EIS Spectrum Analyser
Equivalent Circuit Elements and Parameters
a). Capacitor (C). Capacitor contributes to the imaginary part of the impedance:
Z(ω) = (j*ω*C)-1
where j is imaginary unit, ω is angular frequency, the parameter C is capacitance (F)
b). Resistor (R). Resistor with resistance R (Ohm)
contributes to the real part of impedance:
Z(ω) = R
c). Warburg (W). Warburg element represents the impedance of semi-infinite diffusion to/from flat electrode.
This element contributes equally to the real (ReZ) and imaginary (ImZ) parts of impedance:
ReZ (ω) = AW /ω0.5
ImZ (ω) = - AW /ω0.5
where AW is the Warburg coefficient (Ohm s-0.5):
R - universal gas constant, T - absolute temperature, n - number of electrons, A - electrode surface area, D - diffusion coefficient of the electroactive species, CS,O , CS,R - surface concentrations of oxidized and reduced form.
d). Constant Phase Element (CPE). CPE is an element used often in modelling the ac response of non-homogeneous systems. CPE has two parameters: Q and n and impedance of this element is given by the formula:
Z(ω) = Q-1(j*ω)-n
Q becomes equal to capacitance, when n = 1. Similarly to
capacitor, CPE keeps the phase constant at variable frequency but the
phase shift differs from 90°. CPE is especially helpful for representing slightly distorted capacitances.
e). Inductor (L). Impedance of the inductor is given by the formula:
Z(ω) = j*ω*L
where the parameter L is the inductance.
f). Warburg Short (Ws). Impedance of finite-length diffusion with transmissive boundary. This element has two parameters: WSr and WSc. Impedance of WS element is given by the formula:
where WSr is equal to Warburg coefficient, WSc = d/D0.5, d - Nernst diffusion layer thickness.
g). Warburg Open (Wo). Impedance of finite-length diffusion with reflective boundary. This element has two parameters: WOr and WOc. Impedance of WO element is given by the formula:
where WOr is equal to Warburg coefficient, WOc = d/D0.5.
h). Gerisher (G). Gerisher element has two parameters: Yg and Kg. Impedance of Gerisher element is given by the formula:
Z(ω) = (Yg(Kg + j*ω)0.5)-1
i). User defined element (U-element).
Though the ac response of a real object can be modelled adequately with
a sufficient number of the basic predefined elements, the physical
meaning of some elements in complex equivalent circuits may be not
clear. U-element can be used in such cases to find the model hidden
behind inexpressive elements. U-element allows testing various
mathematical models of ac response. In this program the U-element
with up to five parameters can be added to an equivalent circuit.
Combining U-element with predefined elements can be useful when a part
of ac response has a known origin (resistance of a solution, double
layer capacitance, etc.) but the other part is of unknown origin.
Modelling the remaining part by U-element can help understanding its
origin. See instructions on how to use this element at U-element
EIS Spectrum Analyser, 2008