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Electromagnetic Fields
Electromagnetic Waves in Material Media
In a material medium free charges may be present, which generate
a current under the influence of the wave electric field. The current
Jc is related to the electric field E through the conductivity σ as
Jc = σ E
The material may also have specific relative values of dielectric
permittivity and magnetic permeability
ε = εr εo
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µ = µr µo
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Electromagnetic Fields
Maxwell’s equations become
∇ × E = − jωµ H
σ
∇ × H = σ E + jωε E = jω(ε − j )E
ω
In phasor notation, it is as if the material conductivity introduces an
imaginary part for the dielectric constant ε. The wave equation for
the phasor electric field is given by
∇ × ∇ × E = ∇∇ ⋅ E − ∇ 2 E = − jωµ ∇ × H
= − jωµ(J c + jωε E)
⇒
∇ 2 E = jωµ(σ + jωε )E
We have assumed that the net charge density is zero, even if a
conductivity is present, so that the electric field divergence is zero.
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Electromagnetic Fields
In 1-D the wave equation is simply
∂ 2E x
∂ z2
= jωµ(σ + jωε )E x = γ 2 E x
with general solution
E x ( z) = A exp(−γ z) + B exp( γ z)
1 ∂E x
σ + jωε
H y ( z) = −
=
( A exp(−γ z) − B exp( γ z) )
jωµ ∂ z
jωµ
1
= ( A exp(−γ z) − B exp( γ z) )
η
These resemble the voltage and current solutions in lossy
transmission lines.
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Electromagnetic Fields
The intrinsic impedance of the medium is defined as
η= η e
jτ
jωµ
=
σ + jωε
For the propagation constant, one can obtain the real and imaginary
parts as
γ=
jωµ(σ + jωε ) = α + jβ
2


ω µε 
σ
 
1 +   − 1
α=

2 
 ωε 


2


ω µε 
σ
 
β=
1 +   + 1

2 
 ωε 


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Electromagnetic Fields
Phase velocity and wavelength are now functions of frequency
vp =
λ=
ω
β
2π
β


µε 

2
=
=


µε 

2
f
1+
( )
1+
σ
2
ωε
( )
σ
ωε

+ 1

2
−1 / 2

+ 1

−1 / 2
The intrinsic impedance of the medium is complex as long as the
conductivity is not zero.
The phase angle of the intrinsic
impedance indicates that electric field and magnetic field are out of
phase. Considering only the forward wave solutions
E x ( z) = A exp( −γ z) = A exp( −α z) exp( − jβ z)
H y ( z) =
1
η
A exp( −γ z − jτ ) =
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1
η
A exp( −α z) exp( − jβ z − jτ )
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Electromagnetic Fields
In time-dependent form
E x ( z, t ) = Re { A exp( jθ ) exp( −α z) exp( − jβ z) exp( jω t )}
= A exp( −α z) cos(ω t − β z + θ )
H y ( z, t ) =
=
1
η
1
η
{
}
Re A exp( jθ ) exp( −α z) exp( − jβ z − jτ ) exp( jω t )
A exp( −α z) cos(ω t − β z + θ − τ )
where the integration constant has been assumed to be in general a
complex quantity as
A = A exp( jθ)
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Electromagnetic Fields
Classification of materials
Perfect dielectrics - For these materials σ = 0
Propagation constant
Phase velocity
β = ω ε r ε o µ rµ o
α=0
ω
1
vp = =
β
µ r µ oε r ε o
Medium Impedance
Wavelength
jωµ
µ rµ o
η=
=
jωε
ε rε o
2π v p
1
λ=
=
=
f
β
f µ r µ oε r ε o
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Electromagnetic Fields
Imperfect dielectrics – For these materials σ ≠ 0 but (σ/ωε)<<1
γ=
σ
jωµ(σ + jωε ) = jω µε 1 − j
ωε
σ µ
≈
+ jω µε + …
2 ε
σ µ
α≈
2 ε
1
ω
vp = ≈
β
µε
jωµ
η=
=
σ + jωε
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β ≈ ω µε
2π
1
λ=
≈
β
f µε
jωµ 
σ 
1 − j 
ωε 
jωε 
−1
2
µ
≈
ε
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Electromagnetic Fields
If (σ/ωε)<<1, the errors made in the approximations for α, β, vp
2
and λ are very small, since only terms of order (σ/ωε) or higher
appear in the expansions. The error is slightly higher fo the
medium impedance η since the expansion contains a term of order
(σ/ωε).
The simple rule of thumb is that approximations for imperfect
dielectric can be applied when
σ
≤ 0.1
ωε
When the condition above is verified, the imperfect dielectric
behaves in all respects like a perfect dielectric, except for an
attenuation term in the fields.
The quantity σ/ωε is called Loss Tangent.
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Electromagnetic Fields
Good conductors – For these materials σ ≠ 0 but (σ/ωε)>>1
γ=
jωµ (σ + jωε ) ≈
jωµσ = ωµσ j
π
1 
 1
= ωµσ exp( j ) = ωµσ
+ j
= πf µσ (1 + j )
 2
4
2
β ≈ πf µσ
α ≈ πf µσ
ω
4 πf
vp = ≈
β
µσ
jωµ
η=
≈
σ + jωε
2π
λ=
≈
β
4π
f µσ
ωµ
π
jωµ
=
exp( j )
σ
σ
4
π fµ
ωµ  1
1 
=
+ j
=
(1 + j )
σ  2
σ
2
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Electromagnetic Fields
The simple rule of thumb is that approximations for good conductor
can be applied when
σ
≥ 10
ωε
Note that for a good conductor the attenuation constant
propagation constant β are approximately equal.
α and the
The medium impedance η has nearly equal real and imaginary
parts, therefore its phase angle is approximately 45°.
This means that in a good conductor the electric and magnetic
fields have always a phase difference τ = 45° = π /4.
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Electromagnetic Fields
Also, in a good conductor the fields attenuate very rapidly. The
distance over which fields are attenuated by a factor exp(−1.0) is
1
1
=δ=
= Skin depth
α
π f µσ
A typical good conductor is copper, which has the following
parameters:
σ = 5.80 × 10 7 [S/m]
ε ≈ εo
µ ≈ µo
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Electromagnetic Fields
Copper remains a good conductor at extremely high frequencies.
Another good conductor example is sea water at relatively low
frequencies
σ ≈ 4.0 [S/m]
ε ≈ 80ε o
µ ≈ µo
At a frequency of 25 kHz
σ
≈ 36, 000
ωε
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Electromagnetic Fields
Perfect conductor - For this ideal material σ
→∞
For this material, the attenuation is also infinite and the skin depth
goes to zero. This means that the electromagnetic field must go to
zero below the perfect conductor surface.
General medium - When a material is not covered by one of the limit
cases, the complete formulation must be used. We can classify a
material for which the conditions (σ/ωε)<<1 or (σ/ωε)>>10 are
invalid as a general medium.
The simple rule of thumb for general medium is
σ
10 >
> 0.1
ωε
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