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Although I can understand both of the diagrams above, I am confused about the relations between them.

As marked in the per-phase equivalent circuit of a 3-phase induction motor above, we assume the 3-phase value is the vector sum of three per-phase equivalent circuits separated by 120 degree phase position.

Then according to the vector sum of three separated per-phase equivalent circuits:

  • the red arrow current signal = Ids (which is the magneting current from stator for the rotor to build magnetic field)?
  • Iexc = Iqs (which is the stator magnetic field current)?
  • Finally the cross product of Ids x Iqs is proportional to the torque output?

If it doesn't, how to match the equivalent circuit currents to the vector control dq currents?

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Generally speaking, the torque output is proportional do the active current (Iq) and flux is proportional to the reactive current (Id). You have to consider the machine model in both d and q frames separately. Please see source link below.

MODEL OF A THREE-PHASE INDUCTION MOTOR

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