Electromagnetic Analysis and Experimental Validation of an Ironless Tubular Permanent Magnet Synchronous Linear Motor

Abstract

The ironless tubular permanent magnet synchronous linear motor (TPMSLM) is in high demand for high-precision servo control applications due to its advantages of having zero cogging effect and high dynamic response. However, its electromagnetic field analysis model has not yet been perfected. This paper aims to accurately predict the magnetic field distribution and electromagnetic performance parameters of an ironless TPMSLM. Taking the axially magnetized ironless TPMSLM as an example, and disregarding the influence of the armature magnetic field on the air gap magnetic field, a simplified analytical model of the TPMSLM is established in the cylindrical coordinate system based on the equivalent magnetization current method (EMC), and the analytical formula for the air gap magnetic flux density is then derived. Subsequently, by applying electromagnetic field theory and the analytical formula for the magnetic flux density in the air gap, analytical expressions for the back electromotive force (back EMF) and thrust are derived, reducing analytical complexity while maintaining accuracy. The accuracy and practicality of the proposed analytical formulas are validated through comparisons with finite element analysis (FEA) and experimental prototypes. This analytical approach facilitates the optimization of linear motor parameters and the study of thrust fluctuation suppression, thereby laying the foundation for high-precision servo control of linear motors.

1. Introduction

Currently, the primary methods for achieving linear motion include ball screw drives, synchronous belt drives, rack and pinion drives, and linear motor drives. Linear motors directly convert electrical energy into mechanical energy, producing motion in a straight line. Consequently, their system structure is simpler, and both control accuracy and dynamic response performance are improved. Furthermore, due to the advantages of tubular permanent magnet synchronous linear motors (TPMSLMs) over other linear motor designs—such as high power density, absence of end-winding, and unilateral magnetic force—they have attracted widespread attention in recent years in fields including transportation, office automation, healthcare, and new energy development [1,2,3].

High-precision servo control necessitates permanent magnet linear motors with minimal positioning and normal forces. The TPMSLM is becoming the preferred solution due to its unique structure. The primary permanent magnets in such motors are classified into three types: axial magnetization, radial magnetization, and Halbach magnetization [4,5,6,7,8]. Axial magnetization arrays are selected for the primary permanent magnets to reduce costs and facilitate convenient manufacturing. Meanwhile, the ironless structure of the linear motor secondary offers advantages such as zero cogging effect, high dynamic response, low heat loss, absence of hysteresis, and a uniform magnetic field. It is widely used in applications requiring high precision, low noise, and high dynamic performance. Although the power density is lower than that of traditional secondary structures, its advantages make it an indispensable choice for specific applications.

Ishiyama N first proposed a TPMSLM structure without iron in its primary and secondary [9]. The adjacent axially magnetized permanent magnets in the primary are magnetized in opposite directions and are closely mated. Compared with conventional TPMSLMs, this motor exhibits a higher utilization rate of the permanent magnet flux and fewer harmonic components in the permanent magnet magnetic field. Consequently, it offers advantages such as high efficiency and rapid dynamic response, enabling nanoscale resolution [10,11]. However, due to their high cost and installation difficulties, there have been few studies on the magnetic field analysis and electromagnetic characteristics of such motors, both domestically and internationally.

The air gap magnetic field directly influences the quality, thrust fluctuation, and efficiency of the back electromotive force waveform. Accurate prediction of the air gap magnetic field is a prerequisite for optimizing the electromagnetic performance of the motor, such as reducing thrust fluctuations and achieving high-precision servo control. Currently, the primary methods for predicting air gap magnetic flux density include the magnetic equivalent circuit method (MEC) and finite element analysis (FEA). MEC is widely used due to its high computational efficiency [12,13,14,15,16], but its accuracy is limited when addressing significant magnetic leakage or complex magnetic circuits [17]. Although FEA can accurately solve nonlinear problems [18,19,20,21], its preprocessing is time-consuming and challenging to parameterize, which is not conducive to rapid design optimization. In recent years, analytical methods that combine efficiency and accuracy have attracted attention, especially the equivalent magnetic potential method (EMP) [22] and the equivalent magnetization current method (EMC) [23,24]. Among these methods, EMP offers high accuracy and a straightforward solution [25]. The EMC provides higher accuracy, but the traditional method has high complexity in cylindrical coordinate systems [26], and its applicability in ironless tubular structures (such as TPMSLM) has not been fully verified [27,28]. Comparisons of existing forecasting methods are shown in

Table 1. Comparison of existing forecasting methods.

Method	Prediction Accuracy	Computational Efficiency	Parameter Sensitivity
MEC	±15%	high	low
EMP	±10%	high	medium
EMC	±5%	medium	high
FEA	±2%	low	low

.

It is worth noting that progress has been made in improving the aforementioned methods for specific motor structures. In [29], a 2-D analytical model was established for a Halbach array permanent magnet linear motor based on the EMC. In [5,24,30,31], the EMC was extended to a broader range of permanent magnet geometries. In [32], an analytical formula for the magnetic flux density in the air gap was derived using analytical methods, taking into account the influence of the armature magnetic field. However, there is still a lack of research on the TPMSLM structure that is of interest in this study, especially in the cylindrical coordinate system.

This paper accurately predicts the air gap flux density and electromagnetic performance parameters of an ironless TPMSLM based on the EMC. As shown in Figure 1, the EMC is applied within a cylindrical coordinate system to analyze the air gap flux density of the TPMSLM. Subsequently, formulas for back EMF and thrust force are derived. Furthermore, the accuracy and effectiveness of the analytical method are validated through comparative verification of FEA and prototype experiments.

2. Motor Magnetic Field Analysis and Analytical Solution

The structural schematic of the ironless TPMSLM is shown in Figure 1. A hole is located at the center of the axially magnetized permanent magnet array, and the primary employs a center shaft fixation method to provide greater rigidity and stability. This design reduces installation difficulties and makes the motor suitable for high-load applications. Additionally, a sleeve surrounds the permanent magnet. The sleeve not only protects and secures the primary but also serves as thermal insulation and vibration isolation, thereby enhancing the motor’s precision and stability.

Since the geometry of the TPMSLM is axisymmetric, only the upper part is selected for analysis to simplify the process. Based on this symmetry, the corresponding analytical model is shown in Figure 2, where $\tau$ is the pole pitch, $R_1$ is the inner radius of the permanent magnet (the radius of the primary center axis), $R_2$ is the outer radius of the permanent magnet, $R_i$ is the inner radius of the secondary coil, and $R_s$ is the outer radius of the secondary coil.

2.1. Assumptions

The analytical model shown in Figure 2 was used to analyze the magnetic field of the TPMSLM. To facilitate solving the magnetic field distribution of the TPMSLM, we made the following assumptions [4,28]:

Assumption 1.

The primary of the TPMSLM extends infinitely along the z-direction, causing the permanent magnetic field to vary periodically along the z-axis.

Assumption 2.

Because the primary sleeve, primary center shaft, and secondary shell are made of non-conductive materials, their influence on the permanent magnetic field is negligible.

2.2. Governing Equations

Suppose Assumptions 1 and 2 are considered simultaneously, and the influence of the coil windings on the magnetic field is neglected. In this case, the subdomain model of the motor in the cylindrical coordinate system is shown in Figure 2. It is divided into three regions: the permanent magnet (Region I), the air gap and armature winding (Region II), and the center shaft (Region III).

Based on Assumptions 1 and 2, the air gap magnetic field problem of the TPMSLM can be transformed into a 2-D constant magnetic field problem. For a uniformly magnetized permanent magnet, the internal magnetization currents cancel each other out, so the magnetization current exists only on the surface of the permanent magnet [33,34]. Since all the permanent magnets are axially magnetized, the relationship between the surface magnetization current density ${\mathbf{J}}_{\mathbf{s}}$ and the magnetization strength $\mathbf{M}$ is expressed as follows [35,36,37]:

$$J_s=M\times \widehat{n}=M\left(z\right)$$

(1)

According to the distribution of the permanent magnet array shown in Figure 2, an analytical model of the magnetization strength vector distribution is presented in Figure 3, where $M_0=B_{rem}/{\mu }_0$ is the residual magnetization strength of the permanent magnet and $B_{rem}$ is the remanence. Based on Fourier series theory [38], the square-wave magnetization intensity depicted in Figure 3 is expanded into a Fourier series as follows:

$$\begin{array}{c}\mathbf{M}\left(\mathbf{z}\right)={\displaystyle \sum _{n=1}^{\infty }{M}_n\cos \left(kz\right)}\\ M_n=4{\left(-1\right)}^{n+1}{\displaystyle \frac{M_0}{\left(2n-1\right)\pi }}\end{array}$$

(2)

where $n$ is the harmonic order and $k=\left(2n-1\right)\pi /\tau$.

Due to the unique tubular structure of the TPMSLM, the vector potential $\mathbf{A}$ is independent of the coordinates $\mathit{\theta }$. When the eddy current effect is neglected, based on Maxwell’s equations combined with the Coulomb gauge and constitutive relations, the distribution of the magnetic vector potential satisfies Poisson’s equation in the active region and Laplace’s equation in the passive region, which can be expressed as [39]

$$\begin{array}{c}{\nabla }^2{A}_{\mathbf{I}}=-{\mu }_{0}J\\ {\nabla }^2{A}_{\mathbf{II},\mathbf{III}}=0\end{array}$$

(3)

According to the definition of vector magnetic potential $\nabla \times A=B$, the components of the magnetic flux density in the $r$ and $z$ directions are given by [40]

$$B_r=-{\displaystyle \frac{\partial A}{\partial z}}{i}_r$$

(4)

$$B_z={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \left(rA\right)}{\partial r}}{i}_z$$

(5)

3. Air Gap Magnetic Field Analysis and Simulation Verification

Based on Assumption 2, Region $III$ can be considered equivalent to air. By solving Equation (3) using the method of separation of variables, the general solutions to Laplace’s and Poisson’s equations in each region can be derived as follows:

$$\left\{\begin{array}{l}{\mathbf{A}}_{\mathbf{I}}={\displaystyle \sum _{n=1}^{\infty }\left[a_n^I{I}_0\left(kr\right)+b_n^I{K}_0\left(kr\right)\right]}\left[c_n^I\sin \left(kz\right)+d_n^I\cos \left(kz\right)\right]-{\displaystyle \frac{{\mu }_0{\mathbf{M}}_{\mathbf{z}}}{2}}{z}^2\\ A_{\mathbf{II},\mathbf{III}}={\displaystyle \sum _{n=1}^{\infty }\left[a_n^{\mathit{II},\mathit{III}}{I}_0\left(kr\right)+b_n^{\mathit{II},\mathit{III}}{K}_0\left(kr\right)\right]}\left[c_n^{\mathit{II},\mathit{III}}\sin \left(kz\right)+d_n^{\mathit{II},\mathit{III}}\cos \left(kz\right)\right]\end{array}\right.$$

(6)

where $a_n$, $b_n$, $c_n$ and $d_n$ are with the coefficients to be solved, $I_0\left(kr\right)$ and $K_0\left(kr\right)$ are the first and second types of zero-order modified Bessel functions, respectively, and $I_1\left(kr\right)$ and $K_1\left(kr\right)$ are the first and second types of first-order modified Bessel functions, respectively [41].

3.1. Boundary Conditions

Combined with the TPMSLM magnetic field distribution from finite element simulation shown in Figure 4, the boundary condition satisfied by Equation (6) is

$$\left\{\begin{array}{l}z=0, {\mathbf{B}}_{\mathbf{I}\mathbf{r}}\left(r,0\right)=0\\ z=0, {\mathbf{B}}_{\mathbf{II}\mathbf{r}}\left(r,0\right)=0\\ z=0, {\mathbf{B}}_{\mathbf{III}\mathbf{r}}\left(r,0\right)=0\\ z=\tau /2, {\mathbf{B}}_{\mathbf{II}\mathbf{z}}\left(r,\tau /2\right)=0\\ r=0, {\mathbf{H}}_{\mathbf{III}\mathbf{r}}\left(0,z\right)=0\\ r=\infty , A_{\mathbf{II}}\left(\infty ,z\right)=0\end{array}\right.$$

(7)

In addition, the normal component of the magnetic flux density remains continuous at both the interface between Region I and Region II ($r=R_1$) and the interface between Region I and Region III ($r=R_2$). However, the tangential component of the magnetic field strength exhibits discontinuities at these interfaces due to the abrupt changes caused by the presence of surface currents at the boundary between the permanent magnet and the adjacent regions. The magnitude of this discontinuity equals the tangential surface current density ${\mathbf{J}}_{\mathbf{s}}$ perpendicular to the interface. Therefore, the following boundary conditions apply [37]:

$$\left\{\begin{array}{l}r=R_1, -{\mathbf{H}}_{\mathbf{I}\mathbf{z}}+H_{\mathbf{III}\mathbf{z}}=M\left(z\right)\\ r=R_2, -H_{\mathbf{I}\mathbf{z}}+H_{\mathbf{II}\mathbf{z}}=M\left(z\right)\\ r=R_1, B_{\mathbf{I}\mathbf{r}}=B_{\mathbf{III}\mathbf{r}}\\ r=R_2, B_{\mathbf{I}\mathbf{r}}=B_{\mathbf{II}\mathbf{r}}\end{array}\right.$$

(8)

3.2. Analytic Solution of Laplace’s Equation in Region II

Substituting Equations (7) and (8) into Equation (6) yields the expression for the vector magnetic potential in Region II as

$$A_{\mathbf{II}}={\displaystyle \sum _{n=1}^{\infty }\frac{{\mu }_0{M}_n\mathrm{C}}{k}}{K}_1\left(kr\right)\cos \left(kz\right)$$

(9)

where $A=\left[-K_0\left(k{R}_1\right)I_1\left(k{R}_1\right)-I_0\left(k{R}_1\right)K_1\left(k{R}_1\right)\right]/I_1\left(k{R}_1\right)$, $B=\left[I_0\left(k{R}_2\right)K_1\left(k{R}_2\right)+I_1\left(k{R}_2\right)K_0\left(k{R}_2\right)\right]/K_1\left(k{R}_2\right)$, and $C=\left[A{I}_1\left(k{R}_2\right)+B{K}_1\left(k{R}_2\right)\right]/AB{K}_1\left(k{R}_2\right)$.

Using Equations (4), (5), and (9), the radial and tangential flux density distributions in Region II are as follows:

$$\left\{\begin{array}{l}{B}_{\mathrm{II}r}\left(r,z\right)=-{\displaystyle \frac{\partial A_{\mathrm{II}}}{\partial z}}={\displaystyle \sum _{n=1}^{\infty }{\mu }_0{M}_{k}C{K}_1\left(kr\right)\sin \left(kz\right)}\\ B_{\mathrm{II}z}\left(r,z\right)={\displaystyle \frac{1}{r}}{\displaystyle \frac{\partial \left(r{A}_{\mathrm{II}}\right)}{\partial r}}=-{\displaystyle \sum _{n=1}^{\infty }{\mu }_0{M}_{k}C{K}_0\left(kr\right)\cos \left(kz\right)}\end{array}\right.$$

(10)

3.3. Comparison with Axisymmetric FEA

The above analytical method calculates the air gap magnetic field distribution of an axially magnetized ironless TPMSLM. In this section, we perform the FEA of TPMSLMs with varying permanent magnet sizes and verify the accuracy of the analytical method by comparing its results with multiple sets of FEA data.

First, we verify whether the analytical model can maintain high accuracy in predicting the air gap magnetic field at various spatial positions within Region $II$. Figure 5 shows the surface distribution of the radial and tangential flux densities calculated by the analytical method at different spatial positions. Figure 6 compares the analytical results at six different measurement radii ($r$) with the radial and tangential flux densities obtained by FEA. The results in Figure 5 demonstrate that the radial and tangential flux densities calculated analytically closely align with the FEA results across different spatial positions within Region $II$.

Figure 7 and Figure 8 are used to verify whether the proposed analytical method can maintain high accuracy in predicting the air gap magnetic field at 2 mm from the outer surface of the permanent magnet, compared with FEA results for different permanent magnet outer diameters $R_2$. Figure 7 shows the distributions of radial and tangential flux densities calculated using the analytical method for various permanent magnet outer diameters $R_2$. Figure 8 presents six sets of comparison plots of radial and tangential flux densities obtained from both the analytical calculations and FEA at different permanent magnet radii $R_2$. The results in Figure 8 demonstrate that the radial and tangential flux densities calculated by the analytical method closely agree with the FEA results across different primary sizes. The analysis of Figure 5, Figure 6, Figure 7 and Figure 8 indicates that the air gap magnetic fields calculated by the analytical method are highly consistent with the FEA results at various spatial locations within Region II and for different primary motor sizes, thereby fully validating the accuracy of the analytical method. Furthermore, analysis of Figure 6, Figure 7 and Figure 8 shows that the air gap flux density curves calculated for the motor at different air gap regions and under varying primary sizes remain within a 5% error margin relative to the FEA curves. The primary cause of the deviation is attributed not only to Assumptions 1 and 2 but also to the demagnetization of the permanent magnet due to the influence of adjacent magnets, resulting in an actual magnetization intensity lower than that assumed in the analytical model.

By examining Figure 5 and Figure 6, it is evident that as the measurement radius $r$ decreases and approaches the primary, the tangential flux density waveform exhibits greater fluctuations, the radial flux density waveform becomes steeper, and the harmonic content increases. These factors contribute to fluctuations in the linear motor’s back EMF waveform and thrust force. Additionally, the increased harmonic content generates extra heat and noise, which reduce the motor’s performance and lifespan.

By examining Figure 7 and Figure 8, it is evident that a larger radius of the permanent magnet results in smaller fluctuations in the tangential flux density waveform, a more sinusoidal radial flux density waveform, and reduced harmonic content. This occurs because, when measuring the air gap magnetic field 2 mm from the outer surface of the permanent magnet, a larger magnet radius provides a greater effective magnetic pole area, increasing the magnetic field coverage and reducing local magnetic field gradient variations. Consequently, the tangential flux density changes more uniformly, and the radial flux density waveform becomes more sinusoidal. Analysis indicates that, during the design of linear motors, optimizing the radius of the permanent magnets and the air gap thickness can improve the magnetic flux density waveform, thereby enhancing the back EMF waveform, reducing thrust force fluctuations, and improving motor control accuracy.

According to the above analysis, not only is the accuracy of the proposed analytical model and method verified, but the distribution characteristics of the linear motor’s magnetic field are also explored. This reveals the variation pattern of the flux density waveform, which provides valuable guidance for improving the motor’s operational stability and overall performance.

4. Back EMF and Thrust Force Calculations

In this study, the secondary of the TPMSLM consists of the armature winding and the support skeleton. Each phase winding is composed of several groups of coils connected in series/parallel. Figure 9 presents a schematic diagram of the motor winding.

According to Faraday’s Law of Electromagnetic Induction, the back EMF is induced in each coil as it moves through the radial magnetic field while the primary moves linearly along the z-axis. The back EMF equation for a single-phase winding can be expressed as follows [4,24]:

$$e_c=N_c{B}_{\mathit{II}r}\left(z\right)lv={\displaystyle \sum _{n=1}^{\infty }{K}_n\sin \left(kz\right)v}$$

(11)

where $N_c$ is the number of the series turns per phase, $l$ is the circumference of the coil at radius $r$, and $v$ is the velocity of the secondary. $K_n$ is defined as the $\left(2n-1\right)$th harmonic constant and calculated as

$$K_n=2\pi N_c{b}_0{K}_{rn}{K}_{dn}$$

(12)

where $K_{dn}=\sin \left(k{b}_0/2\right)/\left(k{b}_0/2\right)$ is defined as the winding factor of the $\left(2n-1\right)$th harmonic, and $K_{rn}$ is the coefficient related to the $\left(2n-1\right)$th harmonic in the radial field distribution and calculated as

$$K_{rn}={\displaystyle \underset{R_i}{\overset{R_s}{\int }}r{\mu }_0{M}_{k}C{K}_1\left(kr\right)}dr$$

(13)

Since the coils of each phase winding are positioned relative to the primary in the same manner, assuming a total of $p$ pairs of windings, the back EMF of the single-phase winding can be expressed as follows:

$$E=p{e}_c=p{\displaystyle \sum _{n=1}^{\infty }{K}_n\sin \left(kz\right)v}=K_{e}v$$

(14)

where $K_e$ is the back EMF constant.

Similarly, the thrust force generated by the TPMSLM is the Lorentz force produced by the interaction between the armature magnetic field and the permanent magnetic field. The thrust force equation resulting from energizing the phase A winding, when the winding shown in Figure 8 is activated in this position, is as follows:

$$F_c=N_c{B}_{\mathit{II}r}\left(z\right)Il={\displaystyle \sum _{n=1}^{\infty }{K}_n\sin \left(kz\right)I}$$

(15)

where $I$ is the current through each phase of the winding. Assuming the RMS value of the phase current is $I_m$, the specific equations are as follows [23,24]:

$$\left\{\begin{array}{l}{I}_A=\sqrt{2}{I}_m\sin \left(\omega t\right)\\ I_B=\sqrt{2}{I}_m\sin \left(\omega t-2\pi /3\right)\\ I_C=\sqrt{2}{I}_m\sin \left(\omega t+2\pi /3\right)\end{array}\right.$$

(16)

The total thrust force of the three-phase TPMSLM is as follows:

$$F=F_A+F_B+F_C=\sqrt{2}p{I}_m{\displaystyle \sum _{n=1}^{\infty }{K}_n\left\{\begin{array}{l}\sin \left(kz\right)\sin \left(\omega t\right)\\ +\sin \left[k\left(z-\tau /3\right)\right]\sin \left(\omega t-2\pi /3\right)\\ +\sin \left[k\left(z+\tau /3\right)\right]\sin \left(\omega t+2\pi /3\right)\end{array}\right\}}=K_f{I}_m$$

(17)

where $K_f$ is the thrust force constant.

5. FEA, Analytical, and Experimental Verification

A prototype of an axially magnetized, ironless TPMSLM was developed. The main design parameters are presented in

Table 2. The main parameters of the TPMSLM.

Designation	Symbol	Value/(Unit)
Permanent magnet material	N42	/
Remanence density	$B_{rem}$	1.359 T
Inner radius of permanent magnet	$R_1$	1.5 mm
Outer radius of permanent magnet	$R_2$	5 mm
Outer radius of primary sleeve	$R_3$	6 mm
Pole pitch	$\tau$	24 mm
Inner radius of coil	$R_i$	7 mm
Outer radius of coil	$R_s$	11.5 mm
Winding width	$b_0$	7.5 mm
Number of turns of coil group	$N_c$	152 turns
Total length of coils	$L$	10 m
Magnet wire gauge	$d$	0.3 mm
Input voltage peak	$U_p$	24 V
Output power peak	$P_p$	80 W

. The air gap magnetic field, back EMF, and thrust force were evaluated through both experimental measurements and FEA of the prototype to verify the accuracy of the analytical formulas. The prototype and test equipment are provided by Qinhuangdao Daze Electromechanical Equipment Co. (Qinhuangdao, China).

5.1. Air Gap Magnetic Field Distribution Validation

The air gap magnetic field of the TPMSLM was measured using the device shown in Figure 10. We placed the Gaussmeter probe against the sleeve ($r$ = 6 mm) and read the radial air gap magnetic flux density at that location. In addition, based on the linear motor module, the Gaussmeter probe was fixed to the secondary, and the host computer controlled the secondary to move along the z-axis at 1 mm intervals to measure the magnetic flux density of the air gap at radii $r$ = 6 mm (radial), $r$ = 7.3 mm (radial), and $r$ = 9.3 mm (tangential) in turn. The radial and axial flux density values can be measured by adjusting the orientation of the Gaussmeter probe.

Figure 11 presents a comparison between the measured results and those calculated using the analytical method. As shown, the analytical method’s results closely align with the measured values and waveforms. Apart from inherent errors in the analytical method, deviations may also arise from the Gaussmeter’s measurement accuracy and the experimental conditions.

5.2. Back EMF and Thrust Force Verification

The experimental platform for the ironless TPMSLM is shown in Figure 12 and consists of a complete machine assembly and a servo driver. The machine assembly includes two prototypes and a thrust sensor. Prototype 1 is energized to drive Prototype 2. An oscilloscope was used to observe the linear back EMF waveform of Prototype 2 operating in generator mode. Additionally, the three-phase back EMF waveform was captured in real time by a high-precision current sensor integrated into the linear motor driver and the internal data acquisition system. The force sensor measures the thrust force curve generated as Prototype 1 drives Prototype 2. This measurement platform further validates the analytical equations derived in the study. The test conditions include an ambient temperature of 25 °C, humidity of 40%, and power ripple below 1%, and the force sensor communicates via the RS485 protocol.

Figure 13a compares the back EMF curves of the coil windings of Prototype 2 operating in generator mode as it moves axially at a speed of 2.4 m/s. The back EMF waveforms of the three coils are highly consistent. The observed deviations are attributed to the demagnetization of the permanent magnets, manufacturing variations in the prototype, and measurement errors.

Simultaneously, to verify the model’s accuracy across the entire operating range, the peak magnitude of the back EMF was evaluated over a wide speed range of 0 to 4.8 m/s, as shown in Figure 13b. The back EMF demonstrates strong linearity throughout this range.

Figure 14a compares the thrust force curves generated when the motor is powered by a three-phase symmetrical AC current with an effective value of 1 A. The thrust force waveform calculated by the comparative finding analysis method is close to the FE simulation and the measured curve.

Simultaneously, to assess the impact of the armature magnetic field, the thrust magnitude was evaluated over a wide range, up to four times the nominal current value of I = 1 A. As shown in Figure 14b, the thrust demonstrates strong linearity within this range, confirming that the influence of the armature magnetic field on the air gap flux density in the motor under no-load and light-load conditions can be neglected.

Table 3. Analysis, FEA, and experimental results.

Symbol	Description	Unit	Result	RMS	Force Fluctuations
$K_e$	Back EMF constant	$V_{peak}$	Analytical	4.21	/
			FEA	4.29	/
			Experimental	4.14	/
$K_f$	Thrust force constant	$N/Arms$	Analytical	8.28	±1.01%
			FEA	8.27	±1.02%
			Experimental	8.23	±3.16%

presents the analytical analysis, FEA, and experimental results for the back EMF and thrust force constants of the TPMSLM. Compared to the FEA and experimental data, the errors in the analytically obtained back EMF and thrust force constants are maintained within 2%. The significant fluctuations observed in the measured thrust force may be attributed to various factors, including friction, processing and assembly errors, mechanical coupling vibrations, and sensor accuracy or signal fluctuations within the test platform itself.

In analytical calculations, the Fourier series is expanded up to the 20th harmonic. Including higher harmonics and refining the FEA mesh do not significantly improve accuracy. In the 2-D finite element (FE) model, the area is discretized into 3941 units. FE simulation uses Ansys Electronics Desktop 2022 R1, and MTALAB R2022b is used for analytical calculations. Analytical force calculations are approximately 20 times faster (about 3 s) than FE calculations on an Intel^® Core™ i5-12500H CPU at 2.50 GHz.

6. Conclusions

In this paper, the magnetic field distribution and electromagnetic properties of a novel ironless TPMSLM are thoroughly analyzed, and a TPMSLM model is selected for experimental and FEA verification. First, an analytical model of the TPMSLM is developed based on EMC within a cylindrical coordinate system, neglecting the influence of the armature magnetic field. An analytical expression for the air gap magnetic flux density of the TPMSLM is then derived. Subsequently, an analytical method for calculating electromagnetic performance parameters, such as back EMF and thrust force, is proposed. To validate the accuracy of the analytical formulas, an experimental prototype was fabricated, and a corresponding test platform was established. The proposed analytical method’s accuracy is confirmed through comparison with experimental results and finite element analysis. Currently, linear motor auxiliary design software based on this analytical method has been developed to support early-stage motor development in industry, significantly reducing the motor research and development cycle.

The proposed analytical method applies not only to the electromagnetic design of this type of motor but also to any motor without an iron in the primary structure. For linear motors with iron in the primary, the magnetic field magnetizes the iron. However, analytical calculations cannot determine the iron’s magnetic permeability or magnetization intensity. Therefore, although the analytical method presented in this article cannot analyze the TPMSLM structure with iron in the primary, it remains a valuable reference for the theoretical analysis of this type of permanent magnet synchronous linear motor. Additionally, this paper does not consider the effects of the armature magnetic field, permanent magnet demagnetization, or eddy current losses, which are important aspects for future improvement of analytical methods.