Represent an electrochemical double-layer capacitor

**Library:**Sources

The Supercapacitor block represents an electrochemical double-layer capacitor (ELDC), which is commonly referred to as a supercapacitor or an ultracapacitor. The capacitance values for supercapacitors are orders of magnitude larger than the values for regular capacitors. Supercapacitors can provide bursts of energy because they can charge and discharge rapidly.

You can model any number of supercapacitor cells connected in series or in parallel
using a single Supercapacitor block. To do so, set the
relevant parameter, that is **Number of series cells** or
**Number of parallel cells**, to a value larger than
`1`

. Internally, the block simulates only the equations for a
single supercapacitor cell, but it calculates:

The output voltage according to the number of series-connected cells

The current according to the number of parallel-connected cells

Calculating the output of a multiple-cell supercapacitor based on the output for a single cell is more efficient than simulating the equations for each cell individually.

The figure shows the equivalent circuit for a single cell in the Supercapacitor block. The circuit is a network of resistors and capacitors that is commonly used to model supercapacitor behavior.

Capacitors *C _{1}*,

$${V}_{cn}=\frac{v}{{N}_{series}}-{i}_{n}{R}_{n},$$

*v*is the voltage across the block.*N*is the number of cells in series._{series }*n*is the branch number.*n*= [1, 2, 3].*i*is the current through the_{n}*n*th branch.*R*is the resistance in the_{n}*n*th branch.*V*is voltage across the capacitor in the_{cn }*n*th branch.

The equation for the current through the first branch of the supercapacitor depends on the voltage across the capacitors in the branch. If the capacitors experience a positive voltage, that is

$${V}_{c1}>0,$$

$${i}_{1}=({C}_{1}+{K}_{v}{V}_{c1})\frac{d{V}_{c1}}{dt},$$

$${i}_{1}={C}_{1}\frac{d{V}_{c1}}{dt},$$

*V*is voltage across the capacitors in the first branch._{c1 }*C*is the capacitance of the fixed capacitor in the first branch._{1 }*K*is the voltage-dependent capacitance gain._{v}*i*is the current through the first branch._{1}

For the remaining branches, the current is defined as

$${i}_{n}={C}_{n}\frac{d{V}_{cn}}{dt},$$

*n*is the branch number.*n*=[2, 3].*C*is the capacitance of the_{n }*n*th branch.

The total current through the Supercapacitor block is

$$i={N}_{parallel}\left({i}_{1}+{i}_{2}+{i}_{3}+\frac{v}{{R}_{discharge}}\right),$$

*N*is the number of cells in parallel._{parallel}*R*is the self-discharge resistance of the supercapacitor._{discharge}*i*is the current through the supercapacitor.

[1] Zubieta, L. and R. Bonert. “Characterization of
Double-Layer Capacitors for Power Electronics Applications.” *IEEE
Transactions on Industry Applications*, Vol. 36, No. 1, 2000, pp.
199–205.

[2] Weddell, A. S., G. V. Merrett, T. J. Kazmierski, and B. M.
Al-Hashimi. “Accurate Supercapacitor Modeling for Energy-Harvesting Wireless
Sensor Nodes.” *IEEE Transactions on Circuits And Systems–II: Express
Briefs*, Vol. 58, No. 12, 2011, pp. 911–915.

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