Modeling and simulation verification based on mathematical models using Matlab-simulink

Recently, I have done some study and summary on the control methods of permanent magnet synchronous motor (PMSM). Since there is a lot of content involved, I plan to share it in several notes, mainly focusing on the understanding of SVPWM control algorithm and modeling and simulation verification based on mathematical models using Matlab-simulink.

1. Introduction

At present, the definition of permanent magnet synchronous motors at home and abroad is to define brushless DC motors (BLDCM) and permanent magnet synchronous motors (PMSM) based on the stator current waveform. Brushless motors with trapezoidal or square wave current waveforms are defined as BLDCM, and brushless motors with sinusoidal current waveforms are defined as PMSM.

SVPWM (Vector Pulse Width Modulation) is a current control strategy for PMSM. The basic idea is to control the motor through different switching modes of the inverter bridge so that the generated voltage/current/flux trajectory is as close to the ideal circle as possible, so that the operation of the motor has relatively ideal performance.

The concept is as shown in the figure. When a point moves in an ideal circle at a constant speed in space, the waveform generated in the time domain is a sine wave. Of course, it can also be understood in reverse. The trajectory of the sine wave in space is a circle. The denser the trajectory of the discrete points moving in the time domain, the closer the movement in space is to the ideal circle.

2. Inverter bridge and motor principle

For the two-level three-phase inverter bridge arm, the voltage on the DC side is Udc as shown in the figure below. For the three-phase bridge arms a, b, and c, each bridge arm has two upper and lower switch tubes (such as MOS), and the upper and lower tubes cannot be turned on at the same time. In order to facilitate control calculations, the switch of a group of upper tubes or a group of lower tubes is usually defined as 1/0. According to the switch status of the three-phase bridge arms A, B, and C, 8 states can be combined using three-bit binary coding.

In real motor control conditions, the inverter is a real component. By controlling the conduction and disconnection of these six switches, the required sinusoidal voltage can be generated on the three motor stator phases on the right in conjunction with the DC voltage source on the left. (The switching states of the six switches are discrete, so the algorithm is very suitable for discrete digital systems)

If the three-phase upper tube is (100), it can be calculated;

Solved;

For the remaining seven pairs of combinations, the phase voltage and composite voltage can be calculated in the same way as shown in the following table.

For a three-phase synchronous generator, take the following figure as an example; on the stator side, the stator windings are arranged counterclockwise at intervals of 120°. The excitation current is passed through the excitation winding on the rotor, and the rotor core is magnetized. When the rotor rotates counterclockwise, the rotating magnetic field will cut the A, B, and C phase windings in turn. According to the law of electromagnetic induction, an electromotive force will be induced on the ABC windings, and because the windings are 120° apart, the induced electromotive force also differs by 120°, which is the three-phase symmetrical voltage we are familiar with.

For synchronous motors, its working principle is exactly the opposite of that of generators: a three-phase symmetrical voltage is applied to the stator ABC windings to generate a counterclockwise rotating magnetic field in the air gap. The rotating magnetic field drags the rotor poles, and after overcoming the static friction, the rotor will begin to rotate. The key is how to control the three-phase electromotive force of the stator so that it can generate a standard circular magnetic field. The more circular the magnetic field is, the higher the motor control accuracy.

3. SVPWM control principle

For space vector pulse width modulation, I mainly explain it through the idea of ​​simulation modeling. In order to make the inverter output a three-phase sine wave, the commonly used modulation strategy is: given a reference voltage signal, by modulating the PWM drive switch on and off, the output voltage of the ABC three-phase approaches the reference input voltage.

The instantaneous expression of the reference three-phase voltage is as follows; the three-phase symmetrical voltage formula added to the three-phase coordinate system will synthesize the corresponding voltage vector, which rotates counterclockwise with the change of time and the amplitude remains unchanged. Here we need to build the concept of coordinate system. ABC is a fixed three-phase coordinate system, and sinusoidal quantities are distributed in each coordinate system.

ABC is 120 degrees apart in space. According to the counterclockwise mode, A lags behind B and C. The quantities on the three coordinate axes of ABC differ by 120 degrees in time, and the quantity on the A axis leads the quantity on the B axis and the quantity on the C axis. As the sinusoidal quantity changes with time, the synthetic vector is also constantly changing. The coordinate system here originates from the stator and rotor position of the motor. The schematic diagram is as follows: Synthesized space voltage vector expression; This formula is relatively important, so I will explain it in detail here. The expression is inversely transformed into the Fourier series through the Euler formula;

The final formula has only two components on the imaginary and real axes, and the imaginary axis is 90 degrees ahead of the real axis. In this way, the quantities in the original three directions can be used to express the resulting vector using quantities in two directions. **At this time**** let the real axis be the α axis and the imaginary axis be the β axis. This is what we call the CLARK transformation;

****Uα=Ua-1/2(Ub+Uc)

Uβ=sqrt3/2(Ub-Uc)

**Ualpha, Ubeta are the components in the two-phase stationary coordinate system, and the corresponding coordinate system is as shown in the figure below;

Substituting the three-phase voltage into the calculation, we can get:

** Uα leads Uβ by 90 degrees, and the amplitudes are equal, but the amplitude of the mathematical expression is already 3/2 times the original three-phase voltage, which obviously violates the law of physical quantity, so in the control algorithm, a compensation coefficient of 2/3 is usually multiplied to offset this multiple, which is called equal amplitude transformation. ** (In addition, equal power transformation is used when calculating power, and the derivation is similar, so I won’t go into details)

** Sector Division **

According to the eight switching states of the inverter bridge, six basic non-zero vectors and two zero vectors can be obtained. The amplitudes of the non-zero vectors are equal, both 2/3Udc, and the angles differ by 60°. Therefore, connecting the vertices of the six non-zero vectors is exactly a regular hexagon, and two adjacent vectors and the sides of the hexagon form an equilateral triangle. There are a total of 6 equilateral triangles, which are usually called sectors.

The trajectory of the synthetic vector in the sector is a circle, and the peak trajectory of our phase voltage is the inscribed circle of the six sectors. You can imagine that the equivalent vector trajectory points generated by the modulation switch tube are discrete. The more times the synthetic vector rotates one circle, the smoother the vertex trajectory will be and the closer it will be to a circle.

Here we use the specific example of vector synthesis in the first sector to explain, based on the principle of volt-second balance; in each sector, by selecting two adjacent voltage vectors and a zero vector, any voltage vector in the sector can be synthesized according to the principle of volt-second balance.

As shown in the figure, if the duration of the synthesis voltage vector is 1S, its effect is equivalent to the vector effect produced by U4 acting for 0.5S + U6 acting for 0.5S;

The TPWM period in the formula limits the time for the components to act, so the area outside the circle cannot be synthesized;

Modulation

The amplitude of the maximum phase voltage is the inscribed circle of the six sectors. We can calculate its value based on the trigonometric formula. Regardless of whether equal amplitude transformation is considered or not, the final bus voltage is three times the square root of the phase voltage. Then, from the line voltage = Sqrt(3)*phase voltage, we can conclude that the modulation index of our inverter bridge = 1, which means that the theoretical voltage utilization rate reaches 100%.

Finally, use Mtlab-Smulink to build the Clark transform; use the sine wave production line module to generate three sine waves with a phase difference of 120°, with an amplitude of 40V and a frequency of 50HZ;

After coordinate transformation, two sinusoidal quantities on the αβ axis are generated. After simulation, the waveform is correct and the amplitude is consistent before and after the transformation.