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Kursiver Text

$W={\frac {1}{2}}\cdot C_{\text{total}}\cdot V_{\text{loaded}}^{2}$

$W={\frac {1}{2}}\ C\ (V_{\text{max}}^{2}-V_{\text{min}}^{2})$

$P={\frac {1}{4}}\cdot {\frac {V^{2}}{ESR}}$

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Description of the system type give rise to pseudocapacitance:

• Redox system: Ox + ze‾ ⇌ Red und O²‾ + Hˡ ⇌ in lattice
• Intercalation system: Liˡ in "Ma₂"
• Electrosorption, underpotential deposition of metal adatoms: M^꞊ˡ + S + ze‾ ⇌ SM (S = surface lattice sites) or Hˡ e‾ + S ⇌ SH

Energy density and power density

Supercapacitors in point of stored energy occupy or bridge the gap between electrolytic capacitors and rechargeable batteries. The amount of energy stored in a supercapacitor is called specific energy. The energy W_max of a capacitor is given by the formula

$W_{\text{max}}={\frac {1}{2}}\cdot C_{\text{total}}\cdot V_{\text{loaded}}^{2}$

This formula describes the total amount of energy stored in a capacitor and is often used in science publications to describe new research successes. But in reality only a part of the stored energy is available, the voltage drop and the time constant over the internal resistance reduce the practical available energy specified in datasheets of commercial available components. This effective realized amount of energy W_eff a supercapacitor can deliver is reduced by the used voltage difference between V_max and V_min and can be represented as:

$W_{\text{eff}}={\frac {1}{2}}\ C\cdot \ (V_{\text{max}}^{2}-V_{\text{min}}^{2})$

This formula represent also the energy of supercapacitors with asymmetric voltages like lithium ion capacitors.

Energy density is either measured gravimetrically (per unit of mass) in watt-hours per kilogram (Wh/kg) or volumetrically (per unit of volume) in watt-hours per litre (Wh/l).

As of 2013 the commercial available effective gravimetrically energy densities of supercapacitors range from around 0.5 to 15 Wh/kg. For comparison, an aluminum electrolytic capacitor stores typically 0,01 to 0.3 Wh/kg while a conventional lead-acid battery stores typically 30 to 40 Wh/kg and modern lithium-ion batteries about 100 to 265 Wh/kg. That means, supercapacitors can store 10 to 100 times more energy than electrolytic capacitors but only one tenth of batteries.

Although the energy densities of supercapacitors are insufficient compared with batteries the capacitors have an important advantage, the power density. Power density combines the energy density with the speed at which the energy can be delivered to the load or can be absorbed. The maximum power P_max is given by the formula:

$P_{\text{max}}={\frac {1}{4}}\cdot {\frac {V^{2}}{R_{i}}}$

with V = voltage applied and R_i , the internal DC resistance calculated as described in paragraph "Internal resistance".

Power density the time rate of energy transfer is either measured gravimetrically (per unit of mass) in kilowatt per kilogram (kW/kg) or volumetrically (per unit of volume) in kilowatt per litre (kW/l).

The described maximum power P_max specify the power of a rectangular single maximum current peak of a given voltage. In reality the current peak is not rectangular caused by time constants and the voltage is smaller caused by the voltage drop. The IEC standard 62391–2 therefore proposed a formula to calculate a more reality oriented effective power P_eff for supercapacitors for power applications

$P_{\text{eff}}={\frac {1}{8}}\cdot {\frac {V^{2}}{R_{i}}}$

The power density of supercapacitors is typically 10 to 100 times greater than for batteries and can reach values up to 15 kW/kg for industrial produced types. Special development of a tailored composite electrode have achieved a maximum power rating of 990 kW/kg.

Power density and energy density are usually displayed in a so-called Ragone plot. With such a diagram, the position of a particular storage technology compared with other technologies, is visually clearly represented, see picture.

The peak current flow at maximum power density rate generates internal heat. This heat is related to the specified life time. The average heat generation coming from power pulses should be smaller than 5 to 10  K, which has only minor influence on expected life time. To reach this relatively small heat increase, the heat generated by a single pulse may be distributed over the time until the next pulse occurs. More frequent pulses reduce the allowed peak current respectively the power density because the conditions of the life time specification should not be exceeded.