Extinction angle control for power factor

Extinction angle control is a power factor improvement method in which a forced-commutated switch is turned ON at the zero crossing of the supply voltage ($\omega t = 0$) and turned OFF at $\omega t = \pi-\beta$, where $\beta$ is the extinction angle. The output voltage is controlled by varying $\beta$. Because the current pulse is advanced, the fundamental input current leads the voltage by $\beta/2$.

Circuit

              P (+)
        +-------+-------+-------+
        |       |       |       |
       S1      S2       |       |
        |       |      Dm     Load
 a o----+       +--o b  |    (Ia)
        |       |       |       |
       D1      D2       |       |
        |       |       |       |
        +-------+-------+-------+
              N (-)
 S1, S2: forced-commutated switches (GTO/IGBT)
 Dm: freewheeling diode

Operation (highly inductive load, $i_o = I_a$ constant)

1. At $\omega t = 0$, S1 is turned ON; current path $a \to$ S1 $\to$ load $\to$ D2 $\to b$. $v_o = v_s$, $i_s = +I_a$.
2. At $\omega t = \pi-\beta$, S1 is turned OFF by its gate (forced commutation, since $v_s$ is still positive). Load current freewheels through Dm: $v_o = 0$, $i_s = 0$.
3. At $\omega t = \pi$, S2 is turned ON; path $b \to$ S2 $\to$ load $\to$ D1 $\to a$; $v_o = -v_s$, $i_s = -I_a$.
4. At $2\pi-\beta$, S2 is turned OFF and Dm conducts until $2\pi$.

 vs   ___
     /   \             /
 ---/-----\-----------/---
           \___/
 vo  ___       ___
    /  |      /  |        (sine from 0 to 180-b,
 --/---+-----/---+----     zero from 180-b to 180)
 is
 +Ia +------+
     |      |
 ----+      +--+      +---
               |      |
 -Ia           +------+
     0  180-b 180  360-b 360

Average output voltage

$$\begin{aligned} V_{dc} &= \frac{1}{\pi}\int_0^{\pi-\beta}V_m\sin\omega t\,d(\omega t) = \frac{V_m}{\pi}\big[-\cos\omega t\big]_0^{\pi-\beta} \\ &= \frac{V_m}{\pi}(1+\cos\beta) \end{aligned}$$

$V_{dc}$ varies from $2V_m/\pi$ ($\beta = 0$) to $0$ ($\beta = \pi$).

RMS output voltage

$$\begin{aligned} V_{rms}^2 &= \frac{1}{\pi}\int_0^{\pi-\beta}V_m^2\sin^2\omega t\,d(\omega t) = \frac{V_m^2}{2\pi}\left[\omega t - \frac{\sin2\omega t}{2}\right]_0^{\pi-\beta} \\ &= \frac{V_m^2}{2\pi}\left[\pi-\beta - \frac{\sin(2\pi-2\beta)}{2}\right] = \frac{V_m^2}{2\pi}\left[\pi-\beta + \frac{\sin2\beta}{2}\right] \end{aligned}$$ $$V_{rms} = \frac{V_m}{\sqrt2}\left[\frac{1}{\pi}\left(\pi-\beta+\frac{\sin2\beta}{2}\right)\right]^{1/2}$$

Power factor improvement

• Fundamental input current: $I_{s1} = \frac{2\sqrt2 I_a}{\pi}\cos\frac{\beta}{2}$, leading $v_s$ by $\phi_1 = \beta/2$.
• Displacement factor $= \cos(\beta/2)$ leading, compared with $\cos\alpha$ lagging for a phase-controlled converter.
• The converter can appear as a capacitive load and can partly compensate the lagging reactive power of other loads.
• Self-commutated devices are needed, which adds cost.