Measurement and Modeling of Magnetic Materials under 3D Vectorial Magnetization for Electrical Machine Design and Analysis

: The magnetic properties of magnetic cores are essential for the design of electrical machines, and consequently appropriate mathematical modeling is needed. Usually, the design and analysis of electrical machines consider only the one-dimensional (1D) magnetic properties of core materials, i.e., the relationship of magnetic ﬂux density ( B ) versus magnetic ﬁeld strength ( H ), and their associated power loss under 1D magnetization, in which the B and H are constrained in the same orientation. Some studies have also been performed with the two-dimensional (2D) magnetizations in which the B and H are vectorial, rotating on a plane, and they may not be in the same direction. It has been discovered that the 2D rotational property is very different from its 1D alternating counterpart. However, the magnetic ﬁelds in an electrical machine, in particular claw pole and transverse ﬂux machines, are naturally three-dimensional (3D), and the B and H vectors are rotational and may not lie on the same plane. It can be expected that the 3D vectorial property might be different from its 2D or 1D counterpart, and hence it should be investigated for the interests of both academic research and engineering application. This paper targets at a general summary about the magnetic material characterization with 3D vectorial magnetization, and their application prospect in electrical machine design and analysis.


Introduction
Electrical machines play crucial roles in modern societies, e.g., the majority of electricity is generated by electric generators, and a large number of mechanical loads are driven by electric motors.To overcome the problems of fossil fuel depletion, environmental pollution and global warming, it is anticipated that most of the future electrical power will be produced by clean energy sources and most of the mechanical loads will be driven by electric motors.For example, the transportation electrification has recently attracted extensive research interests and the electric motor is among the key technologies.Various electric motors have been developed for driving various electrified vehicles, such as electric bicycles [1], electric cars [2][3][4][5], electric buses [6,7], electric trains [8][9][10], electric ships [11][12][13] and even electric aircrafts [14,15].In these vehicles the allowed room and weight are generally quite restricted, so the electric motors should be designed with high power density, but the power loss within certain volume and the associated temperature rise would increase, deteriorating the machine performance and reliability.Therefore, it is crucial to properly consider and predict the power loss and temperature rise in the motor design [16][17][18][19][20].
Energies 2023, 16, 417 2 of 11 Magnetic materials, in particular soft ferromagnetic materials, are usually applied as the core of electrical machine and their properties affect the machine performance directly.For electrical machine analysis and design, the core material properties must be available, including the relationship of magnetic flux density (B) versus field strength (H), and associated magnetic power loss, which are usually provided by the material supplier.In general, the magnetic property data are measured on material samples, which usually have magnetic circuits of closed path constructed by stacked lamination steel rings, wound ribbon, solid ring of sintered or bonded powders, or Epstein frame of assembled strips on a frame [21].In all these types of samples, the B and H are restrained in one orientation.
In electrical machines, however, the tips of B and H vectors are generally twodimensional (2D) and rotational, and it has been found by various researchers that the magnetic properties under the 2D magnetizations are very different from those under the one-dimensional (1D) alternating flux densities [22][23][24][25][26].At a low flux density range, the magnetic power loss (core loss) with 2D rotational magnetic fluxes is about twice that with 1D alternating fluxes and goes up with the increase of flux density, but around the saturation point, the core loss may drop greatly if the flux density further increases.This is very different from that the 1D alternating core loss always goes up with the B increase.A number of research works have been performed for modeling the B-H relations under 2D rotation magnetization, such as Stoner-Wohlfarth models [27][28][29], vectorial Preisach models [30][31][32], combined models [33,34], E&S model [35,36], and Jiles-Atherton models [37,38].Some models have also been reported for calculating the 2D rotational core loss in electrical machines [39][40][41][42][43].
In some types of electrical machines, such as claw pole machines and transverse flux machines, the magnetic fields are three-dimensional (3D) and rotational.Naturally, the core material properties under 3D magnetization should be studied and appropriately modeled for the machine design and analysis.This paper targets at an overview about the research works on 3D vectorial magnetic properties of magnetic materials, including the magnetic measurement, mathematical modeling and application prospect.
The rest of the paper is planned as follows.Section 2 introduces a relatively new 3D flux material, soft magnetic composite, which is considered as an ideal core material for designing 3D flux electrical machines.In Section 3, the development of 3D magnetic property testing systems is described.Section 4 discusses the mathematical modeling of the 3D vectorial magnetic properties and how to apply it in the design and analysis of electrical machines.In Section 5, some experimental results of soft magnetic composite (SMC) materials under different 1D, 2D and 3D magnetization patterns are described for implementing and evaluating their application in the electrical machine design and analysis with 3D flux.Finally, Section 6 concludes the paper by discussing possible future research works with the major challenges highlighted.

SMC Materials and SMC Electrical Machines
SMC materials possess a number of merits over traditional electrical steel sheets, and hence the materials and their applications in electrical machines have acquired great popularity of research in the past three decades [44][45][46].The SMC material basis is the iron powder of high purity and compressibility, and the iron particles are covered with thin insulation, so the material features very high electrical resistivity and very low eddy current loss under varying magnetic fields.The insulated powder articles are pressed into magnetic components with desired shape and dimensions by using the powder metallurgical techniques, so the manufacturing cost can be very low when performing mass production.
The most notable advantage might be the 3D magnetic isotropy caused by the powder nature.This creates the key design benefit as now the electrical machines can be designed with 3D magnetic circuits.In conventional electrical machines with laminated steel sheet cores, the magnetic field path must be within the lamination plane, i.e., 2D.Any magnetic flux component in the third direction, i.e., vertical to the lamination, might induce huge eddy current loss.The SMC material removes this restraint, so that electrical machines can be designed with great flexibility, and novel configuration with very high-power density becomes possible.
To study the SMC application in electrical machines, a large number of works have been conducted by different scholars.Nearly all kinds of electrical machines have been investigated with the SMC cores, but the most promising types appear to be those with 3D magnetic flux paths, e.g., claw pole machines [47][48][49][50][51][52][53][54] and transverse flux machines [55][56][57][58][59][60].As a result, the SMC magnetic properties with 3D magnetic fluxes should be obtained and appropriately modeled for the design and analysis of these 3D flux machines.

Measuring System of 3D Magnetic Properties
Led by J. Zhu, the University of Technology Sydney (UTS) Magnetic Testing Group developed the world-first 3D magnetic testing system in 2001, as illustrated in Figure 1 [61].A photo of the 3D magnetic property tester is shown in Figure 2.
sheet cores, the magnetic field path must be within the lamination plane, i.e., 2D.Any magnetic flux component in the third direction, i.e., vertical to the lamination, might induce huge eddy current loss.The SMC material removes this restraint, so that electrical machines can be designed with great flexibility, and novel configuration with very highpower density becomes possible.
To study the SMC application in electrical machines, a large number of works have been conducted by different scholars.Nearly all kinds of electrical machines have been investigated with the SMC cores, but the most promising types appear to be those with 3D magnetic flux paths, e.g., claw pole machines [47][48][49][50][51][52][53][54] and transverse flux machines [55][56][57][58][59][60].As a result, the SMC magnetic properties with 3D magnetic fluxes should be obtained and appropriately modeled for the design and analysis of these 3D flux machines.

Measuring System of 3D Magnetic Properties
Led by J. Zhu, the University of Technology Sydney (UTS) Magnetic Testing Group developed the world-first 3D magnetic testing system in 2001, as illustrated in Figure 1 [61].A photo of the 3D magnetic property tester is shown in Figure 2.  The measurement system consists of a 3D magnetic property tester, a computer data acquisition and control system, and a 3-channel power amplifier.Three pairs of excitation windings wound around the six yokes of the tester are used to produce 3D magnetic flux in the material sample, which is located in the tester center.By controlling the magnetic excitations in three axes, i.e., the magnitudes and phase angles of excitation currents, the tester is capable of producing various flux patterns, such as 1D alternating in any specified orientation, 2D circularly or elliptically rotating in a plane tilted for a specified angle from an axis and rotating in a 3D pattern with the loci of the B vector tip forming a specified surface, according to the measurement requirement [62][63][64][65][66][67][68][69][70][71].

Material Sample in the 3D Measurement System
As seen in Figures 1 and 2, a sample of the testing material is put in the middle of the testing system, in which a 3D vectorial magnetization is generated by the currents passing the three excitation coils.In our studies, the sample of cubic shape was manufactured by cutting the preformed SMC material.The size was determined by pre-analysis, e.g., calculating the magnetic field distribution based on 3D finite element analysis.It is important to design the tester and sample capable of generating the requested flux density magnitude and uniformity in the sample with the measurement range of frequency.

Measurement of 3D Magnetic Field
The magnetic flux density (B) and magnetic field strength (H) sensing coils are employed to measure the B and H components at three axes.As shown in Figure 3, on each surface there are two H coils for measuring the two H components tangential to the surface, and for each axis four coils are connected in series.Three B coils are wrapped around the sample for measuring the B components along three axes.The measurement system consists of a 3D magnetic property tester, a computer data acquisition and control system, and a 3-channel power amplifier.Three pairs of excitation windings wound around the six yokes of the tester are used to produce 3D magnetic flux in the material sample, which is located in the tester center.By controlling the magnetic excitations in three axes, i.e., the magnitudes and phase angles of excitation currents, the tester is capable of producing various flux patterns, such as 1D alternating in any specified orientation, 2D circularly or elliptically rotating in a plane tilted for a specified angle from an axis and rotating in a 3D pattern with the loci of the B vector tip forming a specified surface, according to the measurement requirement [62][63][64][65][66][67][68][69][70][71].

Material Sample in the 3D Measurement System
As seen in Figures 1 and 2, a sample of the testing material is put in the middle of the testing system, in which a 3D vectorial magnetization is generated by the currents passing the three excitation coils.In our studies, the sample of cubic shape was manufactured by cutting the preformed SMC material.The size was determined by pre-analysis, e.g., calculating the magnetic field distribution based on 3D finite element analysis.It is important to design the tester and sample capable of generating the requested flux density magnitude and uniformity in the sample with the measurement range of frequency.

Measurement of 3D Magnetic Field
The magnetic flux density (B) and magnetic field strength (H) sensing coils are employed to measure the B and H components at three axes.As shown in Figure 3, on each surface there are two H coils for measuring the two H components tangential to the surface, and for each axis four coils are connected in series.Three B coils are wrapped around the sample for measuring the B components along three axes.By measuring the induced electromotive force of the sensing coils, the B and H components along each axis can be worked out by where i = x,y,z, KBi and KHi are respectively the constants of the B and H coils, and the constants are determined by calibration [72,73].

Core Loss Testing under 3D Magnetization
When the B and H values have been obtained, the sample core loss Pt in W/kg can be computed according to Poynting's theorem by 0 0 1 1 where T = 1/f is the excitation period, f is the excitation frequency, and ρm is the mass density of the sample material.

3D Vectorial Magnetic Property Modeling and Application
The prediction of B from H or vice versa is necessary for magnetic field analysis of electrical machines.Michelakis et al. proposed a 3D moving vectorial Preisach-type model of hysteresis for magnetic material composed of uniaxial interacting particles [74].Zhong et al. presented a 3D vector magnetization model based on the 3D Stoner-Wohlfarth element operator, in which a phenomenological mean-field approximation was used to consider the magnetic interactions among particles [75].Cardelli et al. extended their study on 2D vector hysteresis operator to a 3D case [76].Li et al. presented a 3D magnetic hysteresis model based on a 3D operator according to the minimum energy principle of a stable magnetization state [77].
The term of magnetic reluctivity or permeability is usually applied to relate the B and H.For 1D alternating magnetic field, the B and H are in the same direction and the constitutive equation can be expressed as By measuring the induced electromotive force of the sensing coils, the B and H components along each axis can be worked out by where i = x,y,z, K Bi and K Hi are respectively the constants of the B and H coils, and the constants are determined by calibration [72,73].

Core Loss Testing under 3D Magnetization
When the B and H values have been obtained, the sample core loss P t in W/kg can be computed according to Poynting's theorem by where T = 1/f is the excitation period, f is the excitation frequency, and ρ m is the mass density of the sample material.

3D Vectorial Magnetic Property Modeling and Application
The prediction of B from H or vice versa is necessary for magnetic field analysis of electrical machines.Michelakis et al. proposed a 3D moving vectorial Preisach-type model of hysteresis for magnetic material composed of uniaxial interacting particles [74].Zhong et al. presented a 3D vector magnetization model based on the 3D Stoner-Wohlfarth element operator, in which a phenomenological mean-field approximation was used to consider the magnetic interactions among particles [75].Cardelli et al. extended their study on 2D vector hysteresis operator to a 3D case [76].Li et al. presented a 3D magnetic hysteresis model based on a 3D operator according to the minimum energy principle of a stable magnetization state [77].
The term of magnetic reluctivity or permeability is usually applied to relate the B and H.For 1D alternating magnetic field, the B and H are in the same direction and the constitutive equation can be expressed as The reluctivity ν is a scalar and its value may vary with the change of B, which is called magnetic nonlinearity.For 2D or 3D rotational magnetization, the reluctivity becomes a full 2D or 3D tensor [78][79][80][81], and the constitutive equation is where ν ij is the reluctivity tensor, i,j = x,y,z in Cartesian coordinates, or r,θ,z in cylindrical coordinates.
The nine elements in the tensor can be obtained by a few measurements under 3D or quasi-3D magnetizations [78].Then they can be used to solve the 3D magnetic field distribution in electrical machines [82].

Implementation and Evaluation of 3D Vectorial Properties of Soft Magnetic Composite Materials in 3D Flux Electrical Machines
Based on the 3D magnetic testing system shown in Figures 1 and 2, some experi-mental results have been obtained with the soft magnetic composite (SMC) samples under various magnetization patterns, e.g., 1D alternating (Figure 4a), 2D circularly rotating (Figure 4b), and 3D spherical flux densities (Figure 4c) at 50 Hz.For 1D case, the relationship of magnetic flux density (B) versus magnetic field strength (H) can be plotted as a series of loops, but the B-H relations under 2D or 3D case become more complex.To work out the corresponding core losses, the Poynting's theorem, given in Equation (3), can be used.
The reluctivity ν is a scalar and its value may vary with the change of B, which is called magnetic nonlinearity.For 2D or 3D rotational magnetization, the reluctivity becomes a full 2D or 3D tensor [78][79][80][81], and the constitutive equation is where νij is the reluctivity tensor, i,j = x,y,z in Cartesian coordinates, or r,θ,z in cylindrical coordinates.
The nine elements in the tensor can be obtained by a few measurements under 3D or quasi-3D magnetizations [78].Then they can be used to solve the 3D magnetic field distribution in electrical machines [82].

Implementation and Evaluation of 3D Vectorial Properties of Soft Magnetic Composite Materials in 3D Flux Electrical Machines
Based on the 3D magnetic testing system shown in Figures 1 and 2, some experimental results have been obtained with the soft magnetic composite (SMC) samples under various magnetization patterns, e.g., 1D alternating (Figure 4a), 2D circularly rotating (Figure 4b), and 3D spherical flux densities (Figure 4c) at 50 Hz.For 1D case, the relationship of magnetic flux density (B) versus magnetic field strength (H) can be plotted as a series of loops, but the B-H relations under 2D or 3D case become more complex.To work out the corresponding core losses, the Poynting's theorem, given in Equation (3), can be used.For implementing the measured properties of SMC material samples to the electrical machines with SMC cores, e.g., calculation of core losses, a few SMC machines have been designed, analyzed and prototyped.According to the literature, it might be sufficient to measure the quasi-3D properties, i.e., the 2D circularly rotational magnetic properties on the three orthogonal planes [83].Based on the 1D alternating and 2D circularly rotating properties, the core losses of SMC electrical machines with 3D flux path have been calculated and the results are found quite close to experiments on motor prototypes, e.g., all within 8% error compared with the measurements [84][85][86][87].By comparison, the calculations using 1D data only would cause a discrepancy of 20-50% from the measurements.

Discussions and Challenges
This paper has presented an overview about the study on 3D vectorial magnetic properties of magnetic materials, specifically SMC, for design and analysis of electrical machines with 3D magnetic flux path.In particular, this paper extends greatly the coverage of previous works [65] and discusses the existing challenges and future research trends in this area from a perspective view.As explored above, it would be significant both theoretically and practically to apply 3D vectorial magnetic properties for proper electrical machine design and analysis, but only a couple of groups have conducted research in this area due to the complexities of 3D testing system setup and measurement.The possible future works and challenges are summarized below.

Standard Masurement and Modeling of 3D Vectorial Magnetic Properties
Although the study on 3D vectorial magnetic properties has already been conducted for two decades, it appears to be still at a very early research stage.For example, the magnetic properties of SMC under true 3D flux density patterns such as a sphere can be measured by using the 3D magnetic testing system, but it is still a pending issue how to apply these properties.
In addition, the sample shape might be an issue.The currently used cubic sample may cause measurement error due to shape factor, but the preferred spherical sample would have difficulties in placing the B and H sensing coils.For implementing the measured properties of SMC material samples to the electrical machines with SMC cores, e.g., calculation of core losses, a few SMC machines have been designed, analyzed and prototyped.According to the literature, it might be sufficient to measure the quasi-3D properties, i.e., the 2D circularly rotational magnetic properties on the three orthogonal planes [83].Based on the 1D alternating and 2D circularly rotating properties, the core losses of SMC electrical machines with 3D flux path have been calculated and the results are found quite close to experiments on motor prototypes, e.g., all within 8% error compared with the measurements [84][85][86][87].By comparison, the calculations using 1D data only would cause a discrepancy of 20-50% from the measurements.

Discussions and Challenges
This paper has presented an overview about the study on 3D vectorial magnetic properties of magnetic materials, specifically SMC, for design and analysis of electrical machines with 3D magnetic flux path.In particular, this paper extends greatly the coverage of previous works [65] and discusses the existing challenges and future research trends in this area from a perspective view.As explored above, it would be significant both theoretically and practically to apply 3D vectorial magnetic properties for proper electrical machine design and analysis, but only a couple of groups have conducted research in this area due to the complexities of 3D testing system setup and measurement.The possible future works and challenges are summarized below.

Standard Masurement and Modeling of 3D Vectorial Magnetic Properties
Although the study on 3D vectorial magnetic properties has already been conducted for two decades, it appears to be still at a very early research stage.For example, the magnetic properties of SMC under true 3D flux density patterns such as a sphere can be measured by using the 3D magnetic testing system, but it is still a pending issue how to apply these properties.
In addition, the sample shape might be an issue.The currently used cubic sample may cause measurement error due to shape factor, but the preferred spherical sample would have difficulties in placing the B and H sensing coils.

Infludences of Various Paramaters on Magnetic Properties
The magnetic properties are also influenced by many other parameters, e.g., operational temperature and mechanical stress, which should be properly measured and modeled [88][89][90][91].This is also true for 1D or 2D measurements, but it appears much more difficult to measure the effects under 3D case, e.g., how to apply 3D compressive or shear forces on the sample which is surrounded by magnetic poles in all three axes.The 3D testing system is generally quite large in size, so a large oven with controllable temperature is needed.
The core loss affects directly the temperature rise as well as mechanical stress, and the temperature and stress also affect the core loss.Therefore, a dynamic modeling considering the two-way effects is desired based on the digital-twin concept [92,93].
Besides the magnetic flux density pattern including waveform, magnitude and frequency, machine operational temperature and mechanical stress, there are many other factors which affect the core loss, such as DC bias of magnetic flux and magnetostriction of the ferromagnetic material.It is very challenging to properly account for all these influencing factors, and in general an appropriate compromise between accuracy and complexity is needed in practice.

Figure 2 .
Figure 2. A photo of the 3D vectorial magnetic property measurement system.

Figure 2 .
Figure 2. A photo of the 3D vectorial magnetic property measurement system.

Figure 3 .
Figure 3.A cubic material sample with B and H sensing coils.

Figure 3 .
Figure 3.A cubic material sample with B and H sensing coils.

Figure 4 .
Figure 4. Experimental results of SMC samples at 50 Hz: (a) B-H relation and associated core loss under 1D alternating sinusoidal B; (b) B-H relation under 2D circularly rotating B; and (c) B-H relation under 3D spherical B.

Figure 4 .
Figure 4. Experimental results of SMC samples at 50 Hz: (a) B-H relation and associated core loss under 1D alternating sinusoidal B; (b) B-H relation under 2D circularly rotating B; and (c) B-H relation under 3D spherical B.