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8. Impedance technique

8.9. Kramers-Kronig transforms

The following criteria must hold in an impedance measurement:

 

1. Causality. The causality criterion means that the system response is only due to the perturbation applied, and does not contain contributions from spurious sources.

2. Linearity. The relation between the system perturbation and response is described by a set of linear differential laws.

3. Stability. The system must be stable in the sense that it returns to its original state after the perturbation is removed.

4. Finite-valued. Impedance must be finite at \( \omega \) → 0 and \( \omega \) → ∞, and must be a continuous and finite function at all frequencies.

 

Under these conditions, a set of transforms between the real and imaginary components are held, known as Kramers-Kronig transforms, the following two being the most important of these:

 \( \displaystyle Z'(\omega)-Z'(\infty)=\frac{2}{\pi} \int_{0}^{\infty}{\frac{xZ''(x)-\omega Z''(\omega)}{x^2-\omega^2}} dx \)

  (8.99)
\( \displaystyle Z''(\omega)=-\frac{2}{\pi} \int_{0}^{\infty}{\frac{Z'(x)-Z'(\omega)}{x^2-\omega^2}} \) (8.100)

In Equations (8.99) and (8.100) x is an auxiliary variable (frequency). The transforms are solely mathematical and generic, and thus they do not contain any physical interpretations. In terms of impedance measurements, the fourth criterion is slightly problematic. At low frequencies, the impedance is controlled by mass transfer, i.e. diffusion. However, the Warburg and coth elements increase boundlessly when \( \omega \) approaches zero. We can avoid the problem when determining the integrals by using an equivalent circuit that gives the same result at the boundary. Consider, for example, the coth element. Figure 8.16 illustrates that at low frequencies, it corresponds to the combination of \( R+1/(j\omega C) \). The term xZ”(x) ≈ -x/(xC) = -1/C in the integral (8.99) is therefore a finite number. Respectively, for the Warburg element, the product \( \omega \)Z”(\( \omega \)) ~  \( \omega \)1/2 → 0, when \( \omega \) → 0.

Kramers-Kronig transform is usually integrated in the control and analysis software of a potentiostat, and there is no need for the user  to know which algorithm is used for the mathematical treatment of the integral or how the boundary values of frequency are treated. As the transform is easily available in the software, it is useful to do the transform from time to time. For example, if the experimental conditions are such that the electrode is passivated easily, it is important to check that the stability criterion is still holding.


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