A Novel Modular Sigma DC/DC Converter with a Wide Input Voltage Range

Abstract

A modular Sigma DC/DC converter with wide input voltage range is proposed in this paper. The proposed converter is combined with a traditional LLC converter and two multi-resonant converters via Sigma architecture. Among them, the traditional LLC converter operates as a DC transformer (DCX) at resonant frequency to achieve maximum efficiency. Meanwhile, one of the multi-resonant converters, the DC to DC (D2D) part of the Sigma structure, is responsible for voltage regulation over a wide input voltage range. In addition, the other multi-resonant converter has two operation modes, including DCX and D2D. When it operates in the DCX mode, the Sigma converter has better performance and higher efficiency. When it operates in the D2D mode, the Sigma converter can handle a wider input voltage range. For each submodule, the operation principle and characteristics of the proposed Sigma converter are analyzed in detail. In addition, some key points of the parameter design for each part are also demonstrated. Finally, an experimental prototype with an input voltage of 540~1100 V is built to verify the effectiveness of the proposed converter.

1. Introduction

With the increase of photovoltaic generation capacity and electric vehicle load capacity, higher DC bus voltage level is considered to be a promising solution to reduce body weight and improve system efficiency [1]. Photovoltaic panels in series connection are often accompanied by the requirement for DC/DC converters with a wide input voltage range [2]. Furthermore, in new electrical vehicle applications, the on-board DC/DC converter is responsible for converting the high voltage of the power battery (generally 200~750 V) to the low voltage of 24 V or 48 V; so, the on-board DC/DC converters still need to have the ability to regulate the output voltage with a wide input voltage range [3,4].

As an important part of energy conversion, a DC/DC converter is an indispensable electronic device that is widely used in various applications. Unfortunately, there are many factors limiting the use of DC/DC converters in high-voltage applications, and one of them is power devices [5]. It is difficult for the rated voltage of a single power device to meet the high voltage requirement. Meanwhile, it is not feasible to connect the power devices in series to adapt to higher voltages because it is difficult to achieve voltage equalization and drive synchronization [6].

Input-series output-parallel (ISOP) converters have been proposed to overcome the limitations of DC/DC converters in high-voltage applications [7,8,9,10]. The voltage stress of power devices can be reduced with input terminals arranged in series connection. The submodules of an ISOP converter mainly include two types: one is a dual-active bridge (DAB) converter [7,8], and the other is a resonant converter [9,10]. However, these submodules of the ISOP converter have the same voltage and current-sharing control, which may lead to low efficiency under light-load and voltage regulation conditions.

The concept of the Sigma converter, as a type of special ISOP converter, was proposed by Dr. Xu [11]. The Sigma converter is composed of an unregulated converter and a regulated converter. The unregulated converter acts as a DC transformer (DCX), and the regulated converter acts as a DC/DC (D2D) converter, as shown in Figure 1. Based on this concept, which is different from the normal ISOP structure with an identical submodule, a hybrid ISOP converter consists of a DAB converter as a D2D converter and several series resonant converters (SRCs) as a DCX [12]. An LLC converter can also be used as a DCX to improve system performance [13,14].

In order to adapt to wide input voltage range applications, a series of measures are implemented on LLC-based resonant DC/DC converters. A variable frequency with phase-shift modulation has been proposed for wide input voltage range applications in [15], and a variable frequency with a duty-cycle-modulated control method has been used to improve the LLC light load efficiency for a wide input voltage range in [16]. Furthermore, switching between full-bridge and half-bridge is achieved by controlling the opening and closing of the switch tubes, thereby expanding the voltage gain [17]. Similarly, a dual-bridge LLC resonant converter is also proposed for wide input voltage range applications in [18]. Another transformer winding can also be introduced to change the system structure to adapt to the large voltage gain. The new transformer winding can be integrated into a single LLC with an LC antiresonant circuit in a parallel branch [19], as well as with the previous winding as two split resonant branches for wide-input-range applications [20]. In addition to the above measures, changing the structure of the resonant tank is also a feasible method. A notch filter can be introduced into the resonant tank to make the voltage gain decrease rapidly with an increase in frequency so as to achieve wider gain regulation [21,22,23]. The results show that these multi-resonant converters cannot only broaden the voltage gain but also transmit the fundamental and third harmonics so as to improve the efficiency.

Based on the above analysis, a modular Sigma ISOP DC/DC converter for high and wide input voltage applications is proposed in this paper. The Sigma converter consists of a traditional LLC converter as a DCX and two novel multi-resonant converters as D2D. The DCX is designed to operate at its resonant frequency to deliver as much power as possible, and the D2D is designed to regulate the output voltage with a wide input voltage range. Moreover, in order to improve system performance, one of the D2D converters can change its operation mode from D2D to DCX based on the input voltage conditions.

The rest of this paper is organized as follows. Section 2 introduces the topology of the proposed Sigma converter and analyzes the characteristics of each submodule. The operation principle of the proposed Sigma DC/DC converter is analyzed in Section 3. Some key points of the parameter design are given in Section 4. Section 5 shows the experimental results. Finally, Section 6 concludes this paper.

2. Topology of Proposed Modular Sigma DC/DC Converter

The topology of the proposed Sigma converter is shown in Figure 2. The Sigma converter combines two multi-resonant converters with an LLC–DCX converter through an input-series output-parallel (ISOP) structure. Among them, the two multi-resonant converters share the same parameters. Figure 2a shows the overall structure. The input voltage of the Sigma converter (V_in) includes three parts, i.e., V_inL, V_inM1, and V_inM2. They are the input voltages of the LLC–DCX converter, multi-resonant converter #1, and multi-resonant converter #2, respectively. Moreover, I_in, I_o, and V_o are the input current, output current, and output voltage, respectively, of the Sigma converter.

2.1. LLC–DCX Converter

Figure 2b shows an LLC–DCX converter whose operating frequency is fixed at a resonant frequency to increase system efficiency. Thus, the LLC–DCX converter is equivalent to an unregulated DC transformer. Power switches S₁~S₄ are driven by switching signals with 50% duty cycle. D_L1~D_L4 are diodes in the rectifier circuit, and C_oL is an output rectifier capacitor. L_rL, L_mL,, and C_rL are the resonant inductor, magnetizing inductor, and resonant capacitor, respectively. In addition, i_LrL, i_LmL,, and i_sL represent the resonant current, magnetizing inductor current, and secondary side current, respectively. n_L is turns ratio of transformer T_L.

Since the LLC converter operates in the DCX mode, it is assumed that the relation between V_inL and V_o can be expressed as

$$V_{inL}=n_L{V}_o$$

(1)

2.2. Multi-Resonant Converter

In addition to LLC–DCX, there is also the D2D part in the Sigma structure to regulate voltage with a wide input voltage range, so a type of wide-input-voltage multi-resonant converter is introduced as the D2D converter, as shown in Figure 2c. Different from traditional LLC converters, a notch filter, composed of C_p, L_p, C_r, and C_r, is introduced, and the excitation branch is changed into L_m and C_m in series.

The topology of the multi-resonant converter can be simplified, as shown in Figure 3, where R_eq is the load resistance reflected to the primary side of the transformer. It can be expressed as R_eq = 8n²R_o/π² [24], where R_o is the load resistance.

The resonant frequency f_r1 and f_r2 and notch frequency f_p can be calculated as follows:

$$f_{r1}=\sqrt{\frac{h+p+1-\sqrt{{\left(h+p+1\right)}^2-4ph}}{2ph}}{f}_{r0}=\eta f_{r0}$$

(2)

$$f_{r2}=\sqrt{\frac{h+p+1+\sqrt{{\left(h+p+1\right)}^2-4ph}}{2ph}}{f}_{r0}$$

(3)

$$f_p=\sqrt{\frac{h+1}{ph}}{f}_{r0}$$

(4)

where p, h, η, and f_r0 are defined as

$$p=\frac{L_p}{L_r},h=\frac{C_p}{C_r},f_{r0}=\frac{1}{2\pi \sqrt{L_r{C}_r}}$$

(5)

$$\eta =\sqrt{\frac{h+p+1-\sqrt{{\left(h+p+1\right)}^2-4ph}}{2ph}}$$

(6)

The multi-resonant converter can operate in the under-resonance mode (f_s < f_r1), over-resonance mode (f_s > f_r1), and resonant point mode (f_s = f_r1).

Based on Figure 2c, the voltage gain M of the multi-resonant converter can be derived as follows:

$$M=\frac{n_M{V}_o}{V_{\mathrm{inM}}}=\frac{1}{\sqrt{A^2+{\left(B+1\right)}^2}}$$

(7)

where A and B in (7) are defined as

$$A=-Q\left[\eta f_n-\frac{1}{h\eta f_n}\left(1+\frac{1}{ph{\eta }^2{f}_n^2-h-1}\right)\right]$$

(8)

$$B=\frac{q}{h}\left[\frac{h{\eta }^2{f}_n^2-1}{kq{\eta }^2{f}_n^2-1}-\frac{1}{\left(ph{\eta }^2{f}_n^2-h-1\right)\left(kq{\eta }^2{f}_n^2-1\right)}\right]$$

(9)

In (8) and (9), k, q, Q and f_n are expressed as

$$k=\frac{L_m}{L_r},q=\frac{C_m}{C_r},Q=\frac{Z_0}{R_{eq}}=\frac{\sqrt{L_r/C_r}}{R_{eq}},f_n=\frac{f_s}{f_{r1}}$$

(10)

According to (7)–(9), the gain curves of the multi-resonant converter can be drawn as in Figure 4. As shown in Figure 4, due to the introduction of the notch filter, the gain curves drop rapidly when the switching frequency is close to the notch frequency fp. In addition, the improvement in the excitation branch brings higher voltage gain in the under-resonance area compared to traditional LLC converters. Therefore, the multi-resonant converter has a good capacity to maintain the output voltage over a wide input voltage range within a narrow frequency range.

3. Operating Principle of Sigma Converter

As mentioned above, the LLC converter works in the DCX mode to enhance overall efficiency, and the multi-resonant converter can handle a wide input voltage range. However, when just one multi-resonant converter is insufficient for the input voltage range regulation, the other one should be changed to a D2D converter to provide assistance. Thus, the Sigma converter in Figure 1 can operate in two different modes in response to different input voltage ranges as follows.

(1) Mode 1 (DCX + DCX + D2D): when multi-resonant converter #2 can handle the input voltage range itself, multi-resonant converter #1 can just operate in the DCX mode to help enhance the overall efficiency.

(2) Mode 2 (DCX + D2D + D2D): when multi-resonant converter #2 cannot handle the input voltage range itself, multi-resonant converter #1 should change its operation mode from DCX to D2D.

Normally, the LLC resonant converter and its variants adopt pulse frequency modulation (PFM). Figure 5 shows a uniform controller of the Sigma converter.

Due to the two multi-resonant converters sharing the same parameters, they have the same rated input voltage V_M. Assuming the input voltage range of the two multi-resonant converters is within (V_M1~V_M2) and the rated input voltage of the LLC–DCX converter is V_L, the rated input voltage of the Sigma converter can be expressed as

$$V_{innom}=V_L+2V_M$$

(11)

3.1. Mode 1

Mode 1 (DCX + DCX + D2D): In this mode, multi-resonant converter #1 operates in the DCX mode to help enhance the overall efficiency, and multi-resonant converter #2 operates as a D2D converter to handle wide input voltage.

At this time, the input voltage of LLC–DCX and multi-resonant converter #1 maintain their own rated voltages, and multi-resonant converter #2 operates between V_M1 and V_M2. Thus, the minimum and maximum input voltages of the Sigma converter can be calculated as follows:

$$V_{1min}=V_L+V_M+V_{M1}$$

(12)

$$V_{1max}=V_L+V_M+V_{M2}$$

(13)

When the overall input voltage changes between the above ranges shown in (12) and (13), the LLC–DCX converter and multi-resonant converter #1 are open-loop control, operating at their own fixed resonant frequencies. Whereas, multi-resonant converter #2 is closed-loop control, and a proportional–integral controller is adopted to regulate the output voltage.

3.2. Mode 2

Mode 2 (DCX + D2D + D2D): When multi-resonant converter #2 cannot handle the input voltage range itself, multi-resonant converter #1 should operate as a D2D converter.

At this time, the input voltage of LLC–DCX maintains its own rated voltage, and multi-resonant converters #1 and #2 both operate between V_M1 and V_M2. Thus, the minimum and maximum input voltages of the Sigma converter can be calculated as

$$V_{2min}=V_L+2V_{M1}$$

(14)

$$V_{2max}=V_L+2V_{M2}$$

(15)

Similar to the control method of mode 1, when the input voltage of the Sigma converter changes between the voltage ranges in (14) and (15) in mode 2, the LLC–DCX converter adopts open-loop control. Whereas, multi-resonant converters #1 and #2 both adopt closed-loop droop control to regulate the output voltage.

3.3. Relationship between Each Modular Input Voltage and Total Input Voltage in Different Modes

Based on the previous analysis, the relationship between the input voltages of the three submodules and the input voltage of the Sigma converter can be summarized as shown in Figure 6.

For each submodule, the input voltage is relevant to V_in, and the relationship curves are as shown in Figure 7. In this case, the LLC–DCX converter always operates at the resonant frequency, and its input voltage V_inL is always constant. The input voltages V_inM1 and V_inM2 of multi-resonant converter #1 and multi-resonant converter #2 vary with the input voltage of the Sigma converter and operate in different modes.

4. Key Parameter Design

4.1. Resonant Inductance of LLC–DCX Converter

The LLC–DCX converter works at the quasi-resonant point with the best efficiency, which means that the gain maintains at 1 and has no relationship with the load. But, in reality, as the operating conditions, such as voltage, current, temperature change, and aging effects, the values of the resonant components will also change, which means that the highest efficiency will be lost.

Assuming that the tolerance of the resonant components will cause a 10% change in the resonant frequency, the worst-case scenario is the normalized switching frequency f_nL being equal to 0.9 and 1.1. Then, the deviation between the actual voltage gain and the ideal gain is as shown in

Table 1. Deviation of voltage gain.

kL	fnL = 0.9	Error1	fnL = 1.1	Error2
kL = 2	1.132	0.132	0.9196	0.0804
kL = 5	1.048	0.048	0.9664	0.0336
kL = 7	1.033	0.033	0.9751	0.0249
kL = 10	1.023	0.023	0.9826	0.0174
kL = 20	1.011	0.011	0.9909	0.0091
kL = 30	1.007	0.007	0.9935	0.0065

.

Here, k_L and f_nL are the inductance ratio and normalized switching frequency, respectively. They can be expressed as

$$k_L=\frac{L_{mL}}{L_{rL}},f_{nL}=\frac{f_{sL}}{f_{rL}}$$

(16)

where f_sL and f_rL are the switching frequency and resonant frequency, respectively, of the LLC–DCX converter.

It can be seen in Table 1 that the inductance ratio k_L increases with the deviation decreases. It means that even if the value of the resonant components changes with k_L, the voltage gain M_L is still close to the ideal value 1. Therefore, an effective method to reduce the impact of changes in resonant components is to select a high k_L during parameter design. The curves of voltage gain M_L changing with k_L are shown in Figure 8. It can be seen that the larger the inductance ratio k_L, the flatter the curve. Due to DCX operation, the converter can choose a large inductance ratio. But, if k_L is excessively large, the reduction in resonant inductance will bring an increase in the resonant capacitor, leading to a large volume and low power density. Taking all these factors into consideration, k_L = 20 is selected in this paper.

4.2. Design of k, q, and Q of Multi-Resonant Converter

Reasonable parameter design contributes to higher efficiency and power density in converters. The design of k, q, and Q of the multi-resonant converter should consider three aspects, i.e., voltage gain, input impedance, and efficiency.

The input equivalent impedance Z_in of the resonant tank can be expressed as

$$\begin{array}{ll}{Z}_{in}=& j{\omega }_s{L}_r+\left(j{\omega }_s{L}_p+\frac{1}{j{\omega }_s{C}_r}\right)\parallel \frac{1}{j{\omega }_s{C}_p}\\ & +R_{eq}\parallel \left(j{\omega }_s{L}_m+\frac{1}{j{\omega }_s{C}_m}\right)\end{array}$$

(17)

where ω_s = 2πf_s is the angular frequency. Based on (17), the imaginary part of Z_in in p.u. value can be calculated as

$$\mathrm{Im}\left(Z_{inu}\right)=f_n\eta +\frac{-\frac{1}{f_{n}h\eta }\left(f_{n}p\eta -\frac{1}{f_n\eta }\right)}{f_{n}p\eta -\frac{1}{f_{n}h\eta }-\frac{1}{f_n\eta }}+\frac{\left(f_{n}k\eta -\frac{1}{f_{n}q\eta }\right)}{1+Q^2{\left(f_{n}k\eta -\frac{1}{f_{n}q\eta }\right)}^2}$$

(18)

Apart from voltage gain M in (7)–(9) and Im(Z_inu) in (18), the RMS value of the resonant current (I_Lr(rms)) and the peak value of the excitation current (I_Lm(max)) can also be affected by k, q, and Q. Meanwhile, I_Lr(rms) and I_Lm(max) have an important impact on system efficiency. The resonant current i_Lr(t) and excitation current i_Lm(t) can be expressed as

$$i_{Lr}(t)=\sqrt{2}{I}_{Lr(rms)}\sin \left({\omega }_{r1}t+\phi \right)+\frac{\sqrt{2}}{3}{I}_{Lr(rms)}\sin \left(3{\omega }_{r1}t+\phi \right)$$

(19)

$$i_{Lm}\left(t\right)=\{\begin{array}{cc}\frac{n_M{V}_o}{L_{eq}}t-\frac{n_M{V}_o}{4L_{eq}{f}_{r1}}& 0\le t<\frac{1}{2f_{r1}}\\ -\frac{n_M{V}_o}{L_{eq}}t+\frac{3}{4}\frac{n_M{V}_o}{L_{eq}{f}_{r1}}& \frac{1}{2f_{r1}}\le t<\frac{1}{f_{r1}}\end{array}$$

(20)

where ω_r1 is the angular frequency and ω_r1 = 2πf_r1; L_eq is the equivalent excitation inductance of the excitation branch. L_eq at the resonant point can be expressed as

$$L_{eq}=L_m-\frac{1}{{\omega }_{r1}^2{C}_m}$$

(21)

Based on (19)~(21), I_Lr(rms) and I_Lm(max) can be calculated as follows:

$$\{\begin{array}{l}{I}_{Lr\left(rms\right)}=\frac{\sqrt{10}n{V}_o}{4R_{eq}}\sqrt{\frac{R_{eq}^2}{32L_{eq}^2{f}_{r1}^2}+\frac{288}{25{\pi }^2}}\\ I_{Lm\left(max\right)}=\frac{n{V}_o}{4L_{eq}{f}_{r1}}\end{array}$$

(22)

The normalized value of I_Lr(rms) and I_Lm(max) with the reference value I_N = n_MV_o/R_eq can be derived as follows:

$$\{\begin{array}{l}{I}_{Lr\left(rms\right)}^{\ast }=\sqrt{\frac{5{\pi }^2{q}^2{\eta }^2}{64Q^2{\left(kq{\eta }^2-1\right)}^2}+\frac{36}{5{\pi }^2}}\\ I_{Lm\left(max\right)}^{\ast }=\frac{\pi q\eta }{2Q\left(kq{\eta }^2-1\right)}\end{array}$$

(23)

(1) Design consideration of k and q

In view of the voltage gain, the curves of M varying with k and q are as shown in Figure 9. It can be seen that the curves almost overlap in the over-resonance area, which means k and q have little effect on M. In the under-resonance area, M becomes larger when k or q is smaller. Thus, k and q should be small so that the multi-resonant converter can meet the voltage gain requirement.

From the perspective of the input impedance, the curves of Im(Z_inu) are as shown in Figure 10. It can be seen that the inductive impedance frequency range gradually decreases as k or q decreases. Because one of the conditions for ZVS is inductive input impedance, k and q should be large enough for the multi-resonant converter to achieve ZVS.

In view of the loss and efficiency, the curves of I*_Lr(rms) and I*_Lm(max) changing with k and q are as shown in Figure 11. On one hand, the larger the value of k, the higher the values of I*_Lr(rms) and I*_Lm(max). On the other hand, the lower the value of q, the higher the values of I*_Lr(rms) and I*_Lm(max). Thus, k should be large, and q should be small so that the system loss can be reduced and the efficiency can be improved.

(2) Design consideration of Q

In view of the voltage gain, the relationship between M and Q is as shown in Figure 12. It can be seen that Q decreases as the peak of M increases. Thus, Q should be small so that the voltage gain requirement can be met.

From the perspective of the input impedance, the curves of Im(Z_inu) are as shown in Figure 13. As the value of Q becomes larger, the frequency range corresponding to the inductive input impedance gradually shrinks, making it harder to realize ZVS. Hence, Q should be small for ZVS.

5. Experimental Results

5.1. Experimental Results of Multi-Resonant Converter

The basic structure and some parameter design considerations of the multi-resonant converter are described in Section 2.2 and Section 4.2, respectively. In order to verify the accuracy of the theoretical analysis and the proposed design methodology, two sets of parameters, k = 7, q = 5.5, Q = 0.5 and k = 6, q = 3, Q = 0.3, are selected for the experiments on the multi-resonant converter. The specifications of the converter and the main parameters are shown in

Table 2. Specifications of multi-resonant converter.

Symbol	Parameter	Value (k = 6, q = 3, Q = 0.3)	Value (k = 7, q = 5.5, Q = 0.5)
Vin	Input voltage	135–415 V	135–415 V
Vo	Output voltage	48 V	48 V
Po	Output power	500 W	500 W
nM	Turn ratio	6:1	6:1
fr1	Resonant frequency	80 kHz	80 kHz
Lr	Resonant inductance	29.4 μH	43 μH
Lp	Resonant inductance	70.6 μH	103.2 μH
Lm	Magnetizing inductance	176.4 μH	301 μH
Cr	Resonant capacitor	37.4 nF	26 nF
Cp	Resonant capacitor	22.4 nF	15.6 nF
Cm	Capacitor	112.2 nF	143 nF

. The effectiveness of the parameter design can be verified by the characteristics of the voltage gain and soft switching of the multi-resonant converter, which are shown in Figure 14.

Figure 14 shows the steady-state experimental waveforms of the multi-resonant converter under different parameter conditions. In Figure 14, V_gs is the gate drive signal of Q₁, V_ds is the drain–source voltage of Q₁, Is is the secondary side current, and V_o is the output voltage. From Figure 14, it can be seen that V_ds is reduced to 0 before V_gs is turned on, indicating that the power switching tube is capable of realizing ZVS. By comparing Figure 14a with Figure 14d, it can be seen that with the quality factor Q and the resonant capacitance ratio, q, remains unchanged; when the input voltage is constant, the output voltage V_o decreases significantly with the increase in k, i.e., the voltage gain M decreases. Similarly, comparing Figure 14b with Figure 14d, it can be concluded that with the resonant inductance ratio k and resonant capacitance ratio q kept constant, when the input voltage is constant, the output voltage V_o also decreases with the increase in Q, i.e., the voltage gain becomes smaller. Similarly, comparing Figure 14c and Figure 14d, it can be seen that the output voltage V_o decreases slightly with the increase in q when the resonant inductance ratio k, the quality factor Q, and the input voltage are kept constant, and the effect of q on the voltage gain is relatively small. Based on the above comparative analysis, the experimental results are consistent with the theoretical analysis in Section 4.2.

Furthermore, in order to observe how the trend of the voltage gain M changes with the frequency f_n increases, a set of experiments have been performed accordingly. The voltage gain M within the normal operation frequencies f_n for k = 6, q = 3 is shown in

Table 3. Voltage gain variation with frequency for k = 6, q = 3.

	Value	Value	Value	Value	Value	Value	Value	Value
fn	0.82	0.85	0.9	0.925	0.95	0.975	1	1.038
M	1.32	1.16	1.04	0.98	0.96	0.93	0.91	0.88
fn	1.0625	1.125	1.1875	1.25	1.3	1.37	1.45	1.5
M	0.868	0.85	0.81	0.8	0.78	0.74	0.68	0.67

. It can be seen that the voltage gain M changes with the increase in frequency fn. We combine the experimental results at different frequencies to make a comparison with the theoretical analysis. The specific comparison results are shown in Figure 15.

It can be seen from Figure 15 that the experimental results are basically consistent with the theoretical analysis, which shows that the theory given in Section 4.2 can provide a good prediction of the practical design of the proposed converter.

5.2. Experimental Results of Sigma Converter

A 2 kW experimental prototype of the proposed Sigma converter is built with 540~1100 V input and 48 V output voltages, as shown in Figure 16. The detailed specifications are listed in

Table 4. Specifications of prototype.

Symbol	Parameter	Value
Vin	Input voltage	540~1100 V
Vo	Output voltage	48 V
Po	Output power	1500 W
nM	Turns ratio	26:5
nL	Turns ratio	22:4
fr1	Resonant frequency	80 kHz
frL	Resonant frequency	150 kHz
LrL	Resonant inductance	13 μH
LmL	Magnetizing inductance	260 μH
CrL	Resonant capacitor	89.8 nF
Lr	Resonant inductance	29.4 μH
Lp	Resonant inductance	70.6 μH
Lm	Magnetizing inductance	176.4 μH
Cr	Resonant capacitor	37.4 nF
Cp	Resonant capacitor	22.4 nF
Cm	Capacitor	112.2 nF

. TMS320F28377D is chosen as the digital signal processor (DSP) for system control.

Figure 17 shows the steady-state experimental waveforms of the proposed Sigma converter when the input voltage is 540 V. From the previous theoretical analysis, it can be concluded that when the input voltage is between 540 V and 645 V, multi-resonant converter #1 operates in the D2D mode, i.e., the Sigma converter operates in mode 2. At this point, the input voltage of the two multi-resonant converters is 1/2 × (V_in − V_inL). As can be seen in Figure 17, the input voltage of the LLC–DCX converter is 270 V, and each of the two multi-resonant converters is approximately 135 V, which is consistent with the theoretical analysis.

Figure 18 shows the steady-state experimental waveforms when the input voltage is 750 V. When the input voltage is between 645 V and 925 V, multi-resonant converter #1 operates in the DCX mode, i.e., the Sigma converter operates in mode 1. As can be seen in Figure 18, the input voltage of the LLC–DCX converter is 270 V, and the input voltage of multi-resonant converter #1 is approximately 235 V; thus multi-resonant converter #2 is approximately 245 V, which is consistent with the theoretical analysis.

Figure 19 shows the steady-state experimental waveforms when the input voltage is 1100 V. When the input voltage is between 925 and 1100 V, multi-resonant converter #1 operates in the D2D mode, i.e., the Sigma converter operates in mode 2. At this point, the input voltage of the two multi-resonant converters is 1/2 × (V_in − V_inL). As can be seen in Figure 19, the input voltage of the LLC–DCX converter is 270 V, and that of the two multi-resonant converters is approximately 415 V, which is consistent with the theoretical analysis.

6. Conclusions

In order to achieve flexible voltage regulation over a high and wide input voltage range, a modular Sigma DC/DC converter composed of an LLC–DCX converter and multi-resonant converter is proposed in this paper. Among them, the LLC–DCX converter operates at the resonant frequency to ensure high efficiency, and one of the multi-resonant converters works as a D2D converter to regulate the voltage. In addition, if the input voltage is too wide, another converter operates as a D2D converter; otherwise, it operates in the DCX mode to enhance the overall efficiency. The key considerations of each submodular and system operation modes are analyzed in detail. Finally, an experimental prototype is built, and the experimental results show that the proposed Sigma converter can maintain a constant output voltage over a wide input voltage range of 540 V to 1100 V. Moreover, the proposed topology and control method can also be extended to more modular and higher-voltage applications.