Full-speed sensorless control system of synchronous reluctance motor with flux saturation model | Scientific Reports

Scientific Reports volume 15, Article number: 9048 (2025) Cite this article

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To solve the problem that the rotor estimation accuracy is greatly affected by parameter nonlinearity and high frequency oscillation, when the synchronous reluctance motor adopts the sensorless control system.This article proposes a new flux saturation model considering cross saturation effect applied to sensorless control of SynRM. The magnetic flux saturation data obtained by the hysteresis voltage injection method is used to estimate the model parameters, and the motor model under accurate magnetic flux saturation is obtained. The proposed magnetic flux saturation model is applied to the position sensorless control system of the synchronous reluctance motor. The proposed magnetic flux saturation model can effectively represent cross saturation and satisfy the reciprocity condition, and solve the nonlinear relationship between SynRM current and flux linkage. Finally, builting an experimental platform to verify the validity and correctness of the theoretical and experimental analysis.

Synchronous reluctance motor has been widely concerned in the industry because of it’s low cost and high temperature resistance1. Because this kind of motor rotor does not contain permanent magnets, high torque output requires it to have. It has high salient ratio characteristics2. References3 and4 conducted targeted research on SynRM flux saturation modeling and identification methods to obtain accurate motor models.

Nowadays, the research on SynRM sensorless control technology at home and abroad can be divided into two categories : zero-low speed and medium-high speed. At zero and low speed, the salient pole characteristics of the motor are mainly used to obtain position information, including pulse high-frequency signal injection method5, rotating high-frequency signal injection method6, high-frequency square wave signal injection method1, etc. In the middle and high speed, the position information is obtained by the back electromotive force model, mainly including the sliding observer method7, the model reference adaptive method8, the flux observation method9, the extended Kalman filter method10 and so on11. By measuring the three-phase currents of the motor, a rotor position estimator is achieved. Then, a velocity estimator is derived from the estimated rotor position by using a state estimating technique.

With the increase of load, the cross saturation effect is enhanced, and the accuracy of rotor position estimation is reduced. A fixed motor reference model cannot complete a high-performance position sensorless control. Reference12 introduced the coefficients related to the cross-saturation effect to reduce the estimation error, but the algorithm cannot correctly calculate the mutual inductance between the d-q axes, and cannot guarantee the accuracy of the coefficients. In13, the position is tracked by the function conversion of current-flux linkage, which essentially overcomes the influence of cross-saturation effect. The use of low-pass filter for signal separation will cause additional estimation error. In References14 and 15, the flux saturation is represented by the model equation. The model equation is divided into two groups. One is the d-q axis current flux expressed by polynomial function15, and the other is the d-q axis current flux expressed by arctangent function and polynomial function 16-18.For the flux function of d-q axis current, although cross saturation is considered, it does not meet the reciprocity condition. When the current is 0 A, the model will deviate and cannot accurately represent the cross saturation characteristics.

There are three common methods to obtain the flux saturation model. One is to obtain the flux saturation model in the rotating state19, which requires an additional load motor for speed control. In addition, the rotor is locked by a physical device20 to identify the flux saturation model at rest. The above two methods require additional systems or devices and are not suitable for use in general drive systems ; the third method uses the free axis to identify the static flux saturation model and does not depend on the rotor position, which is suitable for general systems and sensorless drive systems.

Because synchronous reluctance motor and permanent magnet synchronous motor have many similarities, the research results of permanent magnet synchronous motor can also be applied to synchronous reluctance motor. The traditional high-frequency signal injection method has a large number of filtering processes, which limits the bandwidth of the system and reduces the response speed of position observation. Reference12 proposed an initial position detection method of IPMSM based on filterless square wave signal injection, which eliminated the influence of filter on the system. In reference13, a position sensorless control strategy of PMSM based on phase-locked loop ( PLL) flux estimation method is proposed. The PLL can eliminate the fluctuation of rotor position and speed caused by torque ripple and improve the stability of the system. In reference21, a novel variable structure model reference adaptive observer is proposed for speed identification of speed sensorless vector control system of permanent magnet motor.

In order to achieve full-speed sensorless operation, it is not only necessary to combine the above two methods, but also to design a hybrid observer to obtain the mixed rotor position and rotor speed observations22,23,24. In22, the high-frequency current injection method and the flux observation method are combined and the algorithm is implemented to smooth the transition between the two methods. However, the proposed method relies too much on the motor parameters that change greatly during operation. In Reference23, the speed observation values obtained by the high frequency signal injection method and the model reference adaptive method were synthesized by the weighting function, and the sensorless wide speed range operation was effectively realized. In24, a hybrid observer method is proposed, which combines the high frequency signal injection and the position error information of the back EMF model method. The stability and parameter design of the hybrid observer are analyzed.

In this paper, a new magnetic flux saturation model considering cross saturation effect is proposed. By introducing self-saturation term and cross saturation term, a magnetic flux saturation model including d-q axis current-flux function is constructed, which solves the problem that the existing saturation model does not meet the reciprocity condition or cannot accurately express the magnetic flux saturation. Finally, the model is applied to the sensorless control system of synchronous reluctance motor in full speed range to realize the accurate identification of rotor position under magnetic saturation.

Figure 1 shows the actual synchronous rotating coordinate system and the estimated position relationship diagram. The mathematical model of synchronous reluctance motor is established in the d-q axis coordinate system, and the voltage equation is :

Where ud and uq are d-q axis stator voltages ; id and iq are d-q axis stator current ; ψd and ψq are d-q axis magnetic chains, respectively. We is the electrical angular velocity, Rs is the stator resistance.

The relationship between the actual rotor and the estimated rotor rotating coordinate system.

The magnetic flux relative to the d-q axis current is expressed as follows :

The flux function representing flux saturation is nonlinear. When the inductance neither generates nor consumes electrical energy, the reciprocity condition is :

The electromagnetic torque is expressed as :

where pp is the polar logarithm.

The injected high frequency voltage can be expressed as :

When the high frequency voltage is injected, the resistance voltage drop can be ignored because the injected voltage frequency is much higher than the fundamental frequency of SynRM. At this time, the voltage equation of SynRM can be simplified as follows :

The current response corresponding to the high frequency injection voltage can be obtained by using the mathematical model of SynRM :

For the SynRM drive system, the change of flux linkage in the stator and rotor magnetic circuits leads to the change of magnetic density, which affects the change of yoke permeability and the magnetic saturation of the motor. Under different loads, SynRM has different degrees of electromagnetic saturation and cross-coupling, resulting in motor parameter changes and rotor angle estimation errors.

The typical d-axis flux-current characteristics are shown in Fig. 2. When the q-axis current is set to 0 A, the influence of the d-axis current on the inductance can be clearly observed. In the diagram, the apparent inductance is the ratio of the flux linkage to the current at the working point, and the incremental inductance is the slope of the tangent of the current-flux linkage curve at the working point. The expression is as follows :

d-axis flux-current characteristic diagram.

It can be seen from the figure that the d-axis inductance decreases with the increase of current, and the q-axis flux-current also has the same characteristics as the d-axis flux-current. In order to observe the influence of d-q axis current on inductance more intuitively, the three-dimensional surface diagrams of d-q axis current and flux linkage are drawn respectively, and the characteristics and laws of inductance change are analyzed. Figures 3 and 4 respectively draw the three-dimensional surface diagram including d-axis current, q-axis current and d-q-axis magnetic chain.

d-axis flux-current diagram.

q-axis flux-current diagram.

As shown in the figure, when the q-axis current is constant, the d-axis flux linkage decreases with the increase of the d-axis current, and is also affected by the cross-saturation effect. The d-q-axis flux linkage decreases with the increase of the current amplitude of the other axis. Ignoring the cross-saturation effect will lead to a decrease in the accuracy of rotor position estimation.

The flux saturation model polynomial constructed in this paper contains the d-q axis current flux function, where the d-axis and q-axis self-saturation models have the same root-mean-square function form, and the cross-saturation model has the same arctangent function form. In this section, only the d-axis flux saturation model is described, and the q-axis model can be expressed in a similar form.

The self-saturation model is expressed as follows :

Where Ad, Bd and Cd are non-negative coefficients, Ad and Bd represent the smoothness of the cross section from the linear region to the saturated region and the slope of the linear region, respectively, and Cd represents the slope of the saturated region.

Figure 5 shows the flux linkage curves containing Ad, Bd and Cd, respectively. Curves 2 and 3 show that when Ad and Bd increase, the smoothness between the linear region and the saturated region and the slope of the linear region increase. Curve 4 shows that when Cd increases, the flux-current slope of the saturation region increases. Based on the above characteristics, the self-saturation characteristics of the q-axis can be modeled by equations with different parameters. The q-axis self-saturation model is constructed as follows :

Flux linkage curve including Ad, Bd and Cd parameters.

In order to satisfy the reciprocity condition, the differential of the q-axis saturation model relative to the d-axis current and the d-axis cross-saturation term must include the same d-axis function. The q-axis cross saturation term must include the q-axis function, which is equal to the derivative of the d-axis saturation model with respect to the q-axis current. Therefore, the cross-saturation term of the d-axis saturation model is constructed as kd / id2 + kd2. kd / id2 + kd2 can be obtained by differentiating tan-1 (id/kd), which requires additional modeling of the arctangent function tan-1 ( id /kd) in the cross-saturation term of the q-axis. In summary, the proposed flux saturation model consists of a self-saturation term and a cross-saturation term. The specific manifestations are as follows:

Where kd and kq are positive coefficients and Ddq is negative coefficient. kd and kq represent the cross-saturation effect according to the magnitude of the d-q axis current, and Ddq represents the overall cross-saturation effect. The following formulas are obtained by differential treatment of the above equations :

The proof of formula ( 12 ) shows that formula ( 11 ) satisfies the reciprocity condition.

Figure 6 shows the process of measuring parameters by hysteresis voltage injection method. The sampling period and PWM control period are set to T. For the test of d axis, the control equation is expressed as :

In the formula, Ud, in represents the given value of the d-axis test voltage, Ud, out represents the amplitude of the d-axis test voltage, id, max represents the d-axis current limiting, Uq, in represents the given value of the q-axis test voltage. The q-axis test method is similar to the d-axis.

Parameter identification control block diagram.

The flux linkage calculation formula is as follows :

The current-flux model of the motor under test is fitted by the linear least squares ( LLS ) method.

The traditional high frequency signal injection method needs to separate the carrier signal through the filter, and then form the current closed loop. Excessive filtering process will increase the computational burden of the chip, the complexity of the system structure, and the signal amplitude attenuation and phase lag. In this paper, a filter-free signal separation method is used to avoid the problems mentioned above.

Because the high frequency voltage frequency is much higher than the fundamental frequency. At two adjacent sampling moments, it can be considered that the fundamental frequency current signal remains unchanged, while the injected high-frequency square wave voltage signal remains the original amplitude only changes positive and negative due to its characteristics. Therefore, for the response current, there is the following relationship between the high frequency component and the fundamental frequency component in two adjacent sampling periods :

Where f and h denote the high-frequency component and the low-frequency component, respectively.

The sampling current of SynRM is defined as :

Substituting Eq. ( 15 ) into Eq. ( 16 ), we can get :

The signal separation method realized by simple operation instead of the traditional filter greatly reduces the burden of the system and accelerates the convergence time of the position observation strategy. The filter-free signal separation method is shown in Fig. 7.

Filterless signal separation strategy.

Although the high-frequency current response in the stationary α-β coordinate system contains the rotor position information, considering that the calculation of Iαh and Iβh noise through the arctangent function is more sensitive and less robust. Therefore, this paper uses the vector cross multiplication method to decouple the position error information, and the principle is shown in Fig. 8. The position tracking is realized by PI phase-locked loop, and the rotor position information is observed. The principle block diagram is shown in Fig. 9.

Position error signal decoupling.

In the figure, udh is the high frequency voltage component injected under the observation d axis ; iah and iβh are high-frequency response currents under stationary α and β axes, respectively. Iαh and Iβh can be expressed by the following formulas :

Wh is the frequency of the injected square wave voltage signal ; Lavg = ( Ld + Lq ) /2 is the average inductance ; Lavg = ( Ld-Lq ) /2 is the inductance difference.

It can be seen that Iαh and Iβh contain rotor position information, but the premise is Ld ≠ Lq, that is, the motor has a salient pole effect.

Position error signal decoupling.

Because the high-frequency injection algorithm introduces additional high-frequency harmonics to the motor, it inevitably causes high-frequency oscillations and sharp noise. It is mainly used in the zero-speed and low-speed range. When the speed reaches medium and high speed, it is generally switched to the position algorithm based on the fundamental wave model. Figure 6a,b, c shows the speed error curves of the motor speed from 0r/min to 70r/min, 80r/min and 90r/min respectively under the high frequency injection algorithm. It can be seen that as the speed increases, the high frequency harmonics introduced by the high frequency injection algorithm have a great influence on the system.

The speed change curve of the high frequency injection algorithm at different given speeds. (a) 0r / min-70r / min speed error curve. (b) 0r / min-80r / min speed error curve. (c) 0r / min-90r / min speed error curve.

In order to realize the rotor position estimation of the synchronous reluctance motor in the full speed range, this paper adopts the model reference adaptive method in the medium and high speed operation. The model reference adaptive system has the characteristics of the identification idea. The two outputs of the expected model and the adjustable model with the same physical meaning are subtracted, and then the optimal adaptive law is selected to make the system gradually converge to the ideal reference.

Another mathematical model of the synchronous reluctance motor in the d-q axis coordinate system can be derived by Eq. (1) :

The current and speed in Eq. ( 19 ) are replaced with the estimated current and speed to obtain the adjustable model equation :

The integral operation of the above formula is obtained :

The state errore is defined, and the state equation of the error can be obtained by using Eq. (19 ) minus Eq. ( 20 ) :

Of which :

The adaptive law is designed by using the Popov hyperstability theory. There are two conditions for the superstability : 1 ) the input and output integrals satisfy the Popov integral inequality 2 ) the transfer function matrix satisfies the positive realness. The most common adaptive law of model reference adaptation is proportional integral, which is shown as follows :

Of which :

Bring Eq. (18), Eq. (19), Eq. (20) and Eq. (21) into Eq. (22), and the motor speed is estimated as :

The rotor position can be expressed as :

Where \({\theta _{\text{0}}}\)is the initial rotor position angle of the motor. Figure 11 is the block diagram of model reference adaptive estimation.

Structure block diagram of model reference adaptive method.

The position error information of the above two sensorless methods is fused. The position error signal obtained by the high-frequency square wave signal injection method is used as the input in the zero-low speed range, and the position error signal obtained by the model reference adaptive method is used as the input in the medium-high speed range. The two determine the transition stage together. The control block diagram is shown in Fig. 12.

Rotor position hybrid observer structure diagram.

\(X\left( {{\omega _r}} \right)\)is the weighting function, P is the number of poles of the motor, \({\zeta _1}\)and\({\zeta _2}\)is the gain coefficient.

Through the analysis of the speed error of the high-frequency square wave signal injection method at different given speeds in Fig. 10, it can be seen that when the given speed reaches 80r/min, the system is gradually affected by the high-frequency harmonics. Therefore, the start speed switching point and the end speed connecting point of the hybrid observer are set to 80r/min and 120r/min respectively.

Since these two error signals have been normalized respectively, the error information can be directly fused, and the fused position error comprehensive signal is used to obtain the final position observation value through a phase-locked loop. This scheme realizes the unified design of the position observers of the two sensorless methods. The algorithm is more convenient, and the weighting function is shown in Fig. 13.

Switching control weighting function.

Figure 14 is the control block diagram of position sensorless technology of synchronous reluctance motor based on flux saturation model. In order to verify the effectiveness of the proposed scheme, the experimental platform of the control system is built as shown in Fig. 15. The main parameters of the synchronous reluctance motor are shown in Table 1.

Sensorless control block diagram of synchronous reluctance motor.

Experimental platform.

A square wave voltage with an amplitude of 200 V is injected into the d-axis of the traditional and the two high-frequency square wave voltage models with the new flux saturation model proposed in this paper. The q-axis voltage is 0, and the d-q-axis voltage signal is shown in Fig. 16. The three-phase current waveform is shown in Fig. 17.

The d-q axis injects a square wave signal.

Three-phase current waveform.

Figures 18 and 19 are the rotor position and actual rotor position diagrams of the motor, when using the traditional high-frequency square wave voltage injection method and the high-frequency square wave voltage injection method based on the flux saturation model, respectively. Based on the high frequency square wave voltage injection method with flux saturation model, the actual rotor position of the motor is almost coincident with the estimated position, and the error is very small.

Table 2 shows the experimental results of the traditional high frequency square wave voltage injection method and the high frequency square wave voltage injection method based on the flux saturation model. Figure 20 shows the comparison of the speed and position errors of the two different methods under rated load.

Traditional high frequency square wave voltage injection rotor position diagram.

Rotor position diagram of high frequency square wave voltage injection based on magnetic flux saturation model.

Error comparison diagram of different methods.

In the comparison results of rotational speed error and position error between 0-60r/min and rated load, the maximum peak-to-peak value of rotational speed error of high frequency square wave voltage injection.

method based on flux saturation model is 5.75r/min lower than that of traditional high frequency square wave voltage injection method, and the maximum peak-to-peak value of position error is 0.097 rad lower. The maximum peak-to-peak value is reduced by 5.5r /min in the case of sudden speed change, and the maximum peak-to-peak value is reduced by 7.7 r/min in the case of sudden load.

Under the rated load of the motor, the speed error results of 0-60r/min are shown in Figs. 21 and 22.The waveforms in the figure represent the actual speed, speed error and A-phase current, respectively.

Traditional high frequency square wave voltage injection speed estimation error.

Rotating speed estimation error of high frequency square wave voltage injection based on flux saturation model.

It can be seen from Fig. 21 that the speed error fluctuates greatly under the traditional high frequency square wave voltage injection method. In Fig. 22, the motor based on the high-frequency square-wave voltage injection method with the flux saturation model has a small oscillation when starting, but it can converge quickly. The speed tends to be stable at 0.5 S, and the speed error does not exceed ± 5.1r/min. The high frequency square wave voltage injection method based on the flux saturation model reduces the peak value of the speed error from 10.6r/min to 5.1r/min.

When the speed changes abruptly, the speed error and the A-phase current will fluctuate significantly. The waveform change of the motor starting speed from.

0-50r/min to suddenly accelerating to 200r/min at 0.5 S is shown in Figs. 23 and 24.

Traditional high frequency square wave voltage injection speed mutation error.

Speed mutation error of high frequency square wave voltage injection based on flux saturation model.

From the comparison between Figs. 23 and 24, it can be seen that under the condition of constant load and variable speed, the high-frequency square wave voltage injection method based on the flux saturation model can accurately track the speed in the case of current mutation, and the speed error fluctuates less at the speed mutation. The maximum error of the traditional high frequency square wave voltage injection speed is 12.7 r/min. The high frequency square wave voltage injection method based on the flux saturation model can reduce the error peak to 7.2 r/min.

When the motor is running at 50r/min, the performance of applying 2 N·m load torque disturbance is shown in Figs. 25 and 26. When the load is suddenly added during stable operation, under the high-frequency square wave voltage injection method based on the flux saturation model, the actual speed changes significantly, the speed error has a small fluctuation, and the A-phase current fluctuation is small. Compared with the traditional high-frequency square wave voltage injection method, the peak-to-peak value of the speed error is reduced from 15.8 r/min to 8.1 r/min during the sudden loading process.

Traditional high frequency square wave voltage injection load mutation error.

Load mutation error of high frequency square wave voltage injection based on flux saturation model.

In summary, the dynamic performance of the high frequency square wave voltage injection method based on the flux saturation model is better than that of the traditional high frequency square wave voltage injection method. The proposed flux saturation model of synchronous reluctance motor can effectively reduce the influence of self-saturation and cross-saturation. In the sensorless control of synchronous reluctance motor without filter, the position estimation accuracy is affected by parameter nonlinearity, and high frequency signal injection introduces high frequency harmonics to the motor, which has a significant suppression effect.

In this paper, the synchronous reluctance motor is taken as the research object, and a position sensorless control method of synchronous reluctance motor with flux saturation model is proposed to solve the problem of cross-saturation of quadrature and direct axis inductances accompanying the low-speed operation of the motor. Compared with the traditional high frequency square wave voltage injection method, the nonlinear model of d-q axis flux and current is established, and the model is applied to the high frequency voltage injection method to improve the accuracy of rotor position estimation. The position information separation method without filter is used to eliminate the problem of signal amplitude attenuation and phase lag caused by LPF. The proposed flux saturation model accurately exhibits self-saturation and cross-saturation characteristics and satisfies the reciprocity condition. The parameters of the proposed new flux saturation model are obtained by linear least square method. The proposed flux saturation model is applied to the position sensorless control system of synchronous reluctance motor, which improves the tracking performance of position sensorless control information. The validity and feasibility of the proposed flux saturation model method are verified.

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Changsha University of Science & Technology, Changsha, 410114, China

Hui Cai & Wen-jian Luo

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CAI Hui has carried out the experimental platform construction and data processing.LUO Wenjian wrote the main manuscript text, including the editing of pictures and tables.All the authors have read the manuscript.

Correspondence to Hui Cai.

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Cai, H., Luo, Wj. Full-speed sensorless control system of synchronous reluctance motor with flux saturation model. Sci Rep 15, 9048 (2025). https://doi.org/10.1038/s41598-025-92441-7

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Received: 10 August 2024

Accepted: 27 February 2025

Published: 17 March 2025

DOI: https://doi.org/10.1038/s41598-025-92441-7

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