# 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
From the principle of volts per second balance, to discharge the capacitor back to zero current, the same number of volts per second must be applied with inverse polarity, and it will drain the same amount of current in the discharge process (the end of the first half cycle of the shape of the current) compared to the current drained during the charging process. This means that at the end of the resonant charge interchange (at the end of the semi-sine), capacitor \(C_{1}\) would lose twice the charge expressed in (9) and (10), and their final charge can be expressed as (11)  
\[C_{1}v_{1f}=C_{1}v_{1i}-2(C_{1}v_{1i}-C_{1}v_{x})=2C_{1}v_{x}-C_{1}v_{1i} \tag{11}\]  
where \(C_{1}v_{1f}\) is the final charge in \(C_{1}\), expressed as the initial charge minus the charge lost during the transient (twice the charge expressed in (9)). An analog procedure can be used to calculate the final charge in \(C_{2}\), their initial charge plus twice the charge it wind (expressed in (10) and (12)). \[C_{2}v_{2f}=C_{2}v_{2i}+2(C_{2}v_{x}-C_{2}v_{2i})=2C_{2}v_{x}-C_{2}v_{2i} \tag{12}\]  
By substituting (8) into (11) and (12), the final charge in \(C_{1}\) and \(C_{2}\) can be expressed as (13) and (14), respectively. Figure 23.0  
Figure 23.1  
Let us consider for a moment that both capacitors have the same capacitance (\(C_{1}=C_{2}=C\)), then (13) and (14) would lead, respectively, to (15) and (16). Figure 23.2Figure 23.3  
Equations (15) and (16) bring an interesting fact. If the capacitors have the same capacitance, the capacitors swap their voltage, and the final voltage in \(C_{1}\) is equal to the initial voltage in \(C_{2}\), and vice-versa. Equations (15) and (16) can be used to demonstrate that there is no difference from the initial stored energy to the final stored energy (there are no discharging losses as in the pure SC circuits); both the initial and final stored energy can be expressed as (17). Figure 23.4  
The conclusion is interesting, but this applies only to the case when capacitors have the same capacitance. The case when capacitors are different leads to the same result (as will be shown). There is no difference between the initial and final stored energy; to demonstrate that, let us consider that the capacitance \(C_{1}\) is different from \(C_{2}\), and let us define a ratio among the capacitances (18). Figure 23.5  
By considering (18), Equations (13) and (14) can be rewritten as (19) and (20). Figure 23.6  
Figure 23.7  
The definition of \(k\) in (18) allows for simplifying the mathematical procedure. Only one capacitance, \(C_{1}\), is present in equations, which allows simplifying (19) and (20) as (21) and (22), respectively. Figure 23.8  
Figure 23.9  
Let us consider the following definitions for the sake of simplicity:  
Figure 23.0  
Figure 23.1  
Equations (21) and (22) can be re-written, considering (23) and (24), as (25) and (26). Figure 23.2  
Figure 23.3  
The initial stored energy in the capacitors can be expressed as (27). Figure 23.4while the final energy can be expressed as (28). Figure 23.5  
Now we can substitute (25) and (26) into (28) to investigate if the result is equal to (27). Figure 23.6  
Expanding the quadratic terms in (29) and considering \(C_{2}=kC_{1}\) leads to (30). Figure 23.7  
Multiplying by \(2/C_{1}\) on both sides of (30) leads to (31)  
Figure 23.8  
Now we can group similar terms as in (32). Figure 23.9  
The factor that multiplies \({v_{1}}_{i}^{2}\) can be simplified as (33). Figure 23:0  
The factor that multiplies \({v_{2}}_{i}^{2}\) can be simplified as (34):  
Figure 23:1  
The factor that multiplies \({v_{1}}_{i}{v_{2}}_{i}\) is:  
Figure 23:2  
By substituting (33), (34), and (35) into (32), the stored energy in the capacitors after the charge interchange transient can be expressed as (36). Figure 23:3  
The stored energy at the end of the charge interchange transient equals the initial stored energy calculated in (27). The conclusion of this exercise is also interesting, and there are no discharge losses.