Symmetrical angle control: average and RMS output

Symmetrical angle control is a power factor improvement technique for single-phase converters in which a self-commutated switch (GTO, IGBT, power transistor) conducts for an angle $\beta$ placed symmetrically about the peak of each half cycle of supply voltage: it is turned ON at $(\pi-\beta)/2$ and OFF at $(\pi+\beta)/2$. The fundamental input current then lies in phase with the voltage (displacement factor = 1).

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)

• S1 is turned OFF by a negative gate pulse (forced turn-off), not by the supply; the load current then transfers to Dm.
• The output is a series of "caps" of the sine wave centred at $90^\circ$ and $270^\circ$.

 vo        ___          ___
          |   |        |   |      caps of |vs|, width b
 ---------+---+--------+---+------
 is       +---+
  +Ia     |   |
 ---------+   +--------+   +------
  -Ia                  +---+
      (90-b/2)(90+b/2)(270-b/2)(270+b/2)

Average output voltage

The output repeats every $\pi$:

$$\begin{aligned} V_{dc} &= \frac{1}{\pi}\int_{(\pi-\beta)/2}^{(\pi+\beta)/2}V_m\sin\omega t\,d(\omega t) = \frac{V_m}{\pi}\left[\cos\frac{\pi-\beta}{2} - \cos\frac{\pi+\beta}{2}\right] \\ &= \frac{V_m}{\pi}\left[\sin\frac{\beta}{2} + \sin\frac{\beta}{2}\right] = \frac{2V_m}{\pi}\sin\frac{\beta}{2} \end{aligned}$$

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

RMS output voltage

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

Power factor

• $I_s = I_a\sqrt{\beta/\pi}$; fundamental $I_{s1} = \frac{2\sqrt2 I_a}{\pi}\sin\frac{\beta}{2}$ with $\phi_1 = 0$.
• $PF = \dfrac{I_{s1}}{I_s} = \dfrac{2\sqrt2\sin(\beta/2)}{\sqrt{\pi\beta}}$; e.g. at $\beta = 90^\circ$, $PF = 0.90$, whereas a full converter giving the same output ($\cos\alpha = 0.707$) has $PF = 0.64$.