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 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 exists with a slope of value –1/2.

General equation

The Warburg diffusion element is a constant phase element, with a constant phase of 45° and with a magnitude inversely proportional to the square root of the frequency by:
where
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:
where
where is the thickness of the diffusion layer and is the diffusion coefficient.
There are two special conditions of finite-length Warburg elements: the Warburg Short for a transmissive boundary, and the Warburg Open 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:

Warburg Open (WO)

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