# An Overview of Non-Isolated Hybrid Switched-Capacitor Step-Up DC-DC Converters
## 3 Switched-Capacitor Circuits Controlled by Transistors
\(C_{1}\), in series with the input voltage,  
Figure 11: The _Fibonacci Converter_ is another possible extension to the circuit in Figure 9, (**a**) topology, (**b**) equivalent circuit when the single firing signal is high, and (**c**) equivalent circuit when the single firing signal is low. charges \(C_{2}\) with twice the input voltage at the next switching state when the circuit behaves again as Figure 11b, with \(C_{1}\) in series with \(C_{2}\), which charges \(C_{3}\) to three times the input voltage. Then, back in Figure 11c, \(C_{2}\), in series with \(C_{3}\), charges \(C_{4}\) with five times the input voltage. Finally, back in Figure 11b, \(C_{3}\), in series with \(C_{4}\), charges capacitor \(C_{5}\) with eight times the input voltage. The circuit is called the _Fibonacci Converter_ since capacitors get charged with multiples of the input voltage, and the multiples are equal to the Fibonacci series of numbers (considering the input voltage as the first one), which is 1, 1, 2, 3, 5, 8..., (the following number is the sum of two previous numbers). The circuit in Figure 11 can acquire gains which are different to the Fibonacci series; for example, if group \(A\) is made by the switches \(s_{1}\), \(s_{3}\), \(s_{4}\), \(s_{6}\), \(s_{7}\), \(s_{9}\), \(s_{10}\), and \(s_{12}\), and group \(B\) is made by all the other switches (\(s_{2}\), \(s_{5}\), \(s_{8}\), \(s_{11}\), and \(s_{13}\)), their switching states would be equivalent to Figure 10b,c, and the gain would be linear. As with the _Series-Parallel_, other switching sequences can be tried. One of the last classical SC converters covered is the _Ladder Converter_, see Figure 12a. This converter cannot be considered as an extension of the converter in Figure 9, at least not in a straightforward manner, and this is one of the most popular SC circuits nowadays [22; 23; 24; 25; 26]. Following the same procedure used to discuss the operation of the _Series-Parallel_ and _Fibonacci_ converters, we can divide the switches into two groups: group \(A\), made by all the odd switches, and group \(B\), made by all the even switches. By driving the converter with a single firing signal, a square signal (50% of duty cycle), the switches of group \(A\) are all closed, and the switches of group \(B\) are all open when the firing signal is one (high), which produces the equivalent circuit shown in Figure 12b. On the other hand, when the firing signal is zero (low), switches of group \(A\) are all open while switches of group \(B\) are all closed; this produces the equivalent circuit shown in Figure 12c. The parallel connection between capacitors means that all capacitors are charged to the input voltage, the converter can be extended with more switches and capacitors, and the circuit's output voltage in Figure 12 is four times the input voltage. This converter can be considered the transistorized version of a _Cockcroft-Walton Voltage Multiplier_ or _Ladder Voltage Multiplier_, a rectifier or ac-to-dc converter that utilizes the alternating voltage to activate diodes. Several studies have been done with this converter, such as [22; 23; 24; 25; 26]. Figure 12: The _Ladder Converter_ is another possible extension to the circuit in Figure 9. (**a**) Converter topology, (**b**) equivalent circuit when the firing signal is high, and (**c**) equivalent circuit when the firing signal is low. The resonant version of this converter has been used in several applications, mainly in voltage equalizers for either batteries or capacitors connected in series, as mentioned in Section 5.4. Converters in this section are presented in the form of their step-up topology, but, in all cases, they can also work as a step-down converter by inverting their source and load side. The field of investigating new topologies is very active among the recent contributions to the state-of-the-art in topologies of the SC circuits.