Biomedical Engineering Reference
In-Depth Information
tive direction of the
Y
”/Z axis. It is apparent that the fitting of a
series of four RC meshes to the experimental spectrum shows ap-
preciable deviations at the lower frequencies when the spectrum is
displayed on a Cole-Cole plot.
An equivalent circuit consisting of a series of RC meshes with
relatively close time constants yields calculated semicircles on
Nyquist, M or Cole-Cole plots that overlap to an appreciable ex-
tent. Such an overlapping resembles a single depressed semicircu-
lar arc, namely an arc whose center lies below the horizontal axis.
These arcs are often fitted to an equivalent circuit consisting of a
parallel combination of a resistance R and of a
constant phase el-
ement
(CPE), whose empirical impedance function has the form:
Z
CPE
=-(iZ)
-D
$
The hybrid CPE reduces to a pure capacitive im-
pedance, i/Z$, for D=1, and finds its justification in a continuous
distribution of time constants around a mean. The impedance of
this parallel combination, called ZARC, is given by:
R
D
1
1
Z
R
i
Z
A
o
Z
(11)
ZARC
ZARC
D
1
(
i
Z
)
RA
For D= 1, this expression reduces to the impedance of a RC mesh,
and
A
coincides with the capacitance.
The use of ZARC elements for the fitting to impedance spec-
tra of films deposited on satisfactorily smooth supports, such as
mercury, seems redundant and difficult to relate to the structure of
the film. In this case, a more significant approach is the following.
The experimental spectra are fitted by an equivalent circuit con-
sisting of a progressively increasing number of RC meshes in se-
ries.
8
The quality of the fitting is conveniently monitored on a M
plot. Fitting errors less than 10% for the
R
and
C
values of the dif-
ferent RC meshes can be regarded as acceptable. When the addi-
tion of a further RC mesh does not determine a detectable im-
provement in the agreement between experimental and calculated
M plots, the fitting error for the added RC mesh is normally found
to be close to 100%. The last added RC mesh is, therefore, dis-
carded from the fitting.
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