The system variables needed are used to calculate the size and number of ultracapacitor cells appropriate for any application. In order to get the most accurate assessment of what is necessary, the following parameters need to be defined:

• Maximum charged voltage of the ultracapacitor cell or module (Vmax)
• Working voltage of the ultracapacitor cell or module (Vw)
• Minimum voltage of the ultracapacitor cell or module (Vmin)
• Required current (in amps) to discharge the ultracapacitor cell or module (I)
• Duration (in seconds) of discharge from Vw to Vmin (td)
• Total capacitance of the ultracapacitor cell or module (C)
• Equivalent series resistance of the ultracapacitor cell or module (ESR)

These values can be used in some simple exercises to determine the size and number of ultracapacitor cells required for the application. The size and number of cells required depends upon the total ultracapacitor discharge-cycle voltage drop. During the discharge cycle of an ultracapacitor, there are two voltage-drop parameters to consider: the voltage drop due to equivalent series resistance (ESR) and the voltage drop due to capacitance.

Discharge Curve

As can be seen above, during a discharge cycle the initial voltage drop is due to the ultracapacitor's ESR. The amount of the drop is a function of the ESR and discharge current as indicated by the following equation: dVESR= I * ESR

After the initial voltage drop due to ESR, the capacitor will discharge according to its capacitance and discharge current, as indicated by the following equation: dVcap= I * td/C

By placing these two equations together, the total voltage drop can be calculated per the following equation: dVTotal = I * td/C + I * ESR

The total voltage drop indicates the number and size of the ultracapacitor cells required to meet the demands of the application. Please note that allowing a larger drop in voltage will reduce the necessary capacitance. Typically, by allowing the ultracapacitor cell or module to drop to half of the Vw value, 75% of the ultracapacitor's energy will be discharged.