The total energy stored in a carbon/carbon capacitor can be calculated from the relationship E=1/2CV2. If the voltage of the capacitor is restricted to the range V0 to V0/2, only 75% of the stored energy can be used. Hence, the usable energy density is given by
This simple relationship is often used to calculate the energy density of ultracapacitor. The most reliable approach to determining the energy stored in a device, however, is to measure the Wh stored for a range of constant power densities Wkg−1. In general, tests should be made for power densities between 100 and 1000Wkg−1 or even higher for high-power devices. The plot of Whkg−1 versus Wkg−1 is called the Ragone curve for the device. Typical data for a commercially available 3000 F capacitor are shown in Table 1. Note that the energy density gradually decreases with Wkg−1. This is the case for all ultracapacitor. The value of energy density quoted by the device manufacturer is often calculated from the energy corresponding to 1/2CV2 using the rated voltage and specified capacitance.
This value is too high as it is not the usable energy density and also it corresponds to a low power density of 100Wkg−1 or lower. As shown in Table 1, the effective capacitance Ceff of a device can decrease significantly with Wkg−1 and often agrees with the value claimed by the manufacturer only for relatively low-power densities. Hence, combining the usable energy factor (0.75) and the effective capacitance reduction factor (0.9 from Table 1), the simple calculation of energy density from 1/2CV2 can overestimate the energy density of a device by at least 1/3. As shown in Table 2, for very high-power devices, the effective capacitance can remain constant for power densities almost to 2000Wkg−1.
Table 1: Test data for the 3000F device
—– Quote from：《Supercapacitors: Material, System, and Applications》Energetic Performance and Discharging at Constant Power
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