Guide to field-oriented control: Step 4

Modulate the correction voltages onto the motor windings. Combining Vd and Vq using vector math results in a voltage vector which we need to apply to the stator windings. But Vd and Vq exist on the rotating reference frame. The stator windings exist on the stationary reference frame. (You can probably see where I am going with this.) In order to put these voltages onto the motor windings, we first need to jump off of the rotor, and translate Vd and Vq into equivalent waveforms in the stationary frame. So we take the Park transform (which is what got us up on the rotor to begin with), and rearrange the terms to solve V_α and V_β.

Equation 1: Inverse Park transform.

When finished with this step, we now have two voltage waveforms: one on the α axis, and one on the β axis. When plotted as a function of rotor angle over time, the result will be a cosine wave and a sine wave, respectively. These waveforms simply represent the magnitude of the net voltage vector (which is spinning) when it is reflected onto the stationary α axis and β axis. This is graphically represented in figure 1.

Figure 1: Graphical representation of the inverse Park transform for a spinning rotor.

If we had a two-phase motor, we would be done. All we would have to do is drop the scaled instantaneous values of V_α(t) and V_β(t) into the PWM module, and congratulate ourselves for our motor control prowess. But since we have a three-phase machine, we have to convert V_α and V_β into three voltage values (Va, Vb, and Vc), which can be directly applied to the three motor windings. To do this, Edith Clarke once again rides to the rescue by providing the inverse Clarke transform. Because the zero-sequence (common mode) voltage term is zero, the inverse Clarke transform takes the following form:

Equation 2: Inverse Clarke transform.

We now have our three voltage values! Put them in the PWM module and then exit the Interrupt Service Routine. By the time we get our next interrupt, the three voltages should have done their work on the motor to hopefully drive the motor currents closer to their desired values. We then do the whole process over again. On a microcontroller such as TI's C2000 TMS320F28069, all of these calculations can be done in less than 12µS! The entire FOC process is summarised in figure 2.

Figure 2: Overview of FOC Control.

Hopefully by now you have a good understanding of how FOC can be used to control the torque of a permanent magnet synchronous motor. But what about other motor topologies? In my next article, we will take what we have learned so far and use it to control the torque on an AC induction motor.

About the author
Dave Wilson is Senior Industrial Systems Engineer for C2000 Microcontrollers at Texas Instruments.