Pure capacitance have two types, 1) time constant and 2) phase constant. U can't replace each other. CPE doesn't have any physical significance, Its just away from ideal capacitor.

Generally, non-debye type cases explained with CPE, and debye type explained with capacitor.

U can be related both in the followed expressions,

Capacitance = (CPE*R)^(1/n) /R

R is the resistance

n is the phase change values obtained from the fitting

I have worked with Zsimwin a lot. CPE has no physical meaning. it is just a calculated parameter when circuit element behaving in between capacitor and resistor. Usually, people introduce CPE because the circuit with pure capacitor doesn't fit in simulation.

The physical significance of CPE elements, according to Cottis, is unclear. This is a good resource for EIS analysis: Electrochemical Impedance and Noise, NACE International by R. Cottis and S. Turgoose (1999). It's useful if you have depressed semicircles in your Nyquist plot. So, you can probably replace it with a pure capacitor *if* you have an ideal semicircle.

Due to inhomogeneous distribution of grains, depressed semicircles or arcs will be observed in impedance spectroscopy plots, in these case ideal capacitor is not possible to fit the data, because there should not be any depression of the impedance plots (only exact semi circles are required), this is possible only homogeneous distribution of grains and grain growth takes place,however in practice it is not possible to get exact semi circles.

using CPE it is possible to get C values

By observing n value (1-0.5), one can say depression of semi circles

when n=1, it is an exact semi circle that means it is an ideal capacitor

In general a CPE cano not be replaced by a capacitor. A capacitor can be treated as a pure good capacitor connected in parallel to a high resistance. The impedance is Z*=Z'-j Z" , Z'=R/(1+(wRC)^2 ), Z"=wRC R/(1+(wRC)^2 ) = wRC Z'. So current flowing through such a circuit will be equal to v exp(jwt) / Z* = v exp(jwt)/ (Z'-jZ")

= [v exp(jwt) ] ( Z' +j Z")/ (Z' ^2 + Z" ^2) ) = [ v exp(jwt) ] exp (j theta ) with cos(theta)=( Z' )/ (Z' ^2 + Z" ^2) ) and

sin (theta) = ( Z")/ (Z' ^2 + Z" ^2) ) .THe phase angle is given by Tan(theta)=Z"/Z' = wRC. That means the phase angle is not a constant and is frequency dependent. In the same way you can show that no pure lumped elements or usual combinations of these gives a CPE. A Z" vs Z' plot for a CPE is a straight line ( frequency indepedndant). So where ever experiments show a straight line in immitance ( impedance, admittance, permittivity, modulus) a CPE is assumed to be present in the model equivalent circuit Afterwards various meanings are atrributed to the material. You can see some of our papers by searching at google with name L Pandey or Lakshman Pandey where we have tried to simulate these behaviours. I was using my own Complex Non Linear Least Sqares program that I wrote in BASIC in eraly 90's. If you feel interested pl let me know , I will dig out our papers and tell you.

Lakshman Pandey, Rani Durgavati University, Jabalpur, [email protected]

Dear Yellareswara Rao K,

Further in continuation with the earlier note : A pure C would have R=0 , and thus the phase angle is given by Tan(theta)=Z"/Z' = wRC = infinity leading to theta =90 degrees ie a vertical line in the Z" vs Z' plot. The staright line in case of CPE would be inclined at some angle. Also any function for Z' and Z" , that give Tan (theta)= a frequency independent constant , can be chosen to represent a CPE. If I remember correctly there have been some efforts to simulate CPE behaviour by modeling the system with analogy to transmission line, a fractal interface etc.

Lakshman Pandey

I thank you one and all for kind attention...

Dear Senthil Venkatesan, could you give me the reference how the equation comes from? Thank you.

dear L. Pandey sir, could you inform me the unit of CPE?

The normalized value for Z* should have the unit [F/s].

More explanations about CPE are given in the attachment.

http://www.bio-logic.net/wp-content/uploads/20120817-Application-note-42.pdf

The relation between a constant phase element (CPE) and a pure capacitor (C) :

ZCPE=1/ ((CPE) (jw)^n)

wheren is the characteristic parameter of CPE.

If n=1, only when you can replace CPE with a pure capacitor.

• electrode inhomogeneity and surface roughness (3D)

• electrode porosity (3D)

• variability in thickness and conductivity of surface coating (3D)

• slow, uneven adsorption process (2D)

• nonuniform potential and current distribution at the surface (2D)

• grain boundaries and crystal phases on polycrystalline electrode(2D)

These processes are responsible for the deviations from ideal resistor and capacitor response and the resulting CPE representation. If you are sure that u dont have such processes in your system go ahead and apply pure capacitor in Randles circuit