# Supercapacitor

Implement generic supercapacitor model

**Libraries:**

Simscape /
Electrical /
Specialized Power Systems /
Sources

## Description

The Supercapacitor block implements a generic model parameterized to represent most popular types of supercapacitors. The figure shows the equivalent circuit of the supercapacitor:

The supercapacitor output voltage is expressed using a Stern equation as:

${V}_{SC}=\frac{{N}_{s}{Q}_{T}d}{{N}_{p}{N}_{e}\epsilon {\epsilon}_{0}{A}_{i}}+\frac{2{N}_{e}{N}_{s}RT}{F}{\mathrm{sinh}}^{-1}\left(\frac{{Q}_{T}}{{N}_{p}{N}_{e}{}^{2}{A}_{i}\sqrt{8RT\epsilon {\epsilon}_{0}c}}\right)-{R}_{SC}\cdot {i}_{SC}$

with

${Q}_{T}={\displaystyle \int {i}_{SC}dt}$

To represent the self-discharge phenomenon, the supercapacitor electric charge is modified
as follows (when *i _{SC}* = 0):

${Q}_{T}={\displaystyle \int {i}_{self\_dis}dt}$

where

${i}_{self\_dis}=\{\begin{array}{l}\frac{{C}_{T}{\alpha}_{1}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& t-{t}_{oc}\le {t}_{3}\end{array}\end{array}\\ \frac{{C}_{T}{\alpha}_{2}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& {t}_{3}\prec t-{t}_{oc}\le {t}_{4}\end{array}\end{array}\\ \frac{{C}_{T}{\alpha}_{3}}{1+s{R}_{SC}{C}_{T}}\begin{array}{cc}& if\begin{array}{cc}& t-{t}_{oc}\succ {t}_{4}\end{array}\end{array}\end{array}$

The constants *α1*, *α2*, and *α3* are
the rates of change of the supercapacitor voltage during time intervals
(*toc*, *t3*), (*t3*,
*t4*), and (*t4*, *t5*) respectively, as
shown in the figure:

Variable | Description |
---|---|

A_{i} | Interfacial area between electrodes and electrolyte
(m^{2}) |

c | Molar concentration (mol/m^{3}) equal to c =
1/(8N_{A}r^{3}) |

r | Molecular radius (m) |

F | Faraday constant |

i_{sc} | Supercapacitor current (A) |

V_{sc} | Supercapacitor voltage (V) |

C_{T} | Total capacitance (F) |

R_{sc} | Total resistance (ohms) |

N_{e} | Number of layers of electrodes |

N_{A} | Avogadro constant |

N_{p}
| Number of parallel supercapacitors |

N_{s} | Number of series supercapacitors |

Q_{T} | Electric charge (C) |

R | Ideal gas constant |

d | Molecular radius |

T | Operating temperature (K) |

ε | Permittivity of material |

ε_{0} | Permittivity of free space |

### Examples

The `parallel_battery_SC_boost_converter`

example shows a simple
hybridization of a supercapacitor with a battery. The supercapacitor is connected to a
buck/boost converter and the battery is connected to a boost converter. The DC bus voltage
is equal to 42V. The converters are doing power management. The battery power is limited by
a rate limiter block, therefore the transient power is supplied to the DC bus by the
supercapacitor.

## Assumptions and Limitations

Internal resistance is assumed constant during the charge and the discharge cycles.

The model does not take into account temperature effect on the electrolyte material.

No aging effect is taken into account.

Charge redistribution is the same for all values of voltage.

The block does not model cell balancing.

Current through the supercapacitor is assumed to be continuous.

## Ports

### Conserving

### Output

## Parameters

## References

[1] Oldham, K. B. “A Gouy-Chapman-Stern model of the double
layer at a (metal)/(ionic liquid) interface.” *J. Electroanalytical
Chem*. Vol. 613, No. 2, 2008, pp. 131–38.

[2] Xu, N., and J. Riley. “Nonlinear analysis of a classical
system: The double-layer capacitor.” *Electrochemistry
Communications*. Vol. 13, No. 10, 2011, pp. 1077–81.

## Extended Capabilities

## Version History

**Introduced in R2013a**