DIFFERENTIAL IMPEDANCE ANALYSIS OF CONDUCTIVITY AND ...

the effective resistances R 1 and R 2 , the effective capacitance C and the effective time-constant T =
CR 2 . More detailed information on the principle of DIA can be found in [3-6].
The frequency dependence of the LOM parameters’ estimates ˆP LOM
is an important step of
the analysis, termed temporal analysis, since it provides information for the model structure:
lg ˆP LOM = F (lgT f ), (1)
where T f = 1/f is the sinusoidal period and f is the frequency. For convenience Eqn. (1) is
expressed in a logarithmic form.
R 2
R 1
-Im / Ω
C
Re / Ω
Fig. 1. Local operating model (LOM) – first order inertial system: equivalent circuitry
and impedance diagram
When the structure of the LOM corresponds to that of the impedance object in a given
frequency range, the parameters’ estimates have a constant behaviour, i.e. they are presented as
plateaus (Fig.2b). The number of the plateaus yields the number of the time-constants in the
model. For better visualisation of the results obtained by the temporal analysis, a spectral form of
presentation is introduced [4]. Fig. 2c presents the effective time-constant spectrum. The rest of
the parameters’ estimates have the similar spectra.
- Im / Ω
200
a)
0.1Hz
100
10Hz
10mHz
100Hz 1Hz
1mHz
0
0 100 200 300 400
Re / Ω
lg P^
4
2
0
-2
-4
I III II
^
T / s
^
C / F
b)
-3 -2 -1 0 1 2 3
lg (T / Hz -1 )
F
^
R 2 / Ω
Intensity / dB
45
30
15
I
III
c)
0
-6 -4 -2 0 2 4 6
lg T^
II
Fig. 2. DIA of synthetic model of two steps Faradic reaction: a) impedance diagram;
b) temporal plots; c) effective time-constant spectrum.
The regions where the LOM parameters’ estimates are frequency dependent (segment III
in Figs.2b,c) need an additional examination with the so-called Secondary Differential Impedance
Analysis [7-10]. It works with the derivatives of the LOM parameters’ estimates with respect to
the frequency:
P13-2