1 Introduction
In recent years, sensorless vector control of permanent magnet synchronous motors (PMSMs) has become a research hotspot. Currently, sensorless vector control of PMSMs has achieved good control performance in the medium and high speed ranges, but good control has not yet been achieved in the extremely low speed range (<1Hz). This is because sensorless vector control of PMSMs requires the use of back electromotive force (EMF), which is very small at extremely low speeds and is greatly affected by changes in motor parameters, leading to reduced control performance and making sensorless vector control at extremely low speeds and zero speeds impossible.
To achieve sensorless control of PMSMs in the extremely low speed range, researchers have proposed various control methods. Among them, the high-frequency signal injection method has been studied extensively, which uses the current response generated by the injected high-frequency stator voltage signal to estimate the rotor position [1]-[7]. These high-frequency signal injection-based methods all utilize the non-ideal characteristics of PMSMs, such as electromagnetic salient poles and saturation effects. Therefore, these methods are suitable for embedded permanent magnet synchronous motors (IPMSMs) with rotor salient poles, but the control effect on surface permanent magnet synchronous motors (SPMSMs) is not obvious.
This paper introduces a low-frequency signal injection method [8] and builds a simulation model to realize sensorless control of SPMSM in the extremely low speed range and zero speed range. This method estimates the motor speed by injecting a low-frequency d-axis stator current signal and using the generated back EMF response. It only uses the fundamental wave model of PMSM and does not depend on various non-ideal characteristics, so it is suitable for SPMSM control. This paper conducts a large number of simulations and analyzes the simulation results, which not only proves the effectiveness of the method, but also puts forward problems and directions that need further research.
2 PMSM Mathematical Model
The q-axis back electromotive force is defined as:
The electromagnetic torque is:
Where P is the pole pair number. The system's equation of motion is:
Where J is the moment of inertia and TL is the load torque.
3. Principle of Low-Frequency Signal Injection Method
In practical systems, estimation
From equation (7), the electromagnetic torque response caused by Icq can be obtained:
Substituting equation (8) into equation (7) and assuming a constant load torque, we obtain the speed response caused by harmonics:
Based on equations (5) and (9), the q-axis back EMF response induced by the injected signal can be obtained: The component of this response on the estimated q' axis is:Assuming the error angle is sufficiently small, we can obtain:
From the above derivation, it can be seen that if the control e <sub>cq </sub> is zero, the rotor position can be accurately estimated. Zeroing this value can be achieved by controlling the error angle to be zero. However, since the error cannot be directly obtained, an error function needs to be constructed such that when e<sub>cq</sub> = 0, c<sub>q</sub> = 0. Through derivation, we obtain:
From equation (13), after PI adjustment, the estimated speed value can be obtained:
Where Kp and Ki are the proportional and integral coefficients, respectively. Theoretically, the speed estimate can be obtained from equation (14), but in order to improve the dynamic response speed of the system, the steady-state speed value obtained from equation (4) is used instead.
The final speed estimate is obtained by superimposing the speed estimate obtained from the error signal onto the speed estimate obtained from the error signal.
Therefore, the rotor position angle is:
Figure (2) is a block diagram of the system control principle of the above low-frequency signal injection method.The above analysis shows that the low-frequency signal injection method introduced in this paper does not rely on the rotor salient poles and the non-ideal characteristics of the PMSM, but only utilizes the fundamental wave model of the PMSM. Therefore, from a theoretical analysis perspective, this method is applicable to the extremely low-frequency control of the SPMSM. The simulation results below also prove this conclusion. Figure 2 Block diagram of the control principle of the low-frequency signal injection method system.
4. Simulation Results and Analysis
This paper uses MATLAB/Simulink to simulate the proposed low-frequency signal injection method. The motor parameters used in the simulation are shown in Table 1.
Table 1. SPMSM parameters used in simulation
Based on the motor parameters used in the simulation, the frequency of the injected low-frequency d-axis stator current signal is 62.5Hz and the amplitude is 0.5A.
Figure 3. SPMSM at full load (1.7 Nm) for 1.5 s
The rotational speed changes abruptly (75rpm -> -75rpm).
Figure 3 shows the response waveform of the SPMSM when it suddenly changes from forward to reverse rotation during full-load operation at extremely low speed. Although the sudden change in speed causes a large pulsation, the system can reach steady state relatively quickly, and the steady-state error of the actual speed is very small. Figures 4 and 5 show the response waveforms of the SPMSM's load torque changing from zero to a full-load of 1.7 Nm at extremely low speed and zero speed, respectively. As can be seen from the figures, the SPMSM can operate stably in the extremely low speed or even zero speed range, regardless of no-load or full-load conditions. When the load changes abruptly, although there are large fluctuations, the system can recover to stability quickly, and the steady-state error of the actual speed is very small.Figure 4. Sudden change in load torque at 75 rpm (0->1.7 Nm)
Figure 5. Sudden change in load torque at zero speed (0->1.7Nm)
Figure 6 shows the response waveform of the SPMSM when the load torque changes abruptly from zero to -1.7 Nm at zero speed. As can be seen from the figure, the SPMSM can operate stably in the zero-speed region regardless of load changes. When the load changes abruptly, although there are large fluctuations, the system can quickly recover to stability, and the steady-state error of the actual rotational speed is very small.
Figure 6. Sudden change in load torque at zero speed (0->-1.7Nm)
The simulation results show that the low-frequency signal injection method proposed in this paper can achieve sensorless vector control of SPMSM in the extremely low-speed range and even the zero-speed range, with small steady-state error and good steady-state performance. However, some problems also exist. The simulation results show that when the speed or load changes abruptly, the speed pulsation is large, and the dynamic response speed of the system is also slightly slow. Therefore, in order to improve the dynamic response speed of the system and reduce the pulsation, further research is needed to combine the low-frequency signal injection method proposed in this paper with a more advanced observer to improve the control performance of SPMSM in the extremely low-speed range.5. Conclusion
This paper introduces a sensorless vector control method for ultra-low-speed permanent magnet synchronous motors (PMSMs) based on low-frequency signal injection. Theoretical analysis and simulation verification show that this method does not rely on the non-ideal characteristics of the PMSM and can be obtained solely from the fundamental frequency model. Therefore, it is applicable not only to embedded PMSMs but also to surface-mount PMSMs without salient poles. Compared with methods based on high-frequency signal injection, it has wider applicability. However, further research is needed to accelerate its dynamic response and reduce the significant speed and torque ripples during dynamic processes.
