Inductance

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Inductance and Inductive Loops
It is generally easy to imagine what real world features might correspond to a resistance or a capacitance in an equivalent circuit model of an electrochemical cell. It is much harder to explain the presence of an inductor, though.  Here we will examine a few of the explanations which have been proposed.

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An inductor is formed by a wire wound in a coil, such as an electromagnet or transformer. An electrical inductor stores energy in the magnetic field formed when current flow through the coil. This energy is released when the current decreases and the magnetic field collapses.  The impedance of an inductor is characterized by the equation, below, in which L is the inductance in Henrys (H).

Z_L = j L

Current and voltage are +90° out of phase, in contrast to the -90° phase shift of a capacitor, and the impedance of an inductor increases with increasing frequency. While the Nyquist plot of a common Randles cell (representing a simple slow electron transfer) shows a semicircle in the first quadrant, the Nyquist plot for circuits involving inductors show impedances below the x-axis, generally in the fourth quadrant.
 

A Nyquist plot showing high frequency inductive behavior.

The effects of inductances are often seen at the highest frequencies. The impedance of an inductor increases with frequency, while that of a capacitor decreases. High frequency inductive behavior has several possible causes. The easiest to visualize is the actual physical inductance of the wires and, possibly, of the electrode itself. I was surprised to find out the inductance of a straight piece of wire! Since the working electrode assembly often has a long wire or rod between the actual electrode surface and the potentiostat connections, this can be important if the electrode impedance is low.

Some batteries, formed by rolling a thin anode-electrolyte-cathode "sandwich" into a compact cylinder, may show these effects. These "stray" inductances are generally only a few microHenry (µH). However, since battery impedances are often low, these strays can be important.

Instrumental Artifacts

High frequency inductive behavior can also be caused by instrumental artifacts, notably capacitance associated with the current measuring resistor (see references).   Many potentiostat manufacturers have already made corrections for this effect in their EIS software. This artifact can be identified by the fact that the apparent inductance changes every time the current range is changed.

Another source of high frequency inductive behavior is the mutual inductance of the wires connecting potentiostat to the cell. The AC current that flows in the counter electrode and working electrode leads generates a magnetic field surrounding the wires.  This time-varying magnetic field induces a voltage in the reference and working-sense ( called RE2 by some potentiostat manufacturers ) leads. This is observed as high frequency inductive behavior. Because the effect is magnetic, using shielded wires or coax does not eliminate the effect. The inductances are often in the 50-500 nH range. These effects can be minimized by twisting the wires, and by their proper placement. The remaining effect is difficult to "calibrate out" unless the positions of the wires are strictly controlled and reproducible. Moving the wires will change the observed inductance.

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REFERENCES
"Corrections for Potentiostat Response.", Technical Note 201, Princeton Applied Research.
"Correction for Potentiostat Response in Impedance Measurements", RS Rodgers and WJ Eggers, Paper #228, 183rd Meeting of the Electrochemistry Society, Honolulu, HI, May 1993.
"Impedance Measurements with Real Potentiostats: Corrections for Potentiostat Response", RS Rodgers, Paper #134, 2nd International Symposium on Electrochemical Impedance Spectroscopy, Santa Barbara, CA, July, 1992.

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