A 3-phase grid-connected voltage source converter with DC link voltage of 1000 V is switched using sinusoidal Pulse Width Modulation (PWM) technique. If the grid phase current is 10 A and the 3-phase complex power supplied by the converter is given by (– 4000 – j3000) VA, then the modulation index used in sinusoidal PWM is ______. (round off to two decimal places).

To find the modulation index used in the sinusoidal Pulse Width Modulation (PWM) technique, we can follow these step-by-step derivations:

1. Identify the given parameters:
- DC link voltage, $V_{\mathrm{dc}}=1000\text{V}$
- Grid phase current, $I_{\mathrm{ph}}=10\text{A}$ (RMS value)
- 3-phase complex power supplied by the converter, $S=-4000-j3000\text{VA}$

2. Calculate the magnitude of the 3-phase apparent power ($\left|S\right|$):
$\left|S\right|=\sqrt{{(-4000)}^2+{(-3000)}^2}=5000\text{VA}$

3. Calculate the RMS value of the fundamental phase voltage ($V_{\mathrm{ph}}$):
The relation between the 3-phase apparent power, phase voltage, and phase current is given by:
$\left|S\right|=3V_{\mathrm{ph}}{I}_{\mathrm{ph}}$
Substitute the values into the equation:
$5000=3\cdot V_{\mathrm{ph}}\cdot 10$
$V_{\mathrm{ph}}=\frac{5000}{30}=\frac{500}{3}\text{V}$

4. Relate the phase voltage to the modulation index ($m$) and DC link voltage ($V_{\mathrm{dc}}$):
For a 3-phase voltage source converter using sinusoidal PWM, the peak value of the fundamental phase voltage is:
$V_{\mathrm{ph,peak}}=m\frac{V_{\mathrm{dc}}}{2}$
The RMS value of the fundamental phase voltage ($V_{\mathrm{ph}}$) is:
$V_{\mathrm{ph}}=\frac{V_{\mathrm{ph,peak}}}{\sqrt{2}}=\frac{m{V}_{\mathrm{dc}}}{2\sqrt{2}}$

5. Calculate the modulation index ($m$):
Substitute $V_{\mathrm{ph}}=\frac{500}{3}\text{V}$ and $V_{\mathrm{dc}}=1000\text{V}$ into the equation:
$\frac{500}{3}=\frac{m\cdot 1000}{2\sqrt{2}}$
Simplify the equation:
$\frac{500}{3}=m\cdot \frac{500}{\sqrt{2}}$
$\frac{1}{3}=\frac{m}{\sqrt{2}}$
$m=\frac{\sqrt{2}}{3}\approx \frac{1.4142}{3}\approx 0.4714$

Rounding off to two decimal places, the modulation index is 0.47 (or 0.471 if rounded to three decimal places).