# Simulation - the Debye circuit

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The Debye circuit is often used to interpret the impedance spectra of polymer electrolyte films sandwiched between blocking electrodes (i.e., electrodes that don’t undergo any reaction themselves), such as steel, or gold. This is a common method for calculating ionic conductivity of these materials. The circuit looks like this:

where Ri is the ionic resistance, Cdl is the double layer capacitance (represented by a constant phase element), and Cd is the dielectric capacitance.

It is important to account for the dielectric capacitance when making these measurements, because of the small inter-electrode distance (giving a higher dielectric capacitance) and a much higher ionic resistance of a polymer electrolyte than a liquid electrolyte (often 3 orders of magnitude higher, or more).

You should now start to see how the relaxation frequency associated with the semicircle that arises from the parallel Ri-Cd combination might well appear in the high frequency (1 kHz – 1 MHz) part of an impedance measurement, where it would normally be well outside the measurable range if the electrolyte was a more conductive liquid. Failure to account for this can lead the experimentalist to mistakenly assign the ionic resistance to the highest-frequency intercept of the real axis (it does happen, I’m afraid).

Below is an app which allows you to simulate the impedance spectrum associated with this circuit. However, unlike the other apps I’ve included in this guide, the input values are physical parameters, such as the electrolyte conductivity, electrolyte thickness, and so on. The app will then calculate resistance and capacitance values from these, and plot the Nyquist plot. Have a go at changing some of the important values – electrolyte conductivity and thickness especially, but also the frequency range of the simulation. How much of the semicircle do you see? As with the other apps, you may find it more convenient to open it in a new window.

Some typical conductivities as a guide: polymer electrolyte: 10-6 S cm-1; Li-ion battery electrolyte: 4 x 10-3 S cm-1; aqueous electrolyte: 10-1 S cm-1.

Next: the Kramers-Kronig transform >>