180° three-phase inverter and fundamental phase voltage

In 180° conduction each switch of the six-switch bridge conducts for half a cycle, three switches are ON at a time (one per leg), and firing is in the order S1 to S6 at 60° intervals. With a star resistive load this gives a six-step phase voltage.

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)

Operation

In mode 1, a and c are on the positive bus and b on the negative bus: total resistance $R/2 + R = 1.5R$, so $v_{bn} = -2V/3$ and $v_{an} = v_{cn} = V/3$. Other modes follow similarly.

 van 2V/3    ___
      V/3 __|   |__
        0 ----------+----------- wt
     -V/3           |__     __
    -2V/3              |___|
 vab   V  _______
        0        |___       ___
       -V            |_____|
          0  60 120 180 240 300 360

Line voltage $v_{ab}$: $+V$ for 0–120°, 0 for 120–180°, $-V$ for 180–300°, 0 for 300–360°.

Fundamental component of phase voltage

Method 1 — Fourier analysis of the six-step wave (odd, half-wave symmetric):

$$\begin{aligned} b_n &= \frac{2}{\pi}\int_0^{\pi}v_{an}\sin n\theta\,d\theta = \frac{2V}{3n\pi}\Big[2 + \cos\tfrac{n\pi}{3} - \cos\tfrac{2n\pi}{3}\Big] \\ b_1 &= \frac{2V}{3\pi}\big[2 + 0.5 + 0.5\big] = \frac{2V_{dc}}{\pi} \end{aligned}$$

Method 2 — from the line voltage: the 120° quasi-square $v_{ab}$ has fundamental peak $\frac{4V_{dc}}{\pi}\cos 30^\circ = \frac{2\sqrt3V_{dc}}{\pi}$; dividing by $\sqrt3$ gives the same phase value.

$$\begin{aligned} \hat V_{ph1} &= \frac{2V_{dc}}{\pi} = 0.6366\,V_{dc} \\ V_{ph1,rms} &= \frac{\sqrt2\,V_{dc}}{\pi} = 0.4502\,V_{dc} \end{aligned}$$

Full series: $v_{an} = \sum_{n=6k\pm1}\frac{2V_{dc}}{n\pi}\sin n\omega t$ (triplen harmonics absent). Total rms phase voltage $= \frac{\sqrt2}{3}V_{dc} = 0.4714V_{dc}$. Example: $V_{dc} = 600$ V gives a fundamental of 381.97 V peak (270.09 V rms).