# 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 (see double layer (interfacial)), 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| versus 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:

${\displaystyle {Z_{W}}={\frac {A_{W}}{\sqrt {\omega }}}+{\frac {A_{W}}{j{\sqrt {\omega }}}}}$
${\displaystyle {|Z_{W}|}={\sqrt {2}}{\frac {A_{W}}{\sqrt {\omega }}}}$

where AW is the Warburg coefficient (or Warburg constant), j is the imaginary unit and ω is the angular frequency.

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:

${\displaystyle {Z_{O}}={\frac {1}{Y_{0}}}\tanh(B{\sqrt {j\omega }})}$

where ${\displaystyle B={\frac {\delta }{\sqrt {D}}}}$,

where ${\displaystyle \delta }$ 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 (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:

${\displaystyle Z_{W_{S}}={\frac {A_{W}}{\sqrt {j\omega }}}\tanh(B{\sqrt {j\omega }})}$

### Warburg Open (WO)

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

${\displaystyle Z_{W_{O}}={\frac {A_{W}}{\sqrt {j\omega }}}\coth(B{\sqrt {j\omega }})}$