DQ0 coordinate transforms

The DQ0 transform is a space vector transformation of three-phase time-domain signals from a stationary phase coordinate system (ABC) to a rotating coordinate system (DQ0). 

Every machine that is modeled in the DQ0 coordinate frame is interfaced to the ABC coordinate frame (the real world) using the following ABC↔DQ0 transform structure: 

The matrix representing the expressions for the V_D and V_Q controlled voltage sources is: 

The matrix representing the expression for the Ia, Ib, and Ic controlled current sources is: 

If the reference frame is stationary, then θ is always zero. This is the case for the Induction machine squirrel cage model in Multisim. 

If the reference frame is rotating synchronously with the rotor, then θ is the instantaneous mechanical angle of the rotor. This is the case for the Synchronous permanent magnet model in Multisim.