Demagnetization Analysis of Ferrite Magnet
Using Two-line Approximation Based on Reluctance Network Analysis
D. Momma, Y. Yoshida*, and K. Tajima
Department of Cooperative Major in Life Cycle Design Engineering, Akita Univ., 1-1, Tegata Gakuen-machi, Akita 010–5802,

Japan*Department of Electrical and Electronic Engineering,

Akita Univ., 1-1, Tegata Gakuen-machi, Akita 010–5802, Japan

This study presents a method for determining the demagnetization characteristics of ferrite magnets based onreluctance network analysis (RNA). First, an RNA model for determining the operating points of a magnet
considering demagnetization using a two-straight-line approximation of the demagnetization curve is discussed.
Then, using the proposed model, demagnetization characteristics are determined. Experimental results demonstrate
the validity of the proposed method.
Key words: reluctance network analysis, ferrite magnet, demagnetization analysis, two line approximation
1. Introduction
Permanent magnet (PM) motors based on powerfulrare-earth magnets are widely used in various
applications because of their high-performance
characteristics. However, rare-earth magnets may besubject to price rises as the production of such metals isconcentrated in a single country. Therefore, thedevelopment of highly PM motors without rare-earthmagnets is required.
Although the maximum magnetic energy product offerrite magnets is one tenth that of rare-earth magnets1), high-efficiency ferrite magnet motors have beenreported 2). Ferrite magnets in motors are exposed to alarge reverse magnetic field to obtain performanceequivalent to that of rare-earth magnet motors.
Therefore, ferrite magnets are at a risk ofdemagnetization because of their low coercive force.
Thus, it is necessary to consider the demagnetization offerrite magnets for ferrite magnet motor design.
Reluctance network analysis (RNA) is a useful
method to save calculation time in the estimation of thecharacteristics of PM motors, as reported in previousstudies 3)-6). However, a demagnetization analysis
method using RNA has not yet been proposed.
Because demagnetization analysis using finite
element analysis (FEA) has been previously discussed7)-8), this method is now applied to RNA. This study
presents an RNA model for determining the operating
points of ferrite magnets; this model uses a two-line
approximation of the demagnetization curve. To verify
the accuracy of the proposed model, the calculated
results are compared to values obtained fromtwo-dimensional (2D) FEA and experimental results.
2. Derivation of the RNA Model
Fig. 1 shows the shape and specifications ofanalytical and experimental objects under consideration.
Exciting coils are wound around U-shaped cores. Ferrite
magnets with a thickness of 2 or 3 mm are sandwiched
between the cores. The number of winding turns of
exciting coils per leg is 80, and the stack length of thecores and ferrite magnets is 20 mm.
Fig. 2 illustrates the division of a 2D RNA model.
The RNA model is derived in a quarter region of theanalysis model shown in the figure, taking advantage ofthe analytical model’s symmetry. In the model, theelements around the air gap are divided with a size of
1.0 mm  1.0 mm. When the 3-mm-thick magnet is used,the element size of the magnet in the y-axis direction is
1.5 mm.
Each divided element is expressed in a 2D-unit
magnetic circuit. Fig. 3(a) shows the unit magnetic
circuit of the core and air region. In the figure, l is thelength of the element. The reluctance, Rm, in the unit
magnetic circuit is obtained fromRm l2s 0 l  ls ,(1)where  s is the relative permeability,  0 is thepermeability of vacuum, and ls is the stack length of themodel. The elements of a ferrite magnet can be expressedas the reluctance and magnetomotive force (MMF), Fm,
shown in Fig. 3(b). The reluctance of a ferrite magnet,
24 mm

20 mm

20 mm

Core35 mm
10 mm
Ferrite magnet(Y30BH)

Excitingcoil

NS
2 mm

Magnets with a thicknessof 2 mm or 3 mmyz

2 mm
SNx

Fig. 1 Specifications of analytical and experimentalobjects.

Rmag, is expressed byRm ag l2r 0 l  ls ,(2)where r is the magnet’s recoil. By connecting all unit
magnetic circuits together, the RNA model is obtained.
Fig. 4 illustrates the demagnetization curve of aferrite magnet, which is approximated by two lines. Inthe figure, Br, Br’, Hc, and Hc’ are the residual magnetic
flux density before demagnetization, residual magnetic
flux density after demagnetization, coercive force before
demagnetization,
andcoerciveforce
afterdemagnetization, respectively. If there is nodemagnetization in a ferrite magnet element, Fm isgiven byBr l .

2r 0(3)When an operating point of a ferrite magnet is changedby an external magnetic field and becomes less than theknee point, the MMF after demagnetization, Fm’, can be
expressed as

Fm ' 

Br ' l .
2r 0yzx
Fig. 2 Division of the RNA model.

(4)

To obtain the residual magnetic flux density afterdemagnetization (Br’), B–H and J–H characteristics,
where J = B–0H as shown in Fig. 5, are used 9). First,the operating point without considering an external
magnetic field, plotted as point a in the figure, isdetermined by the derived RNA model. The magnetic
field at this time is diamagnetic, Hd. At the same time,
point b, where a perpendicular line from point aintersects with J–H characteristics, is determined Then,line l0 is given as a straight line passing through point band the origin. Next, the operating point considering anexternal magnetic field, plotted as point c in the figure,is determined. The external magnetic field, Hex, iscalculated as the difference between the magnetic fieldat point c and Hd. The line, l0, is shifted toward thenegative direction by Hex without changing the slope toobtain line l1. The intersection of J–H characteristics
and line l1 is plotted as point d. When point d is lessthan the knee point of J–H characteristics, irreversible
demagnetization occurs. Br’ is determined as the point
where the vertical axis intersects the straight line,
which passes through point d and is parallel to J–Hcharacteristics before demagnetization.

lRml(a) Core and airlRmagl

Fm(b) Magnet
Fig. 3 Unit magnetic circuit.

BrB  r 0 H  Br
 1.40 106 H  0.385

Br’

3. Demagnetization Analysis of the Ferrite Magnets
Using the proposed RNA model, the demagnetizationof the ferrite magnets is determined with an exciting
current that produces an external magnetic field
opposed to the magnetization direction. Three cycles ofa sawtooth wave, shown in Fig. 6, are applied toexciting coils. The maximum value of the current is 4.0
A when the magnet thickness is 2 mm, and 5.5 A when
the thickness is 3 mm.
Fig. 7 illustrates the direction of the magnetic flux
flowing through the core. This direction is caused by theKnee point

Hc Hc’ Hcb

Parallel lines

B  0.6 104 H  13.7

Flux density (T)Fm 0Magnetic field (A/m)
Fig. 4 Demagnetization curve approximated by twolines.

Knee point( J-H )

l1

Bl0HexbBr

Parallel lines

J-H characteristics( J = B － µ0H ) dBr’aB-H characteristics
approximated by two lines

cHexHex

Hd0H

Fig. 5 Calculation method for the residual magnetic
flux density after demagnetization.

Fig. 8 Mesh distribution of the 2D-FEA model.

4.0A or 5.5A

0.4
0.4

Br = 0.385 T

RNA

0.35
0.35

2D-FEA

0.3
0.3

Current (A)

0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1

Br’ = 0.283 T

0.05
0.05

Flux density (T)Knee point( B-H )

00−-0.050.05

Time (s)

−-0.1
0.1

Fig. 6 Waveform of exciting current.

-250000−250000-200000−200000-150000−150000-100000−100000-50000−5000000

Magnetic field (A/m)(a) Graph for the 2-mm-thick magnet.
0.4
0.4

mag

0.35
0.35

2D-FEA

0.3
0.3

0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1

Br’ = 0.284 T

0.05
0.05

Flux density (T)i

Br = 0.385 T

RNA00−-0.050.05

NSElement A B C

Fig. 7 Direction of magnetic flux and detection points.
winding current and is the inverse of the flux of theferrite magnet. In the figure, elements A, B, and C are
the detection points of demagnetization.
To compare the results from the RNA model withthose obtained using FEA, the same calculation as thatof RNA was performed using the JMAG-Designer
Ver.14.1 software. Fig. 8 shows the mesh distribution of
the 2D FEA model. The number of elements of the FEA
model is 3,429.
Fig. 9 shows the operating points of element B
determined by the RNA and 2D FEA models. It isobserved that the data obtained by the proposed RNAmodel almost agree with the corresponding 2D FEA

−-0.1
0.1

-250000−250000-200000−200000-150000−150000-100000−100000-50000−5000000

Magnetic field (A/m)(b) Graph for the 3-mm-thick magnet.
Fig. 9 Calculated operating points of element B.
calculation results. The residual flux density of themagnet before demagnetization, Br, is 0.385 T. The
calculated residual magnetic flux density after
demagnetization by RNA, Br’, is 0.283 T when themagnet thickness is 2 mm, and 0.284 T when it is 3mm.
Table 1 shows a comparison between thedemagnetization factors calculated by RNA and 2D FEA
for the 2-mm-thick magnet, and Table 2 shows acomparison of the calculated demagnetization factors byRNA and 2D FEA for the 3-mm-thick magnet In thetables, the demagnetization factor, Dfac, is defined asTable 1 Demagnetization factors for the 2-mm-thickmagnet.

10 mm

Element
A [%]

B [%]

C [%]

RNA

14.9

26.6

14.9

2D-FEA

14.2

30.4

14.6

Ferrite magnets

U-shaped cores

Table 2 Demagnetization factors for the 3-mm-thickmagnet.
Element
A [%]

B [%]

C [%]

RNA

12.7

26.3

12.7

2D-FEA

7.9

31.0

8.0

Exciting coils(a) Exposed view

Table 3 Interlinkage magnetic flux of ferrite magnets.
2 mm

Supporters

Cores3 mmbefore [Wb]after [Wb]before [Wb]after [Wb]RNA

0.02360.01820.03080.0239Exciting coil

2D-FEA

0.02360.01730.03080.0229SupporterFerrite
magnetfollows: B '
D fac  1  r  100 .
 Br (5)yDemagnetization factor at the center of the magnet isgreater than that at either end. The calculated values
obtained from the proposed model almost agree withthose obtained from 2D FEA.
Table 3 shows a comparison of the interlinkage
magnetic flux of the ferrite magnets calculated via RNAand 2D FEA. After demagnetization, the interlinkage
magnetic flux decreased by approximately 25 % for bothmagnet thickness.
The number of time steps of the proposed model and2D FEA model is 64 for three cycles of the sawtoothwave. The calculation times of the proposed and 2D
FEA models are 17 s and 58 s, respectively.

(a)4. Comparison with Experimental Results

xTo verify the calculation accuracy of the RNA model,
the calculated values were compared to the measured
ones. Fig. 10 shows a photograph of the experimental
apparatus; Fig. 10(a) is an exposed view. The
supporters are sandwiched between the cores to fix thepositions of the cores and magnets. Fig. 10(b) shows anassembled experimental setup. Fig. 11(a) shows theconditions of the experiment, and Fig. 11(b) shows thedetection point of the magnetic flux density of theferrite magnet. The magnetic flux density wasdetermined using a gauss meter (GM-301, Denshijiki
Kogyo); the average values of the surface magnetic flux
density were obtained from 10 measurements withmagnets of the same size.
Fig. 12 shows the comparison of the magnetic flux
density of the magnet surface before and afterdemagnetization. The DC input current is 4.0 A whenthe magnet thickness is 2 mm and 5.5 A when it is 3zx(b) Assembled setup.
Fig. 10 Photographs of the experimental apparatus.
(b)
Detection point
Gauss meter
probe

yyz

Ferrite magnet

10 mmzx20 mm

Fig. 11 (a) Measurement setup. (b) Detection point of
the magnetic flux density.
mm. In the figure, the error bars denote the maximum
and minimum the measured values. It is concluded thatthe magnetic flux density determined by the proposed
RNA model almost agrees with the corresponding 2D
FEA calculated and measured values.
5. Conclusions
This study presented the demagnetization analysis
of a ferrite magnet based on RNA. In the proposedmodel, the demagnetization curve of the magnet wasapproximated by two lines to estimate the magnetic flux
density before and after demagnetization. The validityof the proposed RNA model has been demonstrated bycomparing the calculated results with 2D FEA
calculation results and experimental results.

References

0.14

Flux density (T)

0.12
0.1

0.08
0.06
0.04

0.02
0

before

after

RNA

0.120

0.0872

2D-FEAmeasured

0.119

0.08270.111

0.0812Flux density (T)(a) Graph for the 2-mm-thick magnet.
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0

beforeafter

RNA
2D-FEA

0.156

0.114

0.155

0.106

measured

0.151

0.110

(b) Graph for the 3-mm-thick magnet.
Fig. 12 Comparison of magnetic flux densities ofmagnets.

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Received Oct. 20, 2015; Revised Jan. 10, 2016; Accepted Mar.
21, 2016