3-Phase Inverters

3-Phase Systems

Consists of three sets of voltages or currents, each 120 degrees out of phase with the others

• Voltage or current waveforms for each phase (a, b, and c) are sinusoidal and displaced by 120 degrees
• Benefits include educed power losses, and allow creating of rotating magnetic field machine

Loss Reduction

$3I_L^2R_L$ rather than $6I_L^2R_L$

voltages in each phase.

• Balanced phase currents:

  $$ \overline{I_A} = I_L \angle \theta_{A} \newline \overline{I_B} = I_L \angle (\theta_{A} - 120^\circ) \newline \overline{I_C} = I_L \angle (\theta_{A} + 120^\circ)

  $$

• KCL at neutral point

  $$ \overline{I_N} = - (\overline{I_A} + \overline{I_B} + \overline{I_C}) = 0 $$

Smooth Rotating Magnetic Field

The second benefit of 3-phase systems is the ability to generate a smoothly rotating magnetic field, which is crucial for the operation of AC motors. The rotating field results from the 120-degree phase shift between each phase, allowing the magnetic field to rotate consistently within the motor

Generating 3-Phase Voltages from a DC Source Using an Inverter

The switching waveforms for each phase (A, B, C) are generated by using Sinusoidal Pulse Width Modulation (SPWM)

• SPWM (Sinusoidal Pulse Width Modulation):

  • SPWM is used to control the inverter switches. A triangular waveform (carrier signal) is compared with the desired sinusoidal reference signal (modulating signal).
  • The points where the reference signal intersects the triangular waveform determine the switching states of the inverter transistors, producing a PWM signal that approximates a sinusoidal output.

• Modulating signals for each phase

  $$ m_A(t) = k \cos(\omega_{out} t + \theta_A) \newline m_B(t) = k \cos(\omega_{out} t + \theta_A - 120^\circ) \newline m_C(t) = k \cos(\omega_{out} t + \theta_A + 120^\circ)

  $$

Output Voltage Calculations

• Average phase voltages

  $$ km $$

• Line to line voltage

  $$ \langle V_{AB}(t) \rangle = \langle V_{AN}(t) \rangle - \langle V_{BN}(t) \rangle $$

• Balanced line to line voltage

  $$ \langle V_{AB}(t) \rangle = \frac{k V_{in} \sqrt{3}}{2} \cos(\omega_{out} t + \theta_A + 30^\circ) \newline \langle V_{BC}(t) \rangle = \frac{k V_{in} \sqrt{3}}{2} \cos(\omega_{out} t + \theta_A + 30^\circ - 120^\circ) \newline \langle V_{CA}(t) \rangle = \frac{k V_{in} \sqrt{3}}{2} \cos(\omega_{out} t + \theta_A + 30^\circ + 120^\circ) \newline \text{Final line-to-line voltage in polar form: } \overline{V_{AB}} = \frac{k V_{in}}{2 \sqrt{2}} \left( \sqrt{3} \angle 30^\circ \right)

  $$