3500 IEEE TRANSACTIONS ON MAGNETICS, VOL 31, NO. 6, NOVEMBER 1995
Numerical Analysis of Permanent
Magnet DC Motor Performances
Ovidiu Craiuf, Nicolae Dan+, Eugene A. Badea++
+Polytechnic University of Bucharest,
313 Splaiul Independentei, 77206, Bucharest, Romania
6503 Taimer Ct., Sugarland, Tx, 77479, USA
Abstract-The paper presents a numerical procedure The risk of demagnetization is bigger at low temper-
based on the finite element method to determine the atures, where the remanent induction B,,, is high and .
performance o ferrite-type permanent magnet DC
f the coercive force H,,,, low. The B - H loop changes
motors. Magnetic saturation, armature’s displace- are reversible and linear over a limited range (0 - 120)’ C.
ment and temperature’s effect were considered in the Correspondingly, a linear approximation was used to de-
computation Qfthe torque’s variation with respect t o termine the temperature coefficients for both remanence
angular position. and coercivity [l]:
Reliable calculation methods are required for design 11. FINITE
purposes of permanent magnet DC motors which are man-
ufactured in many different types and large quantities.
The performance of ferrite-type permanent magnet DC The thermal and electromagnetic fields are linked
motors are temperature dependent. As shown in Fig.1, through permanent magnet’s B,,, and H,,,, dependence
the demagnetizing curves could be approximated as hav- in respect with the temperature. All other active materi-
ing a linear variation for a wide range of positive temper- als used for the core and windings have insignificant vari-
atures. ation with the temperature’s gradient. The plan-parallel
model was chosen assuming that both magnetic and ther-
mal fields are uniform along the motor’s axial direction
+: The magnetic vector potential formulation A
A, (x, uz was used to describe Maxwell’s equations -
curl(vcurl(A)) = J s + Jm
where JS - is the source current density, J m =
curl(Hcoer) - is the equivalent current density corre-
0.3 sponding t o the coercive force of the magnet, and v (B,T)
- is the magnetic reluctivity, which depends on the mag-
0.2 netic flux density B and the temperature T .
The finite element solution was obtained by minimizing
the energy functional associated with ( 2 ) . The Newton-
0.28 0.24 0.20 0.16 0.12 Raphson technique was used to solve the magnetic nonlin-
-mil IT) earities. The eddy-currents induced in the iron core and
the coils were neglected.
The two dimensional heat transfer problem character-
ized by anisotropic thermal conductivities I C x , ky and de-
Fig.1 Demagnetizing Curves of Ferrite-Type Magnets scribed by Fourier’s equation:
d dT d dT
Manuscript received February 17, 1995. Ovidiu Craiu, e-mail
ocraiuQalpha.amotion.pub.ro, phone 40-16-53-18-00; Nicolae Dan,
ax ax + -(k
ay + Qs = 0 (3)
e-mail email@example.com, phone 919-660-5295; Eugene A.
Badea, e-mail firstname.lastname@example.org, phone 713-565-2154, was using Dirichlet and convection-type boundary
fax 713-565-2231. conditions. The ohmic and iron core losses determined
0018-9464/95$04.00 0 1995 IEEE
from the electromagnetic field solution, were considered Fig.3 shows the finite element mesh, the periodic
as inputs to the heat source strength Qs: boundary conditions used for the electromagnetic prob-
lem, and the magnetic flux distribution corresponding to
Qs = q f 2 B 2 + C 2 f B 7 + p [ 1 + a(Tk+l-Tk)] J 2 (4) rotor’s last position of movement at full-load.
f - is the fundamental frequency; c1, cp and y - are mate-
rial constants determined from the experimental loss char-
acteristic; Q - is the resistivity temperature coefficient,
and p - is the electrical resistivity.
The new temperature distribution was used to correct
the remanence and coercive force of the magnet.
FLOWCHART OF THE METHOD
1. Magnetic field’s computation for succesive rotor
2. Torque calculation.
3. Determination of the ohmic and iron core losses.
4. Temperature’s computation.
5. Reset the magnet’s demagnetizing curve.
111. TESTPROBLEM AND RESULTS
The method was tested for the case of a high energy
ferrite-type permanent magnet motor, having the follow-
ing characteristics: - rated torque A = 0.25 Nm , - input
voltage U = 24 V , - rated current I = 7.5 A , - number
of poles 2 p = 2 , - frequency f = 62.5 Hz , - air gap
length S = 0.5 mm , - length of the rotor L = 89 mm.
Fig.2 presents the geometrical configuration of the an-
alyzed motor, together with the graphical display of the
finite element mesh for the thermal problem , the mag-
netic flux plot at no-load, and the corresponding isother-
mal lines. Fig.3 F E Mesh - Electromagnetic;
Magnetic Flux Distribution
The moving band method  was used t o determine
the torque’s variation in respect with rotor’s displace-
ment. To avoid numerical errors, the rotor’s displacement
was chosen to be the distance between two consecutive
nodes placed on the sliping surface. Thus, the shape of
the air gap triangles remained the same for all finite ele-
ment meshes . The magnetic problem was solved for 12
successive rotor positions between two consecutive slots.
The torque was determined using Arkkio’s method :
L - is the motor’s length; B, and B+ - are the radial and
tangential components of the magnetic flux density in the
air gap surface S comprised between r, and rs radii.
The torque’s dependence with respect to the rotor’s
displacement for two different temperatures is shown in
Fig.4. The calculation was performed assuming an ideal
Fig.2 a) Computational Domain; b) FE Mesh - Thermal; linear commutation, considering AB,,, = -0.19 % C-‘
c) Flux Plot at No-Load; d ) Isothermal Lines and A H,,,, = 0.2 % C-I. Correspondingly, the current
density 5 2 varies from 5 to 53 ( where 53 = -51 ) from
1 IV. CONCLUSION
the first to the last position of movement.
The present paper illustrates the importance of tem-
perature’s influence over the global performances of the
ferrite-type DC motors. The moving band method with
a sufficient number of rotor displacements is proved to
be a good tool for computing the motor torque ripples.
The influence of magnet’s width on torque’s response was
included, assuming that the commutation is linear.
T . J. E. Miller, Brushless Permanent Magnet and Reluctance
Motor Drives, Oxford Science Publications, 1989.
N. Dan, 0. Craiu, “2D-Finite Element Analysis of the Mag-
netic Field Coupled with Heat Transfer Analysis in Perma-
nent Magnet Brushless Motors,” ICEM-94, Proceedings, Vol.
2, Paris, France, Sept. 1994, pp. 39-43
R. Gupta, T. Yoshino, Y . Saito, “Finite Element Solutions
of Permanent Magnetic Field,” Trans. on Magnetics, Vol. 26,
No. 2, March 1990, pp. 383-386.
F. A. Fouad, T. W. Nehl, N. A. Demerdash, “Magnetic Field
Modeling of Permanent Magnet Type Electronically Operated
Synchronous Machines Using Finite Elements,” IEEE Trans.
on PAS, Vol. PAS 100, No. 9, Sept. 1981, pp. 4125-4135.
N. Sadowski, Y . Lefevre, M. Lajoie-Mazenc, J. Cross, “Finite
Fig.4 Temperature’s Effect on the Torque Element Torque Calculation in Electrical Machines while Con-
sidering the Movement,” IEEE Trans. on Magnetics, Vol. 28,
No. 2, March 1992, pp. 1410-1413.
Different widths of the magnet were considered to opti-
mize the motor’s geometry. Fig.5 presents the torque’s
variation with respect to the rotor’s displacement for
a1 = 0.854, a = 0.875 and a3 = 0.895, where ai -
is the ratio between the magnet’s width and the pole’s
0 10 20 30 40 50 60 70 80 90
Fig.5 Torque’s Response for Different
Widths of the Magnet