Calculating Capacitance from a CPE


		The CPE - Calculating Capacitance from Q°

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What is the TRUE Capacitance?

If the n-value of a CPE is a little less than 1.0, then the CPE behavior is close to that of a capacitance. Many researchers wish to calculate the "correct" capacitance corresponding to the Q° value of the CPE. The similarity of the equations makes it inviting.

CPE		1 / Z = Y = Q° ( j $omega$ )ⁿ

Capacitor		1 / Z = Y = Q° ( j $omega$ )¹ = C ( j $omega$ )

where Q° is numerically equal to the admittance (1/ |Z|) at $omega$ =1 rad/s. It is tempting to simply associate the value of Q° for a CPE with the capacitance value, C, for an equivalent capacitor. Alas, an examination of the units of C (farad or S-s ) and Q° ( S-sⁿ ) shows that they can not be the same! See ref 1 for unit abbreviations.

For the case of a 'classical' depressed semicircle (CPE in parallel with a resistance) Hsu and Mansfeld (ref 2) have given this equation for calculating the 'true' capacitance, C:

C = Q° ( $omega$ _MAX )^n-1

omega max defined In this equation, $omega$ _MAX represents the frequency at which the imaginary component reaches a maximum. It is the frequency at the top of the depressed semicircle, and it is also the frequency at which the real part ( Z' ) is midway between the low and high frequency x-axis intercepts.

An online calculator implementing this equation is available on this web site. Another calculator uses the R and Q parameters to calculate the "true" capacitance.

The equation proposed by Hsu and Mansfeld is based on the model of a CPE in parallel with a charge transfer resistance.
$omega$ _MAX may not even exist for other circuits, such as a CPE and a resistor in series!

Comments from V. D. Jovic, (Univ. of Belgrade) can be viewed or downloaded as a PDF file. I am indebted to him for bringing the Orazem (ref 3) article to my attention.

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REFERENCES
(1) S-s is the abbreviation for siemens-second ( or second/ohm ). Check the NIST Publication 811 for proper units, abbreviations, and capitalization.
(2) "Concerning the Conversion of the Constant Phase Element Parameter Yo into a Capacitance," CS Hsu, F Mansfeld, Corrosion, 57(2001) 747.
(3) "Extension of the measurement model approach for deconvolution of underlying distributions for impedance measurements," ME Orazem, P Shukla, MA Membrino, Electrochimica Acta, 47(2002) 2027.

The CPE - Calculating Capacitance from Q°

What is the TRUE Capacitance?