Biomedical Engineering Reference
In-Depth Information
Impedance spectra can also be displayed on other types of
plots. To justify their use, we must consider that the impedance
Z
of a single RC mesh is given by:
1
1
Z
R
i
C
(1)
Writing
Z
{
Z
' + i
Z
”, where
Z
' and
Z
” are the in-phase and quadra-
ture components of the impedance
Z
, and rearranging terms, we
obtain:
2
2
2
Z
'
R
1
Z
R
C
(2a)
Z
"
Z
'
Z
RC
(2b)
Eliminating Z
RC
from Eqs. (2a) and (2b) we get:
2
2
2
2
2
Z
"
Z
'
RZ
'
0
o
Z
'
R
/
2
Z
"
(
R
/
2
)
(3)
Equation (3) yields a semicircle of diameter
R
and center of
coordinates (
R
/2,0) on a
Z
” vs.
Z
' plot, called Nyquist plot. Noting
that the maximum of this semicircle is characterized by the equali-
ty of
Z
' and
Z
”, from Eq. (2b) it follows that the angular frequency
Z at this maximum equals the reciprocal of the time constant RC
of the mesh. In the presence of a series of RC meshes, their time
constants may be close enough to cause the corresponding semi-
circles to overlap partially. In this case, if the mesh of highest time
constant has also the highest resistance,
R
1
, as is often the case,
then the Nyquist plot of the whole impedance spectrum exhibits a
single well-formed semicircle,
R
1
in diameter. The semicircles of
the remaining meshes are compressed in a very narrow area close
to the origin of the
Z
” vs.
Z
' plot, and can be visualized only by
enlarging this area. Therefore, the Nyquist plot of the whole spec-
trum can be conveniently employed if one is interested in pointing
out the resistance
R
1
of the dielectric slab of highest resistance.
This is apparent in
Fig. 4
, which shows the Nyquist plot for the
tBLM incorporating valinomycin, whose Bode plot is reported in
Fig. 3
. The whole Nyquist plot displays a single semicircle. How-
ever, the enlargement of the initial portion of the plot in the inset
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