The Kramers-Kronig transform

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I will not go into too much detail about the Kramers-Kronig (or, just K-K) transform other than to say that this is a method for validating impedance data, to give you some idea of whether it’s actually possible to fit an equivalent circuit. This is the equation:

$$\displaystyle -Z''_{KK} = \frac{2\omega}{\pi} \int^\infty_0 \frac{Z'(x) - Z'(\omega)}{x^2 - \omega^2} dx $$

Really, you don’t actually need to know this equation to do a K-K transform, because it is sometimes included as a function in the software for the instrument – but it is useful to know where it comes from. The Kramers-Kronig relations say – in this context – that you can calculate the imaginary part of the impedance from the real part of the impedance, and vice versa, provided the measured impedance response satisfies these conditions:

A response which does not satisfy all of these conditions probably cannot be fitted to an (appropriate) equivalent circuit.

A simpler alternative

Actually, there is an easier way you can check the validity of the impedance response without having to look for a K-K transform function. Typically, when running an impedance measurement, you will likely run through the frequencies from highest to lowest. Do that, but then run the measurement again with the order reversed – start with the lowest frequencies first, and finish the measurement with the highest frequencies. If you get the same impedance spectrum twice, it’s a good indication that the data is reliable.

Next: three- and four-electrode measurements >>

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