Switch states of three-phase VSI, 180° mode

In a three-phase VSI with 180° conduction, each switch conducts for half a cycle, the two switches of a leg are complementary, and three switches (one per leg) are ON at any time. Gate signals are given in the order S1, S2, …, S6 every 60°, giving six switch states (modes) per cycle.

Circuit

     +-----+--------+--------+
     |     |        |        |
     |    S1 D1    S3 D3    S5 D5
    Vdc    |        |        |
     |     +--a     +--b     +--c
     |     |        |        |
     |    S4 D4    S6 D6    S2 D2
     |     |        |        |
     +-----+--------+--------+
      a, b, c -> star load (R per phase)

Switch states

Let the switching function of a leg be 1 when its upper switch is ON (phase on $+V_{dc}$) and 0 when its lower switch is ON.

($V = V_{dc}$.) The states 000 and 111 (zero states) are not used in six-step operation.

Phase voltage in a mode (star resistive load)

Mode 1: a and c are connected to $+V$, and b to the negative bus. Equivalent circuit: $R\parallel R = R/2$ in series with $R$:

$$i_b = \frac{V}{R/2 + R} = \frac{2V}{3R}, \quad v_{bn} = -i_bR = -\frac{2V}{3}, \quad v_{an} = v_{cn} = \frac{V}{3}$$

Mode 2: only a is on the positive bus; b and c are in parallel on the negative bus, giving $v_{an} = 2V/3$, $v_{bn} = v_{cn} = -V/3$. The other modes follow in the same way. In general, using the pole voltages $v_{aO}$, $v_{bO}$, $v_{cO}$ ($\pm V/2$):

$$v_{an} = \frac{2v_{aO} - v_{bO} - v_{cO}}{3}$$

Phase voltage waveforms

 mode       1    2    3    4    5    6
 wt       0    60   120  180  240  300  360
 van 2V/3       ____
      V/3 _____|    |____
        0                |
     -V/3                |____      ____
    -2V/3                     |____|
 vbn 2V/3                 ____
      V/3            ____|    |____
        0           |              |
     -V/3       ____|              |____
    -2V/3 _____|
 vcn 2V/3                           ____
      V/3 _____                ____|
        0      |              |
     -V/3      |____      ____|
    -2V/3           |____|

Each phase voltage is a six-step wave ($V/3$, $2V/3$, $V/3$, $-V/3$, $-2V/3$, $-V/3$), and $v_{bn}$ and $v_{cn}$ lag $v_{an}$ by 120° and 240°.

Results: $V_{ph,rms} = \frac{\sqrt2}{3}V_{dc} = 0.471V_{dc}$; fundamental peak $= \frac{2V_{dc}}{\pi}$; no triplen harmonics.