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

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.


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]. 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.  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 (D 1 ), being D 2 = 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 (D 1 , D 2 ), in this case being D 1 = D 2 , Figure 2c. 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 (D 1 ) and Bridge 2 (D 2 ), and the phase shift (ϕ) between both voltage waveforms, Figure 2d. This modulation strategy improves Energies 2018, 11, 2419 3 of 23 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 V 1 > n·V 2 . 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 V 1 and V 2 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.

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 M 3 , M 5 , M 7 , and M 1 , respectively). By driving the switches in this way, v 11 is generated at the output of the Bridge 1 of amplitude V 1 , v 22 is generated in the primary side of the transformer with an amplitude of V 2 /n, and a current i L flows through the inductor L.

Cases and Switching Modes
The switching modes define according to the profile acquired by the current iL in each operating state. The current iL is defined by the input parameters of the converter (V1, V2, n, L, fsw) and by the parameters of the TPS modulation (D1, D2 and φ). By considering n, L and fsw 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.

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 The switching instants of the voltages v 11 (t 1LH and t 1HL ) and v 22 (t 2LH and t 2HL ), shown in Figure 3, are calculated by Equation (1), considering that the positive part of v 11 centres in T sw /4.

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 (V 1 , V 2 , n, L, f sw ) and by the parameters of the TPS modulation (D 1 , D 2 and ϕ). By considering n, L and f sw as constants, the voltages v 11 , v 22  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 v 11 with v 22 , and D 1 with D 2 . 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.

Switching Modes: Case I and Case II. Boundaries
The total switching modes per Case are seven: SM 1 , SM 2 , SM 2 *, SM 3 , SM 3 *, SM 4 , and SM 5 ; 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 The boundaries in each switching mode are obtained when the switching instants of the voltages v 11 and v 22 occur at the same time. For example, for SM 2 in Figure 4b: t 1HL = t 2HL determines the lower boundary (switching mode from SM 1 to SM 2 ,) and the upper one when t 1HL = t 2LH (switching mode from SM 2 to SM 3 ,), as shown in Equation (2).
The switching modes SM 2 and SM 3 are obtained for D 1 < 1 − D 2 , whereas SM 2 * and SM 3 * are obtained for D 1 ≥ 1 − D 2 . The switching mode SM 2 * is different concerning SM 2 only in the boundaries, whereas SM 3 * is different regarding SM 3 in the boundaries, the current profile and the expression of the power, with respect to those in SM 3 . Table 1 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 D 1 , D 2 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 SM 2 (DEFG), SM 2 * (CEFG), SM 3 * (BCEG) and SM 4 (ABEG), are shown in Figure 6a; and the modes SM 1 (CDEI), SM 3 (ADEG), and SM 5 (ABEH) are shown in Figure 6b. For Case II: the modes SM 2 (DFGJ), SM 2 * (CFGJ), SM 3 * (BCGJ), and SM 4 (ABGJ) are shown in Figure 6c; while, the modes SM 1 (CDJL), SM 3 (ADGJ), and SM 5 (ABJK), are shown in Figure 6d. 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: t1HL = t2HL determines the lower boundary (switching mode from SM1 to SM2,) and the upper one when t1HL = t2LH (switching mode from SM2 to SM3,), as shown in Equation (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 summarises the boundaries of all switching modes for Case I and Case II.

Current Through the Inductor L
The current iL 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.

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.
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, at Case II (column) and SM 1 (row), shows the solution of Equation (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.

Average Power
The input current, i 1 (t), is defined from t 1LH to t 1HL when the power is flowing to V 2 . Without considering losses, the average power (P = V 1 ·I 1 ) 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.

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 M 1 , M 4 , M 6 and M 7 when i L < 0, and in the switches M 2 , M 3 , M 5 and M 8 when i L > 0. ZCS is achieved in all switches when i L = 0, during switching-off. Table 3 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]. Table 3. Soft switching conditions for each switch.

Switch
ZVS ZCS Table 4 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: M 1 has ZVS when (D 1 ·V 1 ·n > D 2 ·V 2 ) and ZCS (D 1 ·V 1 ·n = D 2 ·V 2 ) for SM 1 . Additionally, those conditions are classified into two types: depending on ϕ and non-depending on ϕ.  The non-depending on ϕ conditions (Table 4) summarise in the three expressions shown in Equation (7). The first condition (D 1 ·V 1 ·n ≥ D 2 ·V 2 ) only fulfils when D 1 ·V 1 ·n > D 2 ·V 2 due to the Case I implies v 11 ≥ v 22 and D 1 > D 2 . It means that switches with this condition have ZVS. However, the second condition (D 1 ·V 1 ·n ≤ D 2 ·V 2 ) cannot be satisfied. Therefore, the switches depending on this condition switch with losses (Hard Switching). Finally, the last condition (D 1 ·V 1 ·n + D 2 ·V 2 ≥ 0) can be satisfied for all the possible values of D 1 , D 2 and n.

Depending on ϕ
The conditions that depend on ϕ must be graphically analysed in a cube with the unity side. The switching mode SM 3 * for the Case I is analysed, by considering the voltage ratio shown in Equation (8), to illustrate the procedure.
Equation in (9) show soft switching conditions from Table 4 by considering Equation (8). The plane SS ij (D 1 , D 2 ) represents the soft switching conditions for "i" and "j" switches with any D 1 and D 2 . Figure 6a and soft switching conditions in Equation (9) are plotted in Figure 7a for SM 3 *. In Figure 7a, the switching mode SM 3 * is represented by tetrahedron BCEG, and SS 12 (D 1 ), SS 34 (D 1 ), SS 56 (D 2 ) and SS 78 (D 2 ) are represented by the planes AKWX, DLYZ, DIUT, and AHTU, respectively. Figure 7b-e show the projections of the soft switching conditions and SM 3 * region onto the planes ϕ/π − D 1 and ϕ/π − D 2 . Figure 7b shows that M 1 and M 2 have soft switching for angles ϕ/π ≥ SS 12 (D 1 ,d). It has been indicated by the region in which ϕ/π ≥ SS 12 (D 1 ,d) (in grey), and the values D 1 and ϕ belonging to switching mode SM 3 * (in green). All combination D 1 − ϕ/π, belonging to SM 3 *, fulfil with ϕ/π ≥ SS 12 (D 1 ,d), which means that both switches (M 1 and M 2 ) have soft switching for the entire operating range of SM 3 *. The condition that allows having soft switching in the switches M 3 and M 4 fulfils if ϕ/π ≥ SS 34 (D 1 ,d), Figure 7c. The condition SS 34 (D 1 ,d) takes negative values for the range 0 < D 1 < 1, this means that M 3 and M 4 always switch to soft switching for ϕ/π > 0. Figure 7d shows that the projection of the tetrahedron belonging to SM 3 * onto the ϕ/π − D 2 axes is the BCE plane. For the angles ϕ/π = SS 56 (D 2 ,d) and ϕ contained in the BCE triangle, ZCS is achieved in M 5 and M 6 ; on the other hand, when ϕ/π > SS 56 (D 2 ,d) and the BCE triangle contains to ϕ/π, M 5 and M 6 have ZVS. Similarly, switches M 5 and M 6 , M 7 and M 8 have ZCS when ϕ/π = SS 78 (D 2 ,d) and ZVS for ϕ/π > SS 78 (D 2 ,d), and the BCE triangle contains to ϕ/π, Figure 7e. Finally, Figure 7f shows the values for D 1 , D 2 and ϕ/π, in pink, that allow all the switches to have soft switching for SM 3 *. From Figure 7f, it can be concluded that ZCS is only possible for switches M 7 and M 8 when the plane GTV contains to D 1 , D 2 and ϕ/π; for the rest of the points belonging GBVT volume, all the switches have ZVS.  Table 5 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.  Table 5 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 M 5 to M 8 (conditions in Table 5) when D 1 takes different values (0.3, 0.6 and 0.95). Figure 8a,b depict the power flow and the RMS current for D 1 = 0.3 (D 1 ≤ 0.5), and the switching modes SM 1 , SM 2 , SM 3 , SM 4 , and SM 5 . For D 1 = 0.6 (D 1 > 0.5), two new switching modes appear, SM 2 * and SM 3 *, in the power flow, Figure 8c, and RMS current, Figure 8d. Finally, when D 1 = 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 D 1 is close to 1. For the same power flow, switching modes SM 1 , SM 2 , SM 2 *, and SM 3 * 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 SM 4 , SM 5 , and SM 3 *, but SM 3 * obtains the lowest inductor RMS current when D 1 > 0.5, Figure 8d,f. When D 1 ≤ 0.5 less RMS currents appear in SM 1 and SM 2 , see Figure 8b, but all switches in bridge 2 are in hard switching, see Table 5.

Case II (v11 ≥ v22 and D1 ≤ D2)
Similar to Case I, Table 6 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.

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·V 1· n ≥ D2·V 2 and D1·V 1·n ≤ D2·V 2) 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·V 1· n + D2·V 2 ≥ 0) always

Case II (v 11 ≥ v 22 and D 1 ≤ D 2 )
Similar to Case I, Table 6 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.

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 V 1 ·n ≥ V 2 and D 1 ≤ D 2 , for Case II. So, from the first and the second conditions (D 1 ·V 1 ·n ≥ D 2 ·V 2 and D 1 ·V 1 ·n ≤ D 2 ·V 2 ) 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 (D 1 ·V 1 ·n + D 2 ·V 2 ≥ 0) always fulfils because all its parameters are always positive, which means that the corresponding switches achieve ZVS.
The switching modes depicted in Figure 6c,d, for Case II, are simplified in Figure 9a when expression in Equation (10) (10) implies a limitation to reach the maximum power in the converter due to the maximum value for D 1 = d, which is got when D 2 = 1. In order to reach the maximum power, the expression in Equation (10) has not been considered for d < D 1 < 1 and remaining as a constant D 2 = 1, as shown in Figure 9b. Note that the condition of this last interval coincides with the EPS modulation. fulfils because all its parameters are always positive, which means that the corresponding switches achieve ZVS.
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.

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.

Depending on ϕ
Applying the expression in Equation (10), in Table 6, for 0 < D 1 ≤ d, the four conditions shown in Equation (11) summarise those that depend on ϕ.
That means, for the simple fact of working in each switching mode, it would be fulfilling these conditions and having soft switching.

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 D 2 , 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 (D 1 > d and D 2 = 1) are shown in Table 6, for the switching modes SM 1 , SM 3 * and SM 5 . The boundaries of these three switching modes for the Extended Switching Mode are included in Table 7 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 D 2 = 1, the conditions that do not depend on ϕ are summarised in two equations, as shown in Equation (12). The first condition (D 1 ≤ d) affects to SM 1 , meaning that the switches M 5 -M 8 for D 1 > d lose the soft switching, see Figure 10b. The second condition (D 1 ·V 1 ·n + D 2 ≥ 0) is always fulfilling, and only affects SM 5 , which implies ZVS in switches M 5 -M 8 , see Figure 10b.
Depending and non-depending soft switching conditions are summarised in Table 8 and Figure 10b for D 1 > d and D 2 = 1. All the switches have zero voltage switching in the ZVS zone (blue dashed rectangle). For SM 1 , M 1 -M 2 always have hard switching; and for SM 3 * and ϕ < (1 − d)·π/2, M 5 to M 8 , 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 SM 1 , SM 2 and SM 2 *, compared with SM 3 , SM 4 and SM 5 for the same transferred power.

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 summarises the operating parameters of the converter, and Figure 11 shows a block diagram of the experimental circuit layout.
With these experimental results, the analysis carried out in sections III and IV and summarised in Tables 5 for Case I and Table 8 for Case II, have been validated.  Figure 13 shows six switching modes (SM 1 , SM 2 , SM 3 , SM 4 , SM 5 , and SM 3 *) for Case II and d = 0.677, applying Equation (10) with values D 1 ≤ d. In this case of study, all the switching modes are likely to achieve ZVS or ZCS except for SM 1 and SM 3 *, which may have HS in the Extended Switching Mode, as shown in Table 8 and Figure 10. Figure 13a shows switching mode SM 1 , with the Bridge 1 switches in ZVS and those in Bridge 2 in ZCS for D 1 = 0.44, D 2 = 0.664 and ϕ/π = 0.048. In Figure 13b Table 8 for switching mode SM 3 . Figure 13d-f show all their switches in ZVS, corresponding to switching modes SM 4 (D 1 = 0.312, D 2 = 0.34 and ϕ/π = 0.806), SM 5 (D 1 = 0.221, D 2 = 0.435 and ϕ/π = 0.896), and SM 3 *(D 1 = 0.564, D 2 = 0.838 and ϕ/π = 0.521), respectively.
With these experimental results, the analysis carried out in Sections 3 and 4 and summarised in Table 5 for Case I and Table 8 for Case II, have been validated.

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

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 V 1 and V 2 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 SM 1 , SM 2 , SM 2 *, SM 3 , SM 3 *, SM 4 , and SM 5 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 Tables 5 and 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:

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In Case I, only three (SM 3 *, SM 4 and SM 5 ) of the seven switching modes can achieve ZCS or ZVS for all the switches, although the only SM 3 * has a minimum inductor RMS current when D 1 > 0.5.

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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 D 1 ≤ d and D 2 < 1. High power is got either by operating in extending mode with D 2 = 1 and D 1 > d (EPS modulation), or with D 2 = 1 and D 1 =1 (PS modulation). For SM 1 , SM 2 , and SM 2 *, the lowest RMS current is obtained at the boundary between them, see Figure 10d for the same transferred power. For the highest power, SM 3 * 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.
Author Contributions: C.C. did theoretical analysis, derivation, circuit implementation, experimental testing, data processing and wrote the original draft paper. A.B. is the responsible for funding acquisition, supervision and administration, her contributions were related with the theoretical analysis, data analysis, and the paper reviewing and editing. A.R. wrote and reviewed the paper. P.A. contributed with theoretical analysis and with significant comments on the manuscript structure. A.L., C.F. and P.Z. reviewed and contributed with useful comments on the paper structure and mains paper contributions.

Conflicts of Interest:
The authors declare no conflicts of interest.  11 Output voltage of the Bridge 1. SM x Switching mode "x". v 22 Input voltage of the Bridge 2. M x Switch "x". D 1 Pulse width of v 11 . SS xy Soft switching condition for MOSFET "x" and "y". D 2 Pulse width of v 22