*2.3. MN Principle*

A systematic methodology, the MN principle, is proposed to derive all possible full-bridge topologies with small leakage currents. The MN principle can be described as follows. Let M denote the total number of switches that are turned on in PC mode. Since X1 is the number of switches that connect point P to point A in PC mode, and X2 is the number of switches that connect point B to point N in PC mode, then

$$\mathcal{M} = \mathbb{X}\_1 + \mathbb{X}\_2 \tag{6}$$

Similarly, let N denote the total number of switches that are turned on in NC mode. Since Y1 is the number of switches that connect point P to point B and Y2 is the number that connect point A to point N in NC mode, then

$$\mathbf{N} = \mathbf{Y}\_1 + \mathbf{Y}\_2 \tag{7}$$

According to rule #1, points A and B must be disconnected to points P and N, which means that at least one switch is needed for TPA, TNB, TPB, and TNA. Thus, the minimum values of X1, X2, Y1, and Y2 should be one, as shown in (8).

$$\text{Min}(\mathbf{X}\_1, \mathbf{X}\_2, \mathbf{Y}\_1, \mathbf{Y}\_2) \ge 1 \tag{8}$$

In order to disconnect A, B to P, N in PF and NF modes, according to rule #2, one switch and an extra diode connected in series can be used. Thus, there is one possible way that two switches are in series to implement the equivalent switches TPA, TNB, TPB, and TNA, respectively. For example, two switches, TP1 and TP2, are connected in series between points P and A to implement the equivalent switch, i.e., TPA. Switch TP1 remains o ff to disconnect points P and A, and switch TP2 remains on to construct the freewheeling branch in PF mode. If three switches, TP1, TP2, and TP3, are in series to

implement the equivalent switch TPA, switch TP1 remains off to disconnect points P and A, and two switches TP2 and TP3 are in series to construct the freewheeling branch in PF mode. However, the two switches, TP2 and TP3, can be merged into a single switch. Thus, the maximum value for X1, X2, Y1, and Y2 are less than or equal to 2. Therefore,

$$\text{Max}(\mathbb{X}\_1, \mathbb{X}\_2, \mathbb{Y}\_1, \mathbb{Y}\_2) \le 2 \tag{9}$$

From the above analysis, it may be observed that the MN principle can cover all possible topologies. Some of them can be simplified. Thus, only simplified topologies are introduced in the next section.

## **3. Topology Derivation under USPWM**

In this section, several examples are provided to show how to derive topologies from the MN principle, such as M = 2 and N = 2, M = 2 and N = 3, or M = 3 and N = 2.

#### *3.1. Case 1: M* = *2 and N* = *2*

When M = 2 and N = 2, there is only one possibility to choose the combined values of X1, X2, Y1, and Y2, i.e., X1 = 1, X2 = 1, Y1 = 1, and Y2 = 1.

Figure 8 shows four modes derived from X1 = 1, X2 = 1, Y1 = 1 and Y2 = 1. The PC and NC modes are shown in Figure 8a. One switch, TP1, is adopted to connect points P and A due to X1 = 1; meanwhile, another switch, TP2, is viewed as a connection between points B and N on X2 = 1 at PC mode. Similarly, two other switches, TN1 and TN2, are added in NC mode with Y1 = 1 and Y2 = 1. Figure 8b shows PF mode. According to rule #1, four switches, i.e., TP1, TP2, TN1, and TN2, are off in PF mode. There is no available switch for output current flow. Thus, an extra switch, TP3, and an extra diode, Dp3, are added in series. This is the only choice available for PF mode. Similarly, for NF mode, an extra switch, TN3, and an extra diode, DN3, are added in series for output current flow, as shown in Figure 8c.

**Figure 8.** Four modes under X1 = 1, X2 = 1, Y1 = 1, and Y2 = 1. (**a**) PC and NC modes; (**b**) PF mode; (**c**) NF mode.

Three topologies derived from X1 = 1, X2 = 1, Y1 = 1, and Y2 = 1 are shown in Figure 9. Figure 9a can be achieved from Figure 8b,c in PF and NF modes. In Figure 9b, the body-diodes of switches TP3 and TN3 are used to replace the extra diodes in Figure 9a. Figure 9c is another topology in which four diodes plus one switch are used to replace the two switches, TP3 and TN3. These are well-known HERIC topologies [13,30].

The topologies from the MN principle may be divided into two families. Those in the first family have extra diode for output current flow in PF and NF modes. In contrast, the topologies in the second family don't use the extra diode, and the body-diode of the switch is used to allow current to flow, as in the case in Figure 9b in PF and NF modes. Thus, two corresponding topological families under M = 2 and N = 2 are shown in Table 1.

**Figure 9.** Three topologies under X1 = 1, X2 = 1, Y1 = 1, and Y2 = 1. (**a**) R1 [13]; (**b**) R2 [13]; (**c**) R3 [30].


**Table 1.** Topological families under M = 2 and N = 2.

*3.2. Case 2: M* = *3 and N* = *2 or M* = *2 and N* = *3*

For M = 3, N = 2 or M = 2, N = 3, there are same topologies between M = 3, N = 2 and M = 2, N = 3, as they are equivalent by exchanging the two bridges. Thus, M = 3 and N = 2 is made as an example to explain the derivation method. M = 3 and N = 2 means there are two possibilities to choose the combined values of X1, X2, Y1, and Y2, i.e., X1 = 1, X2 = 2, Y1 = 1, and Y2 = 1, and X1 = 2, X2 = 1, Y1 = 1, and Y2 = 1. Considering the symmetrical characteristics with respect to terminals P and N, the two cases are the same. For the sake of brevity, only the former case is analyzed below.

Figure 10 shows four modes under X1 = 1, X2 = 2, Y1 = 1, and Y2 = 1. The PC and NC modes are shown in Figure 10a. One switch, TP1, is used to connect points P and A due to X1 = 1; meanwhile, switches TP2 and TP3 are viewed as a connection between point B and point N as X2 = 2 in PC mode. Similarly, two switches, TN1 and TN2, are added in NC mode with Y1 = 1 and Y2 = 1. According to rule #1, switches TP1, TP3, TN1, and TN2 are off in PF and NF modes. Switch Tp2 is on according to rule #3, and one diode Dp2 is added based on rule #2 in PF mode, as shown in Figure 10b. In NF mode, an extra switch, TN3, plus the diode DN3 are added in series to flow negative output current. The diode is added or served by the body diode of switch TP2. Thus, there are two circuits to realize NF mode, as shown in Figure 10c,d.

**Figure 10.** Four modes under X1 = 1, X2 = 2, Y1 = 1, and Y2 = 1. (**a**) PC and NC modes; (**b**) PF mode; (**c**) NF mode #1; (**d**) NF mode #2.

According to Figure 10, there are one circuit in PF mode and two circuits in NF mode. There are only two possibilities to combine PF and NF modes. Correspondingly, the two topologies derived from X1 = 1, X2 = 2, Y1 = 1, and Y2 = 1 are shown in Figure 11. Figure 11a shows the topology which combines the PF mode in Figure 10b and the NF mode in Figure 10c, while Figure 11b shows the topology which combines the PF mode in Figure 10b and the NF mode in Figure 10d. Two topological families under M = 3, N = 2 or M = 2, N = 3 are shown in Table 2.

**Figure 11.** Two topologies derived from X1 = 1, X2 = 2, Y1 = 1, and Y2 = 1. (**a**) R4 [31]; (**b**) R5 [27]. **Table2.**Topologicalfamiliesunder M =3andN=2orM=2andN=3.


*3.3. Case 3: M* = *3 and N* = *3*

M = 3, which means there are two possibilities to choose the combined values of X1 and X2: X1 = 1, X2 = 2, and X1 = 2, X2 = 1. Similarly, N = 3 yields two possibilities to combine Y1 and Y2. One is Y1 = 1 and Y2 = 2 and the other is Y1 = 2 and X2 = 1. It should be noted that the same topologies exist between X1 = 2, X2 = 1, Y1 = 1, Y2 = 2, and X1 = 1, X2 = 2, Y1 = 2, Y2 = 1 when two bridges are exchanged. Thus, there are three possibilities: (1) X1 = 2, X2 = 1, Y1 = 2, and Y2 = 1; (2) X1 = 2, X2 = 1, Y1 = 1, and Y2 = 2; and (3) X1 = 1, X2 = 2, Y1 = 1, and Y2 = 2. Considering the symmetry between terminals P and N, case (1) is the same as case (3). For the sake of brevity, only cases (1) and (2) are analyzed below.

Figure 12 shows four modes under X1 = 2, X2 = 1, Y1 = 2, and Y2 = 1. The PC and NC modes are shown in Figure 12a. As shown in Figure 12a, six switches (TP1, TP2, TP3, TN1, TN2, and TN3) are used for X1 = 2, X2 = 1, Y1 = 2, and Y2 = 1. According to rule #1, switches TP1, TP3, TN1, and TN3 are off in PF and NF modes. One rest switch, TP2, is on according to rule #3, and one diode, DP2, is added based on rule #2 in PF mode, as shown in Figure 12b. Diode DP2 can also be served by the body-diode of switch TN2. The reflected PF mode is shown in Figure 12c. For NF mode, switch TN2 is on for negative output current flow. An extra diode, DN2, is added, as shown in Figure 12d. The body-diode of switch TP2 is served as the diode DN2, as shown in Figure 12e.

According to Figure 12, there are two circuits in PF mode and two in NF mode. There are four possibilities to combine PF and NF modes. Correspondingly, four topologies derived from X1 = 2, X2 = 1, Y1 = 2, and Y2 = 1 are shown in Figure 13. Figure 13a shows the topology combining the PF mode in Figure 12b and the NF mode in Figure 12d. Figure 13b shows the topology combining the PF mode in Figure 12b and the NF mode in Figure 12e. Figure 13c shows the topology combining the PF mode in Figure 12c and the NF mode in Figure 12d, and Figure 13d shows the topology combining the PF mode in Figure 12c and the NF mode in Figure 12e.

According to rule #4, the extra diode DP2 can be absent due to the presence of the body-diode of switch TN2 in Figure 13b. Similarly, the extra diode DN2 can be absent owing to the presence of the body-diode of switch TP2 in Figure 13c. In Figure 13b–d, the two switches, TP1 and TN1, are combined into one switch T1. Thus, the topologies in Figure 13b–d are the same.

**Figure 12.** Four modes under X1 = 2, X2 = 1, Y1 = 2, and Y2 = 1. (**a**) PC and NC modes; (**b**) PF mode #1; (**c**) PF mode #2; (**d**) NF mode #1; (**e**) NF mode #2.

**Figure 13.** Four topologies derived from X1 =2, X2 =1, Y1 =2, and Y2 =1. (**a**) R6 [16,31]; (**b**) R7 (circuit one); (**c**) R7 (circuit two); (**d**) R7 (circuit three) [12].

Similarly, one topology derived from X1 = 2, X2 = 1, Y1 = 1, and Y2 = 2 is shown in Figure 14.

**Figure 14.** The topology R8 derived from X1 = 2, X2 = 1, Y1 = 1, and Y2 = 2; [38].

Correspondingly, two topological families under M = 3 and N = 3 are shown in Table 3.


**Table 3.** Topological families under M = 3 and N = 3.

#### *3.4. Case 4: M* = *3 and N* = *4 or M* = *4 and N* = *3*

The same topologies exist between M = 3, N = 4 and M = 4, N = 3, as they are equivalent by exchanging the two bridges. For M = 3 and N = 4, M = 3 means that there are two possibilities to choose the combined values of X1 and X2: one is X1 = 2 and X2 = 1, and the other is X1 = 1 and X2 = 2. Similarly, N = 4 means three possibilities to combine Y1 and Y2. However, only one combination is available according to Equation (9), i.e., Y1 = 2 and Y2 = 2.

Thus, there are two possibilities to choose the combined values of X1, X2, Y1, and Y2: one is X1 = 1, X2 = 2, Y1 = 2, and Y2 = 2, and the other is X1 = 2, X2 = 1, Y1 = 2, and Y2 = 2. Considering the symmetry between terminals P and N, the two cases are the same. Figure 15 shows four modes under X1 = 1, X2 = 2, Y1 = 2, and Y2 = 2.

**Figure 15.** Four modes under X1 = 1, X2 = 2, Y1 = 2, and Y2 = 2. (**a**) PC and NC modes; (**b**) PF mode #1; (**c**) PF mode #2; (**d**) NF mode #1; (**e**) NF mode #2.

The PC and NC modes are shown in Figure 15a. Seven switches (TP1, TP2, TP3, TN1, TN2, TN3, and TN4) are used for X1 = 1, X2 = 2, Y1 = 2, and Y2 = 2, as shown in Figure 15a. From rule #1, the switches TP1, TP3, TN1, and TN4 are off in PF and NF modes. Switch TP2 is on according to rule #3, and one diode DP2 is added based on rule #2 in PF mode, as shown in Figure 15b. Diode DP2 is served by the body-diode of switch TN3; the reflected PF mode is shown in Figure 15c. For NF mode, two switches, i.e., TN2 and TN3, are on for negative output current flow according to rule #3, and two extra diodes, DN2 and DN3, are added from rule #2, as shown in Figure 15d. The body-diode of switch TP2 serves as the diode DN3, as shown in Figure 15e.

According to the above analysis, there are two circuits in PF mode and two in NF mode. Thus, four possible topologies are shown in Figure 16. Figure 16a shows the topology combining the PF mode in Figure 15b and the NF mode in Figure 15d. The other three topologies are shown in Figure 16b–d, respectively; however, they are the same, as diode DP2 in Figure 16b and diode DN3 in Figure 16c can be absent from rule #4. Furthermore, two switches, i.e., TP3 and TN4, are merged into one switch, i.e., T4.

**Figure 16.** Two topologies derived from X1 = 1, X2 = 2, Y1 = 2, and Y2 = 2. (**a**) R9 (new); (**b**) R10 (circuit one); (**c**) R10 (circuit two); (**d**) R10 (circuit three) (new).

Correspondingly, two topological families under M = 3 and N = 4 or M = 4 and N = 3 are shown in Table 4.



*3.5. Case 5: M* = *4 and N* = *4*

In this case, M = 4 and N = 4. According to Equation (9), M = 4 and N = 4 means only one available possibility to choose the combined values of X1, X2, Y1, and Y2, i.e., X1 = 2, X2 = 2, Y1 = 2, and Y2 = 2.

Figure 17 shows four modes under X1 = 2, X2 = 2, Y1 = 2, and Y2 = 2. As shown in Figure 17a, eight switches (TP1~TP4 and TN1~TN4) are used for X1 = 2, X2 = 2, Y1 = 2, and Y2 = 2 in PC and NC modes. With the reference of the above analysis of Cases 1~4, it is easy to make a similar analysis. For the sake of brevity, this is included here.

**Figure 17.** Four modes under X1 = 2, X2 = 2, Y1 = 2, and Y2 = 2. (**a**) PC and NC modes; (**b**) PF mode #1; (**c**) PF mode #2; (**d**) PF mode #3; (**e**) PF mode #4; (**f**) NF mode #1; (**g**) NF mode #2; (**h**) NF mode #3; (**i**) NF mode #4.

It may be observed from Figure 17 that there are four circuits in PF mode and four in NF mode. Thus, sixteen possible topologies may be derived from the MN principle. However, these topologies can be simplified based on rule #4, and the four topologies derived from X1 = 2, X2 = 2, Y1 = 2, and Y2 = 2 are shown in Figure 18.

**Figure 18.** Four topologies derived from X1 = 2, X2 = 2, Y1 = 2, and Y2 = 2. (**a**) R11 (new); (**b**) R12 (new); (**c**) R13 [17].

Two corresponding topological families under M = 4 and N = 4 are shown in Table 5.

**Table 5.** Topological families under M = 4 and N = 4.


#### *3.6. All Simplfied Topologies from MN Principle*

From the above analysis, two corresponding topological families are summarized in Table 6.


**Table 6.** Two simplified topological families from MN principle.

It may be observed from the above analysis that the MN principle can be used to derive all the possible topologies for a single-phase, full bridge, transformerless inverter. Furthermore, thirteen simplified topologies from the MN principle are provided in this paper. All existing topologies (R1–R8, R13) have been covered, and five new topologies marked from R9 to R12 have been found.

For the two topological families with the same M and N, the topologies without an extra diode are of lower cost than those with the extra diode, as the body-diode of switch in the former is used to replace the extra diode; however, the e fficiency of the former will likely be a little lower, as that diode has better performance than the body-diode.

(M + N) is the number of conduction switches in PC and NC modes; the bigger (M + N), the higher the conduction loss. Thus, M = N = 2 is the best choice in terms of low conduction loss.

Although M = 4 and N = 4 means large conduction loss under USPWM, the topologies from M = 4 and N = 4 can also work in DFSPWM, where the equivalent switching frequency is double, and the size of the low pass filter is reduced. A detailed description about DFUSWPM will be given in the next section.

## **4. Topology Derivation under DFUSPWM**

In this section, topology derivation methodology is introduced under DFUSPWM. The unified topology model and MN principle are extended to the topologies under DFUSPWM.
