Abstract: In modern AC servo systems, vector control principles and space vector pulse width modulation (SVPWM) technology enable AC motors to achieve performance comparable to DC motors. Permanent magnet synchronous motors (PMSMs) are complex, coupled nonlinear systems. This paper constructs a simulation model of a PMSM control system in the Matlab/Simulink environment by establishing and combining modules such as the PMSM body and the transformation from the d/q coordinate system to the a/b/c coordinate system. Simulation results demonstrate the effectiveness of the system model.
Keywords: Matlab/Simulink; permanent magnet synchronous motor; voltage space vector pulse width modulation; simulation
Abstract: In today's AC servo system, the vector control theory and SVPWM technique make the AC motor can achieve the performance as good as DC motor when designing the AC servo system. PMSM is a nonlinear system with significant coupling. This novel method for modeling and simulink of PMSM system in Matlab is proposed. In Matlab /Simulink, the isolated blocks, such as PMSM block, coordinate transformation from d/q to a/b/c block, etc, have been modeled. have been testified by the simulated result.
Key words: Matlab/Simulink; PMSM; SVPWM; simulation
0. Introduction
Permanent magnet synchronous motors (PMSMs) use high-energy permanent magnets as rotors and possess advantages such as low inertia, fast response, high power density, low loss, and high efficiency, making them one of the best actuators for high-precision, micro-feed servo systems. Permanent magnet AC servo systems composed of PMSMs have evolved towards digitalization. Therefore, establishing effective simulation models is of great significance. Research on PMSM modeling and simulation methods in Matlab has received widespread attention.
This paper introduces the principle of voltage space vector pulse width modulation and presents the simulation model of coordinate transformation module, SVPWM module and the entire PMSM closed-loop vector control. The simulation model structure diagram and simulation results are also given.
1. Voltage Space Vector Pulse Width Modulation Principle
1.1 Voltage Space Vector
The ultimate goal of inputting a three-phase sinusoidal voltage to a motor is to generate a circular rotating magnetic field in space, thereby producing a constant electromagnetic torque. Directly targeting this goal, the inverter and asynchronous motor are treated as a single unit, and the PWM voltage is controlled by tracking the circular rotating magnetic field. This control method is called "magnetic flux tracking control." The trajectory of the magnetic flux is obtained by adding the voltage space vectors, so it is also called "voltage space vector PWM control."
Space vectors are defined according to the spatial position of the windings to which voltages are applied. In Figure 1, A, B, and C represent the axes of the three-phase stator windings of a motor that are stationary in space. They are 120° apart in space. The three-phase stator voltages UA , UB , and UC are applied to the three-phase windings, respectively. Three voltage space vectors UA , UB , and UC can be defined. Their directions are always on the axes of their respective phases, while their magnitudes change sinusoidally with time, with a time phase difference of 120°.
Figure 1 Three-phase voltage vector
If we consider the plane in Figure 1 as a complex plane, then
(1.1)
The three-phase combined space voltage vector U can be written as...
(1.2)
Since they are all sinusoidal quantities, we can obtain the following using Euler's formula.
(1.3)
We can see that the composite space vector of the three-phase voltage space vector is a rotating space vector with an amplitude that is 1.5 times the voltage value of each phase, and its angular velocity of rotation is equal to the angular frequency of the sinusoidal voltage.
Vector representation of the voltage balance equations for magnetic flux linkage and current space vectors
(1.4)
When the speed is not too low, RI is relatively small, therefore
(1.5)
Equation (1.5) shows that the magnitude of the voltage vector is equal to the rate of change of the magnetic flux linkage, and the direction of the voltage vector is the direction of the magnetic flux linkage motion. In the speed control system, the motor is powered by a three-phase PWM inverter, as shown in Figure 2. To enable the motor to operate symmetrically, all three phases must be powered simultaneously, meaning that at any given moment, three devices in different bridge arms must be turned on simultaneously, while the other three power devices in the corresponding bridge arms are turned off.
Figure 2 Three-phase PWM inverter
The inverter has eight operating states: 001, 010, 011, 100, 101, 110, 111, and 000. Substituting the phase voltage values of the six non-zero switching states into equation (1.2) yields six space voltage vectors, as shown in Figure 3.
Figure 3 Basic space voltage vector
1.2 The role of the zero vector
By inserting a zero vector while a non-zero vector is applied, the motor's flux linkage endpoints "stop and go," thus altering the flux linkage's speed and making its trajectory approximately circular, thereby achieving constant flux variable frequency speed control. Changing the ratio of the non-zero vector's application time to the total application time changes both the frequency and amplitude of the output voltage.
1.3 Space Voltage Vector Control Algorithm
As mentioned above, the control process involves the actions of non-zero vectors and zero vectors. Non-zero vectors are used to control the trajectory of the magnetic flux, while zero vectors are used to change the speed of the magnetic flux.
Now, taking the operating range of U1 and U2 as an example, according to the principle of voltage and time product balance, any reference voltage vector Ur can be obtained.
Figure 4. Composite vector Ur of U1 and U2
2. Coordinate Transformation Module
The basic idea of vector control for three-phase permanent magnet synchronous motors is to control the AC motor as if it were a DC motor, that is, to simulate the control characteristics of a DC motor for the control of the permanent magnet synchronous motor. To simplify the induction motor model, the magnetomotive force generated by the current in the three-phase windings of the motor can be synthesized and decomposed according to the principle of superposition of planar vectors, so that the three-phase windings of the actual motor can be equivalently represented by two-phase orthogonal windings. Due to the orthogonality of the two-phase windings, the coupling between variables is greatly reduced.
The transformations used in vector control include: the transformation from a three-phase planar coordinate system to a two-phase planar rectangular coordinate system (Clarke transformation) and the transformation from a two-phase stationary rectangular coordinate system to a two-phase rotating rectangular coordinate system (Park transformation). The Matlab implementation of the coordinate transformation matrix is shown in Figures 5 and 6.
3. SVPWM module
SVPWM is mainly used to enable the motor to obtain a circular magnetic field with constant amplitude. When the motor is supplied with a three-phase symmetrical sinusoidal voltage, a circular magnetic flux is generated in the AC motor. Using this magnetic flux as a reference, an effective vector is generated by different switching modes of the inverter power devices to approximate the reference circle, and a near-sinusoidal current with a three-phase electrical angle difference of 120° is generated to drive the motor.
3.1 Sector Selection
Figure 7. Determining the sector where the vector is located.
3.2 Calculate X, Y, Z and T. X and T are defined as follows:
Table 1. Assignment Table for T<sub> X</sub> and T<sub> Y </sub>
3.3 Calculate the vector switching points Tcm1, Tcm2, and Tcm3
definition:
Then, Tcm1, Tcm2, and Tcm3 are assigned values according to Table 2 in different sectors.
Table 2. Assignment Table for Switching Points Tcm1, Tcm2, and Tcm3
The implementation in the Matlab Simulink environment is shown in Figures 7, 8, 9, and 10. The calculations of Tcm1, Tcm2, and Tcm3 can be implemented using a multiport switch.
4. PMSM Closed-Loop Vector Control Simulation Model
5. Simulation Results
To verify the correctness and effectiveness of the established simulation model, a simulation experiment was conducted. Given a rotational speed of 400 rad/s, the motor load started at t=0, and at t=0.1s, the load abruptly increased from 2 N·m to 8 N·m. The simulation time was 0.4s. The waveforms are shown in Figures 12, 13, and 14.
The simulation results show that after the motor is powered on, it quickly reaches its maximum torque (30 N·m) and then rapidly returns to a stable value (2 N·m). The speed increases linearly, quickly reaching the given value of 400 rad/s. In 0.1 s, the load torque abruptly changes from 2 N·m to 8 N·m, the speed experiences a slight oscillation before returning to the given value, and the stator current changes within 0.1 s.
6. Conclusion
This paper analyzes the principle and algorithm of voltage space vector control and obtains a mathematical model of permanent magnet synchronous motor. Using Matlab/Simulink software, a model of permanent magnet synchronous motor control system is constructed. The simulation results show that the system can operate smoothly and has good static and dynamic characteristics. The simulation results are consistent with the operating characteristics of permanent magnet synchronous motor and provide new ideas for the design and debugging of actual servo systems.
References
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