## 4 Results and Discussion  
Figure 13 shows experimental results of the proposed circuit for three cases. A higher and a lower voltage of the cells inside the module tend towards each other and all cells tend to the average voltage value of the cells in series. The proposed balancing circuit indeed demonstrates the balancing trend since the voltages of cells gradually reach a balanced state. Furthermore, the results on the different cases indicate that the cells achieved the balanced state regardless of the various arrangement of the initial voltage of all the cells. Energy efficiency of the balancing circuit can be used Equation (20) [16] as follows:  
\[\eta=\frac{E_{residue}}{E_{imbalance}}=\frac{\sum_{i=1}^{N}0.5C_{cell}\left(V_ {Cell\_end}^{2}-V_{min}^{2}\right)}{\sum_{i=1}^{N}0.5C_{cell}\left(V_{Cell\_ start}^{2}-V_{min}^{2}\right)} \tag{20}\]  
where \(E_{imbalance}\) and \(E_{residue}\) are unbalanced energy before cell balancing and remaining energy of \(E_{imbalance}\) after cell balancing, respectively. \(C_{\text{cell}}\) is the capacitance of the battery cell; \(V_{\text{cell\_start}}\) and \(V_{\text{cell\_end}}\) are the cell voltage of the \(i\)th cell before and after balancing, respectively; \(V_{\text{min}}\) and \(N\) are the initial lowest voltage and number of cells, respectively. In case 1 (see Figure 13a), the highest voltage is 3.880 V, the lowest voltage is 3.492, and the initial maximum voltage difference is 388 mV, and then the difference reduces to 115 mV after the balancing circuit runs for 300 min, and from (20) and experimental data and results, the energy efficiency is \(\eta_{1}=76.27\%\). Similarly, in case 2 (see Figure 13b) and case 3 (see Figure 13c), the maximum voltage difference among the cells reduces from 406 mV to 112 mV and from 381 mV to 86 mV after the balancing circuit runs for 150 min, and 200 min, and \(\eta_{2}=83.31\%\) and \(\eta_{3}=81.74\%\), respectively. The energy efficiency in three cases is different due to the difference in balancing time. If the balancing time is short, the energy efficiency will increase. In reality, the initial different voltage arrangement of the cells is set as in cases 2 or 3. These are the cases with the highest energy efficiency of the circuit. From the experimental results in Figure 13, we can see that there are minor residual voltage mismatches because the leakage inductances of the transformer were not uniform, as shown in Table 1. \(L_{\text{lk13}}\) of the transformer was larger than others; hence, the current \(i_{13}\) was smaller. In other words, the balance of the \(Cell_{13}\) was slower than others, as shown in Figure 13a. Figure 13: _Cont._According to Table 2, the initial maximum voltage difference of the cells in three cases is a minor divergence of approximately 25 mV. In addition, the total of the cell voltages is approximate. In particular, cases 1, 2 and 3 are 29.474 V, 29.437 V and 29.343 V, respectively. However, the balancing time for three cases is more different due to dependence on voltage unbalance between the \(M_{1}\) and \(M_{2}\), like the SCC-based balancer theory in Section 2.1. In detail, the initial voltage differences between the \(M_{1}\) and \(M_{2}\) for case 1, case 2, and case 3 are 892 mV, 287 mV, and 381 mV, the balancing times are 300 min, 150 min, and 200 min, respectively. The total voltage and the initial maximum different voltage of the cells are approximate; the voltage difference of the modules is different due to the arrangement of all cells. On the order hand, the balancing time of the circuit depends on the arrangement of the initial voltage of all cells. In order to more clearly show the reduction in the difference of the initial and balanced state voltage, the initial and balanced state voltages of cases 1, 2 and 3 are shown in Figure 14a-c, respectively. Figure 13: The experimental voltage results of eight cells: (**a**) Case 1; (**b**) Case 2; (**c**) Case 3.