Single-phase square-wave inverter rms and fundamental

A single-phase square-wave inverter switches the load alternately to $+V_s$ and $-V_s$ for equal half periods, giving a square-wave ac output whose frequency is set by the switching rate and whose amplitude is fixed by $V_s$.

Circuit and operation (full bridge)

     +-------+-----------+
     |       |           |
     |    S1 D1       S3 D3
    Vs       |           |
     |       A---[Load]--B
     |       |    v0     |
     |    S4 D4       S2 D2
     |       |           |
     +-------+-----------+

• $0 < t < T/2$: S1, S2 ON, $v_0 = +V_s$.
• $T/2 < t < T$: S3, S4 ON, $v_0 = -V_s$.
• With an inductive load, current continues after each reversal through the feedback diodes (D3, D4 or D1, D2) until it changes sign.
• Half bridge version (two switches, centre-tapped source): $v_0 = \pm V_s/2$.

 v0  +Vs |________
         |        |
       0 |--------+--------- t
         |        |________
     -Vs   S1,S2     S3,S4
         0       T/2       T

RMS value of output voltage

$$V_{0,rms} = \left[\frac{2}{T}\int_0^{T/2}V_s^2\,dt\right]^{1/2} = V_s$$

(Half bridge: $V_s/2$.)

Fundamental component

The wave is odd with half-wave symmetry, so $a_0 = a_n = 0$:

$$\begin{aligned} b_n &= \frac{2}{\pi}\int_0^{\pi}V_s\sin n\theta\,d\theta = \frac{2V_s}{n\pi}(1 - \cos n\pi) \\ &= \frac{4V_s}{n\pi}\ \ (n\ \text{odd}), \qquad 0\ \ (n\ \text{even}) \\ v_0(t) &= \frac{4V_s}{\pi}\Big[\sin\omega t + \frac{1}{3}\sin3\omega t + \frac{1}{5}\sin5\omega t + \dots\Big] \end{aligned}$$ $$\begin{aligned} V_{1m} &= \frac{4V_s}{\pi} = 1.273\,V_s \\ V_{1,rms} &= \frac{4V_s}{\sqrt2\,\pi} = 0.9003\,V_s \end{aligned}$$

Half bridge: $V_{1,rms} = \dfrac{2V_s}{\sqrt2\,\pi} = 0.45V_s$.

Performance

• $n$th harmonic: $V_n = V_1/n$ (HF$_3$ = 33.3%).
• $THD = \sqrt{V_{0,rms}^2 - V_{1,rms}^2}/V_{1,rms} = \sqrt{1 - 0.9003^2}/0.9003 = 48.3\%$.
• Output voltage cannot be varied without changing $V_s$; PWM is used for that.

Example: $V_s = 230$ V gives $V_{0,rms} = 230$ V and $V_{1,rms} = 207.1$ V.