Specific capacitance calculation of symmetric or asymmetric supercapacitor device from charge-discharge measurements?

 

Dear Bidhan and Adam,

welcome again,

I will try to derive the relation for nonlinear GCD,

At first we have to make some assumptions:

- We are interested in the electrode capacitance C which has an active material mass= m, then the specific electrode capacitance of the electrode is Cs= C/ m,

This capacitance can be measured directly.

The electrode capacitance is nonlinear such that one defines its value at certain given voltage by  C= dQ/dV, With dQ= Idt, then we have 

C= I dt/ dV,

Since the capacitance is nonlinear it value changes with charging and discharging voltage of the electrode. With constant charging current the voltage can change from an initial voltage Vi to a final voltage to Vf. Then one can define an average capacitance during the charging Cav which can be deified as:

Cav =INTEGRAL I dt/ dV,which can be put in the form by multiplying the numerator and denominator by V :   Cav=  Integral  I V dt / VdV,

Integrating we get from Vi to Vf we obtain:

Cav= 2* I integral Vdt / V^2 from Vi to Vf,

Then Cav= 2* I integral Vdt / (Vf^2- Vi^2),

And the integral is the area under the charging voltage curve from t at V i to t at Vf.

We need  only to divide both sides by the active area mass of the electrode m to get the specific average charging capacitance of the electrode; that is:

Cs av= 2* I integral Vdt / m(Vf^2- Vi^2),

This is the formula which you introduced in your question. Now it is fully derived from the first principles.

 https://www.researchgate.net/post/Specific_capacitance_calculation_of_symmetric_or_asymmetric_supercapacitor_device_from_charge-discharge_measurements