Magnetic Poles Position Detection of Permanent Magnet Linear Synchronous Motor Using Four Linear Hall Effect Sensors

: Magnetic pole position detection is the core of the closed-loop control system of the permanent magnet linear synchronous motor (PMLSM), and its position estimation accuracy directly affects control performance and dynamic response speed. In order to solve the problem of the increased estimation error of magnetic pole position caused by magnetic ﬁeld distortion at the end of PMLSM while also considering the cost of control hardware, the paper uses four linear Hall sensors as magnetic pole position detection components and adopts an optimized estimation algorithm to improve the dynamic performance of the motor. Firstly, a numerical simulation of the magnetic ﬁeld of poles was conducted using Ansoft Maxwell software, and combined with theoretical analysis, the optimal installation position range of four linear Hall orthogonal placements relative to the motor was obtained. Meanwhile, based on the existing vector tracking position observer, an improved observer detection model is proposed. The Matlab/Simulink software was used to compare the Hall-based detection model with the Hall-based improved observer detection model, verifying the feasibility of the improved detection algorithm. Finally, the rationality of the spatial layout design of linear Hall and the feasibility of improving the estimation algorithm were veriﬁed through experiments.


Introduction
The PMLSM has the characteristics of small structural volume, large thrust, and high efficiency.Compared to the linear feed motion provided by the ball screw in traditional machinery, the linear motor can be directly connected to the linear motion components rigidly, which can effectively improve friction, noise, and other problems in mechanical transmission.Therefore, the linear motor control system has better motion performance [1].
Magnetic pole position detection is one of the key links in linear motor control systems, and the accuracy of position estimation is crucial for the control accuracy of the motor.There are two main types of magnetic pole position detection: position sensor detection and sensorless detection.At present, the main detection methods for position sensors include grating, magnetic grating, and Hall sensors [2][3][4].Although magnetic gratings and gratings have high detection accuracy, their high cost, large volume, and high requirements for the working environment severely limit the range of motor use.Low-resolution switch-type and lock-type Hall sensors, despite their low cost, have limited position estimation accuracy and are usually used in square wave-controlled permanent magnet brushless motors with low requirements for position detection accuracy [5,6].
Hall sensors can be divided into two types based on whether the output electrical signal has continuous amplitude changes over a continuous time range: switch-type Hall sensors and linear Hall sensors.A switch-type Hall sensor controls its output tube to turn off and on based on changes in the magnetic field.An ideal switch-type Hall sensor should respond quickly when triggered.However, due to the presence of non-ideal factors, the switching action of the output tube of the switch-type Hall sensor will have a significant delay, resulting in the output high and low levels not being able to flip in time.The switchtype Hall sensor is not suitable for working in situations that require high accuracy.The Hall voltage output by the linear Hall sensor is proportional to the magnetic flux density, which converts the weak magnetic signal into analog output.In the peripheral signal processing circuit of the linear Hall sensor, the corresponding compensation structure is designed to make the linear Hall sensor have high detection accuracy and can detect a high magnetic field range.
In recent years, a new type of sensor (Hall sensor) has rapidly developed in the field of information collection, which can be used as a position sensor instead of a photoelectric encoder in the control process of permanent magnet motors to detect rotor position information.Due to its advantages of small size, light weight, and low price, it is widely used in fields such as automatic monitoring, automatic control, and information detection.It plays an increasingly important role in daily life [7].Compared with the switch Hall sensor, the linear Hall sensor can output a Hall voltage that is proportional to the magnetic flux density, so it is often used in displacement measurement, throttle detection, and other applications that need to respond quickly to changes in magnetic flux density and detect the displacement of objects and liquid surplus, according to the size of the magnetic flux density at different positions [8].For example, in the control process of permanent magnet motors, linear Hall position sensors are used to accurately detect rotor position in order to reduce costs and improve reliability.However, due to the strong offset voltage that accompanies the output of the Hall voltage caused by the induction of magnetic field changes, the use of linear Hall sensors to achieve precise displacement and position measurement can seriously affect the output accuracy of the linear Hall sensor, resulting in a deviation from the actual measurement [9].Therefore, in order to improve the accuracy of the linear Hall sensor in detecting magnetic flux density changes, improve the ability of the linear Hall sensor to convert magnetism into electricity, and reduce the interference of offset voltage and low-frequency noise in the chip on the output accuracy, it is the most important to design the linear Hall sensor.
Linear Hall sensors can output continuous signals, and when there are more than two sensors, they can uniquely and accurately determine the position information of each magnetic pole [10].The reference [11] used 13 linear Halls for the position detection of permanent magnet linear synchronous motors and solved the phase deviation and harmonic problem caused by the increase in the number of Halls through the fast Fourier transform and fixed-point iteration methods.The reference [12] proposes an improved rotor position estimation method in a permanent magnet synchronous motor (PMSM) with low-resolution Hall effect sensors.This method promises to decrease the estimated position errors, which are caused by the Hall position offset.A linear interpolation based on the least squares method is used to estimate the rotor position.
At present, the position sensorless detection technologies include the sliding mode control method [13], back electromotive force calculation method [14], state observer method [15], Kalman and extended Kalman filter method [16], etc.However, these algorithms have a high dependence on motor parameters, poor control robustness, and poor universality, and most of them are complex, making them less suitable for engineering applications.
However, there is still a trend of increasing in the same direction between the measurement accuracy and cost of position sensor detection.Different types of sensors have effects on the structural volume, application scenarios, and control performance of motor control systems.When using low-cost and low-precision position sensors, it is also necessary to consider the impact of installation errors on measurement accuracy.At the same time, it is necessary to select a reasonable position estimation algorithm and optimize the magnetic pole position detection algorithm based on the motor operating conditions to improve the accuracy of motor position detection and thereby improve motor control performance.
This paper comprehensively considers factors such as control cost, position detection accuracy, and engineering applications.Linear Hall sensors are used as magnetic pole position detection components, and the focus is on researching the magnetic pole position detection method of PMLSM based on linear Hall.The installation layout of linear Hall relative to linear motors is analyzed, and an improved algorithm for position estimation is proposed to reduce the volume of the overall drive system of the motor and reduce the cost of control hardware, balancing the control accuracy and stability of the motor.

Detection Method of Magnetic Poles Position
When the Hall sensors are used for measuring the magnetic pole position of rotating motors, they are usually installed at the teeth or slots at the axial end of the stator to detect changes in the permanent magnet magnetic field on the mover or auxiliary extended mover.By demodulating the magnetic field value, the motor mover position and speed information are obtained.Due to the unique structure of permanent magnet linear synchronous motors, there are more ways to install Hall sensors.The more convenient method is to install the Hall sensor above the permanent magnet of the mover, which has a certain constraint relationship with the motor stator to ensure the accuracy of later control, simultaneously avoiding the impact of the magnetic field generated by the stator on the Hall detection permanent magnet.The installation position diagram of the linear Hall sensor is shown in Figure 1.
magnetic pole position detection algorithm based on the motor operating conditions to improve the accuracy of motor position detection and thereby improve motor control performance.
This paper comprehensively considers factors such as control cost, position detection accuracy, and engineering applications.Linear Hall sensors are used as magnetic pole position detection components, and the focus is on researching the magnetic pole position detection method of PMLSM based on linear Hall.The installation layout of linear Hall relative to linear motors is analyzed, and an improved algorithm for position estimation is proposed to reduce the volume of the overall drive system of the motor and reduce the cost of control hardware, balancing the control accuracy and stability of the motor.

Detection Method of Magnetic Poles Position
When the Hall sensors are used for measuring the magnetic pole position of rotating motors, they are usually installed at the teeth or slots at the axial end of the stator to detect changes in the permanent magnet magnetic field on the mover or auxiliary extended mover.By demodulating the magnetic field value, the motor mover position and speed information are obtained.Due to the unique structure of permanent magnet linear synchronous motors, there are more ways to install Hall sensors.The more convenient method is to install the Hall sensor above the permanent magnet of the mover, which has a certain constraint relationship with the motor stator to ensure the accuracy of later control, simultaneously avoiding the impact of the magnetic field generated by the stator on the Hall detection permanent magnet.The installation position diagram of the linear Hall sensor is shown in Figure 1.The performance parameters of linear Hall sensors themselves have a significant impact on the accuracy of magnetic pole position detection signals.This paper selects the MLX90242 series linear Hall sensor produced by Melexis company.Figure 2 is the schematic diagram of the exterior of the Hall sensor.The sensor has three interfaces, namely the power supply input terminal, grounding terminal, and output voltage terminal.The prescription-shaped block in the center of the sensor is the effective monitoring area (Hall sensing chip).Figure 3 shows the schematic diagram of the detection output waveform of the corresponding magnetic pole of the linear Hall sensor during normal operation.When the surface of the Hall sensor faces the S pole of the permanent magnet, the output voltage shows an upward trend, while the output voltage shows a downward trend at the N pole.The position of the Hall sensing chip relative to the overall component has clear dimensional parameters, which should be taken into account during actual installation to achieve the most accurate installation accuracy possible.This Hall sensor has a small overall size, high sensitivity, small temperature drift, wide allowable temperature range, and a wide detectable magnetic field range.As shown in Figure 4, it contains an error correction circuit inside, which can roughly eliminate the analog compensation error caused by Hall effect devices.The partial magnetic specifications of the sensor are shown in Table 1.The performance parameters of linear Hall sensors themselves have a significant impact on the accuracy of magnetic pole position detection signals.This paper selects the MLX90242 series linear Hall sensor produced by Melexis company.Figure 2 is the schematic diagram of the exterior of the Hall sensor.The sensor has three interfaces, namely the power supply input terminal, grounding terminal, and output voltage terminal.The prescription-shaped block in the center of the sensor is the effective monitoring area (Hall sensing chip).Figure 3 shows the schematic diagram of the detection output waveform of the corresponding magnetic pole of the linear Hall sensor during normal operation.When the surface of the Hall sensor faces the S pole of the permanent magnet, the output voltage shows an upward trend, while the output voltage shows a downward trend at the N pole.The position of the Hall sensing chip relative to the overall component has clear dimensional parameters, which should be taken into account during actual installation to achieve the most accurate installation accuracy possible.This Hall sensor has a small overall size, high sensitivity, small temperature drift, wide allowable temperature range, and a wide detectable magnetic field range.As shown in Figure 4, it contains an error correction circuit inside, which can roughly eliminate the analog compensation error caused by Hall effect devices.The partial magnetic specifications of the sensor are shown in Table 1.We placed two linear Hall sensors orthogonal, and their detection waveforms are shown in Figure 5.The two sine and cosine waveforms represent the output signals of two linear Hall sensors with a phase difference of 90°.The position angle can be obtained using the arctangent function, and the expression is as follows:   We placed two linear Hall sensors orthogonal, and their detection waveforms are shown in Figure 5.The two sine and cosine waveforms represent the output signals of two linear Hall sensors with a phase difference of 90°.The position angle can be obtained using the arctangent function, and the expression is as follows:  We placed two linear Hall sensors orthogonal, and their detection waveforms are shown in Figure 5.The two sine and cosine waveforms represent the output signals of two linear Hall sensors with a phase difference of 90 • .The position angle can be obtained using the arctangent function, and the expression is as follows: By using four linear Hall sensors and ensuring a phase difference of 90° between adjacent Hall sensors, the signal can be amplified using two-signal differential processing, and the problem of zero offset caused by inaccurate sensor installation position can be effectively avoided.
The calculation formula for the output voltage of a linear Hall sensor is: where   is the sensitivity coefficient,  is the input current of the Hall element,  is the magnetic flux density, and  is the electrical angle between the horizontal position of the By using four linear Hall sensors and ensuring a phase difference of 90 • between adjacent Hall sensors, the signal can be amplified using two-signal differential processing, and the problem of zero offset caused by inaccurate sensor installation position can be effectively avoided.
The calculation formula for the output voltage of a linear Hall sensor is: where K H is the sensitivity coefficient, I is the input current of the Hall element, B is the magnetic flux density, and α is the electrical angle between the horizontal position of the sensor and the peak of the vertical magnetic field.When the selected sensor operates at 25 • C, K H I = 39 (mV/mT).

Installation Position of Linear Hall Relative Motor in Horizontal Direction
The installation position of Hall is mainly related to the position of the magnetic electromotive force axis of the three-phase winding [17].A PMLSM with 15 slots and 20 poles was selected as the research object.This type of motor is composed of five unit motors (Z 0 /P 0 = 3/2, Z 0 is the number of slots in the unit motor and P 0 is the number of pole pairs in the unit motor), with a motor pole distance of 12 mm, slot width of 10 mm, and tooth width of 6 mm.
Considering the placement of the external linear Hall sensor of the motor based on the condition that the d axis coincides with the magnetic electromotive force FA axis, the installation layout strategy of the two orthogonal linear Hall sensors relative to the motor in the horizontal direction is shown in Figure 6.By using four linear Hall sensors and ensuring a phase difference of 90° between adjacent Hall sensors, the signal can be amplified using two-signal differential processing, and the problem of zero offset caused by inaccurate sensor installation position can be effectively avoided.
The calculation formula for the output voltage of a linear Hall sensor is: where   is the sensitivity coefficient,  is the input current of the Hall element,  is the magnetic flux density, and  is the electrical angle between the horizontal position of the sensor and the peak of the vertical magnetic field.When the selected sensor operates at 25 °C,    = 39 (mV/mT).

Installation Position of Linear Hall Relative Motor in Horizontal Direction
The installation position of Hall is mainly related to the position of the magnetic electromotive force axis of the three-phase winding [17].A PMLSM with 15 slots and 20 poles was selected as the research object.This type of motor is composed of five unit motors ( 0 / 0 = 3/2,  0 is the number of slots in the unit motor and  0 is the number of pole pairs in the unit motor), with a motor pole distance of 12 mm, slot width of 10 mm, and tooth width of 6 mm.
Considering the placement of the external linear Hall sensor of the motor based on the condition that the d axis coincides with the magnetic electromotive force FA axis, the installation layout strategy of the two orthogonal linear Hall sensors relative to the motor in the horizontal direction is shown in Figure 6.When the mover moves horizontally to the right, if the Hall sensor is installed on the left side of the motor stator, the first linear Hall sensor H1 should be placed at a horizontal distance of 22 + 24n (mm) from the end of the motor stator (where n is a positive integer) in order to ensure that the voltage signal output by the first linear Hall sensor is the same as the waveform of the opposite electromotive force of phase A and to achieve zero crossing control of the opposite electromotive force of phase A using the first linear Hall sensor.Similarly, when the Hall sensor is placed on the right side of the motor stator, the first When the mover moves horizontally to the right, if the Hall sensor is installed on the left side of the motor stator, the first linear Hall sensor H1 should be placed at a horizontal distance of 22 + 24n (mm) from the end of the motor stator (where n is a positive integer) in order to ensure that the voltage signal output by the first linear Hall sensor is the same as the waveform of the opposite electromotive force of phase A and to achieve zero crossing control of the opposite electromotive force of phase A using the first linear Hall sensor.Similarly, when the Hall sensor is placed on the right side of the motor stator, the first linear Hall sensor H1 should be placed at a horizontal distance of 2 + 24n (mm) from the end of the motor stator.

Installation Position of Linear Hall Relative Motor in Vertical Direction
The installation position of the linear Hall in the vertical direction can be determined by Equation (3), which is mainly related to the magnetic field strength at the permanent magnet of the motor.The voltage output of the linear Hall is linearly proportional to the magnetic field strength of the permanent magnet at its installation height.This paper uses Ansoft Maxwell magnetic field simulation software to quantitatively analyze the magnetic field strength of the magnetic pole at vertical height.The permanent magnet adopts neodymium iron boron with brand N35.The size specification of a single permanent magnet is 20 × 12 × 2 mm.It magnetizes according to the thickness direction of the magnet, and the magnetization directions of adjacent magnets are opposite.The back iron is composed of ferrite material with suitable magnetic conductivity, ensuring the complete closure of the entire magnetic circuit.The thickness of the back iron is 2 mm.The 2D model is shown in Figure 7.
field strength of the pole at vertical height.The permanent magnet adopts neodymium iron boron with brand N35.The size specification of single permanent magnet is 20 × 12 × 2 mm.It magnetizes according to the thickness direction of the magnet, and the magnetization directions of adjacent magnets are opposite.The back iron is composed of ferrite material with suitable magnetic conductivity, ensuring the complete closure of the entire magnetic circuit.The thickness of the back iron is 2 mm.The 2D model is shown in Figure 7.The simulation distribution of the magnetic field lines in Figure 8 was obtained through a software solution.The magnetic field lines in Region I of Figure 8 are distorted due to the proximity to the end of the motor stator, which is affected by the silicon steel sheet.Region II is located in the middle section of the overall stroke, and the distribution of magnetic field lines is relatively consistent, with rules to follow.Region III undergoes significant distortion of the magnetic field lines due to its proximity to the end of the mover.We took a pair of adjacent magnetic poles in Region II of Figure 8 and set a 23 mm length inspection path on the vertical line to observe the distribution of vertical magnetic flux density on this path.Figure 9 shows the magnetic flux density variation curves and local enlarged images of two permanent magnets with different polarities on their respective perpendicular lines.Considering the hardware circuit configuration, the maximum vertical magnetic flux density that the Hall sensor can detect is 21.15 mT.As shown in the partially enlarged image in Figure 9, the vertical heights of two permanent magnets with different magnetic properties from the surface of the permanent magnet when reaching the specified magnetic flux density are 12.19 mm and 11.97 mm, respectively.The difference between the two is 0.22 mm.Finally, the vertical height installation position of the linear Hall sensor is selected as 12 mm.The simulation distribution of the magnetic field lines in Figure 8 was obtained through a software solution.The magnetic field lines in Region I of Figure 8 are distorted due to the proximity to the end of the motor stator, which is affected by the silicon steel sheet.Region II is located in the middle section of the overall stroke, and the distribution of magnetic field lines is relatively consistent, with rules to follow.Region III undergoes significant distortion of the magnetic field lines due to its proximity to the end of the mover.
dymium iron boron with brand N35.The size specification of a single permanent magnet is 20 × 12 × 2 mm.It magnetizes according to the thickness direction of the magnet, and the magnetization directions of adjacent magnets are opposite.The back iron is composed of ferrite material with suitable magnetic conductivity, ensuring the complete closure of the entire magnetic circuit.The thickness of the back iron is 2 mm.The 2D model is shown in Figure 7.The simulation distribution of the magnetic field lines in Figure 8 was obtained through a software solution.The magnetic field lines in Region I of Figure 8 are distorted due to the proximity to the end of the motor stator, which is affected by the silicon steel sheet.Region II is located in the middle section of the overall stroke, and the distribution of magnetic field lines is relatively consistent, with rules to follow.Region III undergoes significant distortion of the magnetic field lines due to its proximity to the end of the mover.We took a pair of adjacent magnetic poles in Region II of Figure 8 and set a 23 mm length inspection path on the vertical line to observe the distribution of vertical magnetic flux density on this path.Figure 9 shows the magnetic flux density variation curves and local enlarged images of two permanent magnets with different polarities on their respective perpendicular lines.Considering the hardware circuit configuration, the maximum vertical magnetic flux density that the Hall sensor can detect is 21.15 mT.As shown in the partially enlarged image in Figure 9, the vertical heights of two permanent magnets with different magnetic properties from the surface of the permanent magnet when reaching the specified magnetic flux density are 12.19 mm and 11.97 mm, respectively.The difference between the two is 0.22 mm.Finally, the vertical height installation position of the linear Hall sensor is selected as 12 mm.We took a pair of adjacent magnetic poles in Region II of Figure 8 and set a 23 mm length inspection path on the vertical line to observe the distribution of vertical magnetic flux density on this path.Figure 9 shows the magnetic flux density variation curves and local enlarged images of two permanent magnets with different polarities on their respective perpendicular lines.Considering the hardware circuit configuration, the maximum vertical magnetic flux density that the Hall sensor can detect is 21.15 mT.As shown in the partially enlarged image in Figure 9, the vertical heights of two permanent magnets with different magnetic properties from the surface of the permanent magnet when reaching the specified magnetic flux density are 12.19 mm and 11.97 mm, respectively.The difference between the two is 0.22 mm.Finally, the vertical height installation position of the linear Hall sensor is selected as 12 mm.

Mathematical Model for Estimating Magnetic Pole Position
The position and speed information of the mover are obtained through Hall sensors, and the position detection error is obtained based on the back electromotive force of the motor.The error is promptly compensated to the negative feedback input of the closedloop control, and real-time correction is carried out on the motor operating speed and magnetic pole position angle, thereby reducing the estimation error of speed and position.

Mathematical Model for Estimating Magnetic Pole Position
The position and speed information of the mover are obtained through Hall sensors, and the position detection error is obtained based on the back electromotive force of the motor.The error is promptly compensated to the negative feedback input of the closed-loop control, and real-time correction is carried out on the motor operating speed and magnetic pole position angle, thereby reducing the estimation error of speed and position.Figure 10 is the system structure diagram of the vector tracking position observer.

Mathematical Model Estimating Magnetic Pole Position
The position and speed information of the mover are obtained through Hall sensors, and the detection error is obtained based on the back electromotive force of the motor.The error is promptly compensated to the negative feedback input of the closedloop control, and real-time correction is carried out on the motor operating speed and magnetic pole position angle, thereby reducing the estimation error of speed and position.Figure 10 is the system structure diagram of the vector tracking position observer.The voltage equation of the PMLSM can be represented by the following formula:

Arctangent algorithm
where   * ,   * ,   ,   ,  �  , and  �  are the observed values of stator voltage, current, and back electromotive force in the orthogonal  −  stationary coordinate system, respectively.
The angular velocity of the motor magnetic pole output by the observer is: where   is the motor speed obtained by solving the linear Hall signal through arctangent operation, which serves as the feedforward input of the observer, and  − is the speed error correction value obtained by using the back electromotive force.In Figure 10, ̂ and  * are the back electromotive force values that are subjected to unit processing.The phase angles of the two are the same, but there may be some errors in the estimation results in actual situations.Therefore, the position estimation error can be obtained by detecting ̂ and  * : The voltage equation of the PMLSM can be represented by the following formula: where u * α , u * β , i α , i β , Êα , and Êβ are the observed values of stator voltage, current, and back electromotive force in the orthogonal α − β stationary coordinate system, respectively.
The angular velocity of the motor magnetic pole output by the observer is: where ω Hall is the motor speed obtained by solving the linear Hall signal through arctangent operation, which serves as the feedforward input of the observer, and ω Hall-corr is the speed error correction value obtained by using the back electromotive force.In Figure 10, ê and e * are the back electromotive force values that are subjected to unit processing.The phase angles of the two are the same, but there may be some errors in the estimation results in actual situations.Therefore, the position estimation error can be obtained by detecting ê and e * : Generally, the difference between the two phase angles is very small, which can be further simplified as the magnetic pole position error: Placing the above observer into the entire motor vector closed-loop control system results in a control block diagram, as shown in Figure 11.
Generally, the difference between the two phase angles is very small, which can be further simplified as the magnetic pole position error: Placing the above observer into the entire motor vector closed-loop control system results in a control block diagram, as shown in Figure 11.

Simulation and Comparison of Magnetic Pole Position Estimation
We built a corresponding simulation model in Matlab/Simulink using the linear Hallbased PMLSM vector control system discussed above.A displacement length of 222 mm was selected as the operating stroke of the motor and reached the end of the motor at 222

Simulation and Comparison of Magnetic Pole Position Estimation
We built a corresponding simulation model in Matlab/Simulink using the linear Hallbased PMLSM vector control system discussed above.A displacement length of 222 mm was selected as the operating stroke of the motor and reached the end of the motor at 222 mm to ensure full stroke operation.
Figure 12 shows the estimation error of the magnetic pole position and angle calculated using the linear Hall detection waveform.Figure 13 shows the estimation error diagram of the motor pole position and angle obtained from the simulation model constructed by the observer detection model.Comparing the two, it can be concluded that the actual and estimated positions of the magnetic poles have suitable consistency in the first half of motor operation.The overall angle estimation error of the observer model is smaller than that of the Hall detection model.During the last two magnetic pole cycles of the stroke, the angle estimation error only fluctuates within a small range, and the maximum angle estimation error does exceed 0.125 rad, which is only 20% of the maximum angle error of the Hall detection model in the same region.After simulation analysis and comparison of the two detection models, it was found that the observer pole position detection model based on the linear Hall has stronger adaptability to variable speeds and smaller position estimation error.

Experimental Verification of Magnetic Pole Position Estimation
This experimental platform uses Microchip's dsPIC33EP series main chip and inverter circuit to achieve motor control using a 15-slot, 20-pole, single-sided flat PMLSM with the same model as the simulation.We designed a three-degrees-of-freedom Hall ad- After simulation analysis and comparison of the two detection models, it was found that the observer pole position detection model based on the linear Hall has stronger adaptability to variable speeds and smaller position estimation error.

Experimental Verification of Magnetic Pole Position Estimation
This experimental platform uses Microchip's dsPIC33EP series main chip and inverter circuit to achieve motor control using a 15-slot, 20-pole, single-sided flat PMLSM with the same model as the simulation.We designed a three-degrees-of-freedom Hall ad- After simulation analysis and comparison of the two detection models, it was found that the observer pole position detection model based on the linear Hall has stronger adaptability to variable speeds and smaller position estimation error.

Experimental Verification of Magnetic Pole Position Estimation
This experimental platform uses Microchip's dsPIC33EP series main chip and inverter circuit to achieve motor control using a 15-slot, 20-pole, single-sided flat PMLSM with the same model as the simulation.We designed a three-degrees-of-freedom Hall adjustment device to meet the movement of a linear Hall in three installation directions and assist in using laser displacement sensors to detect the relative movement distance of the Hall installation position.The experimental setup is shown in Figure 14.From Figures 16 and 17, it can be observed that within the first 200 mm of the motor stroke, the maximum error in speed estimation remains within 0.04 m/s.Due to magnetic field distortion at the end of the stroke, the actual speed of the motor is always too high, and the maximum error in speed estimation is 0.043 m/s.At the end of the stroke, the actual displacement of the motor is slightly greater than the estimated displacement, and the actual displacement of the motor exceeds the theoretical displacement by 3.18 mm.At the end of the stroke, the maximum value of the overall displacement error can reach 5.04 mm.From Figures 16 and 17, it can be observed that within the first 200 mm of the motor stroke, the maximum error in speed estimation remains within 0.04 m/s.Due to magnetic field distortion at the end of the stroke, the actual speed of the motor is always too high, and the maximum error in speed estimation is 0.043 m/s.At the end of the stroke, the actual displacement of the motor is slightly greater than the estimated displacement, and the actual displacement of the motor exceeds the theoretical displacement by 3.18 mm.At the end of the stroke, the maximum value of the overall displacement error can reach 5.04 mm.From Figures 16 and 17, it can be observed that within the first 200 mm of the motor stroke, the maximum error in speed estimation remains within 0.04 m/s.Due to magnetic field distortion at the end of the stroke, the actual speed of the motor is always too high, and the maximum error in speed estimation is 0.043 m/s.At the end of the stroke, the actual displacement of the motor is slightly greater than the estimated displacement, and the actual displacement of the motor exceeds the theoretical displacement by 3.18 mm.At the end of the stroke, the maximum value of the overall displacement error can reach 5.04 mm.From Figure 18, it can be observed that the estimated speed of the observer and the actual speed of the motor always follow well throughout the entire motor operation process.Compared to the Hall detection model mentioned above, the speed fluctuation is smaller and smoother.Even at the end of the stroke, the speed estimation error does not exceed 0.02 m/s, and the maximum overall speed estimation error is 0.019 m/s, which is only 42% of the maximum speed estimation error of the Hall detection model mentioned above.

Stator
From the motor displacement diagram shown in Figure 19, it can also be seen that the observer estimated displacement follows the actual motor displacement well as a whole.At the end of the stroke, the actual displacement of the motor is slightly greater than the estimated displacement, and the actual displacement of the motor exceeds the theoretical displacement by 1.90 mm.The displacement error reaches a maximum value of 2.33 mm at the end of the stroke, which reduces the maximum displacement estimation error by more than half compared to the Hall detection model mentioned above.From Figure 18, it can be observed that the estimated speed of the observer and the actual speed of the motor always follow well throughout the entire motor operation process.Compared to the Hall detection model mentioned above, the speed fluctuation is smaller and smoother.Even at the end of the stroke, the speed estimation error does not exceed 0.02 m/s, and the maximum overall speed estimation error is 0.019 m/s, which is only 42% of the maximum speed estimation error of the Hall detection model mentioned above.
From the motor displacement diagram shown in Figure 19, it can also be seen that the observer estimated displacement follows the actual motor displacement well as a whole.At the end of the stroke, the actual displacement of the motor is slightly greater than the estimated displacement, and the actual displacement of the motor exceeds the theoretical displacement by 1.90 mm.The displacement error reaches a maximum value of 2.33 mm at the end of the stroke, which reduces the maximum displacement estimation error by more than half compared to the Hall detection model mentioned above.From Figure 18, it can be observed that the estimated speed of the observer and the actual speed of the motor always follow well throughout the entire motor operation process.Compared to the Hall detection model mentioned above, the speed fluctuation is smaller and smoother.Even at the end of the stroke, the speed estimation error does not exceed 0.02 m/s, and the maximum overall speed estimation error is 0.019 m/s, which is only 42% of the maximum speed estimation error of the Hall detection model mentioned above.
From the motor displacement diagram shown in Figure 19, it can also be seen that the observer estimated displacement follows the actual motor displacement well as a whole.At the end of the stroke, the actual displacement of the motor is slightly greater than the estimated displacement, and the actual displacement of the motor exceeds the theoretical displacement by 1.90 mm.The displacement error reaches a maximum value of 2.33 mm at the end of the stroke, which reduces the maximum displacement estimation error by more than half compared to the Hall detection model mentioned above.Comparing the experimental conclusion with the simulation results, it can also be seen that the results of the two conform to the consistency of the variation pattern.Through experimental comparison of two detection models, it can be found that the observer detection model that compensates and corrects the linear Hall detection signal has superior performance in both motor speed tracking and displacement detection.

Conclusions
This paper comprehensively considers factors such as control cost, position detection accuracy, and engineering application.Firstly, the linear Hall sensor and its internal magnetic characteristics are determined.The theoretical installation layout design of a relative 15-slot, 20-pole PMLSM in sine wave vector control is studied.The specific installation position parameters of the linear Hall are determined by building the corresponding motor model in Ansoft Maxwell.
To solve the problem of increased detection position estimation error caused by magnetic field distortion at the end pole of the motor mover during full stroke operation of PMLSM, a vector tracking position observer method based on the linear Hall is proposed to improve the motor pole position estimation ability.
Two types of magnetic pole position detection models were built in Matlab/Simulink and verified through experiments with the same parameters.The estimation performance Comparing the experimental conclusion with the simulation results, it can also be seen that the results of the two conform to the consistency of the variation pattern.Through experimental comparison of two detection models, it can be found that the observer detection model that compensates and corrects the linear Hall detection signal has superior performance in both motor speed tracking and displacement detection.

Conclusions
This paper comprehensively considers factors such as control cost, position detection accuracy, and engineering application.Firstly, the linear Hall sensor and its internal magnetic characteristics are determined.The theoretical installation layout design of a relative 15-slot, 20-pole PMLSM in sine wave vector control is studied.The specific installation position parameters of the linear Hall are determined by building the corresponding motor model in Ansoft Maxwell.
To solve the problem of increased detection position estimation error caused by magnetic field distortion at the end pole of the motor mover during full stroke operation of PMLSM, a vector tracking position observer method based on the linear Hall is proposed to improve the motor pole position estimation ability.
Two types of magnetic pole position detection models were built in Matlab/Simulink and verified through experiments with the same parameters.The estimation performance of the two models for motor magnetic pole position angle, motor speed, and displacement in trapezoidal variable speed mode was discussed.After comparison, it was found that the observer magnetic pole position detection model based on the linear Hall has stronger adaptability to variable speed and smaller position estimation error.

Figure 1 .
Figure 1.Schematic diagram of the installation position of linear Hall sensor.

Figure 1 .
Figure 1.Schematic diagram of the installation position of linear Hall sensor.

Figure 6 .
Figure 6.Installation diagram of linear Hall in the horizontal direction.

Figure 6 .
Figure 6.Installation diagram of linear Hall in the horizontal direction.

Actuators 2023 ,Figure 9 .
Figure 9. Distribution and local amplification of magnetic flux density at a vertical height of the magnetic pole.

Figure 9 .
Figure 9. Distribution and local amplification of magnetic flux density at a vertical height of the magnetic pole.

Figure 11 .
Figure 11.Control block diagram of position observer based on the linear Hall.

Figure 11 .
Figure 11.Control block diagram of position observer based on the linear Hall.

Figure 12 .Figure 13 .
Figure 12.Hall detection model estimation angle and angle estimation error.

Figure 13 .
Figure 13.Observer model estimation angle and angle estimation error.

Figure 15 Figure 15 .
Figure 15 shows the waveform signal measured at a vertical installation height of 12.8 mm in the motor Region II of the Hall circuit board.It can also be seen from the Lissajous figure that the orthogonality of the waveform is suitable.The experimental verification shows that the detection height value is in suitable agreement with the simulation height value.Figures 16-19 show the variation curves of motor speed and displacement in the Hall detection model and Hall-based position observer model observed in the MPLAB IDE simulation software under trapezoidal variable speed mode.

Figure 15 Figure 14 .
Figure 15 shows the waveform signal measured at a vertical installation height of 12.8 mm in the motor Region II of the Hall circuit board.It can also be seen from the Lissajous figure that the orthogonality of the waveform is suitable.The experimental verification shows that the detection height value is in suitable agreement with the simulation height value.Figures 16-19 show the variation curves of motor speed and displacement in the Hall detection model and Hall-based position observer model observed in the MPLAB IDE simulation software under trapezoidal variable speed mode.

Figure 15 Figure 15 .
Figure 15 shows the waveform signal measured at a vertical installation height of 12.8 mm in the motor Region II of the Hall circuit board.It can also be seen from the Lissajous figure that the orthogonality of the waveform is suitable.The experimental verification shows that the detection height value is in suitable agreement with the simulation height value.Figures 16-19 show the variation curves of motor speed and displacement in the Hall detection model and Hall-based position observer model observed in the MPLAB IDE simulation software under trapezoidal variable speed mode.

Figure 15 .
Figure 15.The best installation position of HALL circuit board oscilloscope waveform and Lissajous graph.(a) Board oscilloscope waveform; (b) Lissajous graph.

Figure 16 .Figure 17 .
Figure 16.Motor speed and error variation curve of the Hall detection model.

Figure 17 .
Figure 17.Motor displacement and error variation curve of the Hall detection model.

Figure 19 .
Figure 19.Motor displacement and error variation curve based on linear Hall observer detection model.

Table 1 .
Magnetic parameters of linear Hall sensors.

Table 1 .
Magnetic parameters of linear Hall sensors.

Table 1 .
Magnetic parameters of linear Hall sensors.