https://doi.org/10.6113/JPE.2018.18.2.332

ISSN(Print): 1598-2092 / ISSN(Online): 2093-4718

A Parameter Selection Method for Multi-Element Resonant Converters with a Resonant Zero Point

Yifeng Wang^*, Liang Yang^†, Guodong Li^**, and Shijie Tu^***

^†,*Key Laboratory of Smart Grid of the Ministry of Education, Tianjin University, Tianjin, China

^**State Grid Tianjin Electric Power Company, Tianjin, China

^***Repair Branch, State Grid Jiangxi Electric Power Company, Jiangxi, China

Abstract

This paper proposes a parameter design method for multi-element resonant converters (MERCs) with a unique resonant zero point (RZP). This method is mainly composed of four steps. These steps include program filtration, loss comparison, 3D figure fine-tuning and priority compromise. It features easy implementation, effectiveness and universal applicability for almost all of the existing RZP-MERCs. Meanwhile, other design methods are always exclusive for a specific topology. In addition, a novel dual-CTL converter is also proposed here. It belongs to the RZP-MERC family and is designed in detail to explain the process of parameter selection. The performance of the proposed method is verified experimentally on a 500W prototype. The obtained results indicate that with the selected parameters, an extensive dc voltage gain is obtained. It also possesses over-current protection and minimal switching loss. The designed converter achieves high efficiencies among wide load ranges, and the peak efficiency reaches 96.9%.

Key words: High efficiency, Multi-element resonant converter, Parameter selection, Resonant zero point, Soft-switching

Manuscript received May 27, 2017; accepted Oct. 15, 2017

Recommended for publication by Associate Editor Yan Xing.

^†Corresponding Author: zyangliang@tju.edu.cn Tel: +86-022-27406033, Tianjin University

*Key Lab. Smart Grid of Ministry of Education, Tianjin Univ., China

**State Grid Tianjin Electric Power Company, China

***Repair Branch, State Grid Jiangxi Electric Power Company, China

Ⅰ. INTRODUCTION

Due to advantages such as high efficiency, low EMI, high power density, the resonant power converters (RPCs) are popular among various applications including the electric vehicle chargers, renewable energy generation and switched mode power supplies [1], [2]. Presently, a large number of studies have been carried out that focus on two-element or three-element RPCs. Among them, the LLC converter, as one of the most advanced three-element RPCs, has been developed in depth. The circuit modelling [3], [4], parameter design [5]-[9], topology morphing [10]-[14], high-frequency performance [15]-[17] and control strategy [18]-[20] have all been investigated. LLC-RPCs harvest higher efficiency and broaden voltage ranges. However, the inherent contradiction between the voltage range and the efficiency is detrimental to the performance of LLC converters.

Fortunately, this problem can be addressed by using MERCs. Generally, a MERC has at least four passive components in the resonant tank. The multi-element characteristic ensures that MERCs have more resonant points than two-element or three-element RPCs. Therefore, this kind of converter exhibits diverse resonant features in different operating frequency ranges. It also shows advantages over the conventional RPCs. Several studies have already conducted research on MERCs, and they successfully improved the conversion performance when compared with traditional RPCs. Nevertheless, due to the complexity of the multiple elements, the parameter selection issue has become an obstacle to the fabrication of MERCs.

Dual-transformer MERCs were discussed in [21]-[27]. In [21]-[23], an auxiliary transformer, often controlled by auxiliary switches, was introduced into a resonant tank. This transformer does not function at the rated state. As result, the MERC operates as a normal LLC converter. Meanwhile, in certain situations, the auxiliary switches enliven the auxiliary transformer to realize different purposes, such as restrained conduction losses, wide voltage ranges and high dc voltage gain. By the same token, the dual-transformer converters in [24], [25] employ two resonant tanks and have successfully obtained des  irable features such as low THD and current balancing. However, although nearly all of the resonant parameters are selected properly for these MERCs, the relevant design processes are more like those of traditional LLC converters. Hence, they show less instructiveness for MERCs.

Four-element RPCs are another type of MERC and they show considerable merits [28]-[31]. A CLLC converter, reported in [28], achieves a wide voltage range as well as a high efficiency with a peak value of over 98%. It employs a symmetrical structure, where two resonant capacitors and two resonant inductors are pre-set identically and the turns ratio of the transformer is set at 1. This contributes to the simplicity of the parameter design, and results in an attractive symmetry for bidirectional operation. However, the design process also loses representativeness for general MERCs. In [29], a CLCL circuit is designed for a LED driver, in which the switching losses are minimized and high efficiency is guaranteed. The parameters are selected using a simple and effective method through 3D figures. However, the parameter boundaries of the 3D figures are given with no clear motivation. LCLC and LLCC topologies are discussed in [30], [31] for current balancing and parasitic effect elimination. However, some of the parameters are calculated in an intricate mathematic way, while some parameters are given for no obvious reason.

For five-element and higher order RPCs, although more resonant points help the converter in attaining the desirable multi-function characteristic, the difficulty of parameter selection increases progressively. In [32], a five-element LCLCL converter was reported, where its dc voltage gain curve contains four resonant frequency points. Through properly arranging these resonant points, this MERC successfully extends the voltage range, restricts the circulating energy and achieves a desirable over-current protection capability.

The dominant reason that the LCLCL converter is able to possess outstanding performances, can be attributed to its unique RZP. At the RZP, the dc voltage gain is kept zero regardless of load variations. A RZP allows the RPC to modulate the voltage gain flexibly from zero to its maximum value within a narrow frequency range. Consequently, wide voltage ranges can be easily realized. Another benefit from the zero-gain feature is that a RZP-MERC is capable of operating at the RZP for output current restriction, which shows the great merit of the current safeguard. In spite of the many advantages of the RZP-MERC [32]-[35], this kind of converter also suffers due to the intricacy of its parameter selection. In addition, most of the existing RZP-MERCs fail to provide a distinct design process. Some of the selection methods are too specific and not available throughout. With these considerations in mind, a simple, effective and widely applicable parameter design method is of great importance for the RZP-MERCs.

In this paper, a parameter selection method is proposed for RZP-MERCs. A MATLAB-aided program, comprehensive consideration of the conduction losses and turn-off losses, 3D figures selection, and the priorities of the resonant characteristic variables (RCVs) are combined to provide a simple and reliable parameter design method. To facilitate comprehension, a dual-CTL converter with a RZP is taken for an example to explain the proposed method in detail. In addition, the generation mechanism of the RZP is also analyzed. At last, based on a 500W prototype, the designed resonant parameters are tested by experiments. The obtained results demonstrate that the converter achieves a high efficiency, wide voltage gain range and inherent over-current protection.

Ⅱ. GENERATION MECHANISM OF THE RESONANT ZERO POINT

This section analyzes the mechanism of RZP converters. Generally, these converters are classified into two types as shown in Fig. 1. These two types are the parallel architectures and the serial architectures. The resonant tank for each of the two types is made up of a resonant capacitor C and a resonant inductor L.

Fig. 1. Two main RZP structures. (a) Parallel type. (b) Series type.

(a)

(b)

For the parallel architecture in Fig. 1(a), the expression of the impedance Zpara is derived as (1), where vpara and ipara are defined as the relevant voltage and current.

                    (1)

Substitute s = j·ω_s = j·2πf_s into (1), where ω_s and f_s are the operating angular frequency and operating frequency of the converter. Thus, (1) is expressed as:

    (2)

From (2), if f_s reaches a certain frequency as presented in (3), conceptionally, Z_para tends to infinite. Thus, the parallel architecture is analogous to the “open circuit” state. The corresponding voltage gain is reduced to and stays at almost zero regardless of load variations. In addition, the converter is operated at the RZP.

       (3)

The impedance Z_seri of the serial structure in Fig. 1(b) is calculated as:

     (4)

Similarly, when (3) is reached, Z_seri is equal to zero, which means the serial architecture exhibits the “short circuit” characteristic. Consequently, the output voltage is also restricted at zero.

Two RZP-MERCs have been reported in previous studies [24,25], [28] as shown in Fig. 2. This indicates that the RZP-MERC topologies also belong to the parallel or serial architectures as shown in Fig. 1. Hence, each of them possesses a RZP in their voltage gain curves, which provides the benefits of over-current protection and a wide voltage gain. However, the parameter design issue is a big problem for this kind of RPC, since almost all of them have more than three resonant elements. The lack of a universal and practical parameter selection method makes the RZP-MERCs hard to be applied.

Fig. 2. RZP-MERC topologies: (a) topology 1; (b) topology 2.

(a)

(b)

Ⅲ. PARAMETER SELECTION METHOD

A design block diagram of the proposed parameter selection method is shown in Fig. 3. This method features simplicity, effectiveness and ease of implantation. As a result, it is suitable for RZP-MERCs. The general design procedure includes the following steps:

1) Enter the rating dc input voltage V_in, the output voltage V_out and the operating frequency f_sr.

2) Establish a first harmonic approximation (FHA) model, and calculate expressions of the dc voltage gain M_gain, the resonant points and the RZP.

3) Confirm the initial ranges and steps of the resonant parameters that need to be selected, and utilize MATLAB to filter these parameters.

4) Consider the comprehensive conduction losses and turn-off losses, and confirm the reasonable parameters within narrow ranges.

5) Use 3D figures to fine-tune the parameters and determine the optimal parameter range for each separate RCV.

6) Set priority for all of the RCVs and compromise to determine the chosen resonant parameters.

Use simulation or experimental results to verify if the design results meet the desired requirements. If it does not, then go to Step 4 and re-select the parameter.

For RZP-MERCs with different architectures, this method is applicable by simply changing the selection constraints of MATLAB. Hence, this method is both effective and practical, and can be applied broadly.

Fig. 3. Block diagram of the proposed parameter design method.

To facilitate the explanation of the parameter design method, a dual-CTL RZP converter is taken as an example in the following sections.

Ⅳ. DESIGN EXAMPLE

The topology of the proposed dual-CTL RZP converter is presented in Fig. 4. The resonant tank contains two resonant capacitors C₁ and C₂, two resonant inductor L₁ and L₂, and two high frequency transformers T₁ and T₂. The parallel RZP structure, including C₂, L₂ and T₂, ensures that this topology has a unique RZP f₀. In addition, the leakage inductors of T₁ and T₂ are integrated into L₁ and L₂, respectively. As a result, the side effects caused by parasites are also weakened.

Fig. 4. Topology of the proposed CLTCL converter.

A. Circuit Modelling

A FHA model of the dual-CLT converter is constructed as shown in Fig. 5. E_i and E_o are the fundamental components of the input and output voltages of the resonant tank. I₁, I₂ and I_C2 are the currents through L₁, L₂ and C₂. In addition, I_s1 and I_s2 are the secondary currents of T₁ and T₂. Furthermore, R_eq is the equivalent ac load of the resistor R_o. Meanwhile, V_T1 and V_T2 are the primary voltages of T₁ and T₂. List the KCL / KVL equations as (5), and the dc voltage gain M_gain can be deduced.

Fig. 5. Equivalent FHA circuit.

              (5)

Then, based on s = j·ω_s, where ω_s is the relevant angular frequency of the operating frequency f_s, it is possible to express M_gain as:

       (6)

                    (7)

From (6)-(7), the M_gain curves with different loads are drawn in Fig. 6. This indicates that the dual-CTL converter has two resonant frequency points f_r1 and f_r2 as well as a RZP f₀, at which M_gain is kept constant. The converter should be operated within the frequency scope from f_r1 to f₀, where ZVS turning on is guaranteed for the power switches. This narrow range (f_r1, f₀) also contributes to a broadened voltage gain range.

Fig. 6. Voltage gain curves at different loads.

In addition, from (6), the expressions of the resonant points are calculated as:

          (8)

       (9)

        (10)

B. MATLAB Program Filtration

Based on a FHA model, MATLAB is employed to filter the resonant parameters for the first step. From (6)-(10), total of six parameters needs to be selected, including C₁, C₂, L₁, L₂ and the turns ratios N₁ and N₂ of T₁ and T₂.

Since the secondary sides of T₁ and T₂ are connected in parallel, as shown in Fig. 1, for most of the time, their voltages V_T1 and V_T2 are clamped by the dc output voltage V_out and do not participate in the resonance. Thus, in this paper, the magnetizing inductors L_m1 and L_m2 of T₁ and T₂ are chosen for a comprehensive consideration of the circulating energy and the core volume. Small values of L_m1 and L_m2 increase the transformer losses, while larger values lead to a larger size and a lower power density.

Then, depending on the practical engineering issues, each of these six parameters is given a pre-set wide range. These ranges cover all of the reasonable parameter values. Every variable varies within its given range from the lowest to the highest at a small fixed step. The scopes and steps of these variables are listed in Table I.

TABLE I RANGES AND STEPS FOR THE RESONANT PARAMETERS

Resonant Parameter	Range		Step
Turns Ratios N1	1	8	0.5
Turns Ratios N2	1	8	0.5
Inductor L1	10μH	300μH	10μH
Inductor L2	10μH	300μH	10μH
Capacitor C1	3nF	30nF	3nF
Capacitor C2	3nF	30nF	3nF

Define GRi (i = 1, 2, 3, …) as a parameter group. Every GRi is a combination of the six variables {L1, L2, C1, C2, N1, N2}, and each variable has a certain value. Every GRi is unique and different from the others. Thus, the GRi acts as the fundamental element for the following design procedures.

MATLAB mainly verifies if a certain GR_i is consistent with the selection constraints. Group GR_Si (i = 1, 2, 3, …), which satisfies the constraints, are recorded, while others are automatically abandoned by the program. As a result, all of the satisfactory groups of GR_Si are able to ensure acceptable MERC resonant performances.

The selection constraints for the dual-CTL circuit are listed below.

1) f_r1 < f₀ < f_r2. Based on (8)-(10), the resonant points should to be arranged properly, so that the M_gain curves can match well with Fig. 6. The converter achieves a widely adjustable voltage gain within the narrow frequency range from f_r1 to f₀.

2) 98kHz < f_r1 < 102kHz. The rating point f_sr is set at 100kHz. Because f_sr is supposed to closely approach f_r1 for loss reduction, f_r1 should meet this inequation.

3) 150kHz < f₀ < 180kHz. The location of f₀ must be considered carefully, since it is significant for a RZP-MERC. If f₀ is near f_r1, M_gain can be modulated flexibly with a small frequency scope. In addition, the circulating energy increases as the detrimental outcomes when f₀ is too closed to f_r1. Hence, f₀ is set among (150kHz, 180kHz) for a compromise [24].

4) Rating voltage gain requirement. To meet the demands of the rating voltages, M_gain(f_r1) needs to be designed. Here V_in = 400V, V_out = 52V. Thus, M_gain(f_r1) is located around 0.13.

5) Leakage inductor requirement. For practical engineering considerations, leakage inductors are generally 3% to 5% of their magnetizing inductors [15]. Hence, L₁ and L₂ must be larger than 5% of L_m1 and L_m2, respectively.

A flow chart of MATLAB is shown in Fig. 7. After filtration, the satisfactory groups GR_Si are picked out from the large parameter ranges of Table I to the more limited ranges.

Fig. 7. Flow chart of MATLAB.

C. Conduction Loss and Turn-Off Loss

For every GR_Si, the conduction loss and turn-off loss are investigated in this section to determine a trade-off group GR_S-TO.

The conduction losses are mainly composed of the primary- side loss and the secondary-side loss. For the latter, the square- shape voltage waveform of the SR is exactly in phase with the SR’s sine current waveform. This is because the equivalent topology of the SR is altered when its current is equal to zero and flows to the opposite direction. Therefore, the secondary- side conduction loss is positively proportional to the output current, which is only decided based on the load condition, and has a weak correlation with the resonant parameters. On the other hand, for the primary side, the voltage and current of the resonant tank are not exactly in phase with each other due to the input impedance Z_in. Z_in is decided by the resonant tank. Therefore, the relation between the primary-side conduction loss P_conS and resonant parameters should be analyzed. P_conS is expressed as (11), where T_s is the switching period, r_DS is the conduction resistance of the MOSFET switch and i_S1(t) is the current through S₁.

             (11)

Based on the FHA analysis, when S₁ turns on, a sinusoidal current can be used to substitute i_S1(t) for the approximation in [6]. Hence, (11) is converted to:

      (12)

Where I_1-rms represents the RMS current of L₁. From (12), it can be inferred that a smaller I_1-rms is preferred in order to limit P_conS. I_1-rms and Z_in are deduced as (13)-(15).

      (13)

   (14)

                    (15)

The turn-off loss is dominated by the turn-off current I_turnoff, which is derived as:

           (16)

Where φ_in is the input impedance angle of the resonant tank. φ_in is

   (17)

Equation (16) demonstrates that the turn-off loss presents a positive correlation with φ_in. Thus, to limit the turn-off loss, φ_in should be confined. In this case, φ_in at f_s = 1.1f_r1 is employed to reflect the turn-off loss for the primary-side switches.

Calculate I_1-rms and φ_in for each of the GR_Si. Then, rank the groups, from the group GR_S-LCL with the lowest P_conS to the group with the highest P_conS. Due to a limited article space, only a part of the results is listed in Table II. From this table, the trade-off group GR_S-TO can be achieved, and the corresponding parameters are listed as L₁ =200μH, L₂ = 140μH, C₁ = 6nF, C₂ = 6nF, N₁ = 1.5 and N₂ = 1.5. Although GR_S-TO elevates I_1-rms slightly resulting in a 1% promotion of P_conS, φ_in reduces 37% when compared with GR_S-LCL. This contributes to much lower turn-off losses. Therefore, GR_S-TO is chosen in this step.

TABLE II PARTIAL SELECTION RESULTS OF MATLAB

L1 (μH)	L2 (μH)	C1 (nF)	C2 (nF)	N1	N2	I1-rms (A)	φin
110	70	12nF	12	1.5	1.5	2.8913	0.83
120	90	9nF	12	2	1	2.8916	0.76
250	90	6	9	2	1	2.8921	0.71
50	60	18	15	1.5	1.5	2.8924	0.74
30	90	15	9	1.5	1.5	2.8928	0.78
200	140	6	6	1.5	1.5	2.8930	0.60
210	50	9	18	2	1	2.8933	0.82
20	220	6	3	1.5	1.5	2.8934	0.71
290	90	6	9	1.5	1.5	2.8938	0.69
80	120	9	9	2	1	2.8940	0.75
60	90	12	12	2	1	2.8943	0.70
70	60	15	18	2	1	2.8946	0.89

Since the GR_S-TO has already been decided, the reasonable ranges for the resonant parameters are further confined. Centering on the parameter values of the GR_S-TO, each of the six resonant parameters is limited within the range from the upper step to the lower step. Nevertheless, the large number of parameters makes it very difficult to conduct the following optimization. Hence, for simplification, some of the parameters are determined first and the remaining parameters are further fine-tuned through 3D figures. By this means, although deviations are inevitably introduced, they can be slight and negligible due to the very limited parameter scopes. Here, the parameters of the transformers are preset. At the same time, the ranges for the remaining parameters are obtained as: L_{1 (200μH} ± 10μH), L₂ (140μH ± 10μH), C₁ (6nF± 3nF) and C₂ (6nF± 3nF).

D. 3D Figure Fine-Tuning

The 3D figures are drawn according to the deduced expressions of different RCVs, including φ_in, M_gain and the ac voltage stresses V_C1 and V_C2 of the capacitor C₁ and C₂. These 3D figures reflect the variation trends between the RCVs and the resonant parameters. Since the borders of these figures were determined at the end of Part C, the main objective of this step is to fine-tune the parameters.

From the φ_in aspect, on one hand, φ_in must keep above zero for ZVS operation. On the other hand, φ_in is supposed to be closed enough to zero so that the turn-off current is limited and the primary-side switches achieve quasi-ZCS turning off.

For M_gain, since M_gain(f_r1) is located at around 0.13, the M_gain of the rating point f_sr should be equal to or above 0.13 in consideration of the rating voltages. In addition, the varying slope of M_gain is expected to be mild to weaken the impacts brought by parasites.

Taking the ac voltage stresses into consideration, both V_C1 and V_C2 must be restricted. In addition, from (5), V_C1 and V_C2 are derived as:

           (18)

           (19)

           (20)

         (21)

Before drawing the 3D figures, the evaluation of C₂ is implemented. From equation (10) it can be known that the RZP f₀ is mainly decided by C₂ and L₂. f₀ is of great significance for a RZP-MERC, since it directly influences the voltage gain range and the operating frequency scope. Thus, f₀ should be in accordance with inequation (3) as analyzed above. However, concerning a small value of C₂, a slight variation of C₂ leads to a large deviation of f₀. When C₂ varies by 1nF, a 11% deviation of f₀ occurs. Meanwhile when L₂ changes by 10μH, only a 5.6% deviation is generated. This phenomenon is illustrated in Fig. 8, where M_gain curves are drawn at C₂ = 4.5nF, 3nF. The corresponding RZPs are located at 217kHz and 265kHz, and they both contradict with inequation (3). Under this condition, the proposed converter loses the attractive over-current protection and widely adjustable voltage gain within a narrow frequency band. As a consequence, to avoid a large deviation of f₀, the value of C₂ is fixed at 6nF, which simplifies the following processes.

Fig. 8. RZP positions for different values of C₂: (a) C₂ = 4.5nF; (b) C₂ = 3nF).

(a)

(b)

Fig. 9. Relations of φ_in and the resonant parameters. (a) φ_in versus L₁ and L₂ at different values of C₁. (b) φ_in versus C₁ and L₂ at different values of L₁.

(a)

(b)

The relations between φ_in and the resonant parameters L₁, L₂ and C₁ are shown in Fig. 9. In Fig. 9a, five surfaces corresponding to C₁ from 3nF to 9nF are presented. Since φ_in needs to be above and closed to zero for ZVS operation and low turn-off losses, the demand is only satisfied when C₁ is 6nF, L₂ is from 145μH to 150μH, and L₂ is from 190μH to 200μH. Meanwhile, Fig. 9b illustrates the relation of φ_in versus various values of L₁. φ_in is kept lower than zero unless C₁ is higher than 6nF, and no obvious relation between φ_in and L₁ is found. Thus, in terms of the above analyses, for φ_in alone, C₁ should be set at 6nF, and L₁ and L₂ are expected to be located within (190μH, 200μH) and (145μH, 150μH).

Fig. 10 shows M_gain curves versus L₁, L₂ and C₁. It is clearly shown that M_gain reaches its peak values around 0.13 in the scope from 5nF to 7nF. In addition, there are less clues  indicating the correlations between M_gain and L₁ & L₂. As a result, the C₁ range of 5nF to 7nF is preferred in consideration of M_gain.

From the voltage stress aspect, the V_C1 and V_C2 curves versus L₁, L₂ and C₁ are shown in Fig. 11 and Fig. 12, respectively. V_C1 and V_C2 both reach their peaks during the C₁ range of 5nF to 7nF. For V_C1, it rises along with the augment of L₁, and has weak correlation with variations of L₂. Similarly, V_C2 shows a positive relation with L₂, and it is barely influenced by L₁.

Fig. 10. Relationships of M_gain and the resonant parameters. (a) L_{2 = 130μH}. (b) L₂ = 140μH. (c) L₂ = 150μH.

Fig. 11. Relationships of V_C1 and the resonant parameters. (a) L₂ = 130μH. (b) L₂ = 140μH. (c) L₂ = 150μH.

Fig. 12. Relationships of V_C2 and the resonant parameters. (a) L₂ = 130μH. (b) L₂ = 140μH. (c) L₂ = 150μH.

(a)

(b)

(c)

(a)

(b)

(c)

(a)

(b)

(c)

E. Priorities for RCVs

To conclude the 3D figures, for each RCV, and an optimal parameter range can be found without concerning other RCVs. Unfortunately, it is impossible to meet all of the demands of the RCVs synchronously, since these optimal ranges may contradict with each other. Accordingly, to deal with this problem, the priority for each of the RCVs is taken into account. The RCV with a higher priority ought to be satisfied at first, since it is more important when compared with the other RCVs. The optimal parameter ranges and priorities for all of the RCVs are listed in Table. III.

TABLE III OPTIMIZED PARAMETERS AND PRIORITIES

Objective	Optimized Parameters	Priority
Impedance angle φin	C1 sets at 6nF	1
	L1 is from 190μH to 200μH
	L2 is from 145μH to 150μH
Voltage gain Mgain	C1 is from 5nF to 7nF	2
Voltage Stress VC1	Small L1	3
	C1 is not within (5nF, 7nF)
Voltage Stress VC2	Small L2	4
	C1 is not within (5nF, 7nF)

φ_in has the highest priority since the efficiency issue is of the most importance for dual-CTL converters. Meanwhile, φ_in represents the dominant influence on the ZVS turn-on and the quasi-ZCS turn-off losses for the primary-side switches. M_gain has the second highest priority, since the rating input and output voltages must be set appropriately. The priorities of the voltage stresses are lower since by applying advanced switching devices with higher withstand voltages, high voltage stresses can be tolerated despite the rising costs. V_C1 is much higher than V_C2. Therefore, the former ranks higher. At last, concerning Table III, the final resonant parameters are selected and listed in Table. IV.

TABLE IV LIST OF OPTIMIZED PARAMETERS

Parameter	Value
Magnetic inductor Lm1	300μH
Turns ratio N1	1.5:1
Magnetic inductor Lm2	300μH
Turns ratio N2	1.5:1
Inductor L1	190μH
Inductor L2	145μH
Capacitor C1	6nF
Capacitor C2	6nF
Rated operating frequency fsr	100kHz
Rated load Ro	5.4Ω

Ⅴ. EXPERIMENTS

To verify the practicability of the proposed design method, a 500W dual-CTL prototype was fabricated in the laboratory. Key waveforms and efficiency conditions are tested experimentally.

Fig. 13 presents waveforms under the rating situation, where the voltage v_S1 and current i_S1 of the switch S₁, and the voltage v_SR1 and current i_SR1 of the SR switch SR₁ are given. Fig. 13(a) indicates that with the chosen parameters, the input impedance angle φ_in is confined to a small value above zero. i_S1 lags almost half a switching cycle behind v_S1, which infers the i_S1 is nearly in phase with the drive signal of S₁. In addition, S₁ fulfills the ZVS turning on and the quasi-ZCS turn-off, since i_S1 resonates to almost zero when S₁ is turned off. At the same time, SR₁ maintains its inherent ZCS turn-off, and achieves quasi-ZCS turning on due to the small turn-on current. Accordingly, the switching losses is minimized, which contributes to highly efficient conversions.

Fig. 13. Waveforms at the rating condition: (a) voltages and currents of S₁ and SR₁; (b) voltages of C₁ and C₂.

(a)

(b)

Fig. 14. Waveforms at different operating frequencies: (a) f_s = 110kHz; (b) f_s = 140kHz; (c) f_s ≈ f₀ = 183kHz.

(a)

(b)

(c)

Fig. 15. Comparison of M_gain curves.

Fig. 16. Efficiency curves with different output voltages.

The voltages across C₁ and C₂ are shown in Fig. 13(b) as v_C1 and v_C2. v_C1 is made up of two parts. The two parts are a dc component, which is equal to half of the input voltage V_in, and an ac component, whose amplitude is V_C1. V_C1 is measured as 1106V and calculated as 1080.9V from (19). The amplitude V_C2 of v_C2 is measured as 522V and calculated as 558V. The deviation of V_C2 is caused by the distortion of v_C2.

Waveforms at different operating frequencies are given in Fig. 14, where the input voltage V_in is rated at 400V. In Fig. 14(a) and 14(b), f_s is set at 110kHz and 140kHz, respectively. In addition, the corresponding output voltages are 37.2V and 18.3V. φ_in increases because f_s is away from the rating point f_sr. S₁ loses the quasi-ZCS but still possesses the ZVS turning on. SR₁ maintains the quasi-ZCS turn-on and ZCS turn-off soft-switching.

In Fig. 14(c), with the rating input voltage, the output voltage is only 4.1V at f_s = 183kHz. The voltage gain M_gain is around 0.01. Therefore, this frequency is regarded as the RZP f₀. It ensures that the dual-CTL converter possesses the inherent over-current protection and the widely adjustable voltage gain.

The calculations, simulations and experiment results of the dc voltage gain curves are compared in Fig. 15. It can be clearly seen that the threes curves basically match each other from f_r1 to f₀. When f_s approaches f_sr, these curves almost converge to a single point. Meanwhile, when f_s is near f₀, small deviation occurs. This deviation is caused by the intrinsic error of the FHA analysis and parasitic parameters. Fig. 15 shows that the proposed parameter design method has excellent accuracy.

Efficiency curves with the rating input voltage and different output voltages are given in Fig. 16. Under the rated condition where V_out = 52V, the efficiency is at its highest, and the peak value is 96.9% at 300W. This higher efficiency, when compared with similar high step-down RPCs [16], [18], [20], verifies the effectiveness of the designed dual-CTL converter. For other operating states, the efficiency curves inevitably drop due to deviations from the rating point. Despite of this, the converter still maintains a relatively high efficiencies for a wide voltage gain range.

Ⅵ. CONCLUSIONS

A parameter selection method is proposed for RZP-MERCs in this paper. The main objective of this method is to narrow the acceptable range for all of the variables. Through MATLAB, loss comparisons and 3D figures, the first three steps seek out reasonable parameters within very limited scopes. Then, the last step determines the final resonant parameters by a priority compromise. For different topologies, by altering the selection constraints in the program, it is easy to implant this method on most of the current RZP-MERCs. In addition, a novel dual-CTL MERC is taken as an example to explain the design process. A 500W prototype is established to test the performance. Experiments verify that the proposed converter harvests a wide voltage gain range, limited switching loss, over-current protection and a relatively high efficiency along the entire load range.

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Yifeng Wang was born in Hubei, China, in 1981. He received his B.S., M.S. and Ph.D. degrees in Electrical Engineering from the Harbin Institute of Technology, Harbin, China, in 2005, 2007 and 2011, respectively. Since 2011, he has been an Associate Professor in the Department of Electrical and Electronics Engineering, Tianjin University, Tianjin, China. His current research interests include the high frequency and soft-switching power converters used for special power supplies, EV chargers, residential photovoltaic grid- connected generation systems, distributed smart wind power generation systems, and the application of power conversion technology to hybrid AC/DC microgrids.

Liang Yang was born in Henan, China, in 1989. He received his B.S. degree in Electrical Engineering from Tianjin University, Tianjin, China, in 2013, where he is presently working towards his Ph.D. degree in Electrical Engineering. His current research interests include small-scale wind generation and high frequency planar magnetic based DC-DC converters.

Guodong Li was born in 1978. He received his B.S. degree in Electrical Engineering from the Northeast Electric Power University, Jilin City, China; and his M.S. degrees in Electrical Engineering from Tianjin Uni- versity, Tianjin, China. His current research interests include new energy grid-connected detection technologies.

Shijie Tu was born in Jiangxi, China, in 1991. He received his B.S. degree in Electrical Engineering from Tianjin Uni- versity, Tianjin, China, in 2013. His current research interests include DC-DC converters.