## 4 Analysis of the Basic Unit of GSMLI
### Capacitance Selection  
The optimal value of capacitances for both switched capacitors (\(C_{1}\) and \(C_{2}\)) is calculated based on the longest discharge time (LDT) over a complete cycle of the fundamental output voltage. The maximum quantity of charges from switched capacitors is discharged during LDT. Figure 4 shows the output voltage of this proposed topology (asymmetric) along with the LDT for both switched capacitors. The LDT for \(C_{2}\) is high compared to \(C_{1}\). The amount of discharge during LDT for switched capacitor \(C_{2}\) can be calculated as  
\[Q_{C2}=\ 2\times\int_{t_{6}}^{T/4}i_{0}(t)dt \tag{1}\]  
For resistive load, the output current during LDT can be expressed as  
\[i_{0}(t)\ =\ \frac{6V}{R_{L}}for\ t_{6}<\ t\ <\ t\gamma i_{0}(t)\ =\ \frac{7V}{R_{L}}for\ t_{7}<\ t\ <\ t_{8} \tag{2}\]  
\[i_{0}(t)\ =\ \frac{8V}{R_{L}}for\ t_{8}<\ t\ <\ T/4\]  
Due to the application of a fundamental frequency scheme, the time \(t_{8}\), \(t_{7}\), and \(t_{6}\) can be given as  
\[t_{6}=Sin^{-1}(11/16)\ *\frac{1}{2\pi f}t\gamma=Sin^{-1}(13/16)\ *\ \frac{1}{2\pi f}t_{8}=Sin^{-1}(15/16)\ *\ \frac{1}{2\pi f} \tag{3}\]  
From Equations (1)-(3), we get \(Q_{C2}\) as  
\[Q_{C2}\ =\ \frac{12V}{2\pi fR_{L}} \tag{4}\]  
The value of optimal capacitance for switched capacitor \(C_{2}\) can be calculated as  
\[C_{2 opt}\geq\ \frac{QC1\ or\ QC2}{p\times V} \tag{5}\]  
From Equations (4) and (5),  
\[C_{2 opt}\geq\ \frac{12}{2\pi f\times R_{L}\times p} \tag{6}\]  
Figure 4: Waveform of output voltage (inverted negative half-cycle) along with the longest discharge time (LDT) of capacitors in asymmetric configuration. The amount of discharge during LDT for switched capacitor \(C_{1}\) can be calculated as  
\[Q_{C1}=\ 2\times\int_{t_{\rm{S}}}^{T/4}i_{0}(t)dt \tag{7}\]  
From Equations (2), (3), and (7), we get \(Q_{C1}\) as  
\[Q_{C}\ =\ \frac{6V}{2\pi fR_{L}} \tag{8}\]  
From Equations (5) and (8), the optimal capacitance (\(C_{1 opt}\)) for switched capacitor \(C_{1}\) is given:  
\[i_{0}(t)\ =\ \frac{6V}{R_{L}}for\ t_{6}<\ t\ <\ t\gamma i_{0}(t)\ =\ \frac{7V}{R_{L}}for\ t_{7}<\ t\ <\ t_{8} \tag{2}\]0  
where \(p\) is the maximum allowable output voltage ripple in percentage, \(R_{L}\) is load resistance, and \(V\) is input source voltage. From Equations (6) and (9), it can be observed that the optimal value of capacitances depends on the ripple in voltage, load resistance, and operating frequency. The variation of the optimal capacitance \(C_{2 opt}\) with frequency (at \(R_{L}\) = 100 \(\Omega\)) for different values of voltage ripples are shown in Figure 5a. The variation in the optimal capacitances (\(C_{1 opt}\) and \(C_{2 opt}\)) with load resistance for different values of voltage ripples is shown in Figure 5b,d, respectively. From Figure 5b,d, it is clear that, for a particular value of voltage ripple, the values of optimal capacitances (\(C_{1 opt}\) and \(C_{2 opt}\)) decrease as the load resistance increases. \[i_{0}(t)\ =\ \frac{6V}{R_{L}}for\ t_{6}<\ t\ <\ t\gamma i_{0}(t)\ =\ \frac{7V}{R_{L}}for\ t_{7}<\ t\ <\ t_{8} \tag{2}\]1 Variation in optimal capacitance \(C_{2 opt}\) with (**a**) frequency (_f_), (**b**) load resistance (\(\rm{R_{L}}\)), (**c**) and phase angle for different values of ripple in capacitor voltage and variation in optimal capacitance \(C_{1 opt}\) with (**d**) load resistance (\(\rm{R_{L}}\)) and (**e**) phase angle for different values of ripple in a capacitor voltage. For R-L load, the output current can be expressed as  
\[i_{0}(t)\ =\ \frac{6V}{R_{L}}for\ t_{6}<\ t\ <\ t\gamma i_{0}(t)\ =\ \frac{7V}{R_{L}}for\ t_{7}<\ t\ <\ t_{8} \tag{2}\]2  
where \(\varphi\) is the phase angle between the fundamental output voltage and output current. From Equations (1), (3), and (10), we get \(Q_{C2}\) as  
\[ \begin{array}{c}Q_{C2}\ =\ 2\times\int_{I_{t6}}^{T/4}I_{0 max}sin(\omega t- \varphi)dt\\ Q_{C2}\ =\ \frac{I_{0 max}}{2\pi f}[cos\ (0.75-\ \varphi\ )\ -sin\ (\varphi)]\end{array}