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This paper proposes ways to reduce motor losses and improve efficiency. The purpose of this paper is to review the magnetic properties of vector magnetic characteristic techniques and to suggest the use of magnetic characteristic analytical methods. In this shows the vector magnetic properties of the non-oriented electrical steel sheet of the motor core material. An engineering model considering vector magnetic characteristics in the magnetic characteristic analysis has been proposed. The utility of this technique and examples of its use is shown.

This paper presents measured results of the 2-D vector magnetic characteristics of ultra-thin electrical steel sheets for high-efficiency high-speed motors. Generally, Non-oriented electrical steel sheets with 0.2–0.5 mm of thickness are used for motor cores. However, their thicknesses are not appropriate to use for cores of high-speed motors. We newly developed “Ultra-thin electrical steel sheet” with 0.08 mm of thickness to reduce the core loss of high-speed motors. The magnetic characteristics of the Ultra-thin electrical steel sheets are almost equal to NO (35A300) under exciting frequency of 50 Hz. Since the thickness of the steel sheet is very thin as 0.08 mm, eddy currents are hardly generated under high frequency. As the result, it was confirmed that the iron loss was decreased by 53% when we compare the iron loss of the motor used ultra-thin electrical steel sheets (0.08 mm) to that of the motor used conventional electrical steel sheets (0.35 mm) under the condition of 10 000 rpm.

The magnetic properties of electrical steel sheets, which are used as core materials in numerous electric-powered machines, tend to deteriorate due to residual stress and external loads during manufacturing processes. Previously, we reported our measurement results for the residual stress distribution in a motor core and clarified that sheer stress, along with tensile and compressive stress, exists in the core’s plate and surface, which makes it necessary to evaluate the magnetic properties of the core material under various stress conditions. In this paper, we report the results of an investigation into the effects of an arbitrary shear stress on the vector magnetic properties of a non-oriented electrical steel sheet under both alternating and rotating conditions.

In this paper, we consider the improvement of the stator structure of the self-propelled rotary actuator, whose rotational axis is parallel to the exciting magnetic field. The stator of the actuator is constructed from the grain-oriented electrical steel sheet. The relation between the rolling direction of the steel sheet and the direction of the exciting magnetic field is important for the starting torque and rotational stability of the actuator’s rotor. Therefore, we analyze the rotor in consideration of the rolling direction using the finite-element method that introduced the complex E&S model, which is able to express the 2-D vector magnetic properties. As a result, we estimate the suitable setting direction of the rolling direction in the steel sheet of the actuator’s stator by this analysis.

The viscosity of a magnetorheological fluid (MRF) changes with the magnetic field; however, the behavior of magnetic particles in an MRF under the application of a magnetic field has not been fully clarified. In our previous work, we proposed a real-time observation system in which the torque is simultaneously measured. Textured poles concentrated on a magnetic field, increasing the transfer of torque. In this paper, the behavior of magnetic particles, such as the formation, fracture, and sliding of clusters, in the presence of textured poles as well as the transfer of torque was investigated by in situ observation using our torque measurement system. With increasing depth of the pole texture, the magnetic-field gradient increased, the magnetic field at the edges of the texture increased, the clusters became concentrated in a narrower region, and the maximum torque increased. These results are expected to lead to the design of more efficient torque transfer devices using MRFs.

MR (Magnetic Rheological) fluid is a kind of functional fluid, which can be hardened by impressing magnetic flux. MR fluid has the high speed response to the external magnetic field and a big yield stress in comparison with the ferrofluid. In recent years, various devices utilizing MR fluid have been developed. Such developments are enabled with knowledge based on measured viscoelastic properties of MR fluid. However, precise measurement to obtain effect of the magnetic field on viscoelasticity is very difficult. The difficulty exists in generating a uniform magnetic field and evaluating the effective magnetic field. Accurate measurements become possible by solving these problems. In this paper, we propose a new magneto-viscoelasticity measurement system of MR fluid, which can generate a uniform magnetic field.

We have reported the measured results for the magnetic properties of a non-oriented electrical steel sheet under various magnetic flux and stress conditions and proposed a new vector magnetic characteristic analysis technique to consider the stress effect on the motor core. In this paper, the stress effect on the magnetic flux density vector, magnetic field strength vector, and magnetic power loss is examined to improve the performance of electric motors. Thus, the assumed stress distribution is applied to the local part of a stator core, and the vector magnetic characteristic analysis is performed using stress-complex $E$ and $S$ modeling. From these results, the loci of the magnetic field strength vector and the magnetic power loss differed by changing the stress distribution. It was possible to decrease the magnetic power loss by applying a controlled local stress distribution.

This paper presents a magnetic coupling for a large-sized mixer, which is separated by a vessel wall. In order to improve the transmission-torque, the magnetic material attached magnetic-flux concentrated surface permanent magnet (MCSPM) arrangement is applied to the magnetic coupling and the construction is optimized. The results show that the MCSPM arrangement is very effective to improve the air-gap magnetic flux density and the transmission torque even quantity of very few permanent magnets.

In general, the magnetic properties of electrical steel sheets are strongly influenced by mechanical stress. We have reported the measurement results of vector magnetic properties of a non-oriented electrical steel sheet under stress conditions. From these results, it is possible to control the vector magnetic property by applying local mechanical stress. Therefore, we next focus on an induction heating technique that applies mechanical stress in a non-oriented electrical steel sheet. It is very important to control the eddy current distribution inside the non-oriented electrical steel sheet. This paper presents a magnetic flux concentration plate structure to control the eddy current in steel sheets by using an induction heating technique. The magnetic flux concentration plate structure is investigated in a three-dimensional magnetic field analysis by the finite element method.

In electrical equipment, such as transformers and motors, rotating magnetic fluxes occur at T-joint parts and back yoke parts. It is therefore necessary to understand the magnetic properties and magnetostrictions under several rotating flux conditions. This paper presents measured vector magnetic properties including 2-D magnetostriction of a nonoriented electrical steel sheet (including a 6.5%SiFe sheet), a grain-oriented electrical steel sheet, and a permendur sheet under rotating flux conditions. The relationship among the magnetic flux density vector, the magnetic field strength vector, and the 2-D magnetostriction is clarified in comparison of the various electrical steel sheets. Furthermore, from the measurement results, suitable materials are suggested for equipment.

Magnetic properties of nonoriented electrical steel sheets are influenced by mechanical stress because they exhibit the phenomenon of magnetostriction. Therefore, it is necessary to clarify the relationship between magnetic properties and magnetostriction under mechanical stress. This paper presents the results of measurements of the vector magnetic properties and the magnetostriction of a nonoriented electrical steel sheet under biaxial stress. The loci of the magnetic field strength vector and the 2-D magnetostriction are measured under both biaxial stresses. The magnetic power loss and the peak-to-peak value of the magnetostriction in an arbitrary direction decreased due to the applied biaxial tensile stress. It can be concluded that it is possible to improve the magnetic properties and reduce the magnetostriction of an electrical steel sheet by applying biaxial tensile stress.

To reduce power loss of transformers further from now on and to raise efficiency, we have measured magnetic property distributions in a three-phase tripod V-connection transformer model core. For the measurement, we have developed a localized vector magnetic property measurement system consisting of a three-axis positioning system, a vector magnetic sensor (V-H sensor), and PC. The measured results showed that rotational magnetic fluxes with a large axis ratio were generated in the V-junction parts of the transformer model core, and large iron loss was observed. To reduce the iron loss near the V-junction parts, we have tried to change local magnetic characteristics of the electrical steel sheet near the V-junction parts by using laser irradiation. The effect of the laser irradiation on the local magnetic properties is evaluated by measuring distributions of the local magnetic properties.

Electrical steel sheet that is used as a core in electrical machines deteriorates due to residual stress and external stress. The magnetic properties and stress need to be clarified in order to effectively utilize electrical steel sheet in electrical machines. We have measured the residual stress distribution in a motor core using an X-ray stress measurement device and clarified not only the tensile and compressive stress distributions, but also the shear stress distribution. In this study, a new measurement system is developed for determining vector magnetic properties under shear stress. The vector magnetic properties of the non- oriented electrical steel sheets under the alternating magnetic flux and rotating magnetic flux conditions were measured with the developed system. As the result, it was clear that the magnitude of the magnetic field strength vector and the magnetic power loss along the direction of about 135 ◦ increased by applying the shear stress.

This paper presents a modeling of vector magnetic properties considering residual stress effects in electrical steel sheets. Because the practical phenomenon is too complicated, complex variable expressions are applied to the modeling used in the characteristic analysis with the FEM. We call this stress-complex-E&S modeling. The validity of the developed modeling was verified in comparison with the measurement. The result shows that the developed method is very effective to reduce necessary conditions in the approximate expression of the material coefficients depending on stress.

In this paper, vector magnetic properties measurement and magnetic characteristics analysis of a non-oriented electrical steel sheet under high magnetic flux is presented. At first, the vector magnetic property of the non-oriented electrical steel sheet under the high magnetic flux condition is measured. Next relationships between the magnetic flux density vector $B$ and magnetic field strength vector $H$ of ring core model is simulated by using the database of measured vector magnetic property and a dynamic integration type E&S modeling. As the results, the difference of the maximum magnetic flux density $\vert B\vert_{\max}$ , the maximum magnetic field strength $\vert H\vert_{\max}$ distribution and the local hysteresis loops in the ring core model is obtained by increasing the exciting voltage. In particular, it was clarified that $\vert H\vert_{\max}$ distribution and locus of $H$ vector was effected due to the magnetic anisotropy of the non-oriented electrical steel sheet strongly.

The Barkhausen signal was used for the nondestructive evaluation methods in order to evaluate residual stress levels in ferromagnetic materials. However, it cannot be said that this technique has been established completely because the evaluation is usually not done under controlled magnetic flux conditions. When the magnetic field is applied into the materials by the exciting coil, material characteristics will change, therefore, controlling the magnetic flux condition is required for the precise evaluation of magnetic materials. Some results of the residual stress level evaluation using Barkhausen signals under the controlled magnetic flux condition measured from different tensile stressed specimens are shown and discussed them for high efficiency motors in this paper.

This paper presents measured local vector magnetic properties in a transformer type model core made of a laser-scratched grain-oriented electric steel sheet by using a vector-hysteresis (V-H) sensor. The V-H sensor consists of B-needle probes for measurement of the magnetic flux density vector B and double H-coil for measurement of the field strength vector H. The distance between the B-needle probes was shortened to be 2.5 mm and a 2 mm × 2 mm-sized double H-coil was mounted on the sensor head to realize higher resolution. The measurement accuracy of the B-needle probes was confirmed with a spatial probe, which can change the needle distance from 2 mm to 20 mm. The results show that the detailed local two-dimensional vector magnetic properties depending on laser-scratches are measurable with the developed new sensor.

In the conventional eddy current testing (ECT), it is difficult to estimate wall thinning from outside of pipes without removing the covered materials due to the large lift-off between a pipe surface and a sensor. In order to overcome this difficulty, a long-pulsed excitation system has been developed with a magnetic sensor, by which the magnetic flux can be penetrated deeply inside pipes because of low frequency and small induced eddy current. In this paper, the numerical results and measured results of the newly developed long-pulsed excitation system are shown. The results show the effectiveness of the proposed long-pulsed method for a large lift-off evaluation to wall thinning.

Magnetic gears are a force transmitter consisting of permanent magnets. The mechanical input can be transmitted to an output shaft without contact by magnetic forces. The magnetic gears are not worn out because there is no friction. As a result, the running costs such as the maintenance fee can be suppressed and the resources can be saved. However, the transmission torques of the conventional magnetic gears, which have so far been developed, are very low. Besides, new structure models designed for high torque density need a lot of permanent magnets and multi-pole constructions. Those structures are complex and the manufacturing is difficult. In this research, we applied a flux concentration type surface permanent magnet arrangement to a surface permanent magnet type magnetic gear in order to improve the transmission torque and to reduce the amount of permanent magnets. The magnetic flux distribution, the gap flux density and the transmission torque of the developed new models are numerically analyzed by using the two-dimensional finite element method. In this paper, a permanent magnet structure optimized to reduce its amount and influence of the flux concentration type surface permanent magnet arrangement on the gap flux density distribution and transmission torque are reported.

Electrical steel sheet in actual rotating machines is magnetized under alternating and rotating magnetic flux conditions. In addition, the magnetic properties of the electrical steel sheet are affected by mechanical stress strongly and are deteriorated due to the residual stress during the manufacturing process. Therefore, it is important to know the magnetic properties under various stress and magnetic flux conditions. In this paper, we investigated the effect of the mechanical stress on vector magnetic property under alternating magnetic flux conditions. Basing on the results, we can conclude that the vector magnetic property of the electrical steel sheet depends on the mechanical stress and the excitation direction. Therefore, it is important to consider the vector magnetic property with the mechanical stress in the design of the electrical machine.

This paper presents measured 2-D vector magnetic properties of a Fe-Si-B amorphous sheet. The relationship between the magnetic flux density and field strength vector under alternating and rotating flux conditions is clarified by using the 2-D vector magnetic property measurement system. The magnetic anisotropy in plane was observed and it was changed depending on the flux density level. Moreover the iron loss depending on the maximum flux density is shown.

This paper presents results of iron-loss evaluation of an induction motor model core by using the finite element method considering two-dimensional vector magnetic properties. We propose a new iron-loss calculation method considering higher harmonic components of the flux density waveforms caused by the secondary slot ripples. Influence of the harmonic components on the total iron loss is clarified numerically. In addition, reduction of rotational iron losses is investigated by making holes at the tooth base parts. The results show that the rotational magnetic flux distribution is effectively controlled by making holes. In this paper, obtained knowledge about iron-loss control with holes in the stator core is presented and discussed.

This paper presents an improved integral-type E&S modeling for applications under high magnetic flux density conditions. In the conventional integral-type dynamic E&S modeling, we assumed linear increment of higher harmonic components including in the magnetic field strength waveforms versus the magnetic flux density. However the higher harmonic components decrease after getting a peak value in rotational magnetization process around saturation region (1.6T – 1.8T). Improvement of the integral-type dynamic E&S modeling for the high magnetic flux conditions was investigated in comparison with the measured one and verification with a simple problem was demonstrated in this paper.

This paper presents comparison of feedback control algorisms of magnetic flux conditions in two-dimensional vector magnetic property measurements under high magnetic field over 1.6 T. In order to develop a fast measurement algorism, several PID feedback control methods were tested. The results show that the proportion-derivative precedence type (I-PD) control is suitable for reduction of measurement times in vector magnetic property measurements.

This paper presents the measured magnetostriction of electrical steel sheets under alternating magnetic flux conditions. In the measurements, we used a two-dimensional vector magnetic property measurement system, and a three-axial strain gauge. In order to know the magnetostriction in an arbitrary direction, the strain tensor was evaluated. In addition, we try to evaluate the magnetostrictions in the thickness direction. In this paper, the three-dimensional magnetostriction in nonoriented and grain-oriented silicon steel sheets are compared. © 2012 Wiley Periodicals, Inc. Electr Eng Jpn, 179(4): 1–9, 2012; Published online in Wiley Online Library (wileyonlinelibrary.com). DOI 10.1002/eej.21257

Self-bearing motor has combined function of AC motor and magnetic bearing. Usually levitation force is produced by Maxwell force while the motor torque is produced by Lorentz force. The authors have proposed Lorentz type self-bearing motor where strong magnetic flux is the key technology for good characteristics. In this paper three types of rotors are designed and compared: a flux concentrated PM rotor, Halbach rotors with iron back yoke and aluminum back yoke. The high density flat coils are designed and used for the stator. The experimental setup is fabricated and tested. The results obtained are compared and discussed in details.

In this paper, the vector magnetic property is explained first. And this paper also presents the magnetic characteristic analysis of the electrical machines by using the vector magnetic hysteretic E&S model. Furthermore, the dynamic E&S model, which can analyze the eddy current effect due to the distorted magnetic flux waveform, is presented. From this method, we can express the various kinds of magnetic properties directly, for example, vector hysteresis loops (Bx-Hx, By-Hy) magnetic power loss and the behavior of H and B vectors by using this model.

Purpose – The purpose of this paper is to show formulation of a dynamic E&S model, which enables analysis of the effects of eddy currents under vector magnetic behavior in numerical simulations and to demonstrate its usefulness.Design/methodology/approach – When a magnetic flux waveform is distorted, effects of eddy currents increase due to harmonic flux components. In such a case, the result calculated by using the conventional E&S model does not agree with the measured one. The conventional E&S model is improved by considering magnetic flux waveform distortion. The harmonic components of the magnetic field strength waveform were estimated with the classical eddy current model.Findings – In the verification of the dynamic E&S model, it was found that the magnetic field was suppressed by the effect of the eddy current. The conventional analysis overestimates the magnetic field, because the magnetic flux waveform cannot distort. In the magnetic characteristic analysis of a three‐phase transformer model cor...

This paper presents rotational power loss properties of magnetic steel sheets under high flux density conditions using two-dimensional vector magnetic properties measurement. Recently it was reported by some research groups that the magnetic power loss measured in a counter clockwise (CCW) rotating field differed from that in a clockwise (CW) rotating field. This phenomenon was only observed in case of higher magnetic flux density excitation condition. We call this the CCW/CW problem. To clarify the reasons why the disagreement exists in the CCW/CW direction, we have examined angle errors of H - and B -coils by using geometrical, optical and magnetic methods. Then we compensated the measured vector components including different signals due to the angle errors. In the components of irremovable small angle error, we have also used the measured field strength waveforms in CCW/CW conditions. We have applied the compensation method to measurement of a grain-oriented electrical steel sheet.

This paper presents a 3-D variable magnetization and Stoner-Wohlfarth (VMSW) method developed by expanding the conventional 2D-VMSW model. The 3-D VMSW method enables us to estimate distributions of remanent magnetization in magnetized permanent magnets in three dimensions.

In the development of high-efficiency actuators, high-magnetic flux density conditions are applied to achieve higher torque, lower weight and smaller size. It is therefore very important to understand the magnetic properties under high-magnetic flux conditions in the electrical steel sheets. However, it is very difficult to measure the two-dimensional vector magnetic properties under high-flux density conditions due to nonlinearity and anisotropy of the magnetic materials. The measured magnetic power loss under clockwise rotating fields differs from that of the counter-clockwise direction, and the value in counter-clockwise rotating fields sometimes becomes negative. It can be considered that angle errors of H-coils and B-coils are the main causes of the measurement errors. In this paper, the angle errors of the double B-coil (search coils) and the double H-coil (cross-type H-coil) are evaluated with a projector-type magnifying scope and a calibration device. We have compensated the measured magnetic field vector components caused by the angle errors and made clear their influence on the two-dimensional vector magnetic property measurements. Furthermore, we proposed a method to cancel out small angle errors that remained in the saturated region. Copyright © 2008 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

This paper presents a dual-rotor electrical machine, designed with grain-oriented electrical steel sheets and Nd-Fe-B permanent magnets. In order to achieve high performance within limited volume, we have tried to increase the magnetic loading by using rare earth permanent magnets in the dual-rotor construction. The use of grain-oriented electrical steel sheets is preferred instead of usual non-oriented sheets in order to let flow high magnetic flux through the core material, avoiding the magnetic saturation. The two-dimensional vector magnetic properties in the dual-rotor core model are analyzed by using the finite element method with the E&SS modeling. The calculated results are compared with measured results on the prototype machine for verification of the designing.

The two-dimensional vector magnetic properties are proposed as a method to express the magnetic anisotropy and iron loss precisely by using the relationship between the flux density vector and the magnetic field strength vector. The ES Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20446

Recently, two-dimensional vector magnetic measurements have become popular and many researchers have been trying to develop a more accurate measuring system and a standard measurement device. In development of high-efficiency actuators, use of high magnetic flux density designs can assist high torque, weight reduction and size down. It is therefore very important to understand the magnetic properties under high magnetic flux conditions in the electric steel sheets. However, there are still some difficulties in the measurements, due to nonlinearity and anisotropy of magnetic materials. The measured magnetic power losses of the clockwise and counter-clockwise rotating fields become different for magnetic flux density over 1.2 T and the value in counter-clockwise case sometimes became negative. It can be considered that angle errors of H and B-coils are the main reasons of the measurement errors. In this paper, the angle errors of the double B-coil (search coils) and the double H-coil (cross-type H-coil) are evaluated with a calibration device, which consists of a special solenoid coil and a high-precision turntable. The magnetic properties are compensated with the measured angle errors.

Recently, two-dimensional vector magnetic measurement has become popular and many researchers concerned with this field have attracted to develop more accurate measuring systems and standard measurement systems. Because the two-dimensional vector magnetic property is the relationship between the magnetic flux density vector B and the magnetic field strength vector H , the most important parameter is those components. For the accurate measurement of the field strength vector, we have developed an evaluation apparatus, which consists of a standard solenoid coil and a high-precision turntable. Angle errors of a double H-coil (a cross-type H-coil), which is wound one after the other around a former, can be evaluated with this apparatus. The magnetic field strength is compensated with the measured angle error.

In this paper, the two-dimensional vector magnetic properties depending on the exciting frequency are made clear by using the finite element method (FEM) and the Enokizono, Soda, and Shimoji (E&SS) modeling as an anisotropic material modeling. In the investigations, a 3-phase 3-leg transformer model is analyzed under some exciting frequency conditions

This paper describes the utilization of two-dimensional vector magnetic property for the advanced magnetic field analysis which is called the magnetic characteristic analysis. There are some possibilities to develop new applications using the magnetic characteristic analysis and the vector magnetic property. This is due to the high density and the high efficiency of electric machines. It is very important to use the vector magnetic property in such cases

In this paper, influence of the magnetic properties on rolling direction of the nonoriented electrical steel sheets were examined. We have carried out the magnetic field analyses considering the two-dimensional vector magnetic properties in the stator core of a condenser motor. The finite element method (FEM) considering Enokizono, Soda, and Shimoji modeling was used in these analyses

In order to detect the amount of fatigue damage in SUS304 steel, we were investigating the relationship between fatigue damage and the perpendicular leakage magnetic flux density caused by martensitic structure induced by cyclic stress. Plane bending fatigue tests were carried out at the temperate of 0°C, 30°C and 65°C. The main results obtained were as follows : (1) The magnetic flux density of the center in the specimen clearly increase with the increase of the bending stress and number of cycles. However it is difficult to estimate the fatigue damage with using method when the temperature changed during the test, since the martensitic transformation also depends on the temperature. (2) When repeating the stress below a fatigue limit, the value of magnetic flux density will not increase from a certain value. This indicates that fatigue damage is not advancing. (3) The final value of saturated magnetic flux density and σa/σω correlated each other even when the temperature differs.

The paper describes the vector magnetic characteristic analytical method using the vector magnetic hysteretic engineering model. By this method, various kinds of vector magnetic properties, for example vector hysteretic loop, coercive force, magnetic power loss, vector behavior of magnetic flux density B, and magnetic field strength H, can be obtained on electrical machines and apparatus.

This paper proposes the improved magnetic characteristic analysis by using the magnetic property is called “vector magnetic property”, which means the relationship between the magnetic flux density B vector and the magnetic field strength H vector. In order to express the relation, “E&S 2 model” as engineering model has been proposed, which can introduce to the numerical simulation technique such as the finite element method. By using this method, the various kinds of magnetic properties, for example vector hysteresis loop, iron loss, vector coercive force, rotational iron loss and so on can be obtained. This method is useful for the development of the high efficiency electrical machines using the high quality electrical steel sheets.

Actual magnetic materials such as the electrical steel sheet have magnetic anisotropy and hysteretic behavior. The magnetic characteristic is defined from the vector relation between the magnetic field strength vector H and the magnetic flux density vector B . It is very difficult to express the vector relation by using the conventional constitutive equation B = μ H . Then, we propose an engineering model instead of the conventional constitutive equation for the expression.

This paper presents the iron loss analysis of three-phase induction motors. It is necessary to consider the detailed magnetic characteristics inside induction motors to minimalize and optimize the losses. Therefore, the authors examined the loss distribution by the finite element method considering the two-dimensional magnetic properties to develop higher efficiency induction motors.

This paper presents magnetic flux distribution and iron loss distribution in ferrite cores under pulse excitation by using the finite element method. The authors propose the numerical analysis method considering the frequency dependence of magnetic characteristics in a ferrite core. It is difficult to consider the magnetic characteristics in numerical analysis under the pulse excitation. The magnetic characteristics under the pulse excitation are examined by using the magnetic characteristics of each frequency component after Fourier expansion.

This paper presents a numerical modeling of vector (two-dimensional) magnetic properties considering hysteresis of silicon steel sheets. We define the magnetic reluctivity tensor newly and calculate it from the measurement data obtained by two-dimensional magnetic measurement. Our new modelings, which are "E&S Model" and "E&S2 Model", can be introduced into a finite element formulation for magnetic field analysis and can get the distribution of the magnetic field strength H-vector and the magnetic flux density B-vector.