# An Overview of Non-Isolated Hybrid Switched-Capacitor Step-Up DC-DC Converters
## 4 The Charge Interchange between Capacitors
### Energy Analysis in the Resonant Charge Interchange  
Recently, many SC topologies have been studied with a resonant charge interchange, with a relatively large efficiency at a relatively large power [40; 41; 42; 43; 44; 45; 102; 103; 104; 105; 106], some of them for a large power [44], which seems unexpected considering SC circuits have these \"discharging losses\". This is because when the resonant charge interchange is introduced,  
Figure 21: Store energy ratio among the final and the initial condition. the difference in the stored energy that we have called the discharging losses seems to disappear. This has been studied and proven, e.g., in [43]. Here, we present different proof based on the volts-second balance and charge conservation principles, which may be simpler to understand by readers involved in the power electronics field. Figure 22 shows a resonant switched-capacitor cell. It has a switch that connects the capacitors through the resonant inductor \"\(L_{res}\)\" (small size). The charge interchange is different than the pure SC case. The main differences are that _(i)_ the current tends to be sinusoidal, and _(ii)_ we can stop the current before it finds an equilibrium (preferably during a zero crossing), meaning the final voltage of capacitors would be different \(v_{1f}\
eq v_{2f}\). In other words, the inductor is chosen in a way that the current has a resonant behavior. This requires a very small inductor, much smaller than the energy storage inductors. If we let the circuit reach the equilibrium, parasitic resistances would damp the current, and it will become zero when capacitors are charged to the same voltage, which would lead to the same result of pure switched-capacitor circuits. However, we can also open the switch before that happens. When capacitors have different voltages, the most common choice is to allow only a half-sine wave in the current, which leads to a current as shown in Figure 19, which is more detailed in Figure 23.  
The current through the inductor starts rising when the switch closes, at an instantaneous rate of \((v_{1i}-v_{2i})/L\); the current of the inductor stops rising when both capacitors have the same voltage, that is, the moment in which the inductor has its maximum current \(I_{max}\), see Figure 23.  
Figure 23: Voltage and current signals during the resonant charge interchange. Figure 22: A basic resonant switched-capacitor circuit. During the x-point or maximum current point, the voltage in both capacitors is the same, and it can be called \(v_{x}\). The voltage \(v_{x}\) can be calculated by using the principle of charge conservation, with the same procedure as in (2), in other words:  
\[v_{x}=\frac{C_{1}v_{1i}+C_{2}v_{2i}}{C_{1}+C_{2}} \tag{8}\]  
We can also calculate the amount of charge that capacitors have interchanged from the beginning of the transient to the x-point. The amount of charge \(C_{1}\) has transferred to \(C_{2}\) can be expressed as (9). \[\Delta_{q}=C_{1}v_{1i}-C_{1}v_{x}=C_{1}(v_{1i}-v_{x}) \tag{9}\]  
In other words, from the beginning of the transient to the x-point, \(C_{1}\) has provided charge (it has lost charge), and the amount of lost charge reduces their voltage from \(v_{1i}\) to \(v_{x}\). Again, we are considering that \(C_{1}\), initially, has a larger voltage; on the other hand, \(C_{2}\) has gained some charge, the same amount of charge, which can be expressed as (10). \[\Delta_{q}=C_{2}v_{x}-C_{2}v_{2i}=C_{2}(v_{x}-v_{2i}) \tag{10}\]  
Note that we consider that \(C_{2}\) has gained some charge, and its voltage has increased, meaning that \(v_{x}\) is larger than \(v_{2i}\). In Figure 23, we assumed that both capacitors have the same capacitance, which makes the figure symmetrical, but the equations can be applied to different capacitances. The inductor was initially discharged. It had zero current at the beginning of the transient, and now, at the x-point, it has its maximum current \(I_{max}=I_{x}\).