*4.1. Principle of DFUSPWM*

The principle of DFUSPWM is shown in Figure 19. Di fferential-mode voltage *V*AB is a three-level waveform.

The levels are <sup>+</sup>*V*PN, 0, and −*V*PN. There are six modes in total line-frequency period. The inverter is working in positive conduction (PC) mode and negative conduction (NC) mode when *V*AB equals <sup>+</sup>*V*PN and <sup>−</sup>*V*PN*,* respectively. There are four modes if *V*AB equals 0.

**Figure 19.** Principle of DFUSPWM.

To achieve double frequency of voltage *V*AB, there are two interleaved freewheeling modes when the grid voltage is positive: one is used to refer to the freewheeling current flowing through top switch, which is defined as PFT mode. The other one, called "PFB mode" is used to flow freewheeling current through bottom switch. Similarly, two interleaved freewheeling modes, called "NFT mode" and "NFB mode", are used in NF mode. The freewheeling current flows through the top switch in NFT and through the bottom switch in NFB mode.

The modes rotate in the sequence of PFT, PC, PFB, and PC in the positive half cycle of the grid voltage. Similarly, the modes rotate in the sequence of NFT, NC, NFB, and NC in the negative half cycle. Thus, the frequency of output voltage *V*AB is double the switch frequency.

The six modes based on H4 topology in DFUSPWM are shown in Figure 20. The PC and NC modes are shown in Figures 20a and 20b, respectively. In the positive half cycle of grid voltage, two PF modes, i.e., PFT and PFB, are shown in Figure 20c,d. Similarly, two NF modes, i.e., NFT and NFB, are shown in Figure 20e,f in the negative half cycle of the grid voltage.

**Figure 20.** Six modes based on H4 topology under DFUSPWM. (**a**) PC mode; (**b**) NC mode; (**c**) PFT mode; (**d**) PFB mode; (**e**) NFT mode; (**f**) NFB mode.

#### *4.2. Unified Topology Model of DFUSPWM*

Figure 21 shows six modes from the unified topology model under DFUSPWM.

**Figure 21.** Six modes based on unified topology under DFUSPWM. (**a**) PC mode; (**b**) NC mode; (**c**) PFT mode; (**d**) PFB mode; (**e**) NFT mode; (**f**) NFB mode.

All rules under USPWM mentioned in Section 2 are also suitable for DFUSPWM. According to rule #1, points A and B must be disconnected from points P and N in the four freewheeling modes, i.e., the two PF modes and the two NF modes. From rule #2, the branch between points A and B is short-circuited for output current flow. Two new controlled branches, BCA ˆ and BEA ˆ , are added to flow the positive current in Figure 21c,d. Two controlled branches, ADB ˆ and AFB ˆ , are added to flow the negative current in Figure 21e,f.

For DFUSPWM, there are two PC modes in a switching period. Thus, according to rule #1, there are at least two couple switches to alternately turn on/off to achieve double frequency. For example, there are two switches, i.e., TP1 connected to point P and TP2 connected to point A, between point P and point A, and two, i.e., TP3 connected to point B and TP4 connected to point N, between point B and point N. Switches TP1 and TP2 are connected in series, as are TP3 and TP4. Switches TP1 and TP3 turn on/off at the same time, and TP2 and TP4 are kept on or off at the same time. Thus, there are two possibilities to disconnect points P and A, and points B and N: one is that switches TP1 and TP3 turn <sup>o</sup>ff. The other is that switches TP2 and TP4 turn <sup>o</sup>ff. Once switches TP1 and TP3 turn off in PFT mode, switch TP2 is kept on and can be used to construct a freewheeling branch because it connects to point A. Similarly, switch TP3 is kept on and serves to construct a freewheeling branch in the PFB mode when switches TP2 and TP4 turn <sup>o</sup>ff. Thus, two PF modes can be achieved when X1 = 2 and X2 = 2. The same is true with Y1 = 2, Y2 = 2.

Figure 22 shows six modes under X1 = 2, X2 = 2, Y1 = 2, and Y2 = 2. The PC and NC modes are shown in Figure 22a. Eight switches (TP1~TP4 and TN1~TN4) are used. The same driving signals are provided for couples of switches TP1 and TP3, TP2 and TP4, TN1 and TN3, and TN2 and TN4.

**Figure 22.** Six modes of M10 topology. (**a**) PC and NC modes; (**b**) PFT mode #1; (**c**) PFT mode #2; (**d**) PFB mode #1; (**e**) PFB mode #2; (**f**) NFT mode #1; (**g**) NFT mode #2; (**h**) NFB mode #1; (**i**) NFB mode #2.

In PFT mode, switches TP1, TP3, TN1, TN2, TN3, and TN4 are <sup>o</sup>ff, and switches TP2 and TP4 are on. One diode, DP2, is added for the positive output current flow, as shown in Figure 22b. Diode DP2 is served by the body-diode of switch TN2. The reflected PFT mode is shown in Figure 22c.

In PFB mode, switches TP2, TP4, TN1, TN2, TN3, and TN4 are <sup>o</sup>ff, and switches TP1 and TP3 are on. One diode, DP3, is added for positive output current flow, as shown in Figure 22d. Diode DP3 is served by the body-diode of switch TN3, as shown in Figure 22e. It is easy to make a similar analysis for NFT and NFB modes. Figure 22f,g show the two NFT modes, while the two NFB modes are shown in Figure 22h,i.

It should be noted from the above analysis that there are two PFT modes, two PFB modes, two NFT modes, and two NFB modes. There are eight possible topologies through choosing one PFT mode, one PFB mode, one NFT mode, and one NFB mode. According to rule #4, the four typical topologies in Figure 23 are derived from the MN principle.

**Figure 23.** Four topologies derived from MN principle under DFUSPWM. (**a**) R11 (new); (**b**) R12 (new); (**c**) R14 [17].

Two corresponding topological families under DFUSPWM are shown in Table 7.


**Table 7.** Two topological families from MN principle under DFUSPWM.
