# An Overview of Non-Isolated Hybrid Switched-Capacitor Step-Up DC-DC Converters
## 5 Hybrid SC Converters
### Demonstrative Experimental Results  
As a demonstrative example, a traditional multilevel boost converter was built. The experimental results are provided in this section. Figure 33 shows the 2\(\times\) multilevel (or multiplier) boost converter, Figure 33a shows the circuit schematic, while Figure 33b,c show the equivalent circuits of the converter when the switch is closed and open, respectively. When the transistor is closed, the inductor is connected to the input voltage \(v_{g}\), when the switch opens, the voltage across the inductor is \(v_{g}-v_{\text{C1}}\). From the circuits shown in Figure 33, and by applying the averaging theory [9], it is possible to write the inductor voltage along the entire switching period, as in (37). \[\frac{d}{dt}i_{L}=d\frac{v_{g}}{L}+(1-d)\frac{v_{g}-v_{\text{C1}}}{L} \tag{37}\]  
where \(d\) is the duty cycle of the converter, i.e., the time in which the converter is closed divided by the total switching period. By considering the small ripple approximation and the steady state condition [9], it is possible to obtain the voltage across the capacitor and express it in terms of the duty cycle and the input voltage. The steady state or equilibrium voltage across \(C_{1}\) can be expressed as (38). \[V_{\text{C1}}=\frac{V_{g}}{1-D} \tag{38}\]  
The use of upper-case letters indicates steady state (or equilibrium) values. From Equation (38), we can see that capacitor \(C_{1}\) has the same voltage of a traditional boost converter. Then, we can see from Figure 33b, that there is a switched-capacitor action in which capacitor \(C_{1}\) transfers charge to capacitor \(C_{2}\). This is a non-resonant charge interchange. Despite this, the charge interchange may lead to discharging losses. If the voltage in \(C_{2}\) is 65% of the voltage in \(C_{1}\) before it is in parallel, the efficiency of the charge interchange would be around 95% (see Figure 21). At the next switching action, \(C_{2}\) transfers charge to \(C_{3}\), in a similar action. The amount of charge transferred from \(C_{2}\) to \(C_{3}\) can be calculated as (39). \[\Delta q_{\text{C2 to C3}}=I_{out}D\frac{1}{f_{S}} \tag{39}\]  
where \(f_{S}\) represents the switching frequency. Equation (39) calculates the amount of charge lost by \(C_{3}\) when the switch is off, and this charge must be recovered from \(C_{2}\) in a steady  
Figure 33: A 2\(\times\) multilevel boost converter, (**a**) schematic, (**b**) equivalent circuit when the transistor is closed, and (**c**) equivalent circuit when the transistor is open. state. The voltage drop in \(C_{3}\) can be calculated by dividing the charge in (39) over the capacitance \(C_{3}\). This amount of charge transferred from \(C_{2}\) to \(C_{3}\) comes from \(C_{1}\) when the switch is closed, and then the same calculation can be performed to calculate the voltage drop in the capacitor \(C_{2}\). Table 4 contains the experiment parameters. Figure 34 shows important waveforms, the current through the inductor, the output voltage (voltage across the resistor), the voltage across \(C_{1}\), and the switch voltage; the voltages are displayed at 50 V/div and the current at 1 A/div. Figure 34a corresponds to an input voltage of \(v_{g}\) = 50 V and a duty cycle of \(D\) = 0.5, the output voltage was measured as 189 V, the output power measured as 357 W, and the efficiency as 93%. Figure 34b corresponds to the operating point in which the input voltage \(v_{g}\) = 50 V, the duty cycle was \(D\) = 0.6, the output voltage was measured as 238 V, the output power measured as 362 W, and the efficiency was 93%. Figure 34: Important waveforms of the converter operation, from top to bottom: the current through the inductor, the output voltage, the voltage across capacitor \(C_{1}\) and switch voltage. (**a**) For \(D\) = 0.5, and (**b**) for \(D\) = 0.6.