Regular extension

The Warburg diffusion element is a common diffusion circuit element that can be used to model semi-infinite linear diffusion, that is, unrestricted diffusion to a large planar electrode. 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 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:

$Z_W=\frac{A_W}{\sqrt{\omega }}+\frac{A_W}{j\sqrt{\omega }}$

$\left|Z_W\right|=\sqrt{2}\frac{A_W}{\sqrt{\omega }}$

where A_W is the Warburg coefficient (or Warburg constant), j is the imaginary number and ω is the angular frequency. The presence of the Warburg element can be recognised if a linear relationship on the log of a Bode plot (log|Z| versus log(w)) exists with a slope of value –1/2.

References

• http://www.consultrsr.com/resources/eis/diffusion.htm