## 3 Applied Methods of Analysis and Assessment of Supercapacitors
on the basis on the recording of currents and voltages occurring during the test. The constant current charging/discharging method that is used in this test, despite the necessity of long-term observation, is most closely related to the actual operating conditions of the capacitors and is subjected to fluctuations in the temperature field of the tested capacitor due to its operating conditions. In supercapacitors' tests, calorimetric systems are generally used to determine maximum temperature increments, however, despite their high accuracy, they do not allow the heterogeneity of the sample to be displayed as described in articles [3; 4; 17]. Throughout tests, in which the proposed method was used, the following parameters of the power supply system were also monitored:  
* the power stored during the test expressed in (mAh);
* voltage on the test sample expressed in (V); and,
* current in the course of test expressed in (mA). The above parameters allowed to determine in the off-line processing the influence of the conducted test on the change of capacities (\(C\)) and \(ESR\) and to verify whether the sample was subject to change as a result of the test. Those parameters were determined on the basis of recorded data. Figure 7 illustrates the theoretical voltage waveform present during the test with the use of constant current charging/discharging method. The collected data allowed to estimate the capacity of the test sample on the basis of the Equation (2):  
\[C=\frac{I_{d}\cdot\Delta t}{\Delta U_{2}} \tag{2}\]  
and \(ESR\) based on the Equation (3):  
\[ESR=\frac{\Delta U_{1}}{|I_{d}|} \tag{3}\]  
where:  
\(I_{d}\)--test current (charging/discharging),  
\(\Delta t\)--discharging time,  
\(\Delta U_{1}\)--voltage drop on \(ESR\), and  
\(\Delta U_{2}\)--voltage drop during the discharging process. It should be noted here, that in real systems, the voltage waveforms are not described accurately by linear waveforms, although, in most studies the nonlinearity is omitted. The condition necessary to estimate the quality is to adopt criteria for its evaluation. The basic criteria that are currently used for  
Figure 7: An image of an ideal charging/discharging process on the basis of which the parameters of the supercapacitor were determined. assessment are: the change in capacity (C) and \(ESR\) value. The change in capacity (C) by 20% and/or \(ESR\) increase by +100% over the initial value, under the recommendations of [4; 5], has been taken as the criterion of exclusion. Such types of tests are characterized by a very long period of performance, while performed with ratings (they are classified as long-term tests). Currently produced capacitors are characterized, inter alia, by guaranteed durability of at least 500,000 charging/discharging cycles at rated current and voltage [17]. In order to accelerate the occurrence of the aging phenomena in the tests, it was decided to exceed the nominal values that were recommended by the manufacturer of the capacitor, assuming as a critical value the average temperature increment (\(\Delta T_{Av}\)) that should be less than 15 \({}^{\circ}\)C for the observed supercapacitor surface [13].