡ |η|ejτ so that |η| = |
σ + jω$
#
jωµ
| and τ = ∠
σ + jω$
#
(b) Snaphot at t>0, with t=0 waveform
for comparison
z
jωµ
.
σ + jω$
–
1. In the special case of a perfect dielectric with σ = 0, we find
!
√
µ
γ = jω µ$ ≡ jβ and η =
,
$
e−αz cos(ωt − βz)
β appears within cosine
argument and determines the wavelength
λ=
and propagation speed
and, therefore,
vp =
∓jβz
Ẽ = x̂Eoe
ŷEoe∓jβz
and H̃ = ±
η
2π
β
–
ω
.
β
α controls wave attenuation by
e∓αz
as before. In this case α = τ = 0.
factor in propagation
5
direction.
2. Another case of imperfect dielectric (or “lousy” conductor) occurs
when σ is not zero, but it is so small that are justified in using
(1 ± a)p ≈ 1 ± pa, if |a| + 1,
with p =
1
2
as follows: For
σ
ω$
+ 1,
"
√
σ
√
σ
σ
)=
γ = (jωµ)(σ + jω$) = jω µ$(1−j )1/2 ≈ jω µ$(1−j
ω$
2ω$
2
hence
Ẽ ≈ x̂Eoe∓(α+jβ)z with α =
also in the same case
ŷEoe∓(α+jβ)z
with η =
H̃ ≈ ±
η
such that
!
σ
2
!
!
µ
√
+jω µ$;
$
µ
√
and β = ω µ$;
$
µ
σ ≈
$(1 − j ω$
)
!
µ
σ
(1+j
)≈
$
2ω$
!
µ j tan−1 σ
2ω# ,
e
$
!
µ
σ
and τ = ∠η ≈
.
$
2ω$
Note: γ and η both are complex valued, the consequences of which
will be examined later on.
|η| ≈
σ
, 1. In that case,
3. A third case of good conductor corresponds to ω$
!
!
!
!
!
!
ωµσ
µ
jωµ
ωµ jπ/4
σ
σ
γ = jω µ$(1 − j ) ≈ ω jµ = (1+j)
=
and η ≈
=
e .
ω$
ω
2
−j ωσ
σ
σ
6
Hence,
α≈β≈
(a) Damped wave snapshot at t=0
together with exponential envelope
!
ωµσ "
= πf µσ while |η| =
2
!
e−αz
ωµ
and τ = ∠η = 45o.
σ
4. Finally, perfect conductor case corresponds to σ → ∞, in which case
Ẽx → 0 as we will show later on. Wave fields cannot exist in perfect
conductors.
z
e−αz cos(ωt − βz)|t=0
(b) Snaphot at t>0, with t=0 waveform
for comparison
• Summarizing, in a homogeneous medium with arbitrary but constant µ, $, and σ, time-harmonic plane TEM waves are in terms of
z
E = x̂Re{Eoe∓(α+jβ)z ejωt} = x̂|Eo|e∓αz cos(ωt ∓ βz + ∠Eo)
e−αz cos(ωt − βz)
and accompanying magnetic fields
Eo
|Eo| ∓αz
H = ±ŷRe{ e∓(α+jβ)z ejωt} = ±ŷ
cos(ωt ∓ βz + ∠Eo − ∠η).
e
η
|η|
• Propagation velocity
ω
ω
"
,
vp = =
β Im{ (jωµ)(σ + jω$)}
now depends on frequency ω and it describes the speed of the nodes
(zero-crossings, not modified by the attenuation factor) of the field
waveform. Subscript p is introduced to distinguish vp — also called
phase velocity — from group velocity vg discussed in ECE 450 (velocity
of narrowband wave packets).
7
•
β appears within cosine
argument and determines the wavelength
λ=
2π
β
and propagation speed
vp =
•
ω
.
β
α controls wave attenuation by
e∓αz
factor in propagation
direction.
(a) Damped wave snapshot at t=0
together with exponential envelope
• Wavelength
2π vp
λ=
=
β
f
now depends on frequency f via both the numerator and the denominator, and measures twice the distance between successive nodes of the
waveform.
• Penetration depth (also called skin depth if very small)
1
1
"
δ≡ =
α Re{ (jωµ)(σ + jω$)}
e−αz
z
e−αz cos(ωt − βz)|t=0
(b) Snaphot at t>0, with t=0 waveform
for comparison
is the distance for the field strength to be reduced by e−1 factor in its
direction of propagation.
– For a fixed σ, and a sufficiently large ω, the penetration depth
2
δ ≈ " µ Imperfect dielectric formula
σ $
z
e−αz cos(ωt − βz)
–
which can be very small if σ is large — with small δ the wave is
severely attenuated as it propagates.
– For a fixed σ, and a sufficiently small ω,
!
2
1
δ≈
=√
Good conductor "skin depth" formula
µωσ
πf µσ
which, although small with large σ, increases as ω decreases, making low frequencies to be preferable in applications requiring propagating through lossy media with large σ, such as in sea-water.
8
β appears within cosine
argument and determines the wavelength
λ=
2π
β
and propagation speed
vp =
–
ω
.
β
α controls wave attenuation by
e∓αz
factor in propagation
direction.