Warburg element

The Warburg diffusion element is an equivalent electrical circuit component that models the diffusion process in dielectric spectroscopy. That element is named after German physicist Emil Warburg.

A Warburg impedance element can be difficult to recognize because it is nearly always associated with a charge-transfer resistance (see charge transfer complex) and a double-layer capacitance, but is common in many systems. The presence of the Warburg element can be recognised if a linear relationship on the log of a Bode plot ($log |Z|$ vs. $log ω$) exists with a slope of value –1/2.

General equation
The Warburg diffusion element ($Z_{W}$) is a constant phase element (CPE), with a constant phase of 45° (phase independent of frequency) and with a magnitude inversely proportional to the square root of the frequency by:


 * $${Z_\mathrm{W}} = \frac{A_\mathrm{W}}{\sqrt{\omega}}+\frac{A_\mathrm{W}}{j\sqrt{\omega}}$$


 * $${|Z_\mathrm{W}|} = \sqrt{2}\frac{A_\mathrm{W}}{\sqrt{\omega}}$$

where
 * $A_{W}$ is the Warburg coefficient (or Warburg constant);
 * $j$ is the imaginary unit;
 * $ω$ is the angular frequency.

This equation assumes semi-infinite linear diffusion, that is, unrestricted diffusion to a large planar electrode.

Finite-length Warburg element
If the thickness of the diffusion layer is known, the finite-length Warburg element is defined as:


 * $${Z_\mathrm{O}} = \frac{1}{Y_0} \tanh\left(B \sqrt{j\omega}\right) $$

where $$B=\tfrac{\delta}{\sqrt{D}},$$

where $$\delta$$ is the thickness of the diffusion layer and $D$ is the diffusion coefficient.

There are two special conditions of finite-length Warburg elements: the Warburg Short ($W_{S}$) for a transmissive boundary, and the Warburg Open ($W_{O}$) for a reflective boundary.

Warburg Short (WS)
This element describes the impedance of a finite-length diffusion with transmissive boundary. It is described by the following equation:


 * $$ Z_{W_\mathrm{S}} = \frac{A_\mathrm{W}}{\sqrt{j\omega}} \tanh \left(B \sqrt{j\omega}\right) $$

Warburg Open (WO)
This element describes the impedance of a finite-length diffusion with reflective boundary. It is described by the following equation:


 * $$ Z_{W_\mathrm{O}} = \frac{A_\mathrm{W}}{\sqrt{j\omega}} \coth\left(B \sqrt{j\omega}\right) $$