Digital Signal Processing Reference
In-Depth Information
and the electric susceptibility is defined from (6-11):
P
ε
0
E
0
N(q
2
/ε
0
m)
χ
=
=
ω
0
−
ω
2
+
j(b/m)ω
leading to an expression for the relative dielectric constant from (6-13) [Huray,
2009]:
N(q
2
/ε
0
m)
ε
r
=
1
+
χ
=
1
+
(6-24)
ω
0
−
ω
2
+
j(b/m)ω
The real and imaginary parts are shown as [Balanis, 1989]
N(q
2
/ε
0
m)(ω
0
−
ω
2
)
ε
r
=
1
+
(6-25a)
(ω
0
−
ω
2
)
2
+
(ω(b/m))
2
q
2
ε
0
m
ω(b/m)
(ω
0
−
ω
2
)
2
ε
r
=
N
(6-25b)
+
ω
m
2
Equations (6-24) and (6-25) calculate the frequency response for a material that
exhibits an atomic or molecular structure with only one natural or resonant fre-
quency, an example of which is shown in Figure 6-6. At the natural (i.e., resonant)
frequency of the harmonic oscillator, the imaginary portion of the complex per-
mittivity will peak, which in turn dramatically increases the dielectric losses,
which are quantified with the loss tangent (6-16). Also note that the real por-
tion of the dielectric permittivity is almost constant until the operating frequency
approaches the resonant frequency of the oscillator, which in this case is an
atomic structure. In the vicinity of the natural frequency, the real portion begins
e
r
′
1.5
1.0
anomalous
dispersion
0.5
e
r
′′
0.5
1.0
5.0
Frequency, GHz
10.0
50.0 100.0
Figure 6-6
Frequency response for a pure material that exhibits an atomic or molecular
structure with only one natural or resonant frequency.
Search WWH ::
Custom Search