Biomedical Engineering Reference
In-Depth Information
Double Resistor and Capacitor Parallel in Series
Figure 4.5 shows the impedance spectrum for the double resistor and capacitor parallel in
series. The circuit has a direct relationship to the Cole-Cole plot. The highest frequency is
located at the origin. The resistances
R
1
and
R
2
can be obtained from the diameters of the
semicircles in this plot. Here the resistor
R
represents either an ionic or an electronic con-
duction mechanism, whereas the capacitor
C
represents the polarizability of the diamond.
The symbols
R
1
,
R
2
,
C
1
, and
C
2
have the same meaning as before for two-layer dielectrics.
The measured complex impedance
Z*
can be expressed as a function of
R
1
,
R
2
,
C
1
, and
C
2
in the following way:
Z
* =
Z
→
−
j
Z
→
(4.30)
R
R C
R
=
1
+
2
(4.31)
Z
+
2
2
2
+
2
2
2
1
ω
1
ω
R C
1
1
2
2
2
2
ω
ω
R C
R C
ω
ω
R C
R C
=
1
1
+
2
2
(4.32)
Z
+
2
2
2
+
2
2
2
1
1
1
1
2
2
where
Z
→
and
Z
→
represent the real and imaginary portions of the impedance, respectively,
and
ω
is the angular frequency. When plotted in a complex plane, Z
→
versus
Z
→
takes the
form of two semicircles. In such a representation, a two-layer contribution is easily identi-
fied. Note that critical to the identification of each single resistor and capacitor in parallel
is the simulated capacitance value for each semicircle. This model is well suited to simulate
the resistance and capacitance associated with the grain boundary and grain interior.
Formulae for Capacitance
It is known that the electrical contribution from grains, grain boundaries, and/or elec-
trodes for a certain polycrystalline material system can be extracted and separated. As
-8 × 10
6
-6 × 10
6
-4 × 10
6
R1
R2
-2 × 10
6
C1
C2
0
0
5 × 10
6
1 × 10
7
1.5 × 10
7
2 × 10
7
2.5 × 10
7
3 × 10
7
Re
Z
(ohms)
FIGURE 4.5
Double
R
-
C
parallel in series. (From Ye, H.T., PhD thesis, University College London, 2004. With permission.)
