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 (ZW) 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:

where

This equation assumes semi-infinite linear diffusion,[1] 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[2] is defined as:

where

where 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 (WS) for a transmissive boundary, and the Warburg Open (WO) for a reflective boundary.

Warburg Short (WS)

This element describes the impedance of a finite-length diffusion with transmissive boundary.[3] It is described by the following equation:

Warburg Open (WO)

This element describes the impedance of a finite-length diffusion with reflective boundary.[4] It is described by the following equation:

References


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