Three-phase SPWM inverter switching instants

A three-phase sinusoidal PWM inverter is a six-switch bridge in which each leg is switched by comparing its own sinusoidal reference (120° apart from the others) with a common triangular carrier. The switching instants are the crossing points of the reference and the carrier.

Circuit

      +----+--------+--------+
      |    |        |        |
 Vdc/2   S1 D1    S3 D3    S5 D5
  ---  |    |        |        |
  O    +-a  +--b     +--c     |
  ---  |    |        |        |
 Vdc/2   S4 D4    S6 D6    S2 D2
      |    |        |        |
      +----+--------+--------+
        a, b, c -> 3-phase load

Operation

• References: $v_{ra} = A_r\sin\omega t$, $v_{rb} = A_r\sin(\omega t - 120^\circ)$, $v_{rc} = A_r\sin(\omega t - 240^\circ)$.
• Carrier: triangle of peak $A_c$, frequency $f_c = m_f f_r$ ($m_f$ odd multiple of 3).
• Leg a: $v_{ra} > v_c \Rightarrow$ S1 ON, $v_{aO} = +V_{dc}/2$; $v_{ra} < v_c \Rightarrow$ S4 ON, $v_{aO} = -V_{dc}/2$.
• Line voltage $v_{ab} = v_{aO} - v_{bO}$ is a train of pulses of $\pm V_{dc}$ and 0 whose widths vary sinusoidally.
• Fundamental (linear range $m_a = A_r/A_c \le 1$): $\hat V_{aO1} = m_a V_{dc}/2$, $V_{L1,rms} = 0.612\,m_a V_{dc}$.

 vc  /\  /\  /\  /\  /\  /\
    /  \/  \/  \/  \/  \/  \
 vra    .--''''--.
 ------'----------'----------
 vaO   _  ___ _____ ___  _
 +   _| ||   |     |   || |_
 -           (S1 ON when vra>vc)

How the switching instants of one leg are found

1. Natural sampling (analogue comparison). In one carrier half-period the carrier is a straight line. For a rising slope starting at $t_k$:

$$v_c(t) = -A_c + \frac{4A_c}{T_c}(t - t_k), \qquad t_k \le t \le t_k + \frac{T_c}{2}$$

The switching instant $t_s$ is the root of

$$A_r\sin\omega t_s = -A_c + \frac{4A_c}{T_c}(t_s - t_k)$$

which is transcendental and is solved numerically (or by an analogue comparator in hardware). At $t_s$ the leg changes state: S1 OFF → S4 ON on the rising slope (as the carrier rises above the reference), and S4 OFF → S1 ON on the falling slope.

2. Regular (uniform) sampling (used in microcontrollers/DSP). The reference is sampled once per carrier period at $t_k$ and held. Then the crossing is found from a straight-line equation:

$$\begin{aligned} \text{duty of S1: } d_k &= \frac{1}{2}\left(1 + m_a\sin\omega t_k\right) \\ t_{on,S1} &= d_k T_c, \qquad t_{on,S4} = (1-d_k)T_c \end{aligned}$$

The ON pulse of S1 is centred in the carrier period (symmetric sampling). For example, $m_a = 0.8$ at $\omega t_k = 90^\circ$ gives $d = 0.9$: S1 is ON for 90% of that carrier period.

3. Complementary gating with dead time. The pair in a leg (S1–S4) receives complementary signals; a short dead time $t_d$ (both OFF) is inserted at every transition to avoid a shoot-through short circuit. During $t_d$ the load current flows through a diode.

The same procedure applies to legs b and c using $v_{rb}$, $v_{rc}$, which gives three balanced pole voltages.