Simplification of Indirect Resonant Switched-Capacitor Converter Based on State-Space Average Model Method

Abstract

This paper simplifies indirect resonant switched-capacitor (ReSC) converters using the state-space average model method. The operation principles of the 4:1 and 5:1 ReSC converters derived from the Dickson (4:1) circuit are analyzed, and the corresponding state-space average matrices are derived based on their equivalent circuits. The resonant inductor of the specific resonant branch is eliminated by analyzing the composition of the state-variable matrix, thereby obtaining the simplified topologies of 4:1 and 5:1 indirect ReSC converters. The simplified topologies are simulated and experimentally verified. The results prove the correctness of the state-space average modeling method and the effectiveness of the simplified topologies.

1. Introduction

With the sharp increase in the global data processing volume and the rapid development of the digital economy driven by artificial intelligence, the strategic position of data centers as important equipment has significantly improved [1,2,3,4,5]. However, the power supply system is also confronted with major challenges caused by problems such as low efficiency and large volumes. In order to reduce the volume of the converters and increase the power density, conducting system modeling analysis on converters to simplify the circuit topology has become a research hotspot.

Most power converters are significantly affected in terms of efficiency and size due to the presence of magnetic components such as transformers and power inductors [6,7,8,9,10,11,12,13,14].

Switched-capacitor converters have the advantages of a high power density, high efficiency and a high voltage conversion ratio (VCR) due to the absence of power inductors [15,16,17]. However, due to the difference in values of the flying capacitors, a large transient current will be generated during mode switching, which in turn causes charging and discharging losses [18,19,20]. To solve the above problem, resonant branches can be formed by introducing one or several small inductors to operate in a resonant state [21,22,23]. The circuit topology of several inductors cascading with flying capacitors is called the indirect resonant switched-capacitor (ReSC) converter, which significantly improves efficiency while having a very limited impact on power density [24,25,26]. For the application of the indirect ReSC in data centers with high VCRs, the number of passive components increases significantly with the rise in the VCR, therefore, simplifying the circuit topology while ensuring the resonant operating mode can effectively reduce losses and increase power density. The above aim can be achieved by analyzing the circuit topology structure through modeling methods. Refs. [27,28] conduct a mathematical analysis on the switched-capacitor converter for power supply optimization. The average modeling approaches are widely used to average out the switching dynamics in power converters [29,30,31]. The state-space average model transforms complex switch circuits into linear models that are easy to analyze. For circuit topologies with limited switching states, it can effectively obtain equivalent circuit diagrams under different switching states. By analyzing the circuit topology through the state-space matrix composition, circuit simplification can be achieved.

In this paper, the operation principle of the 4:1 and 5:1 indirect ReSC converters is briefly shown in Section 2. Their state-space average models are built, and the simplified circuit topology is obtained through the matrix composition analysis in Section 3. And the simulation and experiment results are shown in Section 4.

2. Operation Principle of Indirect ReSC Converter

This section takes the ReSC converters with VCRs of 4:1 and 5:1 as examples, respectively, to introduce their operation principles. They are convenient for the subsequent derivation of the state-space average model corresponding to the indirect ReSC converter.

The operation principle of the indirect ReSC converter with a voltage conversion ratio of 4:1 is shown in Figure 1. The duty cycle of each switching devices is set to close to 50%. The working mode of the relevant circuit can be divided into two stages. In stage 1, as shown in Figure 1a, the switching devices (S_n, n = 1, 3, 5, 7, 9) marked red are turned on, and the switching devices (S_n, n = 2, 4, 6, 8, 10) marked blue are turned off. The input DC source charges C₁ and C_out, and C₂ charges C₃ and C_out. In stage 2, as shown in Figure 1b, the odd number switching devices (S_n, n = 1, 3, 5, 7, 9) marked red are turned off, the even number switching devices (S_n, n = 2, 4, 6, 8, 10) marked blue are turned on, C₁ charges C₂ and C_out, and C₃ charges C_out. If the output voltage is V_out, the maximum voltages on the switching devices S₂ and S₃ are 2 × V_out, and the maximum voltages on the other switching devices are all V_out.

The operation principle of the indirect ReSC converter with a voltage conversion ratio of 5:1 is shown in Figure 2. The working mode of the relevant circuit can also be divided into two stages. In stage 1, as shown in Figure 2a, the switching devices (S_n, n = 1, 3, 5, 6, 8, 10, 12) marked red are turned on, and the switching devices (S_n, n = 2, 4, 7, 9, 11, 13) marked blue are turned off. The input DC source charges C₁ and C_out, C₂ charges C₃ and C_out, and C₄ charges C_out. In stage 2, as shown in Figure 2b, the switching devices (S_n, n = 1, 3, 5, 6, 8, 10, 12) marked red are turned off, the switching devices (S_n, n = 2, 4, 7, 9, 11, 13) marked blue are turned on, C₁ charges C₂ and C_out, and C₃ charges C₄ and C_out. If the output voltage is V_out, the maximum voltages on switching devices S₂, S₃, and S₄ are 2 × V_out, and the maximum voltages on the other switching devices are all V_out.

3. State-Space Average Model of Indirect ReSC Converter

Due to the certain differences in the circuit composition of ReSC converters with odd voltage conversion ratios and even voltage conversion ratios, it is necessary to analyze them separately during modeling. The state equation of the circuit is shown in (1), where x(t) is the state vector; u(t) is the input vector; and K, A, and B are constant matrices determined by the circuit.

$$K{\displaystyle \frac{dx(t)}{dt}}=Ax(t)+Bu(t)$$

(1)

First, model derivation is carried out for the ReSC converter with an even conversion ratio of 4:1, and the state vector and input vector are shown in (2) and (3), respectively.

$$x(t)={\left[\begin{array}{ccccccc}{i}_1(t)& i_2(t)& i_3(t)& v_1(t)& v_2(t)& v_3(t)& v_0(t)\end{array}\right]}^T$$

(2)

$$u(t)=\left[v_{in}(t)\right]$$

(3)

The equivalent circuit diagrams of the 4:1 indirect ReSC converter under different switching states are shown in Figure 3. The reference positive directions of the corresponding state vectors are also marked in Figure 3. It can be seen that i₂(t) = −i₃(t), C₃dv₃(t)/dt = −C₂dv₂(t)/dt in Figure 3b, and i₂(t) and v₂(t) can be represented by i₃(t) and v₃(t), respectively. Therefore, (2) can be simplified to obtain the independent state vector as shown in (4).

$$x(t)={\left[\begin{array}{ccccc}{i}_1(t)& i_3(t)& v_1(t)& v_3(t)& v_0(t)\end{array}\right]}^T$$

(4)

The equivalent circuit diagram of state 1 is shown in Figure 3a,b. For circuit (a), there are (5)–(8). For circuit (b), there are (9)–(11).

$$\begin{array}{l}{L}_1{\displaystyle \frac{d{i}_1(t)}{dt}}=v_{in}(t)-v_1(t)-v_0(t)\\ -i_1(t)(R_{S1}+ES{R}_{C1}+ES{R}_{L1}+R_{S9})\end{array}$$

(5)

$$i_{in}(t)=i_1(t)$$

(6)

$$v_0(t)=v_{out}(t)$$

(7)

$$i_1(t)=C_1{\displaystyle \frac{d{v}_1(t)}{dt}}$$

(8)

$$\begin{array}{l}(L_2+L_3){\displaystyle \frac{d{i}_3(t)}{dt}}=v_2(t)-v_3(t)-v_0(t)-i_3(t)*\\ (R_{S7}+ES{R}_{C3}+ES{R}_{L3}+R_{S5}+ES{R}_{C2}+ES{R}_{L2}+R_{S3})\end{array}$$

(9)

$$i_3(t)=C_{out}{\displaystyle \frac{d{v}_0(t)}{dt}}+{\displaystyle \frac{v_0(t)}{R_{load}}}$$

(10)

$$i_3(t)=C_3{\displaystyle \frac{d{v}_3(t)}{dt}}$$

(11)

The equivalent circuit of state 2 is shown in Figure 3c,d. For circuit (c), there are (12). For circuit (d), there are (13) and (14).

$$\begin{array}{l}(L_1+L_2){\displaystyle \frac{d{i}_1(t)}{dt}}=v_2(t)-v_1(t)+v_0(t)-i_1(t)*\\ (R_{S2}+ES{R}_{C1}+ES{R}_{L1}+R_{S10}+R_{S8}+ES{R}_{C2}+ES{R}_{L2})\end{array}$$

(12)

$$\begin{array}{l}{L}_3{\displaystyle \frac{d{i}_3(t)}{dt}}=v_0(t)-v_3(t)\\ -(R_{S4}+ES{R}_{C3}+ES{R}_{L3}+R_{S6})i_3(t)\end{array}$$

(13)

$$-i_3(t)=C_{out}{\displaystyle \frac{d{v}_0(t)}{dt}}+{\displaystyle \frac{v_o(t)}{R_{load}}}$$

(14)

$${\displaystyle \frac{dx(t)}{dt}}=(D_1{K}_1^{-1}{A}_1+D_2{K}_2^{-1}{A}_2)x(t)+(D_1{K}_1^{-1}{B}_1+D_2{K}_2^{-1}{B}_2)u(t)$$

(15)

It is known that indirect ReSC converters have two switching states. The switching devices in the two states are complementary in conduction at a duty cycle of 50%; that is, D₁ = D₂ = 0.5. The weighted average state-space equation of the system is shown in (15). K₁⁻¹ and K₂⁻¹ are shown in (16) and (17).

$$K_1^{-1}=\left[\begin{array}{ccccc}{\displaystyle \frac{1}{L_1}}& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{L_2+L_3}}& 0& 0& 0\\ 0& 0& {\displaystyle \frac{1}{C_1}}& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{C_3}}& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{C_{out}}}\end{array}\right]$$

(16)

$$K_2^{-1}=\left[\begin{array}{ccccc}{\displaystyle \frac{1}{L_1+L_2}}& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{L_3}}& 0& 0& 0\\ 0& 0& {\displaystyle \frac{1}{C_1}}& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{C_3}}& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{C_{out}}}\end{array}\right]$$

(17)

The equivalent circuit diagrams of the 5:1 indirect ReSC converter under different switching states are shown in Figure 4. The reference positive directions of the corresponding state vectors are also marked in Figure 4. According to the derivation of the 4:1 indirect ReSC converter, it can be known that the independent state vector of the 5:1 indirect ReSC converter can be obtained as shown in (18).

$$x(t)={\left[\begin{array}{ccccccc}{i}_1(t)& i_3(t)& i_4(t)& v_1(t)& v_3(t)& v_4(t)& v_0(t)\end{array}\right]}^T$$

(18)

For switching state 1 of the 5:1 indirect ReSC converter, there are three equivalent circuits:(a), (b), and (c). For circuit (a), there are (A1)–(A4). For circuit (b), there are (A5)–(A7). For circuit (c), there are (A8) and (A9). For switching state 2 of the 5:1 indirect ReSC converter, there are two equivalent circuits: (d) and (e). For circuit (d), there are (A10). For circuit (e), there are (A11). In order to save space in the main text, (A1)–(A11) can be found in the Appendix A.

The 5:1 indirect ReSC converter also has two switching states with complementary and equal duty cycles (D₁ = D₂ = 0.5). Therefore, according to (15), the K₁⁻¹ and K₂⁻¹ can be obtained as shown in (19) and (20).

$$K_1^{-1}=\left[\begin{array}{ccccccc}{\displaystyle \frac{1}{L_1}}& 0& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{L_2+L_3}}& 0& 0& 0& 0& 0\\ 0& 0& {\displaystyle \frac{1}{L_4}}& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{C_1}}& 0& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{C_3}}& 0& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1}{C_4}}& 0\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{C_{out}}}\end{array}\right]$$

(19)

$$K_2^{-1}=\left[\begin{array}{ccccccc}{\displaystyle \frac{1}{L_1+L_2}}& 0& 0& 0& 0& 0& 0\\ 0& {\displaystyle \frac{1}{L_3+L_4}}& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& {\displaystyle \frac{1}{C_1}}& 0& 0& 0\\ 0& 0& 0& 0& {\displaystyle \frac{1}{C_3}}& 0& 0\\ 0& 0& 0& 0& 0& {\displaystyle \frac{1}{C_4}}& 0\\ 0& 0& 0& 0& 0& 0& {\displaystyle \frac{1}{C_{out}}}\end{array}\right]$$

(20)

In the mathematical expressions, the inductors L_n (n = 1, 2, 3, 4……) all represent the inductance of the actual inductor. According to (16) and (17) of the 4:1 indirect ReSC converter and (19) and (20) of the 5:1 indirect ReSC converter, it can be noted that when L₂ = 0, only the inductance of the inductor is changed. This does not affect the composition of the state matrix. Through the above analysis, it can be seen that the resonant inductor in the resonator of the indirect ReSC with VCRs of 4:1 and 5:1 can be eliminated under the following condition: the same circuit topology exists in different switching states, and the inductor with the same label number in the topology can be eliminated, as shown in Figure 5.

The retained resonant block generates a resonant current by switching approximately at the resonant frequency; the remaining capacitor in another resonant block after eliminating the resonant inductor is used to stabilize the voltage of the input and output terminals by connecting it to the ground or to another stable voltage source.

Considering practical implications for design in the indirect ReSC topologies, the electrical characteristics are often associated with resonant capacitor values. While Class II ceramic capacitors with large variations are mostly seen in the existing literature, this leads to a deviation between the resonant frequency and the switching frequency during the actual operation, making it impossible to ensure that the circuit operates in the resonant modes. Class I ceramic capacitors are rarely evaluated and used in indirect ReSC topologies. Class I ceramic capacitors use very low dielectric constant and low-loss factor dielectric material to offer very stable capacitance, a low tolerance (less than ±5%), and low ESRs across all operating conditions. They are ideal capacitor candidates for indirect ReSCs that require tight capacitor matching, resonant operation, and high currents. At the same time, in order not to affect the normal operation of the other resonant branch, the capacitor value of the resonant branch with the resonant inductor eliminated should be much greater than that of the resonant capacitor in the other resonant branch.

The state-space average model uses mathematical equations to express the physical meaning of the actual circuit model. Therefore, inductor L₂ in the 4:1 indirect ReSC converter and the 5:1 indirect ReSC converter can be eliminated, thereby simplifying the circuit topology. The circuit topologies of the simplified 4:1 indirect ReSC converter and simplified 5:1 indirect ReSC converter are shown in Figure 6 and Figure 7, respectively.

A comparison of similar modeling methods is shown in

Table 1. Comparison of similar methods.

Model Name	Model Accuracy	Model Complexity	Main Application Areas
State-space average model	Low- frequency accuracy	Low	Circuit topology and simulation analysis
Generalized average model	Medium accuracy	Medium	Simulation and large-signal analysis
Average small-signal model	Low- frequency accuracy	Low	Controller design and stability analysis
Describing function method	Both low- frequency and high- frequency accuracy	High	Loop gain phase delay and subharmonic oscillation

. Compared with other similar methods, the state-space average modeling method has a very low modeling complexity and is suitable for circuit topology analysis.

4. Simulation and Experiment Results

To prove the validity of the analysis and the correctness of the simplified circuit topology, simulation models of the 4:1 indirect ReSC converter and the 5:1 indirect ReSC converter are, respectively, built. The relevant parameters of the simulation and experiment are shown in

Table 2. Parameters used in the simulation.

Description	Item	Value
Input voltage	Vin	48 V/60 V
Output voltage	Vout	12 V
Output current	Iout	30 A
Resistor load	Rload	0.167 Ω
Resonant capacitor	Cr	2.82 μF
Tank capacitor	Ct	100 μF
Resonant inductor	Lr	100 nH
Resonant frequency	fr	300 kHz
Switching frequency	fs	300 kHz
Deadtime	td	40 ns

.

The simulation results of the simplified 4:1 and 5:1 indirect ReSC converter are shown in Figure 8 and Figure 9, respectively. The simplified topologies can achieve resonant operating modes and realize voltage conversions of 48 V–12 V and 60 V–12 V according to the VCR. The voltage V_ds that each switching device withstands is also the same as that analyzed in the operation principle.

For the simplified 4:1 indirect ReSC converter, except for S₂ and S₃, the V_ds voltage is equal to twice the output voltage, and the V_ds voltage of the remaining switching devices is equal to the output voltage. And for the 5:1 indirect ReSC converter, except for S₂, S₃, and S₄, the V_ds voltage is equal to twice the output voltage, and the V_ds voltage of the remaining switching devices is equal to the output voltage.

In order to save space, the relevant experimental waveforms of the simplified 4:1 indirect ReSC converter are taken as an example, which are shown in Figure 10. Figure 10a shows the voltage waveforms of the GaN power switches V_ds1 and V_ds4, respectively. Figure 10b shows the waveforms of switching device S₁’s V_ds voltage and resonant inductor current, respectively. Figure 10c shows the transient response of the converter. Figure 10d shows the waveforms of the input voltage and output voltage, respectively. Figure 10e shows the voltage waveforms of GaN power switches V_ds1, V_ds2, and V_ds4, and Figure 10f shows the voltage and current waveforms of the resonant capacitor C₂.

From experimental results, the system efficiency from the 48 V to 12 V 4:1 simplified indirect ReSC converter in different load conditions is summarized in Figure 11. The results show that the 48 V to 12 V simplified indirect ReSC converter can achieve a high peak efficiency of 98.6%. And the bode diagram of the prototype is shown in Figure 12. The crossover frequency of the converter is 13.84 kHz, the phase margin is 58.41°, and the gain margin is 26.62 dB, which means the system has good stability.

5. Conclusions

This paper uses the state-space average modeling method to simplify the topology of indirect ReSC converters after analyzing the operation principle of the 4:1 and 5:1 indirect ReSC converters. After obtaining the state-space matrix and analyzing the matrix composition of the 4:1 and 5:1 indirect ReSC converters, the simplified topology of the 4:1 and 5:1 indirect ReSC converters can be obtained by eliminating the specific resonant inductors. The simplified topologies are simulated and experimentally verified. The results prove the correctness of the state-space average modeling method and the effectiveness of the simplified topologies. However, the key focus of this paper is the topological simplification, and a steady-state analysis and a dynamic analysis of the simplified circuits have not been carried out. In future work, we will conduct small-signal modeling of the indirect ReSC converter for stability analysis based on the content of this paper to make the research more complete.