Biomedical Engineering Reference
In-Depth Information
where Re
Z
and Im
Z
are the real part and the imaginary part of the impedance, respec-
tively. The relationships between these quantities are:
→
Z
→
2
= (Re
Z
)
2
+ (Im
Z
)
2
(4.7)
φ= Arc tan
Im
Re
Z
Z
(4.8)
Re
Z
=
→
Z
→
cos
→
(4.9)
Im
Z
=
→
Z
→
sin
→
(4.10)
In Figure 4.1a, the equivalent circuit of an electrochemical interface is depicted; its
impedance is:
2
E
I
1
R
j CR
C
ω
ω
Z
(
ω
=
=
R
+
=
R
+
2
−
2
(4.11)
1
1
1
2
2
2
2
2
R
2
1
+
C R
1
+
ω
+
jC
ω
2
R
2
where
E
is the voltage,
I
is the current,
R
1
and
R
2
are the resistance values of the equivalent
circuit, and
C
is the capacitance value of the equivalent circuit.
Furthermore,
2
2
2
2
R
C R
ω
ω
C R
C R
(4.12)
(
ω
=
+
2
+
2
Z
R
1
2
2
2
2
2
2
1
+
1
+
ω
2
2
When
ω
tends to zero,
→
Z
(
ω
)
→
equals to
R
1
+ R
2
. When
ω
tends to infinite,
→
Z
(
ω
)
→
equals
to
R
1
.
Note that the difference between the two limits is
R
2
. Therefore, the high-frequency
intercept determines
R
1
(the series resistance), whereas the low-frequency intercept yields
the sum of
R
1
+ R
2
. In simple terms, this means that at high frequencies, the capacitor
conducts the current easily. Consequently, the impedance is solely due to the resistance
R
1
, whereas at low frequencies, the current low via the capacitor is impeded. The current
therefore flows through
R
1
and
R
2
, and the impedance is given by the sum of the two resis-
tors. At intermediate frequencies, the impedance takes a value somewhere between
R
1
and
R
1
+ R
2
and thus has both real and imaginary components. This gives rise to the Cole-Cole
plot semicircular shape, which corresponds to the equation as follows:
2
2
R
R
(4.13)
−
+
2
+
2
=
2
Z
R
Z
1
2
2
It has been shown that Equation 4.13 is analogous to the equation of a circle, with a radius
of
R
2
2
R
+
,
0). In all the materials studied,
ω
,
R
1
,
and
R
2
are greater than
zero, thus resulting in a semicircle on the axis when plotted as function of frequency.
Z
(
ω
)
and a center at (
R
2
1
