A modeling technique of the minor loop using typical hysteresis loops is shown. The effect of the minor loop and eddy current in the pole piece of a permanent magnet type of MRI on the residual flux density of the probe coil is examined. It is illustrated that the change &#916;B of residual flux density occurs due to the minor loop of the pole piece. It is also pointed out that the choice of time interval &#916;t is important in a nonlinear analysis considering the minor loop</p

Kayano, Takeshi

Miyata, Koji

Ohashi, Ken

Takahashi, Norio

N. Takahashi

T. Kayano

K. Miyata

K. Ohashi

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Effect of minor loop on magnetic characteristics of permanent magnet type of MRI

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PhysicsElectricity & Magnetism fieldsOkayama University Year 1998Effect of minor loop on magneticcharacteristics of permanent magnet typeof MRINorio Takahashi∗ Takeshi Kayano†Koji Miyata‡ Ken Ohashi∗∗∗Okayama University†Okayama University‡Shin-Etsu Chemical Corporation Limited∗∗Shin-Etsu Chemical Coporation LimitedThis paper is posted at eScholarship@OUDIR : Okayama University Digital InformationRepository.http://escholarship.lib.okayama-u.ac.jp/electricity and magnetism/19IEEE TRANSACTIONS ON MAGNETICS, VOL. 35, NO. 3, MAY 1999 1893 Effect of Minor Loop on Magnetic Characteristics of Permanent Magnet Type of MRI Norio Takahashi and Takeshi Kayano Department of Electrical and Electronic Engineering Okayama University, Okayama 700-8530, Japan Koji Miyata and Ken Ohashi Magnetic Materials R&D Center Shin-Etsu Chemical Co. Ltd., Takefu 915-0802, Japan Abstrac t  - A modeling technique of minor loop using typical hysteresis loops is shown. The effect of minor loop and eddy current h pole piece of a permanent magnet type of MRI on the residual flux density of probe coil is examined. It is illustrated that the change AB of residual flux density occurs due to the minor loop of pole piece. It k also pointed out that the choice of time interval At k important in the nonlinear adysk d e r i n g  minor loop. Index terms - Hysteresis, minor loop, MRI, finite element method I. INTRODUCTION The permanent magnet type of MRI [l] for whole body provides a viable alternative to resistive and superconducting MRI [2]. As the permanent magnet assembly conkins pole pieces and yokes which are made of steel, the minor loop of B and H of steel due to the pulse excitation and the eddy current induced in the steel affect the magnetic characteristics of permanent magnet assembly [3]. In this paper, the nonlinear iteration method for the analysis considering minor loop is discussed, and also the modeling technique of minor loop using typical major hysteresis loops is shown. The behavior of magnetic characteristics of permanent magnet assembly is examined considering hysteresis and eddy current. 11. ANALYZED MODEL Fig. 1 shows the cross-section of a permanent magnet assembly for MRI device. The yoke is composed of two steel plates (530mm), other two steel plates ($33Omm) with a hole ($6Omm) and four columns. The permanent magnet has a hole ($6Omm). Although the actual assembly is three-dimensional having four columns, it is simplified to a two-dimensional one to d u c e  the CPU time and memory requirements. The yoke and pole piece are made of steel (SS400), and its conductivity is 7.5 lx106S/m. The magnetization of Nd-FeB magnet is 1.21 T. The gradient coil having 15 turns is located on the surface of the pole piece. Fig. 2 shows the current of the gradient coil. The flux density B in this model is produd by the permanent magnet and the gradient coil. Therefore, B and H are located on the initial magnetization curve and on the minor loop. Manuscript received June 3, 1998 N. Takahashi, tel: +81-86-251-8115, fax: +81-86-251-8258 e-mail: norio@eplab.elec.okayama-u.ac.jp . Y ient coil (15 turns) 0 X pole piece yoke Fig. 1. Model of permanent magnet assembly for MRI device. -10 1 U (a) pulse series h o.Ooo1 -10 (b) enlarged pulse series Fig. 2. Current of gradient coil. 0018-9464/99$10.00 0 1999 IEEE 1894 111. METHOD OF ANALYSIS A. Choice of B-H curve in Newton-Raphson Iteration The choice of minor loop in the Newton-Raphson nonlinear iteration (NR method) is carried out as follows: (a) Let us assume that B and Hat  a step is located at the point a on the initial magnetization curve 0 shown in Fig. 3. If the obtained flux density Blf(k) of the k-th iteration at the instant t is larger than B1(f-At)*) at the instant t-At, the operating point of B and H exists on the initial magnetization curve Q. (b) If the flux density Blt(k) is reduced (Blt(k)cBl(t-At)*)), the operating point of B and H moves on the upper minor loop Q as shown in Fig. 3. In Fig. 3, the case when the operating point is reached to the point b(Blt(3)) after three NR iterations (B;(1)>B,t(Z)>B;(3)) is shown. (c) When the flux density Blt(k) at the k-th step ( b 3 )  of NR method is larger than Blf, Bit@) moves on the lower minor loop 0 shown in Fig. 3. In the NR iteration, the operating point moves on loop 0 when B is reduced, the operating point moves on loop 0 when B is increased. (d) If the operating point is located at the point h on the lower loop in Fig. 3, the operating point moves on small upper minor loop @ and upper hysteresis loop 0 when B is reduced, and the operating point moves on loop 0 when B is increased. B. Modeling of Minor Loop The upper and lower minor loops can be determined as follows: I) Upper minor loop: The upper minor loop Q is represented by the upper hysteresis loop shown in Fig. 4, which is approximated by a cubic spline function. Typical hysteresis loops are measured using a permeameter [4]. The specimen is steel ‘SS400’ and the dimension is 10x50x300mm. The upper hysteresis loop used is interpolated by the measured loops which are stored as I I I 0 * H Fig. 3. Minor Loop. follows: (a) The upper loop “2” having the maximum flux density Bm2 shown in Fig. 5 is interpolated from two nearest typical loops “1” and “3” satisfying the following relationship: (b) If the required loop is smaller than the smallest loop which is stored, the required loop is obtained by scaling down the smallest one similarly. (c) If the required flux density is larger than that of the largest loop, the initial curve is used. The small upper minor loop CQ which was denoted in Section IIIA.(d) can be obtained from the lower minor loop b- h in the same way that the lower minor loop 0 is obtained from the upper minor loop Q as shown in the following Section III.B.2). 2) Lower minor loop: The lower minor loop 0 is obtained from the upper minor loop Q, assuming that the lower loop is symmetric with the upper one with respect to the middle point of the line a-b. Fig.6 shows the comparison of simulated minor loop and measured minor loop. The minor loop of steel is measured using a permeameter. The calculated minor loop is obtained using the curves in Fig. 4. Fig. 6 suggests the validity of the modeling method of minor loop. ::::I 1.2 \ initial magnetization l.O curve 0.8 ::: 1 0.2 L 1 0 2000 4000 Fig. 4,’Measured upper hysteresis loops (steel). HWm) I ,  & Fig. 5. Interpolated loop. H 0.920r 4 calculated g 0.220 v -rq -1895 / kith eddy current) / - / / Table I Discretization data and CPU time 0.1205 J .  6. Simulated and measured minor loops. U - IV. AND DISCUSSION A. Flux Density Fig.7 shows the operating points of 14 and 14 on the initial magnetization curve and minor loops at the point T (70,88) in the pole piece shown in Fig. 1. The range of minor loop is changed due to eddy current. Fig. 8 shows the time variation of the z-component B, of flux density at the point s(0,O) in the gap. The abscissa denotes the time step. The selection of time interval At is important in the nonlinear analysis taking account of minor loop. When At is chosen as 2 sec during all period of q-t in Fig. 2, the NR iteration was not converged. Therefore, the time increment At is chosen as sec.during the period of p q-r, 2 sec during r-s and sec during s-t. This may because the change of flux density becomes to large when large At is chosen. The difference between the ordinary nonlinear analysis without minor loop and the present analysis with minor loop is that the operating point of B and H moves on the loops having large different slopes, for example, between the upper minor loop Q and the lower minor loop 0. The introduction of the relaxation factor a [5] as shown in the following equation is necessary in the nonlinear analysis taking account of minor loop: where A' is the vector potential at the k-th iteration. 6Ak is the increment of A. In this case, ct is chosen as 0.5. Table I shows the discretization data and CPU time. Fig. 9 shows the change of flux density B, with time at the point S(0,O). The flux densities at the instants Q, @,e**, when the current becomes equal to zero and the transient phenomenon of current is finished, are plotted. The flux density with eddy current is smaller than that without eddy current due to the opposing magnetic field produced by the eddy current. B. Change of Residual Flux Density The change ABz of residual flux density at the point S(0,O) in the gap is examined. ABz corresponds to the output of the probe coil of MRI. ABz is given by ~ Computer used : IBWAT(49.7 M W P S )  Convergence criterion for NR iteration : AB=I x 10-7 (TI 0.230 r bithout eddy currend 0.225 / / 0.215 0.210 I i /  \ initid magnetization curve > t=O ( B  produced by magnet only) 0.205 180 182 184 186 188 190 IHI (Ah) Fig. 7. B and H on minor loop (point T in pole piece). I without eddy cumnt I 0.1230 0.1225 0.1220 Q 0.1210 where B, is the flux density at the instant t=O ( H A ) .  Bzi is the flux density at the instant t=i (MA) shown in Fig. 2. If there is no eddy current and no minor loop in the pole piece, 1896 0.121959 [ E o a i t h o u t  eddy current] 0.121958 1.“ 0.121954 1 1 with minor loop and eddy current 1 / 0.121953 0.121952 0.121951 0 10 20 30 35 t (=I Fig. 9. Flux densities at instants Q, 0 etc (point S in gap). 0.005406 r 0.0054F .0054 qN 0. 5403 8,@.116553(T) E /I I i n  , I -10 -5 0 5 10 1 (A) (a) with minor loop, without eddy cumnt n 0.0054025 hB, is always equal to zero, because B, does not change with time. Fig. 10 shows the change ABz of residual flux density with time. The tendency of the shape of locus of AB, is not so much different from the measured one. Figs. 9 and 10 suggest that the major cause for the residual flux density is the minor loop, and the shape of locus of AB, is affected by the eddy current. CONCLUSIONS The obtained results can be summarized as follows: (1) The minor loop can be represented using upper hysteresis loop, and the validity of the modeling of minor loop is shown by comparing with measurement. (2) The selection of time interval A t  is important in the nonlinear analysis taking account of minor loop. (3) The change ABz of residual flux density is produced due to the minor loop of pole piece. Although the tendency of the shape of locus of ABz is not so much different from the measured one, a more precise analysis taking account of multi-minor loops, namely many minor loops occur in a minor loop, should be carried out. REFERENCES [ I ]  T.Myamoto, H.Sakurai, H.Takabayashi and M.Aoki, “A development of a permanent magnet assembly for MRI devices using Nd-Fe-B material”, IEEE Truns. Mugn., vo1.25, no.5, pp.3907-3909, Sept. 1989. [2] P.R.Locher, “Proton NMR tomography”, Philip Technicul Review, vo1.41,no.3, pp.73-88, 1983i84. [3] K.Miyata, K.Ohashi, N.Takahashi and H.Ukita, “Analysis of magnetic characteristics of permanent magnet assembly for MRI devices taking account of hysteresis and eddy current”, IEEE Truns. Mugn., vo1.34, no.5, 1998. [4] IEC, “Method of measurement of the d.c. magnetic properties of solid steels in a closed magnetic circuit”, 404-4 TC68 WG2 N63, Sept., 1991. [SI K.Fujiwara, T.Nakata and N.Okamoto, “Method for determining relaxation factor for modified Newton-Raphson method”, IEEE Trum. Mugn., vo1.29, no.2, pp.1962-1965, March 1993. I ~n , 1 -10 -5 0 5 10 1 (A) (a) with minor loop and eddy current Fig. 10. Change of residual flux density. 

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Effect of minor loop on magnetic characteristics of permanent magnet type of MRI

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