Abstract : Addressing the challenges of understanding and modeling the direct torque control (DTC) theory of permanent magnet synchronous motors (PMSMs) during teaching, this paper details the MATLAB/Simulink modeling methods for each component of a PMSM DTC system. Based on the mathematical model of the PMSM in the αβ coordinate system, the sampled three-phase stator current and voltage are transformed and fed into the flux linkage and torque estimation model. Combining the motor rotor position, flux linkage, and torque error signals, the inverter switching vector is rationally selected to achieve motor speed regulation. Simulations are conducted under varying speeds and sudden load conditions. The results demonstrate that the system exhibits excellent speed, flux linkage, and torque responses, thus validating the model's effectiveness and providing a theoretical foundation for the hardware and software design of PMSM DTC.
Keywords : PMSM; DTC; simulation
Abstract: In order to deal with these problems about permanent magnet synchronous motor (PMSM) direct torque control (DTC) theory in the teaching process, which is difficult to understand, modeling difficulties and other issues, all aspects of the DTC system MATLAB / Simulink modeling method is introduced in detail. Based on the mathematical model in the αβ coordinate system, sampling to the three-phase stator current and voltage are admitted the flux and torque estimation model through the coordinate transformation, it is a reasonable choice of the inverter switching vector, combined with the rotor position, flux and the torque error signal, to achieve the purpose of the motor speed. The simulation is carried out in the case of changing the speed and suddenly increasing in load, the results show that the DTC system gives a good response to the rotation, flux and torque, which show that the model is effective, and the method provides a base for software and hardware design of an actual PMSM DTC.
Key words: PMSM; DTC; Simulation
introduction
With the rapid development of power electronics technology, microcomputer technology, rare earth permanent magnet materials and control theory, PMSM has become increasingly widely used due to its advantages such as small size, light weight, high efficiency, small moment of inertia and high reliability. Applying DTC strategy to PMSM control to improve the motor's fast torque response has become a research topic of interest.
Direct torque control theory was first proposed for asynchronous motors by German scholar M. Depenbrock and Japanese scholar I. Takahashi in the 1980s, and PMSMDTC theory was proposed by scholars such as Zhong.L, RahmanMF, and Hu.YW in the 1990s[6]. Its basic idea is to use the error between the given speed and the actual speed of the motor as the torque given signal output by the PI regulator; at the same time, the system calculates the magnitude of the motor flux linkage and torque by using the flux linkage model and torque model respectively based on the detected three-phase current and voltage values of the motor, and calculates the position of the motor rotor, the error between the given flux linkage and torque flux linkage and the actual value; finally, the inverter switching voltage vector is selected according to their state so that the motor can adjust the output torque according to the control requirements, and finally achieve the purpose of speed regulation. Since the calculation of motor speed and flux linkage has a great influence on the performance of the control system, simulation research is the most effective tool and means to obtain satisfactory torque calculation.
This paper uses MATLAB/Simulink simulation tools to simulate the PMSMDTC system, and details the construction of each control unit in the DTC system, providing a theoretical basis for the realization of digital control of PMSM AC servo system.
1. Direct Torque Control of Permanent Magnet Synchronous Motors
1.1 Mathematical Model of Permanent Magnet Synchronous Motor
The following assumptions are made about PMSM:
- The stator windings are three-phase symmetrical, with the axes of each phase winding differing from each other by 120 electrical degrees in space;
- There are no damping windings on the rotor, and the permanent magnets have no damping effect.
- Ignoring the effects of magnetic circuit saturation, hysteresis, and eddy currents, the superposition principle can be used for analysis;
- The back electromotive force is sinusoidal, and the stator current generates only a sinusoidal distributed magnetomotive force in the air gap, ignoring higher harmonics.
The PMSM voltage equations in the α-β coordinate system are as follows:
By shifting the direction and integrating, we obtain the flux linkage equation as follows:
The electromagnetic torque equation is
Equations of motion for electric drive systems
In the formula: α and β axis components of stator flux linkage; α and β axis components of stator current; α axis component of stator voltage; β axis component of stator winding resistance; p is the differential operator; ω is the rotor mechanical angular velocity; ω is the electromagnetic torque; ω is the number of pole pairs of the motor; ω is the load torque; J is the moment of inertia of the motor; B is the viscosity coefficient.
1.2 Direct Torque Control System
The principle block diagram of the DTC system is shown in Figure 1. It consists of modules such as voltage source inverter, PMSM, voltage calculation, sampling current 3s/2s, flux estimation, torque estimation, rotor position estimation, PI regulator, hysteresis comparator, and switching meter.
Figure 1. Block diagram of direct torque control system
1.3 Voltage Vector
Figure 2 shows a simplified diagram of the three-phase voltage-source inverter and motor connection principle. The DC bus voltage is [value missing]. Six switching transistors are used to form three bridge arms a, b, and c, respectively, using ideal switches 1 to 6. The upper and lower switching transistors of each bridge arm are interlocked for conduction. The switching status of the switching transistors in the three bridge arms a, b, and c is represented by the switching variables.
The corresponding inverter output voltage space vector can be expressed as:
Assuming the six switching transistors cycle in the following pattern: 456, 561, 612, 123, 234, 345, then the three phases of the inverter have a total of 8 possible state combinations. These include 6 non-zero voltage vectors (1, 2, 3, 4, 5) and 2 zero voltage vectors (1, 2, 3, 4, 5), distributed as shown in Figure 3. From the perspective of normal inverter operation, the first six states are valid, while the last two are invalid because the inverter has no voltage output in these states.
Figure 2. Simplified structure of a three-phase voltage-source inverter. Figure 3. Output space voltage vector.
2. Simulink Simulation Components of the System
This simulation system mainly consists of sub-modules such as a speed loop PI regulator, sampling current 3/2 conversion, stator flux estimation (including voltage calculation), torque estimation, section judgment, reasonable selection of switch meters, and definition of switch vector output.
2.1 Coordinate Transformation
In practical direct torque control systems, the sampled current is the three-phase current of the motor. Coordinate transformation is required to obtain the two-phase current in the two-phase stationary coordinate system for calculation. This requires the use of the 3/2 transformation.
The coordinate transformation formula is as follows:
Figure 4. 3/2 transformation of the sampling current
2.2 Stator flux linkage and torque estimation model
The three-phase voltage of the PMSM stator is calculated using equation (7):
This is the switch state matrix.
The stator flux linkage and torque estimation models can be constructed using equations (2), (3) and (7), as shown in Figures 5 and 6.
Figure 5. Stator flux linkage estimation model
Figure 6 Torque estimation model
2.3 Calculation Model for Magnetic Flux Amplitude and Angle
The calculation models for flux amplitude and angle can be obtained from formulas (8) and (9), as shown in Figures 7 and 8.
Figure 7 Magnetic flux amplitude calculation model Figure 8 Torque calculation model
2.4 Section Judgment Model
Referring to Figure 3, the following partitions are made as shown in Table 1. The segment where the stator flux linkage vector is located can be determined based on the component of the flux linkage in the α-β coordinate. The quadrant of the stator flux linkage vector is determined by the positive and negative signs of the components. Then, the specific position of the stator flux linkage vector is determined by equation (9). The implementation module is shown in Figure 9.
Table 1 Relationship between flux linkage segment and angle
Figure 9 Section Determination Model
2.5 Torque flux linkage error signal model
In the torque control system, the torque is given by the given speed through the output of the PI regulator. The error signals of flux linkage and torque are sent to the switching vector table along with the feedback segment value through the hysteresis comparator. The required voltage vector is output by looking up the table.
Figure 10 Torque and flux linkage error signal model Figure 11 Speed loop PI regulator
2.6 Switch selection model and other models
In a direct torque control system, when the angle between the applied voltage vectors is greater than a certain value, the flux linkage amplitude decreases; when the angle is less than a certain value, the flux linkage amplitude increases. When the voltage vector lags behind a certain value, the torque decreases; when the voltage vector leads a certain value, the torque increases. Let and represent the error states of the motor flux linkage and torque, respectively. A state of 1 is given when the given value is greater than the actual value, and a state of 0 otherwise. The states of and , along with the feedback segment values, allow for the selection of the switching voltage vector according to Table 2. S in Table 2 is a variable set to facilitate table lookup in Simulink, where .
Table 2 DTC System Switch Table
Figure 12 Switch selection table, inverter and motor model
The Lookup Table (2-D) parameters are set as follows:
Row index input values: int16([1,2,3,4]); Column index input values: uint16([1,2,3,4,5,6]);
Table data: uint16(reshape([1,2,5,6,5,3,4,2,4,1,6,3,6,5,2,1,2,4,3,5,3,6,1,4],4,6));
In an embedded MATLAB function, the voltage switching vector M is defined as follows:
function y = fcn(u)
y = [ 0; 0; 0; 0; 0; 0];
switch u
case 1
y = [ 0; 1; 0; 1; 1; 0]; % Corresponding voltage vector
case 2
y = [ 0; 1; 1; 0; 0; 1]; % Corresponding voltage vector
case 3
y = [ 0; 1; 1; 0; 1; 0]; % Corresponding voltage vector
case 4
y = [ 1; 0; 0; 1; 0; 1]; % Corresponding voltage vector
case 5
y = [ 1; 0; 0; 1; 1; 0]; % Corresponding voltage vector
case 6
y = [ 1; 0; 1; 0; 0; 1]; % Corresponding voltage vector
end
3. Simulation Results and Analysis
The system parameters are set as follows: sampling time is 10 seconds, linear bus voltage is 310V, dead time is 2ms, limit value is [-3, +3], number of motor pole pairs, stator resistance, given flux linkage, moment of inertia, viscosity coefficient, equivalent inductance of the direct and quadrature axes, and given speed is 50 rad/s. The motor starts under no-load at t=0s, the given speed is suddenly increased to 60 rad/s at t=0.2s, and a load torque of 0.7 Nm is suddenly applied at t=0.3s. The simulation time is 0.4s. The waveforms of motor speed, stator flux linkage vector, torque, and flux linkage are shown in Figures (a), (b), (c), and (d), respectively.
(a) Rotational speed waveform (b) Stator flux vector waveform
(c) Torque waveform (d) Flux flux waveform
The simulation results show that the motor starts up very quickly and can quickly track the given speed. When the given speed suddenly changes from 50 rad/s to 60 rad/s at t=0.2s, the speed can also track quickly. The torque is affected to some extent, but it can quickly stabilize automatically. When the load is suddenly applied at t=0.3s, the speed decreases within the allowable range but then stabilizes. The electromagnetic torque also quickly stabilizes and fluctuates around the set value of 0.7 Nm.
4. Conclusion
Based on the PMSM DTC theory, a PMSM DTC system was built using MATLAB/Simulink. Simulation results show that the waveform conforms to the theoretical analysis, and the system has good static and dynamic characteristics. Before building an actual system, simulation studies can significantly reduce manpower and material resources, and in particular, provide a theoretical foundation for the hardware and software design of PMSM DTC.
References
[1] Liu Yingpei, Wan Jianru, Liang Pengfei. Direct torque control of permanent magnet synchronous motor based on extended Kalman filter and space voltage vector modulation [J]. Proceedings of the CSEE, 2009, 29(27): 67-74.
[2] Zhong L, Rahman MF Analysis of Direct Torque Control in Permanent Magnet Drives [J]. IEEE Transactions on Power Electronics, 1997, 12(3): 528-535.
[3] Tian Chun, Hu Yuwen. Research on the theory and control scheme of direct torque control system for permanent magnet synchronous motor [J]. Journal of Electrical Engineering, 2002, (2): 8-11.
[4] Yang Jianfei, Hu Yuwen. Design of torque regulator for direct torque control of permanent magnet synchronous motor [J]. Proceedings of the CSEE, 2011, 31(9): 76-81.
[5] Tong Kewen, Zhang Xing, Zhang Yu, et al. Sliding mode variable structure control of permanent magnet synchronous motor based on novel reaching law [J]. Proceedings of the CSEE, 2008, 28(21): 102-106.
[6] Zhou Yangzhong, Hu Yuwen. Direct Torque Control of AC Motors [M]. Beijing: China Machine Press, 2009, 10.
[7] Xie Yunxiang, Lu Zhuqiang. Simulation modeling of direct torque control of permanent magnet synchronous motor based on MATLAB/Simulink [J]. Journal of South China University of Technology, 2004, 32(1): 19-23.
[8] Wang Chengyuan, Xia Jiakuan, Yang Junyou, et al. Modern Control Technology of Electric Machines [M]. Beijing: China Machine Press, 2009, 1.
[9] Zhou Yangzhong. Theoretical Research and Practice of Direct Torque Control for Electrically Excited Synchronous Motors [D]. Nanjing: Nanjing University of Aeronautics and Astronautics, 2006.
