Embedded Motor Control

While FOC algorithm explains how motor can be controlled, it doesn’t explain how electric motor control optimization should be done. More specifically, what are the i_d and i_q commands shall be given during different operation conditions. Using permanent magnet AC machine as an example, following equations describe the machine model:

The meaning of the equations has been explained in FOC Overview. Based on these, following conclusions can be obtained:

• From torque equation, it can be seen that there are different i_d i_q combinations that could achieve the same torque value. For a given torque value, a set of i_d i_q command with the lowest current magnitude shall be chosen. A curve showing the i_d i_q combinations with the minimal current magnitude for different torque values is called Maximum Torque Per Ampere (MTPA).
• Meanwhile, based on u_d and u_q voltage equations, MTPA commands cannot always be used. As explained in PWM Generation the maximum output of the u_d u_q is limited by dc link voltage. Plus, even to achieve the same i_d i_q commands, higher the motor speed, bigger the u_d u_q are needed. For a given speed and dc link voltage, the possible i_d i_q command values are limited by voltage ellipse. On each of the voltage ellipse, there is one i_d i_q set that could provide the maximum torque. The curve showing these i_d i_q combinations is called Maximum Torque Per Volt(MTPV).

Following shows an example of MTPA, MTPV, and voltage ellipse curves.

There are 3 different operation sections based on motor speed:

Low Speed Operation

As torque command goes up, the current trajectory goes from O to A, till it reaches the maximum current which is limited by power electronics device rating and motor winding limitation. Since the speed is low, current will hit current limitation first before reaching the voltage ellipse.

Mid Speed Operation

In this section, the current command trajectory can follow the MTPA when current values are low, but it will hit the voltage ellipse as current command goes up. It will then follow the voltage ellipse till it reaches the MTPV curve. As shown in following, with a fixed speed and dc link voltage, current command moves from O to A, and end up with B, which is the maximum torque that voltage can achieve.

High Speed Operation

In this section, the speed is so high that even with zero torque, a minus i_d current is needed to compensate the back EMF from the rotor permanent magnet field. The current command values are always on the voltage ellipse. Following shows an example for a high-speed operation, current command trajectory starts at O, and ends at A.

From the analysis above, optimized i_d and i_q command for a given torque can be obtained based on MTPA, MTPV, and also voltage ellipse. These values can be calculated be calculated by machine parameters which can be measured by Characterization. A common approach for software implementation is to build several 2d lookup tables for i_d(speed, torque command), and i_q(speed, torque command) for different dc link voltage values. id and iq values can be calculated using interpolation when actual voltage is within the predefined values.

Also, notice that all the discussions here are based on the assumption that copper loss is the biggest loss for the system operation. So, the target is always to use the smallest current to generate the same torque. While this is true for most of the time, the motor iron loss may also become significant especially for high speed operation, while there is a big flux-weakening current. If that is the case, the current command needs to be further tuned for different operation conditions.

Future Reference

More details of current command optimization are explained in following paper:

Bon-Ho Bae, etc. “New field weakening technique for high saliency interior permanent magnet motor“, 38th IAS Annual Meeting on Conference Record of the Industry Applications Conference, 2003.