General Analysis of Switching Modes in a Dual Active Bridge with Triple Phase Shift Modulation

Abstract

This paper provides an exhaustive analysis of the Dual-Active-Bridge with Triple-Phase-Shift (DAB-TPS) modulation and other simpler ones, identifying all the possible switching modes to operate the DAB in both power flow directions, and for any input-to-output voltage range and output power. This study shows four cases and seven switching modes for each case when the energy flows in one direction. That means that the DAB operates up to fifty-six different switching modes when the energy flows in both directions. Analytical expressions for the inductor current, the output power, and the boundaries between switching modes are provided for all cases. Additionally, the combination of control variables to achieve Zero-Voltage-Switching (ZVS) or Zero-Current-Switching (ZCS) is provided for each case and switching mode, by showing which switching modes obtain ZVS or ZCS for the whole power range and all switches—independent of the input-to-output voltage ratio. Therefore, the most interesting cases, switching mode and modulation for using the DAB are identified. Additionally, experimental validation has been carried out with a 250 W prototype. This analysis is a proper tool to design the DAB in the optimum switching mode, reducing the RMS current and achieving to increase efficiency and the power density.

1. Introduction

Currently, the Dual-Active-Bridge (DAB) converter is commonly found in different sectors such as in electric vehicles, in which DAB converters are used as battery charges [1,2] or as active balancing systems [3]. The aeronautics industry is betting on improving emissions and reducing fuel consumption by replacing mechanical and pneumatic systems with electrical systems. In Reference [4] is shown a DAB working in harsh environments with high temperature as a component of an electric actuator; Reference [5] shows a DAB as an interface between the battery storage system and DC bus. Additionally, for electric ships [6,7] and smart grids [8,9,10], DAB converters can be seen as an interface component in the medium-voltage grid. DAB converters are also an alternative for electrochemical energy storage as shown in References [11,12].

The conventional DAB topology consists of two active bridges, a high-frequency transformer (T) and a series inductor (L)—Figure 1. The main characteristics of the DAB are bi-directionality, galvanic isolation, high power density, and soft switching in some operating conditions. Additionally, in the state-of-art, a variant of the DAB without transformer can be found, for application in mobile phones and computer chargers [13].

The most basic modulation applied to DAB is the Phase-Shift (PS), also known as Single-Phase-Shift (SPS), where simplicity its main advantage. In this case, it is only necessary to control the phase shift (φ) between the output voltage (v₁₁ and v₂₂) of the bridges, with D₁ = 1 (pulse width of voltage v₁₁) and D₂ = 1 (pulse width of voltage v₂₂)—Figure 2a. However, this kind of modulation has some disadvantages such as reduction of the operating range with Zero-Voltage-Switching (ZVS) or Zero-Current-Switching (ZCS), if the input-to-output voltage ratio moves away from the unity, and high currents at low power. Therefore, PS is not a proper modulation for wide output and input voltage ranges in the converter [14].

Another modulation scheme applied in DAB is the Extended-Phase-Shift modulation (EPS). This modulation operates by using the Phase-Shift (φ) between output voltages of the bridges, as in the PS modulation, along with the pulse width variation of the Bridge 1 output voltage (D₁), being D₂ = 1, Figure 2b. EPS modulation reduces the circulating energy and the conduction losses for medium power, therefore improving the performance compared to the PS modulation [15,16,17,18,19], although with a reduced impact for low power.

Additionally, with two degrees of freedom, the Dual-Phase-Shift modulation is well known (DPS) [20]. This modulation uses, once more, the Phase-Shift (φ) between output voltages of the bridges, as in the EPS modulation, along with the pulse width variation of both Bridges output voltages (D₁, D₂), in this case being D₁ = D₂, Figure 2c.

A more enhanced alternative is Triple-Phase-Shift modulation (TPS), which involves three control variables: The pulse width of the output voltages of Bridge 1 (D₁) and Bridge 2 (D₂), and the phase shift (φ) between both voltage waveforms, Figure 2d. This modulation strategy improves the converter’s performances, with a more significant impact at low power, reduces RMS current, and presents a higher probability of soft switching operation [21,22,23]. However, the complexity increases due to the higher number of parameters to be controlled, which results in a higher number of possible switching modes of the converter [24,25,26,27,28]. There are different combinations of the three control variables that satisfy the same requirements of transferred power between the input and output ports of the converter. However, not all the combinations imply the same performance from switching conditions and circulating currents.

Many works can be found in the state-of-the-art that are focused on the study of the DAB switching modes, ZVS and ZCS operation, or RMS current reduction, among other topics. In References [26,27], the authors identify five switching modes (considering positives mismatches); in Reference [28], the number of switching modes increases to twelve (considering positive and negative mismatches), all of them for condition V₁ > n·V₂. In Reference [29], the authors analyse the charge and discharge of the parasitic MOSFET capacitances to get ZVS. On the other hand, Reference [30] analyses the reduction of the transformer's coupling to achieve the same effect. However, all these works show useful partial solutions, but without doing a general analysis of all operation possibilities.

Therefore, the contribution of this paper is oriented to provide an exhaustive analysis of the different DAB switching modes when TPS, EPS, and PS modulation (DPS can be considered a particular case of TPS) are applied to get the best performance for whole output power range, and considering each V₁ and V₂ ratio (that includes buck and boost modes). Thanks to this in-depth analysis, the best switching modes are identified as well as the most combinations of the modulation variables to guarantee Zero-Voltage-Switching (ZVS) or Zero-Current-Switching (ZCS). This analysis is a tool to design the DAB converter, ensuring the soft-switching operation, and to improve the efficiency and the power density.

This paper organises as follows: Section 2 presents the basic operation of the TPS modulation applied to the DAB. Section 3 defines the Cases of study (based on bridges output voltages and their duty cycles) and the switching modes when TPS modulation is used, along with the inductor current expressions and the transmitted power for each switching mode. Section 4 analyses and calculates the expressions to get soft switching in the converter for each Case and switching mode. Section 5 validates the analysis for Case I and Case II with a 250 W prototype developed in the laboratory. Finally, Section 6 summarises the conclusions of the work carried out.

2. Triple-Phase-Shift Modulation

Triple-Phase-Shift (TPS) consists of shifting the driving signals v_g3, v_g5 and v_g7 with respect to v_g1 (corresponding to the switches M₃, M₅, M₇, and M₁, respectively). By driving the switches in this way, v₁₁ is generated at the output of the Bridge 1 of amplitude V₁, v₂₂ is generated in the primary side of the transformer with an amplitude of V₂/n, and a current i_L flows through the inductor L.

Three control parameters define TPS modulation: D₁ (0 < D₁ ≤ 1) representing the pulse width of voltage v₁₁, D₂ (0 < D₂ ≤ 1) representing the pulse width of v₂₂ and φ (−π < φ < π) measures the phase shift between v₁₁ and v₂₂, as shown in Figure 3.

The switching instants of the voltages v₁₁ (t_1LH and t_1HL) and v₂₂ (t_2LH and t_2HL), shown in Figure 3, are calculated by Equation (1), considering that the positive part of v₁₁ centres in T_sw/4.

$$\begin{array}{c}\begin{array}{c}{\mathrm{t}}_{1\mathrm{LH}}=\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}·\left(\frac{1}{2}-\frac{{\mathrm{D}}_1}{2}\right)\\ {\mathrm{t}}_{1\mathrm{HL}}=\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}·\left(\frac{1}{2}+\frac{{\mathrm{D}}_1}{2}\right)\end{array}\\ \begin{array}{c}{\mathrm{t}}_{2\mathrm{LH}}=\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}·\left(\frac{\mathsf{\phi }}{\mathsf{\pi }}+\frac{1-{\mathrm{D}}_2}{2}\right)\\ {\mathrm{t}}_{2\mathrm{HL}}=\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}·\left(\frac{\mathsf{\phi }}{\mathsf{\pi }}+\frac{1+{\mathrm{D}}_2}{2}\right)\end{array}\end{array}$$

(1)

3. Cases and Switching Modes

The switching modes define according to the profile acquired by the current i_L in each operating state. The current i_L is defined by the input parameters of the converter (V₁, V₂, n, L, f_sw) and by the parameters of the TPS modulation (D₁, D₂ and φ). By considering n, L and f_sw as constants, the voltages v₁₁, v₂₂ and the pulse widths D₁ and D₂, four cases of study can be defined:

Case I: v₁₁ ≥ v₂₂ and D₁ > D₂

Case II: v₁₁ ≥ v₂₂ and D₁ ≤ D₂

Case III: v₁₁ < v₂₂ and D₁ > D₂

Case VI: v₁₁ < v₂₂ and D₁ ≤ D₂.

In Reference [20], the author concluded that the analysis performed for positive φ is equivalent to negative φ; therefore, DAB can operate in eight cases of study. Additionally, for positive φ, it can be observed that there are equivalences between Case I and Case IV, as well as between Case II and Case III; they obtain by exchanging v₁₁ with v₂₂, and D₁ with D₂. It means that only the analysis of two of them is necessary. Therefore, in this paper, the analysis is developed for the Case I and II.

3.1. Switching Modes: Case I and Case II. Boundaries

The total switching modes per Case are seven: SM₁, SM₂, SM₂*, SM₃, SM₃*, SM₄, and SM₅; they can have positive or negative φ angle values (bidirectionality). Therefore, considering bi-directionality, four cases and seven switching modes per case, DAB can operate up to fifty-six switching modes. As aforementioned, Cases I and II are only analysed for positive φ angle values, as shown in Figure 4 and Figure 5, respectively. These switching modes are obtained by increasing the phase shift φ for any value of D₁ and D₂.

The boundaries in each switching mode are obtained when the switching instants of the voltages v₁₁ and v₂₂ occur at the same time. For example, for SM₂ in Figure 4b: t_1HL = t_2HL determines the lower boundary (switching mode from SM₁ to SM₂,) and the upper one when t_1HL = t_2LH (switching mode from SM₂ to SM₃,), as shown in Equation (2).

$$\begin{array}{c}\mathrm{Lower}\mathrm{boundary}:\left(\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }\\ \mathrm{Upper}\mathrm{boundary}:\left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }\end{array}$$

(2)

The switching modes SM₂ and SM₃ are obtained for D₁ < 1 − D₂, whereas SM₂* and SM₃* are obtained for D₁ ≥ 1 − D₂. The switching mode SM₂* is different concerning SM₂ only in the boundaries, whereas SM₃* is different regarding SM₃ in the boundaries, the current profile and the expression of the power, with respect to those in SM₃.

Table 1. Boundaries for switching modes: Case I and Case II.

SMi	Case I	Case II
SM1	$0<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	$0<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM2	$\left(\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	$\left(\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM2*	$\left(\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	$\left(\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM3	$\left(\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	$\left(\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM3*	$\left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	$\left(1-\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM4	$\left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	$\left(1-\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM5	$\left(1-\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }<\mathsf{\pi }$	$\left(1-\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }<\mathsf{\pi }$

summarises the boundaries of all switching modes for Case I and Case II.

From the information in Table 1, the switching modes are plotted in a three-dimensional way depending on the parameters D₁, D₂ and φ, forming a cube with the unity side, as shown in Figure 6. The tetrahedral volumes contain the switching modes obtained with TPS modulation. In Case I: the modes SM₂ (DEFG), SM₂* (CEFG), SM₃* (BCEG) and SM₄ (ABEG), are shown in Figure 6a; and the modes SM₁ (CDEI), SM₃ (ADEG), and SM₅ (ABEH) are shown in Figure 6b. For Case II: the modes SM₂ (DFGJ), SM₂* (CFGJ), SM₃* (BCGJ), and SM₄ (ABGJ) are shown in Figure 6c; while, the modes SM₁ (CDJL), SM₃ (ADGJ), and SM₅ (ABJK), are shown in Figure 6d.

3.2. Current Through the Inductor L

The current i_L in each switching mode is calculated from Figure 4 (Case I) and Figure 5 (Case II), together with Equation (3) for four consecutive switching instants.

$${\mathrm{v}}_{\mathrm{L}}=\mathrm{L}\frac{{\mathrm{di}}_{\mathrm{L}}}{\mathrm{dt}}$$

(3)

As an example, Figure 5a has t_2LH, t_1LH, t_1HL and t_2HL as consecutive switching instants and i_L(t) = −i_L(t + T_sw/2); therefore, i_L(t) must be calculated for half the switching period. Equation (4) shows i_L(t) from t_2LH to t_2HL by applying Equation (3) and the equation systems in Equation (5) are obtained when switching instants are replaced in i_L(t).

Table 2. Inductor current and output power for Case I and Case II.

SMi	Current		Power
	Case I	Case II	Case I	Case II
SM1	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)=-\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}-2·{\mathrm{V}}_1·\left(\mathsf{\phi }/\mathsf{\pi }\right)·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}\right)=\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}+2·{\mathrm{V}}_1·\left(\mathsf{\phi }/\mathsf{\pi }\right)·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}$	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}-2·{\mathrm{V}}_2·\left(\mathsf{\phi }/\mathsf{\pi }\right)-{\mathrm{D}}_1·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)=\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+2·{\mathrm{V}}_2·\left(\mathsf{\phi }/\mathsf{\pi }\right)-{\mathrm{D}}_1·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}\right)=-{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)=\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}$	$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2·{\mathrm{D}}_2·\left(\mathsf{\phi }/\mathsf{\pi }\right)}{2·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}$	$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2·{\mathrm{D}}_1·\left(\mathsf{\phi }/\mathsf{\pi }\right)}{2·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}$
SM2	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)=\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+2·{\mathrm{V}}_2·\left(\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)-{\mathrm{D}}_1·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)=-\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}-2·{\mathrm{V}}_1·\left(\mathsf{\phi }/\mathsf{\pi }\right)·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}$		$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}·\left[\frac{\mathsf{\phi }}{\mathsf{\pi }}·\left({\mathrm{D}}_1+{\mathrm{D}}_2-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)-\frac{{\left({\mathrm{D}}_1-{\mathrm{D}}_2\right)}^2}{4}\right]$
${\mathrm{SM}}_2^{*}$
SM3	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)={\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)=\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}$		$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2·{\mathrm{D}}_1·{\mathrm{D}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}$
${\mathrm{SM}}_3^{*}$	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_1·{\mathrm{V}}_2-2·{\mathrm{V}}_2·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)=\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}-{\mathrm{D}}_1·{\mathrm{V}}_2+2·{\mathrm{V}}_2·\left(\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)=-\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}-{\mathrm{D}}_2·{\mathrm{V}}_2-2·{\mathrm{V}}_1·\mathrm{n}·\left(\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}-\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}\right)=-\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2-2·{\mathrm{V}}_1·\mathrm{n}·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}$		$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2}{2·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}·\left[\frac{\mathsf{\phi }}{\mathsf{\pi }}·\left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)-\frac{{\left({\mathrm{D}}_1-1\right)}^2+{\left({\mathrm{D}}_2-1\right)}^2}{4}\right]$
SM4	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_1·{\mathrm{V}}_2-2·{\mathrm{V}}_2·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)={\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)=\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}-\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}\right)=\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2-2·{\mathrm{V}}_1·\mathrm{n}·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}$		$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}·\left[\left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)·\left({\mathrm{D}}_1+{\mathrm{D}}_2+\frac{\mathsf{\phi }}{\mathsf{\pi }}-1\right)-\frac{{\left({\mathrm{D}}_1-{\mathrm{D}}_2\right)}^2}{4}\right]$
SM5	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}-\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}\right)=-\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2+2·{\mathrm{V}}_1·\mathrm{n}·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}-\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}\right)=\frac{{\mathrm{D}}_2·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2-2·{\mathrm{V}}_1·\mathrm{n}·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}$	$\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)=-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_1·{\mathrm{V}}_2-2·{\mathrm{V}}_2·\left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)=\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_1·{\mathrm{V}}_2+2·{\mathrm{V}}_2·\left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\\ \begin{array}{r}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}-\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}\right)=-{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}-\frac{{\mathrm{T}}_{\mathrm{sw}}}{2}\right)\\ =-\frac{{\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2}{4·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}\end{array}\end{array}$	$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2·{\mathrm{D}}_2·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{2·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}$	$\frac{{\mathrm{V}}_1·{\mathrm{V}}_2·{\mathrm{D}}_1·\left(1-\mathsf{\phi }/\mathsf{\pi }\right)}{2·\mathrm{L}·{\mathrm{f}}_{\mathrm{sw}}·\mathrm{n}}$

, at Case II (column) and SM₁ (row), shows the solution of Equation (5).

$${\mathrm{i}}_{\mathrm{L}}\left(\mathrm{t}\right)=\{\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)-\frac{1}{\mathrm{L}}·\frac{{\mathrm{V}}_2}{\mathrm{n}}·\left(\mathrm{t}-{\mathrm{t}}_{2\mathrm{LH}}\right);{\mathrm{t}}_{2\mathrm{LH}}\le \mathrm{t}<{\mathrm{t}}_{1\mathrm{LH}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)+\frac{1}{\mathrm{L}}·\left({\mathrm{V}}_1-\frac{{\mathrm{V}}_2}{\mathrm{n}}\right)·\left(\mathrm{t}-{\mathrm{t}}_{1\mathrm{LH}}\right);{\mathrm{t}}_{1\mathrm{LH}}\le \mathrm{t}<{\mathrm{t}}_{1\mathrm{HL}}\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)-\frac{1}{\mathrm{L}}·\frac{{\mathrm{V}}_2}{\mathrm{n}}·\left(\mathrm{t}-{\mathrm{t}}_{1\mathrm{HL}}\right);{\mathrm{t}}_{1\mathrm{HL}}\le \mathrm{t}<{\mathrm{t}}_{2\mathrm{HL}}\end{array}$$

(4)

$$\begin{array}{c}\begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)={\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)-\frac{1}{\mathrm{L}}·\frac{{\mathrm{V}}_2}{\mathrm{n}}·\left(\mathrm{t}-{\mathrm{t}}_{2\mathrm{LH}}\right)\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)={\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{LH}}\right)+\frac{1}{\mathrm{L}}·\left({\mathrm{V}}_1-\frac{{\mathrm{V}}_2}{\mathrm{n}}\right)·\left(\mathrm{t}-{\mathrm{t}}_{1\mathrm{LH}}\right)\end{array}\\ \begin{array}{c}{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)={\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{1\mathrm{HL}}\right)-\frac{1}{\mathrm{L}}·\frac{{\mathrm{V}}_2}{\mathrm{n}}·\left(\mathrm{t}-{\mathrm{t}}_{1\mathrm{HL}}\right)\\ {\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{HL}}\right)=-{\mathrm{i}}_{\mathrm{L}}\left({\mathrm{t}}_{2\mathrm{LH}}\right)\end{array}\end{array}$$

(5)

Using the same procedure for each switching mode, Table 2 gathers the current i_L at the switching instant for Case I and Case II.

3.3. Average Power

The input current, i₁(t), is defined from t_1LH to t_1HL when the power is flowing to V₂. Without considering losses, the average power (P = V₁·I₁) is calculated by Equation (6) and i_L at the switching instant (Table 2). This average power is detailed for each switching mode in Table 2.

$$\mathrm{P}={\mathrm{V}}_1·\frac{2}{{\mathrm{T}}_{\mathrm{sw}}}\underset{{\mathrm{t}}_{1\mathrm{LH}}}{\overset{{\mathrm{t}}_{1\mathrm{HL}}}{{\displaystyle \int }}}{\mathrm{i}}_{\mathrm{L}}\left(\mathrm{t}\right)\mathrm{dt}$$

(6)

4. Soft Switching

In general, soft switching is obtained either by Zero-Voltage-Switching (ZVS) or by Zero-Current-Switching (ZCS) of the converter switches. ZVS is achieved by switching on the switches M₁, M₄, M₆ and M₇ when i_L < 0, and in the switches M₂, M₃, M₅ and M₈ when i_L > 0. ZCS is achieved in all switches when i_L = 0, during switching-off.

Table 3. Soft switching conditions for each switch.

Switch	ZVS	ZCS
M1	iL(t1LH) < 0	iL(t1LH + Tsw/2) = 0
M2	iL(t1LH + Tsw/2) > 0	iL(t1LH) = 0
M3	iL(t1HL) > 0	iL(t1HL + Tsw/2) = 0
M4	iL(t1HL + Tsw/2) < 0	iL(t1HL) = 0
M5	iL(t2LH) > 0	iL(t2LH + Tsw/2) = 0
M6	iL(t2LH + Tsw/2) < 0	iL(t2LH) = 0
M7	iL(t2HL) < 0	iL(t2HL + Tsw/2) = 0
M8	iL(t2HL + Tsw/2) > 0	iL(t2HL) = 0

describes, in detail, the soft switching conditions as a function of the current i_L(t) for each switch in the converter.

For the sake of simplicity, the analysis described in this section is made without taking into account the parasitic inductances, capacitances, and resistances that are in real converters. In particular, MOSFET’s parasitic capacitances that affect the soft switching conditions. The capacitances effect on the DAB has been previously analysed in several papers [26,31,32,33].

4.1. Case I (v₁₁ ≥ v₂₂ and D₁ > D₂)

Table 4. Case I (v₁₁ ≥ v₂₂ and D₁ > D₂): General conditions to obtain Zero voltage switching (ZVS) and Zero current switching (ZCS) for each switch.

SMi	Switch
	M1	M2	M3	M4	M5	M6	M7	M8
SM1	${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}\ge {\mathrm{D}}_2·{\mathrm{V}}_2$		${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}\ge {\mathrm{D}}_2·{\mathrm{V}}_2$		$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_2}{2}·\left(1-\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\le -\left[\frac{{\mathrm{D}}_2}{2}·\left(1-\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$
SM2			$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_1}{2}·\left(1-\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$				${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}\le {\mathrm{D}}_2·{\mathrm{V}}_2$
SM2*
SM3			${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$		${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$
SM3*	$\mathsf{\phi }\ge \left[1-\frac{{\mathrm{D}}_1}{2}·\left(1+\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_1}{2}·\left(1-\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_2}{2}·\left(1-\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\ge \left[1-\frac{{\mathrm{D}}_2}{2}·\left(1+\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$
SM4			${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge$0		${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$
SM5	${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$				$\mathsf{\phi }\le \left[1+\frac{{\mathrm{D}}_2}{2}·\left(1+\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$

is obtained by combining the current i_L(t) through the inductor shown in Table 2 and the information in Table 3. Table 4 collects all the specific conditions to obtain ZVS or ZCS for all the switches in each switching mode for the Case I. ZCS is achieved when the equations are satisfied; ZVS is achieved when the inequalities are satisfied, for example: M₁ has ZVS when (D₁·V₁·n > D₂·V₂) and ZCS (D₁·V₁·n = D₂·V₂) for SM₁. Additionally, those conditions are classified into two types: depending on φ and non-depending on φ.

4.1.1. Non-Depending on φ

The non-depending on φ conditions (Table 4) summarise in the three expressions shown in Equation (7). The first condition (D₁·V₁·n ≥ D₂·V₂) only fulfils when D₁·V₁·n > D₂·V₂ due to the Case I implies v₁₁ ≥ v₂₂ and D₁ > D₂. It means that switches with this condition have ZVS. However, the second condition (D₁·V₁·n ≤ D₂·V₂) cannot be satisfied. Therefore, the switches depending on this condition switch with losses (Hard Switching). Finally, the last condition (D₁·V₁·n + D₂·V₂ ≥ 0) can be satisfied for all the possible values of D₁, D₂ and n.

$$\begin{array}{c}{\mathrm{D}}_1\cdot {\mathrm{V}}_1·\mathrm{n}\ge {\mathrm{V}}_2\cdot {\mathrm{D}}_2\\ {\mathrm{D}}_1\cdot {\mathrm{V}}_1\cdot \mathrm{n}\le {\mathrm{V}}_2\cdot {\mathrm{D}}_2\\ {\mathrm{D}}_1\cdot {\mathrm{V}}_1\cdot \mathrm{n}+{\mathrm{V}}_2\cdot {\mathrm{D}}_2\ge 0\end{array}$$

(7)

4.1.2. Depending on φ

The conditions that depend on φ must be graphically analysed in a cube with the unity side. The switching mode SM₃* for the Case I is analysed, by considering the voltage ratio shown in Equation (8), to illustrate the procedure.

$$\mathrm{d}=\frac{{\mathrm{V}}_2}{\mathrm{n}·{\mathrm{V}}_1}$$

(8)

Equation in (9) show soft switching conditions from Table 4 by considering Equation (8). The plane SS_ij (D₁, D₂) represents the soft switching conditions for “i” and “j” switches with any D₁ and D₂. Figure 6a and soft switching conditions in Equation (9) are plotted in Figure 7a for SM₃*. In Figure 7a, the switching mode SM₃* is represented by tetrahedron BCEG, and SS₁₂ (D₁), SS₃₄ (D₁), SS₅₆ (D₂) and SS₇₈ (D₂) are represented by the planes AKWX, DLYZ, DIUT, and AHTU, respectively.

Figure 7b–e show the projections of the soft switching conditions and SM₃* region onto the planes φ/π − D₁ and φ/π − D₂.

$$\begin{array}{c}{\mathrm{SS}}_{12} \left({\mathrm{D}}_1,\mathrm{d}\right)=\frac{\mathsf{\phi }}{\mathsf{\pi }}=\left[1-\frac{{\mathrm{D}}_1}{2}·\left(1+\frac{1}{\mathrm{d}}\right)\right]\forall {\mathrm{D}}_2\\ {\mathrm{SS}}_{34} \left({\mathrm{D}}_1,\mathrm{d}\right)=\frac{\mathsf{\phi }}{\mathsf{\pi }}=\left[\frac{{\mathrm{D}}_1}{2}·\left(1-\frac{1}{\mathrm{d}}\right)\right]\forall {\mathrm{D}}_2\\ {\mathrm{SS}}_{56} \left({\mathrm{D}}_2,\mathrm{d}\right)=\frac{\mathsf{\phi }}{\mathsf{\pi }}=\left[\frac{{\mathrm{D}}_2}{2}·\left(1-\mathrm{d}\right)\right]\forall {\mathrm{D}}_1\\ {\mathrm{SS}}_{78} \left({\mathrm{D}}_2,\mathrm{d}\right)=\frac{\mathsf{\phi }}{\mathsf{\pi }}=\left[1-\frac{{\mathrm{D}}_2}{2}·\left(1+\mathrm{d}\right)\right]\forall {\mathrm{D}}_1\end{array}$$

(9)

Figure 7b shows that M₁ and M₂ have soft switching for angles φ/π ≥ SS₁₂ (D₁,d). It has been indicated by the region in which φ/π ≥ SS₁₂ (D₁,d) (in grey), and the values D₁ and φ belonging to switching mode SM₃* (in green). All combination D₁ − φ/π, belonging to SM₃*, fulfil with φ/π ≥ SS₁₂ (D₁,d), which means that both switches (M₁ and M₂) have soft switching for the entire operating range of SM₃*. The condition that allows having soft switching in the switches M₃ and M₄ fulfils if φ/π ≥ SS₃₄ (D₁,d), Figure 7c. The condition SS₃₄ (D₁,d) takes negative values for the range 0 < D₁ < 1, this means that M₃ and M₄ always switch to soft switching for φ/π > 0.

Figure 7d shows that the projection of the tetrahedron belonging to SM₃* onto the φ/π − D₂ axes is the BCE plane. For the angles φ/π = SS₅₆ (D₂,d) and φ contained in the BCE triangle, ZCS is achieved in M₅ and M₆; on the other hand, when φ/π > SS₅₆ (D₂,d) and the BCE triangle contains to φ/π, M₅ and M₆ have ZVS. Similarly, switches M₅ and M₆, M₇ and M₈ have ZCS when φ/π = SS₇₈ (D₂,d) and ZVS for φ/π > SS₇₈ (D₂,d), and the BCE triangle contains to φ/π, Figure 7e. Finally, Figure 7f shows the values for D₁, D₂ and φ/π, in pink, that allow all the switches to have soft switching for SM₃*. From Figure 7f, it can be concluded that ZCS is only possible for switches M₇ and M₈ when the plane GTV contains to D₁, D₂ and φ/π; for the rest of the points belonging GBVT volume, all the switches have ZVS.

Table 5. Case I (v₁₁ ≥ v₂₂ and D₁ > D₂): Type of switching and conditions to obtain ZVS and ZCS for each switch.

SMi	Range	Condition	Type of Switching
			M1	M2	M3	M4	M5	M6	M7	M8
SM1	$0<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	${\mathrm{SS}}_{56}\left({\mathrm{D}}_2,\mathrm{d}\right)=\frac{{\mathrm{D}}_2}{2}·\left(1-\mathrm{d}\right)$	ZVS		ZVS		ZVS φ/π > SS56 (D2,d) ZCS φ/π = SS56 (D2,d) HS φ/π < SS56 (D2,d)		HS
SM2	$\left(\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$
SM2*	$\left(\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$
SM3	$\left(\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	Always fulfil					ZVS
SM3*	$\left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$	${\mathrm{SS}}_{78}\left({\mathrm{D}}_2,\mathrm{d}\right)=\left(1-\frac{{\mathrm{D}}_2}{2}·\left(1+\mathrm{d}\right)\right)$					ZVS φ/π > SS56 (D2,d) ZCS φ/π = SS56 (D2,d) HS φ/π < SS56 (D2,d)		ZVS φ/π > SS78 (D2,d) ZCS φ/π = SS78 (D2,d) HS φ/π < SS78 (D2,d)
SM4	$\left(1-\frac{{\mathrm{D}}_1+{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }$						ZVS
SM5	$\left(1-\frac{{\mathrm{D}}_1-{\mathrm{D}}_2}{2}\right)·\mathsf{\pi }<\mathsf{\phi }<\mathsf{\pi }$

summarises the type of turn on each switch for all switching modes, and the condition to get soft switching on the Bridge 2 switches.

Figure 8 shows the switching modes, power flow, RMS current through the inductor, and the boundary between HS and ZVS for M₅ to M₈ (conditions in Table 5) when D₁ takes different values (0.3, 0.6 and 0.95). Figure 8a,b depict the power flow and the RMS current for D₁ = 0.3 (D₁ ≤ 0.5), and the switching modes SM₁, SM₂, SM₃, SM₄, and SM₅. For D₁ = 0.6 (D₁ > 0.5), two new switching modes appear, SM₂* and SM₃*, in the power flow, Figure 8c, and RMS current, Figure 8d. Finally, when D₁ = 0.95 both power flow, Figure 8e, and inductor RMS current, Figure 8f, tend to achieve the maximum levels. In short, higher power is obtained when D₁ is close to 1. For the same power flow, switching modes SM₁, SM₂, SM₂*, and SM₃* have less inductor RMS current than the rest of them, see Figure 8b,d,f. Soft switching in all switches, Table 5, is possible in SM₄, SM₅, and SM₃*, but SM₃* obtains the lowest inductor RMS current when D₁ > 0.5, Figure 8d,f. When D₁ ≤ 0.5 less RMS currents appear in SM₁ and SM₂, see Figure 8b, but all switches in bridge 2 are in hard switching, see Table 5.

4.2. Case II (v₁₁ ≥ v₂₂ and D₁ ≤ D₂)

Similar to Case I,

Table 6. Case II (v₁₁ ≥ v₂₂ and D₁ ≤ D₂): General conditions to obtain ZVS and ZCS for each switch.

SMi	Switch
	M1	M2	M3	M4	M5	M6	M7	M8
SM1	$\mathsf{\phi }\le -\left[\frac{{\mathrm{D}}_1}{2}·\left(1-\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_1}{2}·\left(1-\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$		${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}\le {\mathrm{D}}_2·{\mathrm{V}}_2$		${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}\le {\mathrm{D}}_2·{\mathrm{V}}_2$
SM2	${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}\ge {\mathrm{D}}_2·{\mathrm{V}}_2$				$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_2}{2}·\left(1-\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$
SM2*
SM3			${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$		${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$
SM3*	$\mathsf{\phi }\ge \left[1-\frac{{\mathrm{D}}_1}{2}·\left(1+\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_1}{2}·\left(1-\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\ge \left[\frac{{\mathrm{D}}_2}{2}·\left(1-\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$		$\mathsf{\phi }\ge \left[1-\frac{{\mathrm{D}}_2}{2}·\left(1+\frac{{\mathrm{V}}_2}{{\mathrm{V}}_1·\mathrm{n}}\right)\right]·\mathsf{\pi }$
SM4			${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$		${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$
SM5			$\mathsf{\phi }\le \left[1+\frac{{\mathrm{D}}_1}{2}·\left(1+\frac{{\mathrm{V}}_1·\mathrm{n}}{{\mathrm{V}}_2}\right)\right]·\mathsf{\pi }$				${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2·{\mathrm{V}}_2\ge 0$

summarises the conditions that allow the converter to have soft switching on all switches, considering positive φ. This table is equivalent to Table 4 for Case I. Again, there are two types of conditions that have soft switching: Those depending and those non-depending on φ.

4.2.1. Non-Depending on φ

As in Case I, the non-depending on φ conditions are shown in Equation (7), and all conditions could be fulfilled due to V₁·n ≥ V₂ and D₁ ≤ D₂, for Case II. So, from the first and the second conditions (D₁·V₁·n ≥ D₂·V₂ and D₁·V₁·n ≤ D₂·V₂) is obtained in Equation (10) as the only solution that meets both conditions at the same time, which means that the switches have ZCS. The third condition (D₁·V₁·n + D₂·V₂ ≥ 0) always fulfils because all its parameters are always positive, which means that the corresponding switches achieve ZVS.

$${\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}={\mathrm{D}}_2·{\mathrm{V}}_2$$

(10)

The switching modes depicted in Figure 6c,d, for Case II, are simplified in Figure 9a when expression in Equation (10) is applied, turning the original volumes into planes. On the other hand, the application of the expression in Equation (10) implies a limitation to reach the maximum power in the converter due to the maximum value for D₁ = d, which is got when D₂ = 1. In order to reach the maximum power, the expression in Equation (10) has not been considered for d < D₁ < 1 and remaining as a constant D₂ = 1, as shown in Figure 9b. Note that the condition of this last interval coincides with the EPS modulation.

4.2.2. Depending on φ

Applying the expression in Equation (10), in Table 6, for 0 < D₁ ≤ d, the four conditions shown in Equation (11) summarise those that depend on φ.

$$\begin{array}{c}\mathsf{\phi }\ge -\left(\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }\\ \mathsf{\phi }\le \left(\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }\\ \begin{array}{c}\mathsf{\phi }\ge \left(1-\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }\\ \mathsf{\phi }\le \left(1+\frac{{\mathrm{D}}_2+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }\\ \mathsf{\phi }\ge \left(\frac{{\mathrm{D}}_2-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }\end{array}\end{array}$$

(11)

The first condition (φ ≥ − (D₂ − D₁)·π/2) indicates that the switches achieve soft switching when φ ≥ − (D₂ − D₁)·π /2. In Case II it is only possible to obtain ZVS for φ > 0 due to D₁ ≤ D₂. The second condition means that ZVS is achieved when 0< φ < (D₂ − D₁)·π/2, ZCS for φ = (D₂ − D₁)·π/2 and HS in any other case. The third condition (φ ≥ (1 − (D₂ + D₁)/2)·π) indicates ZVS for φ > (1 − (D₂ + D₁)/2)·π), ZCS for φ = (1 − (D₂ + D₁)/2)·π) and HS for φ < (1 − (D₂ + D₁)/2)·π). The fourth condition (φ ≤ (1 − (D₂ + D₁)/2)·π) indicates ZVS for φ < π, since for φ = π the output power is equal to zero. Finally, the fifth condition means that ZVS is achieved when φ > (D₂ − D₁)·π/2, and ZCS for φ = (D₂ − D₁)·π/2 and HS in any other case.

Therefore, by fulfilling expression in Equation (10), φ > 0 from first condition (lower boundary for SM₁, Table 1, Case II), up to φ ≤ (D₂ − D₁)·π/2 from the second condition (upper boundary for SM₁, Table 1, Case II), all switches for SM₁ get soft switching, see Table 6. By fulfilling expression in Equation (10) and φ ≥ (D₂ − D₁)·π/2 from the fifth condition (lower boundary for SM₂ and SM₂*, Table 1, Case II), all switches for SM₂ and SM₂* get soft switching. Only by fulfilling expression in Equation (10) do all switches for SM₃ get soft switching, see Table 6. In addition, by fulfilling expression in Equation (10) and φ ≥ (1 − (D₂ + D₁)/2)·π from the third condition (lower boundary for SM₃*, Table 1, Case II), all switches for SM₃* get soft switching, see Table 6. By fulfilling expression in Equation (10) and φ ≥ (1 − (D₂ + D₁)/2)·π from the third condition (lower boundary for SM₄, Table 1, Case II), all switches for SM₄ get soft switching. Finally, by fulfilling expression in Equation (10), φ ≥ (1 − (D₂ + D₁)/2)·π from the third condition (φ values less than the lower boundary for SM₅, Table 1, Case II), and φ < 1 from the fourth condition, all switches for SM₅ get soft switching.

That means, for the simple fact of working in each switching mode, it would be fulfilling these conditions and having soft switching.

4.2.3. Extended Switching Modes

As said above, when analysing the non-depending on φ conditions, to overcome the limitation on the power delivered due to the early saturation of D₂, an additional condition that coincides with EPS modulation have to be considered. This operating zone is going to be called Extended Switching Mode, Figure 9b. The soft switching conditions in the Extended Switching Mode (D₁ > d and D₂ = 1) are shown in Table 6, for the switching modes SM₁, SM₃* and SM₅. The boundaries of these three switching modes for the Extended Switching Mode are included in

Table 7. Case II (v₁₁ ≥ v₂₂ and D₁ ≤ D₂): Boundaries for each extended switching modes.

SMi	Case II
SM1	$0<\mathsf{\phi }\le \left(\frac{1-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM3*	$\left(\frac{1-{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }\le \left(\frac{1+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }$
SM5	$\left(\frac{1+{\mathrm{D}}_1}{2}\right)·\mathsf{\pi }<\mathsf{\phi }<\mathsf{\pi }$

and are shown in Figure 10a.

Additionally, soft switching conditions for the Extended Switching Mode are divided into those depending on φ and those non-depending on φ.

From Table 6 and D₂ = 1, the conditions that do not depend on φ are summarised in two equations, as shown in Equation (12). The first condition (D₁ ≤ d) affects to SM₁, meaning that the switches M₅–M₈ for D₁ > d lose the soft switching, see Figure 10b. The second condition (D₁·V₁·n + D₂ ≥ 0) is always fulfilling, and only affects SM₅, which implies ZVS in switches M₅–M₈, see Figure 10b.

$$\begin{array}{c}{\mathrm{D}}_1\le \mathrm{d}\\ {\mathrm{D}}_1·{\mathrm{V}}_1·\mathrm{n}+{\mathrm{D}}_2\ge 0\end{array}$$

(12)

On the other hand, the depending on φ conditions for the same three switching modes (SM₁, SM₃* and SM₅), Table 6, are analysed similarly for D₁ < d. From Table 6 and considering Equation (8), for Case II d ≤ 1 (v₂₂/v₁₁ = V₂/(n·V₁) = d ≤ 1). For SM₁, the first condition (φ ≤ (−D₁/2·(1 − 1/d))·π or φ ≤ (D₁/2·((1/d) − 1))·π) allows soft switching in M₁–M₂, line blue in Figure 10a, and the second condition (φ ≥ D₁/2·(1 − 1/d)·π) is equivalent to consider φ > 0 due to d ≤ 1, this means M₃–M₄ always have soft switching, see Figure 9a.

Soft switching conditions for SM₃* can be simplified as (φ ≥ (1 − D₁·(1 + 1/d)/2)·π), (φ ≥ (D₁·(1 − 1/d)·π/2)) and (φ ≥ (1 − d)·π/2), from Table 6; the third condition (φ ≥ (1 − D₁·(1 + 1/d)/2)·π), red line in Figure 10a) indicates soft switching for M₁–M₂; the fourth condition (φ ≥ (D₁·(1 − 1/d)/2)·π) is equivalent to φ > 0 due to d ≤ 1, this means M₃–M₄ always have soft switching; and the fifth condition (φ ≥ (1 − d)·π/2, horizontal black dashed line in Figure 10a means soft switching for M₅–M₈.

Finally, soft switching conditions for SM₅ are (φ ≥ (1 − D₁·(1 + 1/d)/2)·π) and (φ ≤ (1 + D₁·(1 + 1/d)/2)·π); the sixth condition (φ ≥ (1 − D₁·(1 + 1/d)/2)·π, red line, means soft switching for M₁–M₂, the seventh condition (φ ≤ (1 + D₁·(1 + 1/d)/2)·π) is equivalent to φ < π and it indicates soft switching for M₃–M₄, see Figure 10a.

Depending and non-depending soft switching conditions are summarised in

Table 8. Case II (v11 ≥ v22 and D1 ≤ D2): Boundaries, power, and type of switching for each switch.

	SMi	Range	Power	Type of Switching
				M1–M2	M3–M4	M5–M6	M7–M8
D1 ≤ d D2 = D1/d	SM1	$0<\mathsf{\phi }<\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}-1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot {\mathrm{D}}_1^2\cdot \mathsf{\phi }}{2\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}\cdot {\mathrm{D}}_2\cdot \mathsf{\pi }}$	ZVS	ZVS	ZCS	ZCS
		$\mathsf{\phi }=\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}-1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }$		ZCS
	SM2	$\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}-1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }<\mathsf{\phi }<\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}+1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot \mathrm{d}}{4\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}}\left[\frac{\mathsf{\phi }}{\mathsf{\pi }}\cdot \left(\frac{{\mathrm{D}}_1}{\mathrm{d}}\cdot (\mathrm{d}+1)-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)-\frac{{\mathrm{D}}_1{}^2\cdot {\left(\mathrm{d}-1\right)}^2}{4\cdot {\mathrm{d}}^2}\right]$			ZVS
	SM2*	$\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}-1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }<\mathsf{\phi }<1-\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}+1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }$
	SM3	$\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}+1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }<\mathsf{\phi }<1-\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}+1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot {\mathrm{D}}_1^2}{4\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}}$
	SM3*	$\left(1-\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}+1\right)}{2\cdot \mathrm{d}}\right)\cdot \mathsf{\pi }<\mathsf{\phi }\le \frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}+1\right)}{2\cdot \mathrm{d}}\cdot \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot \mathrm{d}}{2\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}}\left[\frac{\mathsf{\phi }}{\mathsf{\pi }}\cdot \left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)-\frac{{\left({\mathrm{D}}_1-1\right)}^2+{\left({\mathrm{D}}_2-1\right)}^2}{4}\right]$	ZVS			ZVS
	SM4	$\left(1-\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}+1\right)}{2\cdot \mathrm{d}}\right)\cdot \mathsf{\pi }<\mathsf{\phi }\le \left(1-\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}-1\right)}{2\cdot \mathrm{d}}\right)\cdot \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot \mathrm{d}}{4\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}}\left[\left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)\cdot \left(\frac{{\mathrm{D}}_1}{\mathrm{d}}(\mathrm{d}+1)+\frac{\mathsf{\phi }}{\mathsf{\pi }}-1\right)-\frac{{\mathrm{D}}_1{}^2{\left(\mathrm{d}-1\right)}^2}{4\cdot {\mathrm{d}}^2}\right]$
	SM5	$\left(1-\frac{{\mathrm{D}}_1\cdot \left(\mathrm{d}-1\right)}{2\cdot \mathrm{d}}\right)\cdot \mathsf{\pi }<\mathsf{\phi }\le \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot {\mathrm{D}}_1\cdot \left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)}{2\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}}$
D1 > d D2 = 1	SM1	$0<\mathsf{\phi }\le \left(\frac{{\mathrm{D}}_1-1}{2}\right)\cdot \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot {\mathrm{D}}_1^2\cdot \mathsf{\phi }}{2\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}\cdot \mathsf{\pi }}$	ZVS	ZVS	HS	HS
	SM3*	$\left(\frac{{1-\mathrm{D}}_1}{2}\right)\cdot \mathsf{\pi }<\mathsf{\phi }<\left(\frac{1-\mathrm{d}}{2}\right)\cdot \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot \mathrm{d}}{2\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}}\left[\frac{\mathsf{\phi }}{\mathsf{\pi }}\cdot \left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)-\frac{{\left({\mathrm{D}}_1-1\right)}^2}{4}\right]$
		$\mathsf{\phi }=\left(\frac{1-\mathrm{d}}{2}\right)\cdot \mathsf{\pi }$				ZCS	ZCS
		$\left(\frac{1-\mathrm{d}}{2}\right)\cdot \mathsf{\pi }<\mathsf{\phi }<\left(\frac{{1+\mathrm{D}}_1}{2}\right)\cdot \mathsf{\pi }$				ZVS	ZVS
	SM5	$\left(\frac{{3-\mathrm{D}}_1}{2}\right)\cdot \mathsf{\pi }<\mathsf{\phi }\le \mathsf{\pi }$	$\frac{{\mathrm{V}}_1^2\cdot {\mathrm{D}}_1\cdot \left(1-\frac{\mathsf{\phi }}{\mathsf{\pi }}\right)}{2\cdot \mathrm{L}\cdot {\mathrm{f}}_{\mathrm{sw}}}$

and Figure 10b for D₁ > d and D₂ = 1. All the switches have zero voltage switching in the ZVS zone (blue dashed rectangle). For SM₁, M₁–M₂ always have hard switching; and for SM₃* and φ < (1 − d)·π/2, M₅ to M₈, always have hard switching, see HS zone (red dashed rectangle). Table 8 shows the results of the analysis performed for Case II and shows the ranges of each switching mode, the power transferred and the type of switching in the switches.

Figure 10c,d shows the power flow and the inductor RMS current, respectively. As depicted, the lower RMS current can be obtained at the boundary between SM₁, SM₂ and SM₂*, compared with SM₃, SM₄ and SM₅ for the same transferred power.

5. Experimental Results

This section shows the experimental results for Case I and Case II using a 250 W prototype. The prototype has IRFP4468PbF MOSFETs in both bridges, a transformer built with an ETD59 ferrite core and a self-manufactured inductor with a RM12 ferrite core. Additionally, a TMS320F28335 Texas Instrument DSP generates the driving signals.

Table 9. DAB parameters.

Descriptions	Specifications
Port 1 Voltage V1	36 V
Port 2 Voltage V2	72 V
Transformer turns ratio: 1:n	1:3
Inductance: L	3.88 μH
Switching frequency: fsw	100 kHz
Port 1 capacitor: C1	60 μF
Port 2 capacitor: C2	60 μF

summarises the operating parameters of the converter, and Figure 11 shows a block diagram of the experimental circuit layout.

Figure 12 shows six switching modes (SM₁, SM₂, SM₂*, SM₃*, SM₄, y SM₅) for Case I and d = 0.677, which validate the analysis performed about the switching types, detailed in Table 5. Figure 12a–c show ZVS in M₁, M₂, M₃, and M₄; HS in M₇ and M_8; and for M₅ and M₆ the switching type depend on φ and D₂. In Figure 12d–f, the switching type for M₇ and M₈ varies in function of φ and D₂, whereas the rest of switches maintain the same switching type (ZVS). Figure 12a shows the voltages v₁₁, v₂₂ and i_L for the switching mode SM₁, in which the four switches of the Bridge 1 (M₁–M₄) are operating with ZVS. On Bridge 2, switches M₅ and M₆ switch with losses (HS) due to (φ/π = 0.050) < (SS₅₆ = 0.059), as shown in Table 5, for D₁ = 0.5 and D₂ = 0.34. Switches M₇ and M₈ are in HS, as was specified in Table 5. In Figure 12b, the eight switches switch in the same way as shown in Figure 12a, for D₁ = 0.5, D₂ = 0.45 and (φ/π = 0.061) < (SS₅₆ = 0.075). In the switching mode SM₂*, Figure 12c shows ZVS in M₅ and M₆ for (φ/π = 0.222) > (SS₅₆ = 0.077) with D₁ = 0.75 and D₂ = 0.487. Figure 12d shows the switching mode SM₃* with ZVS in M₇ and M₈ when D₁ = 0.75, D₂ = 0.643 and (φ/π = 0.577·π) > (SS₇₈ = 0.494). In Figure 12e, switches M₇ and M₈ achieve ZVS for D₁ = 0.75, D₂ = 0.5 and (φ/π = 0.722) > (SS₇₈ = 0.667) in the switching mode SM₄. For the last switching mode, SM₅, the parameters D₁ = 0.75, D₂ = 0.2, and (φ/π = 0.75) < (SS₇₈ = 0.833) are considered as having HS in M₇ and M₈, as predicted in Table 5, Figure 12f.

Figure 13 shows six switching modes (SM₁, SM₂, SM₃, SM₄, SM₅, and SM₃*) for Case II and d = 0.677, applying Equation (10) with values D₁ ≤ d. In this case of study, all the switching modes are likely to achieve ZVS or ZCS except for SM₁ and SM₃*, which may have HS in the Extended Switching Mode, as shown in Table 8 and Figure 10. Figure 13a shows switching mode SM₁, with the Bridge 1 switches in ZVS and those in Bridge 2 in ZCS for D₁ = 0.44, D₂ = 0.664 and φ/π = 0.048. In Figure 13b, the switching mode SM₂ is shown, achieved ZVS in M₃, M₄, M₅, and M₆, and ZCS in M₁, M₂, M₇, and M₈ for D₁ = 0.42, D₂ = 0.656 and φ/π = 0.206. Figure 13c shows four switches with ZCS (M₁, M₂, M₇ and M₈) and four with ZVS (M₃, M₄, M₅ and M₆) for D₁ = 0.132, D₂ = 0.2 and φ/π = 0.458, as detailed in Table 8 for switching mode SM₃. Figure 13d–f show all their switches in ZVS, corresponding to switching modes SM₄ (D₁ = 0.312, D₂ = 0.34 and φ/π = 0.806), SM₅ (D₁ = 0.221, D₂ = 0.435 and φ/π = 0.896), and SM₃*(D₁ = 0.564, D₂ = 0.838 and φ/π = 0.521), respectively.

With these experimental results, the analysis carried out in Section 3 and Section 4 and summarised in Table 5 for Case I and Table 8 for Case II, have been validated.

6. Conclusions

This paper provides an exhaustive analysis of the different DAB switching modes when TPS, EPS, and PS modulation are applied. This analysis allows the identifying of the best switching modes, among the possible fifty-six different ones, as well as the most suitable combinations of modulation variables to guarantee Zero-Voltage-Switching (ZVS) or Zero-Current-Switching (ZCS), and therefore advance getting the best performance for whole output power ranges and each V₁ and V₂ ratio. With this analysis, it is easier to do further analysis, such as to reduce reactive energy or to define the variables values to get minimum RMS current.

Four cases of study have been established for positive φ, depending on the relative value of input and output voltages and the duty cycle in the bridges voltage waveforms. Two of these four possible cases are considered (Case I and Case II) in this paper since the other two are complementary. Seven switching modes, named as SM₁, SM₂, SM₂*, SM₃, SM₃*, SM₄, and SM₅ have been identified for each case. The analytical expression about boundaries, inductor current and average output power are provided for each analysed switching mode.

The analysis carried out allows knowing the switching in each switch (ZVS, ZCS or HS), detailed in Table 5 and Table 8 for Case I and Case II, respectively. This information is essential to quantify the power losses in each switch (both switching and conduction losses) and to improve the efficiency and the power density of the converter. Some of the most relevant conclusions regarding the soft-switching are the following:

• In Case I, only three (SM₃*, SM₄ and SM₅) of the seven switching modes can achieve ZCS or ZVS for all the switches, although the only SM₃^* has a minimum inductor RMS current when D₁ > 0.5. The remaining switching modes (SM₁, SM₂, SM₂*, and SM₃) operate with hard switching in a leg of bridge 2, since φ < SS₅₆ (D₁, D₂) or φ < SS₇₈ (D₁, D₂), see Table 5, Figure 8d,f.
• In Case II, ZVS and ZCS are reached for all switching modes and the whole power range. For low and medium powers, soft switching is got by applying the expression in Equation (10) with D₁ ≤ d and D₂ < 1. High power is got either by operating in extending mode with D₂ = 1 and D₁ > d (EPS modulation), or with D₂ = 1 and D₁ =1 (PS modulation). For SM₁, SM₂, and SM₂*, the lowest RMS current is obtained at the boundary between them, see Figure 10d for the same transferred power. For the highest power, SM₃* achieves the lowest inductor RMS current.

A 250 W DAB experimental prototype has been built and tested in the laboratory to validate the theoretical analysis and the soft-switching conditions for the switching modes of Case I and Case II. In addition, the switching in each switch has been verified, for each switching mode.