Loss characterization of flux-switched permanent magnet linear motor based on multi-physics field coupling analysis

Due to the progress of science and technology and the continuous development of industrial manufacturing, flux-switching permanent magnet linear motors have been widely used in various fields. Flux switching is the use of basic magnetic principles, by switching the direction of the current to change the direction and size of the motor's flux to achieve the rotation of the motor, usually consists of a power supply, armature windings and permanent magnets [1–2]. When power is applied to the armature winding red, a rotating magnetic field is generated. At the same time, the magnetic field of the water magnets also affects the armature winding. The permanent magnet linear motor, on the other hand, is a type of motor that utilizes permanent magnets and electromagnets to interact with each other to produce linear motion [3–5]. Its principle is based on the interaction of Lorentz force and magnetic field, when the permanent magnets and electromagnets are in a certain electromagnetic field, through the reverse magnetic field interaction, a strong torque is generated to make the motor produce linear motion [6–8]. The main components of the permanent magnet linear motor include permanent magnet, electromagnetic body, suspension parts and control system. The permanent magnet provides a permanent magnetic field for the motor, while the electromagnet generates a strong electromagnetic field by energizing electricity, thus interacting with the permanent magnet [9–11]. The levitation component then enables the motor to produce a levitation effect, thus realizing contactless motion. In permanent magnet linear motors, the direction of current and electromagnetic field can control the direction of motion and speed of the motor [12–14]. By adjusting the size and direction of the current, the direction and strength of the electromagnetic field can be changed, thus realizing the control of the motor such as forward and reverse rotation, acceleration and deceleration. At the same time, permanent magnet linear motors have the advantages of high efficiency, low noise, low vibration, etc., which are widely used in the fields of energy conversion, machinery manufacturing, aviation and railroads, etc. [15–18].

In flux-switching permanent magnet linear motors, loss refers to the various energy consumptions generated during the operation of the motor. In order to improve the efficiency and performance of the motor, it is possible to design the motor can choose low resistivity wire and high performance permanent magnet materials, in order to reduce the copper loss and magnet loss [19–20]. Secondly, the mechanical losses can be reduced by adopting advanced manufacturing techniques and precision machining technology, as well as by optimizing the structure and control algorithm of the motor to reduce the core loss and cooling loss [21–22]. Understanding and analyzing the loss characteristics of motors is of great significance in improving the efficiency and performance of motors, and ultimately realizing the goals of energy conservation, emission reduction and sustainable development.

This paper analyzes the winding topology and electric operation mechanism of LFSPM motor, establishes a thrust model of LFSPM motor considering inductive asymmetry, and reveals the influence law of inductive asymmetry on the thrust characteristics. The loss characteristics of LFSPM motor are also analyzed based on multi-physical field coupling. The influence of the rotational frequency on the motor performance and various types of losses under different working conditions is simulated and analyzed, and a suitable value of the rotational frequency is selected based on the analytical results. Based on the analysis results, a suitable value of the rotational frequency is selected. Taking this value as a given, a simulation system of constant rotational frequency control strategy is constructed, and the performance of the two in terms of various electromagnetic properties and losses is analyzed in detail by combining the field-circuit coupling method and comparing with the vector control system.

2

Theory and modeling

2.1

Flux-switching permanent magnet linear motor related basic theory

2.1.1

Evolution of LFSPM motors

The flux-switching linear motor can be equated to a flux-switching rotary motor along the direction of the rotation direction of the normal direction of the dissection and unfolding of the motor, on the basis of which the unfolding of the motor along the direction of the plumb line of the dissection line straightened to get the flux-switching permanent magnet linear motors, and finally to the actual application of the motor factors such as the conditions to determine the stator and the stator of linear motors, linear motors, that is, the stator and the air gap air to maintain a relatively static part of the linear motors, the air gap to maintain relative motion. Linear motor motor stator, that is, with the air gap air to the relative movement of the part. As shown in Figure 1 the evolutionary process is from a single stator rotary motor evolution into a single side flux switching linear motor process. As shown in Fig. 2, the evolution is from a two-stator rotary motor to a bilateral flux-switching linear motor.

2.1.2

Basic structure and operating principle of LFSPM motors

In order to perform a multi-objective optimization design of the LFSPM motor, it is first necessary to determine the basic structure of the motor and establish the basic dimensions of the motor structural parameters. Secondly, due to the nonlinear and highly coupled characteristics of the electromagnetic field inside the LFSPM, it is necessary to choose a reasonable modeling method with accurate mesh-section modeling in order to analyze the electromagnetic characteristics of the motor. This chapter firstly introduces the topology of LFSPM motor and briefly describes the flux switching principle and the working mechanism of LFSPM motor in the power generation state and electric state. At the same time, the power equation of LFSPM is derived, and the initial structural dimensions of the LFSPM motor are determined based on its mathematical model and the functional relationship between the main dimensions. Combined with the fractional slot winding design theory to determine the motor winding distribution method, a two-dimensional finite element analysis model of the LFSPM motor is established to simulate and validate the basic electromagnetic properties of the motor such as the reverse electromotive force, no-load magnetic chain, electromagnetic thrust, positioning force, motor loss, and air-gap magnetic density, which provides the necessary theoretical basis for the multi-objective optimization design of the LFSPM motor.

The basic structure of a bilateral flux-switching permanent magnet linear motor is shown in Fig. 2, which consists of a primary mover and a long secondary stator on both sides of the mover. Both the primary mover and secondary stator adopt convex stage tooth structure. The primary mover adopts mechanical linkage design, which consists of permanent magnets, iron core and winding, among which the mover core includes two H-type iron cores placed at the end and five double H-type iron cores placed in the middle, and the permanent magnet is placed between each mover core, which is horizontally magnetized and adjacent permanent magnets are magnetized in opposite directions, constituting horizontal alternating forms of permanent magnet magnet magnetization. The armature windings are spanned in the winding slots of the two cores.The LFSPM motor adopts a long stator structure, with less windings and permanent magnets, and the secondary stator consists of magnetic conductive cores only, which makes it inexpensive to manufacture and simple in structural design. The selected stator has a double-edge structure, which can effectively reduce the effect of vertical normal force on the mover and thus improve the operating performance of the motor.

The operating characteristics of LFSPM motors lie in their “flux-switching” nature. Unlike other permanent magnet motors, flux-switching permanent magnet motors have strict slot-pole combinations based on their structural specialties, thus presenting unique power generation and electric operation states of the motors. The topological structure of the motor is shown in Figure 3.

2.1.3

LFSPM motor electric operation mechanism

According to the flux switching principle, both the magnetic chain and the counter electromotive force of the LFSPM motor are close to sinusoidal distribution, so that the motor electric operation can be controlled by controlling the armature reaction current and thus the motor electric operation in order to obtain the maximum output power. If the loss of the secondary stator and mechanical loss are neglected, the motor electromagnetic power can be considered to approximate the output power, and the input power P₁ of the LFSPM motor can be expressed as: 1 $P_{1} = P_{c m} + P_{P M} + P_{C w} + P_{F c} = P_{c m} + P_{a}^{'}$

Where P_em is the electromagnetic power, P_PM denotes the permanent magnet eddy current loss, P_Cu is the copper loss of the armature windings, P_Fe is the core loss of the actuator, and defines P₀′ as the total loss of the electric actuator, which can be expressed as the sum of the three parts of the actuator loss, i.e., P₀′ = P_PM + P_Cu + P_Fe. According to the voltage-current relation equation, the input power P₁ can be expressed again as: 2 $\begin{array}{l} P_{1} & = \frac{m}{T} \int_{0}^{T} e (t) \cdot i (t) d t = \frac{m}{T} \int_{0}^{T} E_{m} \sin (\frac{2 π}{T} t) \cdot I_{m} \sin (\frac{2 π}{T} t) d t \\ = \frac{m}{2} E_{m} I_{m} \end{array}$

Where m is the number of motor winding phases, T represents the armature current period, E_m is the winding induced electromotive force magnitude and I_m represents the winding current magnitude. Assuming that the motor efficiency is η, the output power of the motor P₂ can be expressed as: 3 $P_{2} = η P_{1} = \frac{m}{2} η E_{m} I_{m}$

For the induced electromotive force magnitude E_m, according to the law of electromagnetic induction, the counter electromotive force e of the single phase winding of the motor can be expressed as: 4 $e_{i} = - \frac{d ψ_{p m i}}{d t} = N_{p h i} \frac{d ϕ_{p m i}}{d x} \frac{d x}{d t} = - N_{p h} \frac{d ϕ_{p m i}}{d x} v_{s}$

Where ψ_pmi represents the i nd phase permanent magnet chain, ϕ_pmi is the i th phase magnetic flux, v_s is the synchronous speed of the motor, N_ph is the number of turns of the winding per phase, and x is the distance traveled by the motor. When the motor induced electromotive force e is distributed according to a sinusoidal law, the magnetic flux (ϕ_pmi can be expressed by the following equation: 5 $ϕ_{p m i} = ϕ_{m} \cos (\frac{2 π}{τ_{s}} x)$ where ϕ_m denotes the magnitude of the motor air gap flux, τ_m is the stator pole pitch, and $ϕ_{p m i} = ϕ_{m} \cos (\frac{2 π}{τ_{s}} v_{s} t)$ if the motor runs up to the synchronous speed v_s, where the air gap flux ϕ_m is satisfied: 6 $ϕ_{m} = B_{g \max} S_{a} = B_{g \max} l_{a} c_{s} τ_{s}$

Where, B_gmax is the peak value of the air gap magnetic density, S_a is the area of effective flux circulation of each primary tooth, l_a is the longitudinal depth of the motor, c_s is the ratio of the primary tooth width to the secondary pole pitch of the linear motor, and by substituting Eq. (5) and Eq. (6) into Eq. (4), the induced electromotive force of the single-phase winding is obtained as: 7 $e_{m i} = 2 π N_{p h} B_{g \max} l_{a} c_{s} k_{d} k_{N} v_{s} \sin (\frac{2 π}{τ_{s}} x)$ where k_d and k_N represent the winding coefficient and the leakage coefficient, respectively. Further the winding induced electromotive force magnitude E_m can be obtained as: 8 $E_{m} = 2 π N_{p h} B_{g \max} l_{a} c_{s} k_{d} k_{N} v_{s}$

For winding current amplitude I_m, the current amplitude per phase distributed according to sinusoidal law can be expressed by equation (9): 9 $I_{m} = \sqrt{2} I_{0} = \sqrt{2} \frac{A_{s} \cdot l_{m}}{2 m N_{p l}}$

Where, I₀ denotes the rms value of the phase current, A_s is the line load of the armature winding, and l_m is the transverse length of the motor. In this design, considering the linear structure of the LFSPM motor, the transverse length of the motor l_m can be expressed as: 10 $l_{m} = k_{m} τ_{m}$ where τ_m is the primary pole pitch; k_m is an integer indicating the number of primary phases. Substituting Eqs. (8), (9), and (10) into Eq. (3) further yields the expanded form of the output power P₂: 11 $P_{2} = \frac{\sqrt{2}}{2} η π B_{s \max} l_{a} c_{s} k_{d} k_{N} A_{s} v_{s} k_{m} τ_{m}$

Since the stator losses and mechanical losses are neglected, the electromagnetic power of the LFSPM motor is approximated to the output power, i.e., P_em ≈ P₂, which gives the electromagnetic thrust of the motor: 12 $F_{c m} = \frac{P_{c m}}{v_{s}} = \frac{\sqrt{2}}{2} η π B_{p \max} l_{a} c_{x} k_{d} k_{n} A_{s} k_{m} τ_{m}$

From the results of the above analysis, it can be seen that the electromagnetic power and electromagnetic thrust of the LFSPM motor are closely related to the parameters such as dimensional parameters of the motor parts, peak air-gap magnetic density, and winding coefficients, and are independent of the number of motor phases and winding turns.

2.2

Thrust modeling considering inductive asymmetry

A flux-switched permanent magnet motor is essentially a synchronous motor, so it is subject to the unified mathematical model of synchronous motors, while the mathematical model developed in this section and the related conclusions drawn are also applicable to other types of synchronous motors.

Let the flux-switching linear motor three-phase armature winding current are i_a, i_b, i_c, three-phase armature winding magnetic chain is ψ_a, ψ_b, ψ_c, three-phase armature winding self-inductance coefficient and mutual inductance coefficient are L_aa, L_bb, L_cc and M_ab, M_ac, M_bc, respectively, the equivalent excitation winding and the mutual inductance coefficient of three-phase armature winding is M_af, M_bf, M_cf, the equivalent excitation current is i_f. When the equivalent damping winding is not taken into account, the three-phase armature winding equation of the motor is equation (13). When the equivalent damping winding is not considered, the magnetic chain equation of the three-phase armature winding of the motor is Eq. (13). 13 $[\begin{array}{l} ψ_{a} \\ ψ_{b} \\ ψ_{c} \end{array}] = [\begin{matrix} L_{a a} & M_{a b} & M_{a c} \\ M_{b a} & L_{b b} & M_{b c} \\ M_{c a} & M_{c b} & L_{c c} \end{matrix}] [\begin{array}{l} i_{a} \\ i_{b} \\ i_{c} \end{array}] + [\begin{array}{l} M_{a f} \\ M_{b f} \\ M_{c f} \end{array}] i_{f}$

Influenced by the end effect, cogging effect, and armature reaction of the permanent magnet linear motor, there are asymmetries in the self-inductance coefficient, mutual inductance coefficient, and mutual inductance coefficient between the motor's three-phase armature windings, as well as the mutual inductance coefficients between the equivalent excitation winding and the armature winding. Neglecting harmonics and considering the effect of inductance asymmetry, the self-inductance coefficient of the three-phase winding is: 14 ${\begin{array}{l} L_{a a} = L_{0} + L_{1} \cos (2 θ) \\ L_{b b} = L_{0} + L_{1} \cos (2 θ + 2 π / 3) \\ L_{c c} = (L_{0} + L_{k 0}) + (L_{1} + L_{k 1}) \cos (2 θ - 2 π / 3) \end{array}$ where θ indicates the angle between the winding axis of phase A and the secondary axis of phase d, L_k0 indicates the difference between the average value of the self-inductance coefficient of phase C and that of phases A and B, and L_k1 indicates the difference between the fluctuation amplitude of the self-inductance coefficient of phase C and that of phases A and B.

Neglecting harmonics, the mutual inductance coefficients of the three-phase windings are Eq. (15) where M_h0 represents the difference between the average value of mutual inductance coefficients of phases A and B and the difference between phases A, C, and B and C, and M_A1 represents the difference between the amplitude of mutual inductance coefficient fluctuations of phases A and B and the difference between phases A, C, and B and C. 15 ${\begin{array}{l} M_{a b} = (M_{0} + M_{h 0}) + (M_{1} + M_{h 1}) \cos (2 θ - 2 π / 3) \\ M_{a c} = M_{0} + M_{1} \cos (2 θ + 2 π / 3) \\ M_{b c} = M_{0} + M_{1} \cos (2 θ) \\ M_{b a} = M_{a b} \\ M_{c a} = M_{a c} \\ M_{c b} = M_{b a} \end{array}$

Neglecting harmonics, the mutual inductance coefficient between the equivalent excitation winding and the three-phase armature winding is: 16 ${\begin{array}{l} M_{a f f} = M_{f a} = M_{s 1} \cos (θ) \\ M_{b f} = M_{f b} = M_{s 1} \cos (θ - 2 π / 3) \\ M_{c f} = M_{f k} = (M_{s 1} + M_{j 1}) \cos (θ + 2 π / 3) \end{array}$ where M_j1 represents the difference between the amplitude of mutual inductance fluctuation of C-phase winding and equivalent excitation winding and the amplitude of mutual inductance fluctuation of A-phase and B-phase windings and equivalent excitation winding.

The abc-coordinate system and d – q-coordinate system change matrix is: 17 $C = \sqrt{\frac{2}{3}} [\begin{matrix} \cos (θ) & - \sin (θ) \\ \cos (θ - 2 π / 3) & - \sin (θ - 2 π / 3) \\ \cos (θ + 2 π / 3) & - \sin (θ + 2 π / 3) \end{matrix}]$

Then the inductance coefficient of the 3-phase winding d – q-axis in the d – q-coordinate system is: 18 $[\begin{matrix} L_{d} & L_{d q} \\ L_{q d} & L_{q} \end{matrix}] = C^{- 1} [\begin{matrix} L_{a a} & M_{a b} & M_{a c} \\ M_{b a} & L_{b b} & M_{b c} \\ M_{c a} & M_{c b} & M_{c c} \end{matrix}] C$ d – q Mutual inductance coefficient of the shaft winding and the equivalent excitation winding: 19 $[\begin{matrix} M_{d f} \\ M_{q f} \end{matrix}] = C^{- 1} [\begin{matrix} M_{a f} \\ M_{b f} \\ M_{c f} \end{matrix}]$

Substituting Eqs. (14) and (17) into Eq. (18), then: 20 $\begin{array}{l} L_{d} & = \frac{3}{2} [2 M_{1} + L_{1} + 2 (L_{0} - M_{0})] \\ + [2 M_{k 0} \cos θ \sin (θ - \frac{π}{6}) + L_{k 0} \sin^{2} (θ + \frac{π}{6})] \\ + \cos (2 θ + \frac{π}{3}) [M_{h 1} \cos θ \sin (\frac{π}{6} - θ) - \frac{L_{k 1}}{2} \sin^{2} (θ + \frac{π}{6})] \end{array}$ 21 $\begin{array}{l} L_{q} & = \frac{3}{2} [2 (L_{0} - M_{0}) - 2 M_{1} - L_{1}] \\ + [2 M_{h 0} \sin θ \sin (θ + \frac{π}{3}) + L_{k 0} \cos^{2} (θ + \frac{π}{6})] \\ + \cos (2 θ + \frac{π}{3}) [M_{h 1} \sin θ \cos (\frac{π}{6} - θ) - \frac{L_{k 1}}{2} \cos^{2} (θ + \frac{π}{6})] \end{array}$ 22 $\begin{array}{l} L_{d q} & = \frac{1}{2} (L_{k 0} + 2 M_{h 0}) \sin (2 θ + \frac{π}{3}) \\ - \cos (2 θ + \frac{π}{3}) [\frac{L_{k 1}}{4} \sin (2 θ + \frac{π}{3}) + \frac{M_{h 1}}{2} \cos (\frac{π}{6} - 2 θ)] \end{array}$

It can be seen that the asymmetry of the three-phase windings of the motor leads to the fact that the harmonic content of the self-inductance of the windings of the d and d axes in the d – q-coordinate system is no longer constant, and that the mutual inductance of the d and d axes exists and cannot be fully decoupled. Among them, the asymmetry of the average value of the three-phase winding self-inductance and mutual inductance leads to the 2nd harmonic content in the self-inductance of the d, q-axis windings and the 2nd harmonic content in the mutual inductance of the d, d-axis windings: the asymmetry of the amplitude of the fluctuation of the three-phase winding self-inductance and mutual inductance leads to the 4th harmonic content in the self-inductance of the d, d-axis windings and the 4th harmonic content in the mutual inductance of the d – q-axis windings. Substituting equation (16) into equation (19), then: 23 ${\begin{array}{l} M_{d f} = \frac{3}{2} M_{s 1} + \frac{M_{j 1}}{2} + \frac{M_{j 1}}{2} \sin (2 θ + \frac{5}{6} π) \\ M_{q f} = \frac{1}{2} \sin (2 θ + \frac{1}{3} π) \end{array}$

As can be seen from the equation, due to the mutual inductance imbalance between the motor three- phase windings and the equivalent excitation winding, the mutual inductance of the d-axis winding and the equivalent excitation winding in the d – q-coordinate system contains the 2nd harmonic component, and the mutual inductance of the q 3-axis winding and the equivalent excitation winding is not 0 and contains the 2nd harmonic component.

In the d – q-coordinate system, the d-axis and q-axis windings are magnetically linked: 24 $[\begin{array}{l} ψ_{d} \\ ψ_{q} \end{array}] = [\begin{matrix} L_{d} & L_{d q} \\ L_{q d} & L_{q} \end{matrix}] [\begin{array}{l} i_{d} \\ i_{q} \end{array}] + [\begin{array}{l} M_{d f} \\ M_{q f} \end{array}] i_{f}$

The electromagnetic thrust of the output of the permanent magnet synchronous linear motor is calculated as equation (25). The flux-switching permanent magnet linear motor is essentially a synchronous motor, and in order to establish a unified mathematical model for synchronous motors, Eq. (25) is used here to calculate the thrust. Because the pole pitch τ of the flux-switching motor corresponds to the 360° electrical angle, it is necessary to change 2τ to τ in Eq. (25) when calculating the thrust force of the flux-switching motor. 25 $F = \frac{3 π}{2 τ} (ψ_{d} i_{q} - ψ_{q} i_{d})$

After adopting the current vector control strategy, i_d = 0, i_q = i_max, substituting Eq. (24) into Eq. (25), the electromagnetic thrust output of the motor is obtained when current vector control is adopted: 26 $F = F_{1} + F_{2} + F_{3} + F_{4}$ 27 $F_{1} = \frac{3 π}{2 π} (\frac{3}{2} M_{s 1} + \frac{1}{2} M_{j}) i_{f} i_{q}$ 28 $F_{2} = \frac{3 π}{4 r} (L_{k 0} + 2 M_{h 0}) \sin (2 θ + \frac{π}{3}) i_{q}^{2}$ 29 $F_{3} = \frac{3 π}{2 τ} \cos (2 θ + \frac{π}{3}) i_{q}^{2} [\frac{L_{k 1}}{4} \sin (2 θ + \frac{π}{3}) + \frac{M_{h 1}}{2} \cos (2 θ - \frac{π}{6})]$ 30 $F_{4} = \frac{3 π}{4 r} M_{j} \sin (2 θ + \frac{5}{6} π) i_{i_{q}}$

From the above equation, the electromagnetic thrust output of the flux-switching permanent magnet linear motor can be divided into four parts, of which, F₁ is the average electromagnetic thrust output of the motor, independent of the inductance of the armature winding, and proportional to the armature current i_q and the equivalent excitation current i_f : F₂ is the 2nd harmonic content in the fluctuation of the electromagnetic thrust output of the motor, caused by asymmetric amplitude of the three-phase winding self-inductance and mutual inductance fluctuation, and proportional to the square of the armature current i_q : F₃ The 4th harmonic content in the electromagnetic thrust fluctuation of the motor output, caused by the asymmetry of the three-phase winding self-inductance and mutual inductance fluctuation amplitude, is proportional to the square of the armature current i_q : F₄ is the 2nd harmonic content in the electromagnetic thrust fluctuation of the motor output, caused by the asymmetry of the mutual inductance of three-phase windings and equivalent excitation windings, is proportional to the armature current i_q.

When the current i_q is small, the amplitude of the thrust fluctuation component due to inductive asymmetry is negligible. When the motor is running under load, the average thrust output of the motor is proportional to i_q, and the thrust fluctuation component caused by inductive asymmetry is also proportional to the square of i_q and i_q. As the load current increases, the thrust fluctuation rate of the motor gradually increases, which has a serious impact on the servo control performance such as motor positioning accuracy and machining accuracy.

3

Loss analysis of flux-switched permanent magnet linear motors

3.1

Calculation of motor losses

3.1.1

Motor magnetic density analysis

The motor magnetic density analysis is a prerequisite for the establishment of the motor iron consumption model and the accurate calculation of iron consumption. First of all, the sandwich armature structure of the magnetic field modulated permanent magnet linear motor achieves high thrust density while the existence of the modulating layer makes a large number of space harmonics in the magnetic field of the motor, which is a result of the fact that the modulating layer of the magnetic field modulated permanent magnet linear motor is composed of modulating teeth and modulating layer armature windings alternately, and the magnetic permeability varies periodically along the air gap of the modulating layer. Therefore, the study of magnetically dense harmonics in the motor is of great significance for calculating the iron consumption of the motor. Secondly, the magnetic field of the motor is not only the existence of alternating magnetic field, but also filled with a large number of rotating magnetic field. At present, the most common method of calculating iron consumption in engineering is to only consider the alternating magnetic field, ignoring the rotating magnetic field or using empirical formulas to approximate the rotating magnetic field of the motor to estimate the iron consumption. This method will inevitably bring large errors. Therefore, it is necessary to analyze the magnetic density of each part of the motor to explore the influence of different magnetization methods on the accurate calculation of iron consumption, so as to establish a more accurate mathematical model for the calculation of iron consumption.

The radial and tangential magnetization of each point is not sinusoidal, so it is necessary to carry out the Fourier decomposition of each point to observe the harmonics of the motor magnetization. The fundamental and harmonic amplitudes of the radial and tangential densities at each point are listed in Table 1, and the zero harmonic represents the DC component. The analysis of the table shows that the primary magnetization harmonics are dominated by odd harmonics, and the even harmonics are very low; the magnetization harmonics of points A, B and D are very rich, the radial magnetization amplitude of points A and B is slightly larger than the tangential magnetization amplitude, and the radial and tangential magnetization amplitude of point D is close to the tangential magnetization amplitude; the radial magnetization amplitude of points C and E is much larger than that of the tangential magnetization amplitude and point C is accompanied by large DC bias phenomenon, which is due to the existence of the modulation layer that makes the motor reluctance unevenly. This is due to the presence of the modulation layer that makes the motor reluctance uneven.

Table 1.

The radial and tangential wave at each point and the amplitude of harmonic

Harmonic frequency	0	1	3	5	7	9	11
Point A radial	0.041	2.385	0.382	0.293	0.031	0.058	0.013
point A tangential	0.033	1.952	0.720	0.118	0.026	0.034	0.018
Point B radial	0.054	2.481	0.463	0.067	0.019	0.026	0.007
point B tangential	0.038	1.890	2.061	0.159	0.008	0.046	0.023
Point C radial	0.001	2.267	2.115	0.384	0.036	0.002	0.010
point C tangential	0.072	0.092	2.483	0.182	0.029	0.005	0.020
Point D radial	0.068	1.831	0.820	0.366	0.017	0.013	0.004
point D tangential	0.006	1.505	1.328	0.029	0.073	0.030	0.016
Point E radial	0.064	1.720	1.173	0.290	0.052	0.011	0.008
point E tangential	0.027	0.302	0.005	0.001	0.001	0.002	0.001

3.1.2

Calculation of permanent magnet eddy current loss

The primary magnetic field generated by the permanent magnets is the main cause of iron loss in the motor. Fig. 4 shows the eddy current loss of permanent magnets with time for the motor at rated load and secondary motion speed. It can be seen that the eddy current loss reaches stability after 25ms.

Figure 5 shows the variation of permanent magnet eddy current loss with load current overload multiplier. It can be seen that the total iron consumption of the motor remains basically unchanged when the overload multiplier is one to three times.

3.2

Comparison of Constant Differential Frequency Control and Vector Control under Field-Circuit Coupling

This chapter analyzes the effect of the variation of the rotational frequency on the motor performance, and builds a vector control system with constant rotational frequency based on the analysis results, and then analyzes the electromagnetic dynamic performance of the linear induction motor for rail drive under the control strategy of constant rotational frequency through the joint simulation of the field-circuit coupling and compares it with the vector control.

The simulation working condition of the constant rotational frequency control system: the q-axis current is given as 160A (a larger value is needed to complete the acceleration process, and 360A is given initially), the rotational difference frequency is given as 12Hz, and the load force is 1400N.

The d and q-axis currents and feedback waveforms of the constant differential frequency control system are shown in Figs. 6 and 7. From the figures, it can be seen that after stabilization, the d-axis current is given as 190A, and the feedback is 185.6A, and the q-axis current is given as 150A, and the feedback is 148.7A, and the d- and q-axis currents follow well.

For comparison with the constant differential frequency control, the simulation working condition of the vector control is as follows: the speed is the same as the result of the speed after stabilized operation at constant differential frequency, and the load force is 1400 N. The simulation working condition of the vector control is the same as the result of the speed after stabilized operation at constant differential frequency.

Comparison of the operating speed curves of the two is shown in Figures 8 and 9. The speed of the vector control system is given as 50km/h.

Comparison shows that the constant differential frequency control system, which lacks a speed outer loop, does not stabilize the running speed quickly. q axis current is given as 150A, and the running speed is stabilized at 50km/h with a load force of 1400N. The vector control system can control the speed accurately with less fluctuation when stabilizing, and it can realize the static-independent speed regulation.

Comparison of secondary current losses for constant differential frequency control and vector control is shown in Fig. 10 and Fig. 11. From Fig. 11, it can be seen that during acceleration, the secondary instantaneous loss of vector control is much larger than that of constant differential frequency. After stabilization, the secondary loss is 2.5kW for constant differential frequency and 2.2kW for vector control, and the constant differential frequency also has an advantage in terms of secondary loss.

Comparison of primary iron consumption for constant differential frequency control and vector control is shown in Fig. 12 and Fig. 13. From Fig. 12 and Fig. 13, the primary iron consumption after stabilization is 1.76 kW for constant differential frequency and 2.34 kW for vector control.The iron consumption waveforms of both fluctuate violently due to the voltage source PWM signal.

In conclusion, the constant differential frequency control strategy has a better performance in terms of stable operation of the motor, and the electromagnetic characteristics are not significantly different from those of vector control under the same operating conditions.

4

Conclusion

This paper takes the flux-switching permanent magnet linear motor as the research object, based on the multi-physical field coupling analysis, and carries out in-depth analysis and research from the aspects of motor magnet density, permanent magnet eddy current loss analysis, field and circui-tcoupling, and constant rotational difference frequency control strategy, etc. The specific conclusions are as follows: 1)

The primary magnetization harmonics are dominated by odd harmonics, and the even harmonics are extremely low. Points A, B, and D are very rich in magnetization harmonics; the radial magnetization amplitude at points C and E is much larger than the tangential magnetization amplitude.

2)

The eddy current loss of the permanent magnet under the rated load and secondary motion speed is stabilized after 25ms; the total iron consumption of the motor remains basically unchanged when the overload multiplier is one to three times.

3)

Under the constant rotational frequency control system, the d-axis current is given as 190A and the feedback is 185.6A after stabilization, and the q-axis current is given as 150A and the feedback is 148.7A, which indicates that the d and q-axis currents follow well under the constant rotational frequency control.

4)

The constant rotational frequency control system without speed external loop can not stabilize the running speed quickly. While the vector control system can control the speed accurately, and the fluctuation is smaller when stabilizing, which can realize the speed regulation without static difference.

5)

After stabilization, the secondary loss is 2.5kW under constant differential frequency, and the secondary current loss is 2.2kW under vector control, and the constant differential frequency also has the advantage in secondary loss.

6)

The primary iron loss is 1.76kW for the stabilized constant-differential frequency and 2.34kW for the vector control, and the waveforms of both iron losses fluctuate drastically due to the voltage source PWM signal.

The experimental conclusions show that the constant-differential-frequency control strategy has a better performance in the stable operation of the flux-switched permanent-magnet linear motor, and the electromagnetic characteristics are not significantly different from those of the vector control under the same operating conditions. It realizes the effective verification of the loss characteristic analysis of the flux-switching permanent magnet linear motor based on the multi-physical field coupling analysis carried out in this paper.

Langue:: Anglais

Périodicité:: 1 fois par an
Sujets de la revue:: Sciences de la vie, Sciences de la vie, autres, Mathématiques, Mathématiques appliquées, Mathématiques générales, Physique, Physique, autres

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Loss characterization of flux-switched permanent magnet linear motor based on multi-physics field coupling analysis

Ying Hong

Publié en ligne: 24 sept. 2025

Reçu: 17 janv. 2025

Accepté: 04 mai 2025

DOI: https://doi.org/10.2478/amns-2025-0972

Mots clésMultiphysics field coupling, LFSPM, Motor loss, Vector control, Constant differential frequency

© 2025 Ying Hong, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Mots clés
Multiphysics field coupling, LFSPM, Motor loss, Vector control, Constant differential frequency