# An Overview of Non-Isolated Hybrid Switched-Capacitor Step-Up DC-DC Converters
## 4 The Charge Interchange between Capacitors
### Energy Analysis on the Non-Resonant Charge Interchange  
The energy analysis is well known in the case of a pure switched-capacitor circuit; 50% of the data is in the mind of many researchers, but that 50% applies only in the case of a full-charge interchange of capacitors of the same capacitance, and when one of those capacitors is initially discharged. Let us start from the beginning to understand that value and analyze different cases.  
Let us consider a circuit made only by two capacitors, \(C_{1}\) and \(C_{2}\), as in Figure 20. The capacitances of the capacitors and their initial voltage may be different. In addition, let us recall the well-known analogy between the studied system and a hydraulic system consisting of two interconnected water tanks since capacitors have the same mathematical model as water tanks.  
For the energy-based analysis, it is not necessary to model parasitic resistances, which makes the analysis relatively simple. Let us consider whether capacitors get connected in parallel or whether we open the valve which connects the water tanks.  
Figure 19: Basic resonant SC circuit, (**a**) schematic, (**b**) current between capacitors with resonant charge interchange.  
Figure 20: Pure switched-capacitors circuit and their hydraulic analogy.  
There are many things we can discuss in this experiment; some important comments are:  
1. The stored energy ratio is always smaller than \(1\), which is expected from the efficiency analysis in the power converters.
2. The capacitance of the capacitors seems to disappear from (7), but this is only when the capacitors are equal. If the capacitors are different, the ratio among the capacitances would appear as a factor in Equation (7).
3. The minimum ratio (or efficiency) observed when capacitors are equal is 0.5 of 50%, but this is only the minimum. The efficiency may be larger if the second capacitor initially has more than 50% of the other one. The efficiency remains over 90%. On the other hand, if one capacitor's initial voltage is smaller than 50% of the other, the efficiency decreases rapidly.
4. The efficiency is larger if the initial difference of the voltages is smaller, but if the initial voltage of the capacitors is equal, the charge interchange would be zero. Then, the 100% efficiency has a trade-off versus the amount of charge transferred. This is another way of seeing why pure SC circuits are mainly used in low-power applications.  
We can say the ratio between the final over the initial stored energy has an impact on the efficiency based on the principle of connecting capacitors in parallel.  
Another interesting commend is that the power losses found in this analysis do not depend on the parasitic resistance that connects capacitors, but a low parasitic resistance would lead to a faster charge interchange (which is an advantage) and a larger peak current (which is a disadvantage).  
This kind of analysis is not common in other types of converters, in which losses are analyzed from the power dissipation point of view.  
Before analyzing the resonant switched-capacitor circuit, we would like to emphasize that it is not difficult to transfer charge among capacitors with efficiencies over 95% in pure switched-capacitor circuits.