Digital Signal Processing Reference
In-Depth Information
The relationship between the frequency and its wavelengths is calculated based
on the speed of light, which is the phase velocity (
ν
p
) in a vacuum:
c
λ
f
=
hertz
(2-45)
10
8
m/s), equation
(2-45) can be substituted into (2-44) to obtain a useful formula for
β
in terms of
the wavelength
λ
:
Since
ω
=
2
πf
and
c
is the speed of light in a vacuum (ca. 3
×
ω
β
=
2
πc
βλ
→
β
=
2
π
λ
c
=
rad
/
m
(2-46)
The speed of light in a vacuum is defined as the inverse of the square root of the
product of the permeability and the permittivity of free space:
1
√
µ
0
ε
0
c
≡
m
/
s
(2-47)
Calculation of
λ
in terms of (2-47) allows the phase constant
β
to be rewritten
in terms of the properties of free space:
2
πf
√
µ
0
ε
0
=
ω
√
µ
0
ε
0
β
=
rad
/
m
(2-48)
This is expanded on later in this chapter to include propagation of a wave in a
dielectric medium.
Now that the propagation constant has been defined, (2-43) can be rewritten
in physical terms, assuming free space (which is lossless, so
α
=
0):
Re
(E
x
e
−
jωz
√
µ
0
ε
0
e
jωt
)
=
E
x
cos
(ωt
−
ωz
√
µ
0
ε
0
)
E(z,t)
=
(2-49)
Since (2-49) is a solution to the wave equation, the magnetic field is found simply
by using Faraday's law (
∇×
E
+
jω B
=
0):
∂
∂z
(E
x
e
−
jωz
√
µ
0
ε
0
)e
jωt
=−
jωµ
0
H
y
√
µ
0
ε
0
µ
0
1
E
x
e
−
jωz
√
µ
0
ε
0
e
jωt
η
0
E
x
e
−
jωz
√
µ
0
ε
0
e
jωt
H
y
=
=
(2-50)
1
cos
(ωt
−
ωz
√
µ
0
ε
0
)
η
0
E
x
=
where
η
0
is the
intrinsic impedance of free space
and has a value of 377
:
µ
0
ε
0
η
0
≡
=
377
(2-51)
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