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Modeling a Warburg Impedance in AC Analysis

 

An electrochemical cell can contain a type of impedance referred to as the Warburg impedance which is created by diffusion within the cell. The Warburg impedance represents semi-infinite linear diffusion within the solution and is represented by the following expression:

Z = σ/(ω^1/2) - jσ/(ω^1/2)

where σ is the Warburg coefficient. The value of the Warburg impedance is dependent on the frequency of the perturbation. At higher frequencies, the Warburg impedance is smaller since the reactants do not have as far to move. Conversely, at lower frequencies, since the reactants have a greater length to travel, the Warburg impedance is higher.

In order to model this impedance in an AC analysis, the component used to model it must both be able to vary with frequency and be able to handle a combination of real and imaginary values. One component that can do both of these is the NFI (Nonlinear Function Current) source. The NFI source can handle complex, frequency varying expressions through its FREQ attribute capability. The FREQ attribute is only active during an AC analysis and has no effect in any other type of simulation. An expression defined within this attribute is evaluated at each frequency point of the simuation. The following expression in the FREQ attribute would model the Warburg impedance.

V(G1)/((Aw/sqrt(2*PI*F)) - (j*Aw/sqrt(2*PI*F)))

Since the NFI output is a current, the above expression produces the equivalent current of the Warburg impedance based on the basic I = V/Z relationship. The G1 instance in the expression is the part name of the NFI source in the schematic so it is referencing the voltage across itself. The denominator of the expression is the Warburg impedance equation where Aw is the Warburg coefficient.

Even though the FREQ attribute of the NFI source takes precedence during an AC analysis, the VALUE attribute of the NFI source must also be defined. The value specified for the VALUE attribute will be used during the DC operating point calculation that is performed at the beginning of the AC analysis simulation. The VALUE attribute may need to be set to an appropriate value for some circuits. While the NFI will still work fine no matter what value is used, the rest of the circuit is linearized based on the results of the DC operating point calculation. Therefore, the VALUE attribute can have an effect on what region of operation the circuit is linearized to for the resulting AC analysis run. For this example, the VALUE attribute has been defined as 0 which will create an open circuit during the DC operating point calculation.

In order to plot some of the basic Warburg impedance curves, an NFI source has been defined with the above FREQ expression. The Aw coefficient for the expression is set to 300. The source is in series with a 20 ohm resistor that represents the charge transfer resistance.

The first Warburg impedance curve is shown below. The Y expression is defined as Log(V(Out)/I(G1)). V(Out) measures the voltage across both the NFI source and the series charge transfer resistance. I(G1) is the current through this branch. This expression plots the log value of the impedance of the combination of the Warburg impedance and the series charge transfer resistance. The impedance is plotted versus the log value of the frequency, Log(f). At lower frequencies, where the Warburg impedance dominates, the slope of the impedance plot is -1/2. The -1/2 slope, which also appears as a 45 degree phase shift in phase plots, is characteristic of diffusion impedances.

Impedance Magnitude Plot of the Warburg Impedance

The other two basic Warburg impedance curves are shown below. In this case, the real and imaginary parts of the Warburg impedance are plotted on the Y axis. These impedances are plotted versus the expression 1/sqrt(2*PI*F) which is equivalent to 1/(ω^1/2). Once again, the series charge transfer resistance is also factored into these curves. The real and imaginary parts of the impedance are linear and parallel to each other as can be expected by looking at the standard expression for the Warburg impedance. The imaginary part intersects 0 when the frequency is 0. The real part intersects the value of the series charge transfer resistance when the frequency is 0. In this case, that is 20 ohms. Note that the slopes of both of these curves, which are displayed in the cursor tables below the plot, are equivalent to the Warburg coefficient which has been set to 300 for this AC analysis simulation.

Real and Imaginary Warburg Impedance Plots

The technique of using an NFI source can be used with other frequency varying complex impedances also. As long as one has an expression that models the impedance, the NFI source can be used to model it in an AC analysis.

References:
1) http://www.consultrsr.com/resources/eis/warburg1.htm - Research Solutions and Resources LLC.
2) http://www.gamry.com/App_Notes/EIS_Primer/EIS_Primer.htm#About_The_EIS_Primer - Gamry Instruments

 
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