Digital Signal Processing Reference
In-Depth Information
4.0
0.04
0.035
3.8
0.03
3.6
0.025
Measured
ε
r
′′
Modeled
ε
r
′′
0.020
3.4
0.015
3.2
Measured
ε
r
′′
Modeled
ε
r
′′
0.010
25
40
10
15
20
30
35
Frequency, GHz
Figure 6-9
Measured dielectric response curve fit to equation (6-27).
Subsequently, for a generic material, the most straightforward usage requires a
measured response of
ε
r
and tan
δ
so the damping poles and resonant peaks can
be identified.
Example 6-1
Use equation (6-27) to create an empirical model for the dielectric
measured in Figure 6-9.
SOLUTION Figure 6-9 depicts the measured value of the real and imaginary
permittivity of a dielectric sample between 15 and 35 GHz. Note that the mea-
sured data shows small
ε
r
peaks in the vicinity of 19 and 32 GHz, indicating the
presence of two resonant induced dipoles, which are probably ionic or molecular
in nature. Consequently, ignoring the dc term, the implementation of (6-27) for
this material over the frequency range 15 to 35 GHz is
ε
1
ε
2
ε
=
ε
∞
+
+
j(ω/ω
1
)
−
ω
2
/ω
2
+
+
j(ω/ω
3
)
−
ω
2
/ω
4
1
1
where
ω
1
=
2
π
(20 GHz),
ω
2
=
2
π
(19 GHz),
ω
3
=
2
π
(63 GHz),
ω
4
=
2
π
(32 GHz),
ε
1
3.8. The resonant terms
ω
2
and
ω
4
were chosen at the peaks,
ε
1
and
ε
2
are the variations in the real part
of the dielectric constant in the vicinity of the peaks, and
ω
1
and
ω
3
are damping
terms that were varied until the width and the height of the peaks match the
data measured. In this case, the dielectric model is valid only between 15 and 35
GHz, which are the limits of the measured data [Hall et al., 2007].
=
0.0163,
ε
2
=
0.012, and
ε
∞
=
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