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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 (ZW) is a constant phase element (CPE), with a constant phase of 45¡ã (phase independent of frequency) and with a magnitude inversely proportional to the square root of the frequency by:
Zw=¦Ò(1-j)/[¦Ø(1/2)]
|Zw|=¦Ò*[2(1/2)]/[¦Ø(1/2)]
where ¦Ò is the Warburg coefficient (or Warburg constant), j is the imaginary number and ¦Ø 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 ¨C1/2.
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