ISEF 2005 - XII International Symposium on Electromagnetic Fields
in Mechatronics, Electrical and Electronic Engineering
Baiona, Spain, September 15-17, 2005
Finite Element Analysis of an Outer-Rotor Permanent-Magnet Brushless
DC Motor for Light Traction
Xose M. López-Fernández(1) and J. Gyselinck(2)
(1)
(2)
Department of Electrical Engineering, ETSII – University of Vigo, Spain - xmlopez@uvigo.es
Department of Electrical Engineering, Free University of Brussels, Belgium - Johan.Gyselinck@ulb.ac.be
Abstract- In this paper a new outer-rotor permanent magnet brushless DC motor is studied by means of
the Finite Element Method. Two practical motor versions are considered for using in light traction at low
speed. Special attention is devoted to cogging torque level, magnetic field and performance characteristics.
The different versions of studied motor are dependent on the pole-slot ratio.
Introduction
Although, a cage induction motor (IM) is the most popular traction motor, this motor is not completely
suitable for direct gear-less electromechanical drives. The performance of IMs at low speeds is poor
and the torque density (output torque-to-mass) is low. Better performance of direct electromechanical
drives can be achieved with the aid of permanent magnet brushless motors (PMBMs) which are the
highest efficiency, highest power density and highest torque density traction motors [1].
Traditionally, in electric light traction brush type DC motors, switch reluctance motors (SRMs) and
PMBMs have been used with the aid of a belt or chain gear to drive the rear wheel.
There are some special applications, like wheel where the outer rotor fits ideally because he can be
connected directly to the load. Such is the case of the light traction at low speed applications like
bicycles, wheelchairs, forklift trucks and golf carts [2]. With a slightly more complicated construction
it is possible to use outer rotor motors in classical arrangements with flange by connecting the axis to
the bell type rotor and leading it through the stator.
Therefore, in recent years, attention manufacturers and users in the light traction field has been
focused on the removal of the gearbox and the presence of DC commutator as the next step in
reducing maintenance requirements and increasing system reliability. The recent innovative solutions
consist in coupling the motor directly to the wheel shaft or even incorporating it within the wheel
itself. Either way, the motor speed would be much lower than that of a geared drive and this would
require much higher power-to-weight ratios than hitherto been required. The current powerful
permanent magnet materials are promised to reach these requirements [3, 4].
The purpose of this paper is to analyze by means of finite element software a new outer-rotor PMB
DC hub motor for light traction at low speed. Special attention is devoted to cogging torque level,
magnetic field and performance characteristics. The different versions of studied motor are dependent
on the pole-slot ratio.
ISEF'2005-MSD.2
ISBN 84-609-7057-4
(7-Pages)
Outer-Rotor PMB Drive System
Permanenet magnet brushless (PMB) DC motor is an inside-out version of the brushed PM DC motor.
The PMBDCs are supplied by a trapezoidal back electromotive force (EMF) waveform and requires
rectangular shaped currents to produce smooth torque. As the speed range of a drive is determined by
the peak current ratings, maximum voltage available, motor parameters and the back EMF waveform
where only two phases conduct at any one time.
The PMB drives adapt the stator current to phase current references. Therefore a control loop for
commanding the magnitude of the phase currents to have to be integrated by into drive system for
getting information of the different rotor positions by using an appropriate controller and feedback
device.
To supply DC variable voltages and currents and phase firing control for PMB DC the drive requires
two basic blocks. The first consist of the voltage inverter o power supply which consists of a power
rectifier to convert the AC line voltage into a DC battery voltage, and the inverter which includes the
power switches (usually two per phase) with their current sensors. The second block is the control,
which decodes shaft angle input data and rotor position, and controls DC voltage chopping (PWM).
Fig. 1 shows the typical scheme of a brushless DC drive system.
u
v
w
Fig.1 Typical scheme of a PMB DC motor.
Motor Design Considerations
The schematic geometry of the studied PMB DC motor is shown in Fig. 1. Two versions of the motor
were designed as a low-speed, high-torque outer-rotor type, suitable for direct drive of electric of light
traction where no gear reduction is needed. Both motors have 40 permanent magnet poles with 60
stator-lots in the version-(A), while 57 stator-slots has the version (C).
Permanent magnets
Rotor
Stator teeth
Fig.2 Geometry of the motor.
ISEF'2005-MSD.2
ISBN 84-609-7057-4
(7-Pages)
Desing Parameters
The main parameter of design are as following:
Pair of poles: 20
Rated peed: 150 rpm
Number of phases: 3
Slots per pole: 1.5 (Version A:60 slots / 40 poles), 1.425 (Version B:57 slots / 40 plots),
Windings one pith slot
Air-gap: 0.5, 0,25, 0,125 mm
Outer diameter: 200 mm
Rotor Design
The choice of pole numbers was done considering factor as inertia, torque, cogging torque and speed.
With high pole numbers it became possible to use one short-pitched, winding which can eliminate the
overlap of the winding end turns.
On the inner surface rotor of the proposed machines, 40 pieces of PM (NdFeB) are mounted
alternately to form 40 poles and two adjacent poles make up a pair of poles so that the flux paths of
different pole pairs are independent. The multipole magnetic circuit arrangement enables to reduce the
magnetic iron yoke, resulting in the reduction of volume and weight
Stator Design
The coil span of stator windings is designed to be equal to the slot-pitch, the over-hanging part of the
coil can be significantly reduced, thus resulting in the saving of copper as well as the further reduction
of volume and weight. Fractional number of slots per pole and phase were used, in order to get an
uniform magnetic force between the stator and rotor at any rotating position, and thus minimizing the
cogging torque that usually occurred in PMB DC machines.
Since lamination and winding design are the main factors that influence the ability to automatically
and economically assemble a stator, they were considered as factor of design in order to minimise the
stator assembly cost.
The concentric type winding was chosen in preference to lap windings since the last one cannot be
automatically inserted into the slots.
Mathematical formulation
All results are based on 2D-Finite Element magnetic circuit calculation with a finite element
commercial package FLUX 2D.
Discretization
The difference partial equation governing for a PMB DC motor with the exterior rotor steel shell can
be expressed in a magnetic vector potential A as:
N
N
N
∂ ⎛ 1 ∂A ⎞ ∂ ⎛ 1 ∂A ⎞
1 ⎛ ∂M y ∂M x
⎟⎟ = − I u − I v − I w − ⎜⎜
⎟⎟ + ⎜⎜
⎜⎜
−
∂y
s
s
s
∂x ⎝ µ ∂x ⎠ ∂y ⎝ µ ∂y ⎠
µ ⎝ ∂x
ISEF'2005-MSD.2
ISBN 84-609-7057-4
⎞
∂A
⎞
⎟⎟ + σ ⎛⎜
+ ∇φ ⎟
⎠
⎝ ∂t
⎠
(7-Pages)
(1)
where µ stands for permeability, N for coil turns, s for the cross section area of the winding wire with a
phase current I, σ for electric conductivity, and M for magnetization of permanent magnet. Subscripts
u, v and w represents each phases.
From continuity condition of current, is considered
∂A
⎛
⎞
∇J e = ∇ ⎜ − σ
− σ ∇φ ⎟ = 0
∂t
⎝
⎠
(2)
where Je is the eddy current density in the rotor steel which was neglected.
Equation (1) is discretized into a finite element form and expressed in terms of Galerking equation.
Electric Circuit
As the PMB DC motor is driven by a voltage source inverter. The phase currents in equation (1) have
to satisfy the circuit equations (3,4) at the same time.
dI
dI
dφu dφ w
−
+ Lo u − Lo w + Ru I u − Rw I w = Vu − Vw
dt
dt
dt
dt
(3)
dI
dI
dφu dφv
−
+ Lo u − Lo v + Ru I u − Rv I v = Vu − Vv
dt
dt
dt
dt
(4)
Where φ is the flux linkage of phase winding, I is the phase current and V is square wave phase
voltage supply by inverter. Lo is leakage inductance of the stator coil ends. Because only two of three
phases are kept turn-on state the resistance of the other phase which is kept turn-off state is given an
infinitely large value to solve these equations
The magnetic flux linkage φ can be calculated by a line integral of vector potential in the winding
region expressed as:
n
φ = ∑ ∫ Adl
e =1
(5)
ls
where n is the total number of discretized elements in the winding area, le is the integral path
encompassing elements. Since the winding is a star connection was used.
Iu + Iv + I w = 0
(6)
The electromagnetic torque is with the magnetic co-energy concept based on the principle of virtual
work, on the basis of the symmetric boundary conditions.
Results
The cogging torque due to the slotting of the stator, due to the magnets and due to the distributed
winding was evaluated for two version of motor according with ratio of slots/poles. To take this
influence into account, the electromagnetic torque is calculated for various rotor positions considering
different air gap dimensions (1, 05, 0,25 and 0,125 mm.) as Fig.2, 3 show.
ISEF'2005-MSD.2
ISBN 84-609-7057-4
(7-Pages)
Fig.3 Computed cogging torque vs. electrical degrees with
60 teeth, version (A), under different air gap.
Fig.4 Computed cogging torque vs. electrical degrees with
57 teeth, version (B), under different air gap.
Fig.5 Maximum values of cogging torque 60 teeth (A) v.s. 57 teeth (B)
for different air gaps (1, 0,5, 0,25, 0,125 mm.)
ISEF'2005-MSD.2
ISBN 84-609-7057-4
(7-Pages)
Fig. 4 shows the maximum values of cogging torque of both versions of motor under different values
of air gap (1, 0,5, 0,25, 0,125 mm). The better performance is reached by the version with 57 teeth (B),
and into that with the higher value of air gap. However due to reasons of dropped amper-turns at air
gap the 0,5 mm. was chosen as final parameter of design.
Fig.4 shows the magnetic field pattern of motor version with 60 teeth (A) and 0,5 mm air gap.
Fig.6 Magnetic field with 60 teeth and 0,5 mm air gap distance.
Performance Characteristics
The understanding of the performance of the brushless DC motor is to study the torque vs. speed
curve. This curve represents the steady-state capability of the motor in driving various types of loads.
The two fundamental equations for steady-state analysis are the back-electromotive force (back-EMF)
and torque as following:
E = kE ω
T = kT I
(7)
Fig.5 Performance characteristics either 60 teeth (A) and 57 teeth (B) with 0,5 mm air gap distance:
Torque and current v.s. speed.
ISEF'2005-MSD.2
ISBN 84-609-7057-4
(7-Pages)
Where E is the back-EMF, ω the angular speed, T the electromagnetic torque, I the DC supplied
current, kE is a constant called the EMF constant, and kE torque constant. As an approximation in
practical situation, both constants can be considered equal.
Applying different angular speed to rotor and computing the back-EMF on finite element models was
obtained the constant kE. And therefore, the performance characteristics of speed vs. torque were
obtained for both design versions, which are shown in Fig.5.
Conclusions
A new outer-rotor permanent magnet brushless DC motor was studied by means of the finite element
method. Two practical motor versions were studied for using in light traction at low speed. The
dependence of the ratio stator-slot/pole was evaluated. Special attention was devoted to cogging torque
level, magnetic field and torque speed performance characteristics.
References
[1] J. F. Gieras, M. Wing : Permanent Magnet Motor Technology: Design and Applications. 2nd ed., rev. and
expanded. New York: Marcel Dekker, cop. 2002
[2] Jacek F. Gieras, Nicola Bianchi, “Electric Motors for Light Traction”, EPE Journal Vol. 14, n. 1, pp. 12-23,
February 2004
[3] Duane C. Hanselman : Brushless Permanent Magnet Motor Design. New York : McGraw-Hill 2003
[4] J.R. Hendershot, T.J.R. Miller: Design of Brushless Permanent Magnet Motors. Oxford : Clarendon Press,
1994
ISEF'2005-MSD.2
ISBN 84-609-7057-4
(7-Pages)