Impedance Spectra

  Equivalent Circuit


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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 0.5
                                 ImZ (ω)    = - AW 0.5
where AW is the Warburg coefficient (Ohm s-0.5):

- 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