Motor FOC Control Basics

1 Background

This article uses permanent magnet synchronous motor to introduce its control method. The structure of permanent magnet synchronous motor is: the rotor is permanent magnet and the stator is three-phase AC winding.

The three-phase windings of a typical permanent magnet synchronous motor are arranged at an electrical angle of 120° in space, and three-phase alternating currents with a phase difference of 120° are passed through the windings:

Why do we need three-phase alternating current with a phase difference of 120°? In order to generate a circular rotating magnetomotive force with a constant amplitude.

l

Regarding the above figure: To design a permanent magnet synchronous motor, it is necessary to generate such a magnetomotive force, that is, a three-phase alternating current with a constant amplitude and a phase difference of 120° will generate a circular rotating magnetomotive force with a constant amplitude, which is exactly 1.5 times the maximum amplitude of the single-phase magnetomotive force, as shown in the figure below. Pay attention to the above figure! From the three figures of PhaseA, B, and C, it can be seen that the strength of the rotating magnetic field generated by the change of each phase current is also changing, but the strength of their synthetic rotating magnetic field (Resultant in the lower right figure) remains unchanged, and its speed is the same as the rotor speed. Their changes can also be viewed from the perspective of space vectors (Space vectors in the lower right figure).

It is not difficult to find from the above figure that if the amplitude of the three-phase alternating current is adjusted, the amplitude of the synthetic rotating magnetomotive force will also change accordingly. Of course, its magnitude is still 1.5 times the maximum amplitude of the single-phase magnetomotive force, as shown in the figure below.

What does the change of magnetomotive force mean? The change of motor torque, so we need to find the relationship between three-phase AC and motor torque. It may be complicated to find the relationship between the two directly, and it is difficult to implement control. So is there any way to do some processing first? Note that the three-phase AC passes through the stator coil, and the stator coil is stationary, but the magnetic field generated by the coil interacts with the magnetic field generated by the permanent magnet of the rotor. The rotor is rotating, and the magnetic field generated by the coil is also rotating synchronously. Can we find some clues through this information?

2 Clark and Park transforms

For three-phase alternating current, can we turn it into two phases? Just like when we look at a three-dimensional problem, I always tend to convert it to two dimensions to solve it. The three-phase alternating current has a phase difference of 120°, so can we construct a coordinate system α-β and use it to represent the three-phase alternating current, as shown below:

That is to say, project Ia, Ib, Ic to the coordinate system α-β, and it is not difficult to get the transformation formula:

Written in matrix form:

This is the Clark transformation. The coefficient 2/3 or sqrt(2/3) is usually added before the above matrix, representing equal amplitude transformation and equal power transformation respectively. Through the Clark transformation, the original three-phase alternating current is converted into a two-phase sinusoidal current.

Note: Three-phase AC before Clark transformation

Note: Two-phase sinusoidal current after Clark transformation

If the rotor is rotating at this time, what is the relationship between the rotating rotor permanent magnet and the coordinate system α-β? It is equivalent to the rotor permanent magnet rotating at a speed ω in the coordinate system α-β. If a coordinate system dq is established, the d axis and the magnet NS line completely coincide with each other, and the positive direction is the N pole direction; while the q axis is tangent to the NS line, and the positive direction is consistent with the rotation direction. As shown below:

Then project the coordinate system α-β onto the coordinate system dq. Since the rotor rotates at a speed ω relative to the stator, there is a rotation angle θ (θ=ωt) between the two coordinate systems. Their relationship is as follows:

In fact, the transformation from the coordinate system α-β to the coordinate system dq is the Park transformation, which converts the two-phase sinusoidal wave signal into a two-phase DC signal.

Note: Two-phase DC after Park transformation

Why is the above coordinate system dq established in this way? The reason is that through Park transformation to Id and Iq, the d-axis direction does not cut the magnetic flux lines, so the current Id does not generate the Lorentz force; while the q-axis direction can cut the magnetic flux lines, so the current Iq can generate the Lorentz force.

Combining the Clark transformation and Park transformation above, a three-phase AC can be converted into a two-phase DC, as shown below:

3 SVPWM

3.1 SVPWM Input

To obtain the target electromagnetic torque, we only need to reasonably control Id and Iq. According to the torque equation of the permanent magnet synchronous motor, we know that:

The first part is generated by the interaction between the rotor permanent magnet flux and the three-phase stator winding flux; the second part is the reaction torque generated by the change in magnetic resistance caused by the salient pole effect. It can be seen that because the rotor permanent magnet flux is constant, to adjust the electromagnetic torque of the permanent magnet synchronous motor, only Id and Iq need to be adjusted. According to the definition of the coordinate system dq, Id does not cut the magnetic flux lines, and Iq cuts the magnetic flux lines, that is, the electromagnetic torque is related to Iq, and Iq can be controlled.

Let Id=0, and the target Iq (Iq=T/pψm) can be determined based on the target electromagnetic torque. In this way, the overall FOC control framework of the motor can be obtained:

At this point, the key is the processing of the black box part, which is usually processed using SVPWM. Why use SVPWM? Please refer to: How to deeply understand SVPWM

What is the specific function of SVPWM? In fact, it converts Vα, Vβ into Va, Vb, Vc. The black box part is revealed as follows:

From the above figure, we can know how to obtain the input Vα and Vβ of SVPWM:

• Target Id, Iq (dark blue box): Let the target Id = 0, and the target Iq can be obtained according to the electromagnetic torque;

• Actual Id and Iq (green path): collected by sensors to Ia, Ib, Ic, transformed to Iα, Iβ by Clark, and then transformed to Id, Iq by Park;

• With the target and actual Id, Iq, the target Vd, Vq can be obtained by using PI control;

• Vd, Vq are transformed to Vα, Vβ through inverse Park transformation.

3.2 SVPWM Output

What about the output of SVPWM? Here we need to first understand the circuit principle of the inverter. The inverter converts DC power into AC power. It uses 3 sets of half-bridge MOS circuits to achieve this, as shown below:

Each half-bridge MOS circuit consists of two MOS tubes, an upper bridge arm and a lower bridge arm, with an output line in the middle. By controlling the on and off of the upper and lower bridge arms, the output can be controlled, as shown below:

Note: The MOS tube can be regarded as a switch, and its function is to control the on and off of the bridge arm.

When the upper bridge arm is on and the lower bridge arm is off, the output OUT is connected to the power supply; when the upper bridge arm is off and the lower bridge arm is on, the output OUT is grounded; when both the upper and lower bridge arms are on, this is not used in SVPWM. In this way, an output line is connected to a phase line of the motor, and the most basic motor drive circuit is completed using three sets of half-bridge MOS circuits. That is, the conversion of DC power into three-phase AC power is achieved through the on-off control of the upper and lower bridge arms of the three sets of half-bridge MOS tube circuits of the inverter.

So how to control the on and off of the MOS tube to achieve the conversion of DC into three-phase AC? Here we need to use PWM, which is defined as follows:

That is, the duty cycle of PWM is continuously changed at a certain frequency (i.e., the period T in the figure above). For example, in the figure above, the duty cycle of the first period T is controlled to be 20%, which can reach an average voltage of 20V. The duty cycle of the second period T is controlled to be 50%, which can reach an average voltage of 50V. And so on. As long as the frequency is fast enough, PWM can be used to control the on and off of the MOS tube, thereby generating an approximately sinusoidal current, as shown below:

From the above analysis, we can know that we can get 3 groups of PWM through the SVPWM algorithm, and control 3 groups of half-bridge MOS circuits respectively. In this way, we have clarified the input and output of SVPWM. Next, let's understand what SWPWM is specifically

3.3 What is SVPWM

The overall logic of SVPWM is as follows:

1. According to the inverse Park transformation, we get Vα and Vβ, which can be synthesized into a vector Vref in the coordinate system α-β.

2. By controlling the three groups of half-bridge MOS circuits of the inverter, six basic voltage vectors and two zero voltage vectors can be constructed;

3. Using several of these eight voltage vectors, any vector can be synthesized, such as Vref, which means that the relationship between Vα, Vβ and several voltage vectors can be established;

4. The eight voltage vectors are all related to Va, Vb, and Vc, so the relationship between Vα, Vβ and Va, Vb, and Vc can be established. Of course, as mentioned in the output of SVPWM above, the relationship between Vα, Vβ and Va, Vb, and Vc is actually reflected by Vα, Vβ and the PWM of the half-bridge MOS circuit corresponding to the three-phase power a, b, and c.