Analysis of Field Oriented Control (FOC) Principle

Preface

FOC is indispensable for permanent magnet synchronous motor control. This chapter mainly introduces the basic principles of FOC control, coordinate transformation, and the mathematical model of permanent magnet synchronous motor in the synchronous rotating coordinate system, and simulates and analyzes the permanent magnet synchronous motor FOC control algorithm through Matlab/Simulink.

1. Basic principles of FOC

The basic idea of ​​the Field-Oriented Control (FOC) system is to obtain an equivalent DC motor model in a synchronously rotating coordinate system oriented according to the rotor magnetic field through coordinate transformation, control the electromagnetic torque and flux linkage in accordance with the control method of the DC motor, and then inversely transform the control quantity in the rotor flux linkage oriented coordinate system to obtain the corresponding quantity in the three-phase coordinate system to implement control. The specific process is shown in the figure below:

The most important principle of FOC is: orientation according to the rotor magnetic field, that is, keeping the rotor flux rotation vector always coincident with the d-axis in the dq coordinate system, and the q-axis orthogonal. By orienting according to the rotor magnetic field, the stator current is decoupled into the excitation component id and the torque component iq. The rotor flux is controlled by the current id, and the electromagnetic torque is controlled by the current iq, which is similar to the control of a DC motor. For surface-mounted permanent magnet synchronous motors SPM, the excitation component id = 0 is generally set, and all stator currents are used to generate electromagnetic torque.

The most important task of FOC is to achieve rotor flux orientation by continuously observing the rotor angle, that is, to keep the rotor flux rotation vector always coincident with the d-axis in the dq coordinate system, the q-axis is orthogonal, and the dq coordinate axis rotates synchronously with the rotor flux.

2. Coordinate Transformation

2.1. Clark coordinate transformation

Stationary coordinate transformation Clark transformation:

By adopting equal amplitude transformation, the current in the three-phase stationary coordinate system ABC is converted into the current in the two-phase stationary coordinate system αβ through the following formula:

Transformation results:

Since ia+ib+ic=0, in practice only two-phase currents in the three-phase stationary coordinate system are needed, which can be transformed by the following formula:

2.2.Park coordinate transformation

The current in the two-phase stationary coordinate system αβ is converted into the current in the synchronous rotating coordinate system dq, as shown in the following formula:

Transformation results:

3. Mathematical model of permanent magnet synchronous motor in synchronous rotating coordinate system

The PMSM mathematical model in the three-phase natural coordinate system is converted into a mathematical model in the synchronous rotating coordinate system through coordinate transformation. The d-axis of the synchronous rotating coordinate system is aligned with the rotor flux and keeps synchronous rotation, as shown below:

Stator voltage equation:

Stator flux equation:

Electromagnetic torque equation:

Equations of motion:

Substituting the stator flux equation into the voltage equation, the stator voltage equation can be obtained as follows:

At this time, the electromagnetic torque equation can be written as:

From the above formula, the mathematical model of PMSM in the three-phase natural coordinate system is transformed into a mathematical model in the synchronous rotating coordinate system through coordinate transformation, so that the mathematical model of PMSM is decoupled, and the PMSM can be controlled by imitating the control method of DC motor.

The overall control framework of FOC is shown in the figure below:

4. Matlab/Simulink simulation analysis of permanent magnet synchronous motor field oriented control

4.1. Voltage open-loop control

As shown in the figure above, the Vd and Vq voltages in the synchronous rotating coordinate system are directly given to realize the voltage open-loop control of the permanent magnet synchronous motor field orientation. The overall simulation block diagram of Matlab/Simulink is as follows:

4.1.1. Simulation circuit analysis

The voltage values ​​of Vd and Vq in the synchronous rotating coordinate system are directly given to realize the voltage open-loop control of the magnetic field orientation of the permanent magnet synchronous motor.

A normalization process is performed here to set the output voltage (modulation waveform saddle wave) range of the FOC voltage open-loop control between [0,1].

The main circuit includes an inverter circuit and a permanent magnet synchronous motor. The inverter circuit is shown in the figure below. The Average-Value Inverter module is used to directly generate a three-phase sinusoidal voltage. The permanent magnet synchronous motor uses a BR2804-1700 motor (the motor parameters are measured using ST Motor Proflier), and the parameters are as follows:

4.1.2. Analysis of simulation results

Set the open-loop input voltage Vd and Vq to 0 and 1. The saddle waveform output by the inverse Park transform and SVPWM algorithm is as follows:

Motor speed: 0.2s sudden load increase

Motor stator current:

Stator current value in dq coordinate system:

Stator voltage in dq coordinate system:

4.2. Current closed-loop control

In the voltage open-loop control, after adding the load, the stator current Id in the dq coordinate system is not equal to 0, which is about 0.036, indicating that the stator current is not fully used to generate electromagnetic torque. The current closed-loop control is introduced to accurately control the motor Id and Iq current values. The main function of the current loop is to start the motor with the maximum current during the starting process, and at the same time, to timely resist the fluctuation of the grid voltage, speed up the response speed of the dynamic system, and improve the stability of the system. The control block diagram is shown in the figure above.

The Matlab/Simulink overall simulation block diagram of permanent magnet synchronous motor current closed-loop control is shown below:

4.2.1. Simulation circuit analysis

The difference from voltage open-loop control is that the stator current is fed back, and the stator current in the synchronous rotating coordinate is set to Id_Ref and Iq_Ref. The set value and the feedback value Id and Iq of the stator current are used for PI control. The output of the PI controller is used as the voltage given by the permanent magnet synchronous motor to drive the PMSM.

The rest of the simulation is the same as the voltage open-loop control.

4.2.2. Analysis of simulation results

The current reference values ​​Id_Ref and Iq_Ref are set to 0 and 1, and the errors between the current reference values ​​and the current feedback values ​​of Id and Iq are output via the PI regulator to output voltages Vd and Vq for motor control.

Motor speed: 0.2s sudden load increase

Stator current value in the dq coordinate system: When the motor starts, it starts with the set maximum current of 1A. When the speed reaches the steady-state value, the current drops immediately, achieving an ideal and optimal starting transition process.

Stator voltage in dq coordinate system:

Electromagnetic torque:

4.3. Speed ​​outer loop and current inner loop dual closed loop control

In actual control, we are generally concerned about the change of speed, and expect the motor to change at the set speed. At this time, it is not easy to achieve only by the current closed loop, and the speed closed loop is added to realize the control of the speed. The output of the speed controller is the given of the current controller, and the output of the speed controller must be limited, because the output limit value of the speed controller determines the maximum allowable current of the motor used.

The Matlab/Simulink overall simulation block diagram of the permanent magnet synchronous motor speed outer loop and current inner loop double closed-loop control is as follows:

4.3.1. Simulation circuit analysis

Speed ​​closed-loop control is introduced on the basis of current closed-loop control. The output of the speed controller is used as the input of Iq current to form a double closed-loop control system with speed outer loop and current inner loop.

4.3.2. Analysis of simulation results

4.3.2.1 Set the target speed to 3200r/min

Motor speed: 1s sudden load increase

Motor stator current:

Motor rotor position:

Stator current value in dq coordinate system:

Stator voltage in dq coordinate system:

Electromagnetic torque:

4.3.2.1 Setting the target speed to a variable value

Target speed:

V. Summary

So far, the basic principle of permanent magnet synchronous motor FOC and the simulation part of Matlab/Simulink have been explained. The voltage open-loop control, current closed-loop control, and dual closed-loop control of the speed outer loop and current inner loop of the permanent magnet synchronous motor are consistent with the control concept of the DC motor. The permanent magnet synchronous motor is converted to a synchronous rotating coordinate system oriented by the rotor magnetic field through coordinate transformation in order to achieve the decoupling of the PMSM mathematical model, and to make the PMSM equivalent to a separately excited "DC motor". The PMSM is controlled according to the control concept of the DC motor. The parameter setting of the PID controller, the SVPWM control algorithm, and the engineering implementation of the permanent magnet synchronous motor magnetic field oriented vector control will be supplemented later.