宮城 大輔

The optimization method using the ON/OFF sensitivity analysis has an advantage that an epoch-making construction of magnetic circuit may be obtained. Therefore, it is attractive for designers of magnetic devices. We have already developed the ON/OFF method for the optimization of a static magnetic field problem, and the effectiveness is verified by applying it to the optimization of magnetic recording heads. In this paper, the ON/OFF sensitivity method is extended to the optimization of the eddy current problem using the adjoint variable. The newly developed ON/OFF method is applied to the determination of the optimal topology of the yoke of the billet heater for rolling wire rod. As a result, the optimal shape of yoke, which we could not imagine beforehand can be obtained. It is shown that the local heating of the yoke was reduced without decreasing the heating efficiency.

Magnetic properties of a motor core are affected by the distortion due the cutting and the compression caused by a shrink fitting etc. The report which investigates such behavior systematically is few, although the examination of the detailed behavior of magnetic properties is important. In this paper, the mechanical stress due to the shrink fitting is examined and the mechanical stress of an actual stator core is measured by using strain gauges. Iron losses of actual motor cores made of wire cut and punching, and with and without shrink fitting are compared with each other. It is shown that a considerably large compressive mechanical stress is applied by the shrink fitting and the measured mechanical stress is similar to the analytical one. The iron losses of motor processed by wire cut and press become similar values when the compressive force due to the shrink fitting is large. © 2011 IEEE.

A high-T (c) superconducting (HTS)-coated conductor with a magnetic substrate is promising as a low-cost conductor because Ni alloy can be well texturized by a simple process. However, it has been reported that the magnetism of the substrate makes an AC transport current loss and an AC magnetization loss increase from those losses of the HTS-coated conductor of a nonmagnetic substrate sharply. However, the assembled conductor which bundled many HTS-coated conductors with a magnetic substrate is hardly reported. In this paper, the AC loss and the current distribution in the superconducting layer of the HTS-coated conductor with a magnetic substrate are analyzed using the edge-based hexahedral finite element method (FEM) taking account of both the nonlinear E-J characteristics and the nonlinear magnetic property of the substrate at 77 K. Moreover, AC loss characteristics of HTS-coated conductors with a magnetic substrate arranged with an infinite number of the coated conductors are examined and the effect of the gap length between adjacent conductors on the AC loss is also investigated.

We have already reported the accuracy of measurement of 2D magnetic properties of non-oriented electrical steel sheet using a novel SST. In this paper, magnetic properties under alternating and rotating magnetic flux of grain-oriented sheet are measured. The accuracy of measurement in the case of grain-oriented sheet is examined by comparing the measured results using 2D-SST with those using an ordinary SST under alternating flux excitation. The effect of positioning error of orthogonal B and H coils on measurement accuracy is also discussed.

In order to accurately design magnetic devices which are used at high temperature, it is necessary to understand the behaviour of several magnetic materials at high temperature. In this paper, the magnetic properties of several magnetic materials until Curie temperature are measured. It is shown that the change of iron loss of rolled steel (SPCC and SS400) with the temperature is more remarkable than that of electrical steel sheet.

Purpose - The purpose of this paper is to examine the mechanism of the increase or decrease of eddy current loss of the segmented Nd-Fe-B sintered magnets without insulation, and the effects of parameters on such a phenomenon are discussed. Design/methodology/approach - The measured contact resistance is used in the finite element analysis. Findings - It is shown that the eddy current loss in a magnet shows the peak value when the number of segments are increased at 40 kHz, but this property is changed at low frequency (10 kHz). Its tendency is changed by the contact resistance and the pernieance (surrounding iron core). Originality/value - The reason of a curious property of eddy current loss of segmented magnets is clearly explained by examining the eddy current distribution at various contact resistances.

The multicar elevator system is a revolutionary new technology for high-rise buildings, promising outstanding economic benefits, but also requiring new technology for propulsion, safety, and control. In this paper, we report on experimental results with new components for linear-motor-driven multicar elevators. We show that linear synchronous motors with optimized design and with our new safety and control system can be considered as core components of a new generation of elevator systems. The main new results concern the development of a safety system integrated into the propulsion system, the design methodology of a linear motor optimized for the multicar elevator task, and the motion control system that is expected to be usable for extra high-rise buildings.

In order to improve the induction heating system for an object of magnetic material, it is necessary to clarify the heating property of the object due to the eddy current loss using a magneto-thermal coupled analysis taking account of the temperature dependence of magnetic properties of a magnetic material to be heated. In this paper, the effect of temperature dependence of magnetic properties on heating characteristics of a billet heater is analyzed considering the heat emission, heat conduction and temperature dependence of magnetic properties of the billet. It is clarified that the precise analysis is possible by considering the temperature dependence of the magnetic property and the conductivity of a billet, heat emission, heat conduction etc. Moreover, the effect of temperature dependence of B-H curves on the temperature rise of a billet and the induced voltage of an exciting coil is examined.

The ON/OFF method which was developed for the static field problem is extended to the dynamic field problem. We applied the ON/OFF sensitivity method to the steady state linear eddy current problem in order to determine the optimal topology of the electromagnetic and magnetic shield. As a result, a surprising optimal shape of electromagnetic and magnetic shield, which we could not imagine beforehand can be obtained using the ON/OFF method. A criterion of how to choose the sensitivity with respect to the reluctivity or conductivity is discussed. It is shown that the ON/OFF method can be used for the design of electromagnetic or magnetic shielding, even when the material is the magnetic steel sheet which have two kinds of variables, magnetic permeability and conductivity. The best shielding configuration can be obtained for each frequency by adopting the result with a larger design sensitivity.

The topology optimization by distributing the magnetic material in the design domain, which is called as the ON/OFF method, is attractive for designers of magnetic devices, because an initial conceptual design, that we could not imagined beforehand, may be obtained. We examined how to apply the ON/OFF sensitivity method to the IPM (Interior Permanent Magnet) motor in order to determine the optimal topology of the rotor. A technique to apply the ON/OFF method to the design of motor considering the nonlinearity of B-H curve and rotation of rotor is shown. The effects of the choice of the design region and the phase angle of stator current on the obtained topology of motor core are discussed.

The electrical steel sheet is widely used in iron cores of motors, etc. The magnetic properties of a motor core are affected by the distortion due to the compressive stress in the thickness direction caused by bolt bundles, etc. There have been few reports of magnetic properties being investigated systematically, although the detailed examination of the behavior of magnetic properties is important for the precise design of motor, etc. In this paper, the correlation between the compressive stress and its magnetic properties of a nonoriented electrical steel sheet is measured, using a single sheet compressed in the thickness direction. Moreover, the behavior of hysteresis loss and the eddy current loss under the compressive stress in the thickness direction is examined.

In a practical motor, the compressive stress of more than 100 MPa is sometimes impressed by a shrink fitting etc, as a result, the magnetic properties of iron core deteriorate. Therefore, it is necessary to measure magnetic properties under the compressive stress. However, the measurement under the compressive stress of more than 100 MPa is difficult if a single sheet tester is used, because of the buckling of a single sheet specimen. Then, we measured the magnetic properties under the compressive stress up to 150 MPa without buckling by using a laminated specimen. It is shown that the permeability rapidly decreases even by small stress in both cases of rolling and transverse directions. The iron loss is increased by the stress, but it almost does not increase when the stress is larger than about 50 MPa. Furthermore, the permeability and iron losses of the rolling direction and transverse direction become the same values, and the anisotropy disappears when the stress is larger than about 50 MPa.

For the design of high efficiency and miniature electrical machines, the development of superconducting machines, which can be used at low temperature (less than liquid nitrogen temperature), is considered. In order to examine the design of these machines, magnetic properties of electrical steel sheets, etc. used, such as cores of electrical machines are necessary. It is shown that the permeability and iron loss of nonoriented silicon steel is, in general, increased at low temperature.

The magnetic properties of magnetic materials are considerably changed near the Curie temperature. In order to analyze the magnetic characteristics of magnetic devices which operate at high temperature, such as induction heater of billets, high temperature motor, the B - H curves and iron loss curves are necessary for the precise analysis. But, the reports of the examination of the measuring method and the behavior of such magnetic properties are few. In this paper, the measuring method of magnetic properties at high temperature is investigated. It is shown that the permeability is not changed so much under about 500 degrees C in the case of nonoriented electrical steel. However, it changes drastically above about 500 degrees C. The iron loss decreases with the increase of temperature. The decrease of iron loss is mainly due to the decrease of hysteresis loss with the increase of temperature.

The magnetic property of electrical steel sheet has an anisotropy even in the case of non-oriented material. In order to analyze the operating condition of a synchronous reluctance motor preciously, the magnetic field analysis considering the magnetic anisotropy of electrical steel sheet using the fixed-point method was carried out. © 2010 IEEE.

the design of miniature electrical machines of high power density, it is necessary to measure magnetic properties of electrical steel sheet at high magnetic flux density. The JIS standard single-sheet tester (SST) is insufficient to measure magnetic properties at high magnetic flux density around 2.0 T. In this paper, we have designed and developed an SST, which can excite high magnetic strength field up to about 58 000 A/m. The accuracy of measurement of alternating current (ac) loss at high magnetization (55 000 A/m) is verified by measuring the eddy current loss of a copper plate comparing with the theoretical value. It is shown that the magnetic properties of nonoriented electrical steel sheet can be measured until 2.1 T.

The AC magnetization loss of a HTS coated conductor under the external AC magnetic field is analyzed using the finite element method taking account of the nonlinear E-J characteristic and the nonlinear magnetic property of a substrate at 77 K. The AC loss characteristic including the loss of the substrate under the external AC magnetic field is examined. Furthermore, we compared our numerical results of the AC magnetization loss with experimental results and examined the validity of our calculation.

In large permanent magnet motors, the permanent magnet in the motor is segmented into parts in order to reduce the eddy current loss. It is reported that the insulation between Nd-Fe-B sintered magnets is possible only by contacting the segmented permanent magnets without the insulation. But the mechanism for such a phenomenon has not been clear yet. In this paper, the effect of force on the apparent contact resistance between permanent magnets is examined. The idea of contact resistance coefficient is also discussed. It is shown that the eddy current loss of segmented sintered Nd-Fe-B magnets is almost the same as the loss of the case when an insulating material is inserted between magnets.

The topology optimization, which determines the optimal topology of nonmagnetic material in which the eddy current flows, such as an electromagnetic shield, by distributing the nonmagnetic conducting materials in the design domain, is attractive for designers of magnetic devices, because an initial conceptual design can be obtained. In this paper, the ON/OFF method which was developed for the static field problem is extended to the dynamic field problem. The method makes them particularly useful in topology optimization, where the parameterization enables all feasible shapes of electromagnetic devices to be explored. We applied the ON/OFF sensitivity method to the steady state linear eddy current. problem in order to determine the optimal topology of the electromagnetic shield. As a result, the optimal shape of electromagnetic shield, which we could not imagine beforehand can be obtained using the ON/OFF method.

We performed the stress analysis of a SPM motor core with the shrink fitting, and calculated the iron loss by FEM using the B-H curve and iron loss measured under a compressive stress. We examined the effect of shrink fitting on iron loss of an actual motor and the validity of the estimation of the iron loss by FEM analysis using the measured properties affected by the stress. It is shown that the magnetic properties under the compressive stress should be considered for the precise estimation of the iron loss in a motor with the shrink fitting.

The main issue of a billet heater using induction heating is to avoid billets that were not heated at a desired temperature. In order to improve the induction heating system, it is necessary to clarify the heating property of an object due to eddy current loss and to investigate the temperature distribution in an object by the magneto-thermal coupled analysis. In this paper, the eddy current and temperature distribution of a billet heater is analyzed considering the heat emission, heat conduction, and temperature dependence of magnetic characteristics of the billet. It is shown that the calculated values of temperature in the center and surface of a billet are in good agreement with measured values. The precise analysis is possible by considering the temperature dependence of magnetic characteristics, heat conductivity, etc. The detailed behavior of the heat generation in the billet is clarified. The skin depth is increased because the resistivity of the billet is increased and the permeability is decreased at high temperature. As a result, the flux in the billet is reduced, and then the power (eddy current loss) in the billet is decreased.

Controlling the current distribution in each layer of a co-axial multi-layered high-Tc superconducting (HTS) cable uniformly is important in order to realize a compact and large capacity HTS power cable. The phenomenon of current imbalance should be exactly investigated by the analysis, because the current imbalance causes an increase of AC loss. In this paper, 3-D finite element analysis taking account of the nonlinear E-J characteristic is carried out in order to study the AC loss of a co-axial multi-layered HTS power cable with shield layers. The analysis region is considerably reduced by modeling spirally wound superconducting tapes as conductors having an anisotropic conductivity and using the periodic boundary condition. By using this model, AC transport characteristics of the co-axial three-layers cable (composed of two conductor layers and one shield layer) are studied, and the AC loss is investigated. The structure with low AC loss is also discussed. © 2009 The Institute of Electrical Engineers of Japan.

In order to improve the measurement accuracy of the 2-D SST (2 dimensional single sheet tester), we examined the measurement region of flux density B and magnetic field strength H, and the effect of stress caused by the supporting jig. By considering these factors, the measurement accuracy of iron loss under rotating flux at high flux density condition was improved

In large permanent magnet motors, the permanent magnet in the motor is segmented into parts in order to reduce the eddy current loss in permanent magnet. It is reported that the insulation between sintered Nd-Fe-B magnets is possible only by contacting the segmented permanent magnets without insulation. But the mechanism for such a phenomenon has not been clear. We have already reported that the analysis considering the resistance of the contact part is possible by defining the contact resistance coefficient K. In this paper, the effects of number of segments of magnet, exciting frequency and contact resistance between magnets on the eddy current loss of the magnet are investigated.

Accurate measurement of losses and permeability in electrical steel is often dependent on the performance of surface magnetic sensors such as H-coils. In this paper we compare the accuracy of three H-coils by investigating their performance when used for rotational power loss measurement in grain oriented electrical steel. It is shown that consistent, reasonably accurate results can be obtained provided precautions are made with H-coil design and use.

In order to achieve high-density recording, the detailed behavior of thermal degradation should be investigated. In this paper, the degradation of magnetization of high-density recording medium is examined using the 3-D finite element method (FEM) combined with the modeling of Stoner-Wohlfarth (SW) particles and Neel-Arrhenius switching probability. It is shown that the anisotropy field Hk suppressed the thermal degradation and the saturation magnetization Ms enhances it. The thermal degradation is also changed by the amplitude of magnetization. © 2008 Elsevier B.V. All rights reserved.

Magnetic properties of a motor core are affected by the distortion due to the compression caused by shrink fitting and the distortion caused by punching, etc. In this paper, the B-H curve and iron loss of stator core of actual motor under shrink fitting are measured. It is shown that the maximum permeability is reduced by about 50%, and the iron loss is increased by about 30% due to the shrink fitting. It is illustrated that the loss of motor is increased by about 10%, 4% and 2% due to the shrink fitting, the cutting stress and the eddy current in rotor magnet, respectively. © 2008 Elsevier B.V. All rights reserved.

It is very important to understand AC magnetization loss of coated-conductor with magnetic substrate under the external magnetic field which has an arbitrary angle to the wide face of coated-conductor for designing realistic apparatuses. In this paper, the current distribution and AC magnetization loss in the superconducting layer of the magnetic substrate conductor are analyzed using the finite element method taking account of both the non-linear E-J characteristics and the non-linear magnetic property of the substrate. The influence of magnetic substrate on AC magnetization losses under the external AC magnetic field which has an arbitrary angle to the wide face of coated-conductors is examined. (C) 2008 Elsevier B.V. All rights reserved.

The magnetic properties of magnetic substrates of Ni-alloy at 77 K are measured using a miniature single sheet tester (SST) in liquid nitrogen (LN2) and they are compared with those at room temperature. Moreover, the AC transport current loss of HTS coated conductor is analyzed using FEM taking account of both the nonlinear E-J characteristics and the nonlinear magnetic property of the substrate at 77 K. The numerical results of AC transport current loss are compared with measurement results. Iron loss in the magnetic substrate and AC transport current loss in the superconducting layer are calculated, respectively. (C) 2008 Elsevier B.V. All rights reserved.

It is very important to consider the nonlinear magnetic property to understand and estimate the AC transport current loss characteristic of a conductor with magnetic substrate accurately. In this paper, the current distribution and AC loss of HTS coated conductor with the magnetic substrate are analyzed using the finite element method based on edge elements taking account of both the nonlinear E - J characteristics and the nonlinear magnetic property of the substrate. Moreover, the influence of magnetic saturation of the substrate on AC transport current loss is examined. Our numerical results denote that the AC transport current loss is increased until the magnetic saturation in the substrate, because the current distribution in superconducting layer changes due to the concentration of magnetic flux in the substrate. Moreover, when the magnetic flux in the substrate is saturated, the AC transport current loss is close to that of the Norris' strip model.

We studied the magnetization loss characteristics of the HTS layer of the coated conductor with magnetic substrate. The current distribution and the magnetization loss in the superconducting layer of an HTS coated conductor under the external AC magnetic fields perpendicular or parallel to the wide face of the conductor are analyzed using the finite element method. The nonlinear E - J characteristic and the nonlinear magnetic property of the substrate is taken into account. It is shown that the magnetization losses are hardly affected by the magnetic property of the substrate under the external AC magnetic field perpendicular to the wide face of a conductor. On the contrary, the magnetization losses in the superconducting layer of the HTS conductors having the magnetic substrate are much larger than the losses of that having the non-magnetic substrate under the external AC magnetic field parallel to the wide face of a conductor.

A magnetic recording head for 600 Gb/in' with a gh recording head field, which is called a "standard model," is proposed after deep consideration by utilizing the numerical analysis. In order to improve the topology of the having a higher writing field and higher field gradient, the introduction of a 3-D topology optimization method, which is called the "WoFF method", is indispensable. In this paper, an advanced topology of the standard head model is obtained using the oN/oFF method. The recording field of 1.7 T and the field gradient of 0.0314 T/nm with low stray field of 0.1 T at the center of an adjacent track are obtained. The leakage flux in the adjacent track, which is the cause of the crosstalk, is reduced to 50%, which is the most significant improvement.

AC magnetization losses in subdivided CC (Coated Conductor) with magnetic substrate were experimentally investigated comparing with those in subdivided CC with non-magnetic substrate for an AC external magnetic field perpendicular to the wide face of the CC. It is well known that the subdivision is effective to reduce magnetization losses in CC with non-magnetic substrate. The experimental results show that the subdivision is also effective for the CC with magnetic substrate and that the level of reduction of the losses by the subdivisions is almost the same as that of non-magnetic substrate CCs. It is concluded from the experimental results that the magnetic property of the substrate does not affect the magnetization losses in the subdivided conductor in the range of the experiment where the amplitude of the AC external magnetic field is 0 ∼ 0.1 T and the frequency is 16 ∼ 86 Hz. © 2008 IOP Publishing Ltd.

The eddy current is induced in silicon steel sheets of core block of power reactor due to the fringing flux, and this causes a local overheating. If a bevel edge core is used in the reactor composed of laminated core, the eddy current loss may be reduced. In order to investigate the behavior of flux and eddy current in the reactor, a modeling technique of laminated core for exact and quick practical nonlinear analysis is investigated. It is shown that the practical analysis under distorted flux waveform is possible by subdividing only the region near the surface of lamination, and the inner part is subdivided as a bulk core having anisotropic conductivity. The flux and eddy current distribution of the bevel edge core reactor is investigated using the modeling technique of laminated core. It is illustrated that the maximum eddy current loss is reduced to about 20%, and the total eddy current loss is decreased to about 50% by introducing the bevel edge core. © 2008 The Institute of Electrical Engineers of Japan.

The reduction of the ripple of driving force is required for the practical utilization of linear synchronous motor for rope-less elevator. In addition, it is important to increase the driving force and reduce the weight of motor. In this paper, the Halbach type permanent magnet linear motor is optimized by using the evolution strategy (ES) method in order to reduce the ripple and to increase the driving force. Multi-objective results are also discussed.

AC transport current losses of coated conductors with ferromagnetic substrates are higher than the loss calculated by the Norris equation. In order to reduce the AC transport current loss we propose in this paper a structure of the coated conductor that has wider substrate than the SC (Superconducting) layer. The current distribution and AC loss of the proposed model are analyzed by means of FEM. The AC transport current loss is reduced due to the change of current density distribution near the edge of SC layer, consequent to the high value of magnetic permeability of the ferromagnetic substrate, that is wider than the SC layer. (C) 2007 Elsevier B.V. All rights reserved.

It is very important to understand and estimate the AC loss characteristic of assembled conductors with ferromagnetic substrate. In this paper, the AC loss of cylindrically assembled HTS coated-conductors with ferromagnetic substrate is examined using FEM taking account of the nonlinear E-J characteristics. It is shown that, the current distribution and AC losses depend on the arrangement of the superconducting (SC) layer and the substrate. Moreover, AC loss characteristics depend on the gap length between conductors. The effect of the arrangement of the SC layer and the substrate on the current distribution of SC layer and AC loss is illustrated. (C) 2007 Elsevier B.V. All rights reserved.

HTS coated conductors (CCs) have high potentials as low-cost and long length conductors. However, a question remains as to what influence the magnetic property of the substrates has on the AC losses. In this paper, the influence of magnetic property of substrates on the AC losses in HTS CCs is studied. Based on the study methods to reduce the AC transport current losses and magnetization losses in CCs with magnetic substrates are investigated. It is shown that the losses can be reduced to the same level of those in CCs with nonmagnetic substrates. (C) 2007 Elsevier B.V. All rights reserved.

AC magnetization loss characteristics of an HTS coated tape conductor with magnetic substrate subjected to an external AC magnetic field were investigated. The external magnetic field was perpendicular or parallel to the wide face of the tape conductor. Magnetization losses in the conductor and in the magnetic substrate itself without the superconductor layer, were measured by electric and calorimetric methods. The influence of the magnetic property of the substrate was strongly dependent on the direction of the external magnetic field. When the external magnetic field was perpendicular, magnetic property of the substrate did not affect the magnetization loss characteristics. This result suggests that the magnetization losses can be reduced by subdivisions of the superconducting layers even in the case of magnetic substrate conductors. When the external magnetic field was parallel, the magnetization losses were dominated by the losses in the magnetic substrate. Therefore, to reduce the magnetization losses in this case, reduction of magnetization losses in the substrate is necessary. (C) 2007 Elsevier B.V. All rights reserved.

HTS coated conductors (CC's) with substrates of magnetic materials such as Ni-alloys are expected to be promising for practical applications because Ni-alloys are well-texturized by simple processes and production costs of the conductors are significantly reduced. In the former works, it was shown by experiments that HoBaCuO-123 CC's with Ni-alloy substrates exhibit higher AC transport current losses than CC's with non-magnetic Hastelloy substrates. In this work, the AC transport current loss characteristics of stacked conductors composed of a pair of HoBCO CC's with Ni-alloy substrates are experimentally investigated to study a method to suppress the AC transport current losses. It is found that AC transport current losses can be reduced in the stack arrangement by the minimization of influence of the magnetism of the substrates canceling self-fields in the substrates.

When using a powder magnetic core fabricated from iron powder coated with an insulator in motors, reactors, etc., the effect of stress on its magnetic properties should be considered. In this paper, a system for measuring the magnetic properties of a powder magnetic core under compressive stress was developed and applied to a powder magnetic core, namely a high-density magnetic composite (HDMC). It is illustrated that the iron loss is increased by about 20% when a compressive stress of 100 MPa was applied.

It is reported that the AC transport current loss of coated conductors with ferromagnetic Ni-alloy substrates is larger than the loss calculated by the Norris equation. This means the significant influence of the ferromagnetism of the substrate on AC transport current loss. In this paper, the current distribution and AC loss of HTS coated conductor with ferromagnetic substrate are analysed using the finite element method taking account of the non-linear E - J characteristics. Calculated results show that the AC transport current loss is increased because the current distribution changes in the superconducting layer due to the concentration of magnetic flux in the substrate. The effect of magnetism of substrate on the current distribution of superconducting layer is illustrated and the AC transport current loss characteristics of the HTS coated conductor with ferromagnetic substrate are examined.

By modeling spirally wound superconducting tapes as conductors having an anisotropic conductivity and using the periodic boundary condition, the analysis region is considerably reduced in 3-D edge-based hexahedral FEM analysis. By using this model, the AC transport characteristics of the co-axial five-layer cables (composed of three conductor layers and two shield layers) are studied, and the effect of current distribution among layers on AC loss in shield layers is investigated. It is possible to control the current distribution in the shield layers by adjusting the twist pitches of shield layers. It is shown that the uniform current in each layer dose not give the minimum AC loss and the current should flow in the inner shield layer in order to reduce the AC loss of shield layers.

The detailed analysis of residual magnetization in permanent magnet type of magnetic resonance imaging (MRI) is important for the improvement of MRI image. The behavior of residual magnetization produced by the z-gradient coil is already reported. As 3-D analysis is strictly required for investigating the residual magnetization produced by the x- or y-gradient coil, the report about such a behavior is few. In this paper, the minor loop and residual magnetization produced by the x-gradient coil in a 3-D model are examined. The useful knowledge for understanding the behavior of the disturbance field due to the residual magnetization in the pole piece, etc., is obtained.

Purpose - The paper examines the convergence characteristics of the "minimized residual method based on the MRTR (three-term recurrence formula of CG-type) method" for solving large linear simultaneous equations. Design/methodology/approach - The paper uses an example of magnetic field analysis of permanent magnet type of MRI taking account of the minor loop and eddy current. Findings - It is shown that the preconditioned MRTR method can get a stable and quick convergence for such a relatively ill-conditioned problem. Originality/value - Illustrates that the convergence of the Incomplete Cholesky Conjugate Gradient method is one of the important issues in the practical 3D magnetic field analysis.