Halbach Array Motor

Halbach Array Motor/Generators
A
Novel Generalized Electric Machine
By Bernard T. Merritt, Robert F. Post, Gary R. Dreifuerst, Donald A. Bender;
October 28, 1994; Lawrence Livermore National Laboratory, UCRL-JC-119050
Introduction
Halbach array, and describes the results obtained when
they were tested in the laboratory.
For many years Klaus Halbach has been investigating
novel designs for permanent magnet arrays, using ad
The Dipole Array
vanced analytical approaches and employing a keen in
sight into such systems. One of his motivations for this
Although electric machines can be constructed using
research was to find more e cient means for the utiliza
multipole fields based on the techniques introduced by
tion of permanent magnets for use in particle accelera
Klaus Halbach, the dipole field o ers some unique ad
tors and in the control of particle beams. As a result of
vantages for the construction of a high speed electric
his pioneering work, high power free electron laser sys
machine. Figure 1 shows an end view of a dipole Halbach
tems, such as the ones built at the Lawrence Livermore
array. Shown in Figure 1a are the directions of magneti
Laboratory, became feasible, and his arrays have been
zation of the bars and in Figure 1b, one quadrant of the
incorporated into other particle focusing systems of
computed lines of force produced by the array. Note the
various types. This paper reports another, quite di er
highly uniform field inside the array, and the near
ent, application of Klaus' work, in the design of high
cancellation of the field outside the array.
power, high e ciency, electric generators and motors.
When tested, these motor/generator systems display
some rather remarkable properties. Their success de
rives from the special properties which these arrays,
which we choose to call "Halbach arrays," possess.
In August 1979, Klaus Halbach submitted a paper1 enti
tled "Design of Permanent Multipole Magnets with Ori
ented Rare Earth Cobalt Material." In this paper, he
presented a novel method of generating multipole mag
netic fields using non intuitive geometrical arrange
ments of permanent magnets.
In subsequent publications,2,3,4 he further defined these
concepts. Of particular interest to one of the authors
Richard F. Post was the special magnet array that gen
erated a uniform dipole field. In 1990 Post proposed the
construction of an electric machine a motor/generator
using a dipole field based on Klaus Halbach's array of
permanent magnets. He further proposed that such a
system should be employed as an integral part of "an
electromechanical battery" EMB , i.e., a modular fly
wheel system to be used as a device for storing electrical
Figure 1a -- Halbach Array
energy, as an alternative to the electrochemical storage
battery.
This paper reviews Halbach's theory for the generation
of a dipole field using an array of permanent magnet
bars, presents a simple analysis of a family of novel
"ironless" electric machines designed using the dipole
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
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-1.68
.78
-1.7
.36
.97
-1.28
2.23
-1.2
2.23 2.2
2.2
2.2
2.1
1.10
2.26
1.17
-1.06
-1.01
1.55
1.67
-.82
Figure 1b -- Halbach Array Field Lines
Figure 2 -- Halbach Array Measured Field Values
Except near the ends of the array, or very near its inner
surface, the dipole field inside a such an array is given by
Construction of the Generalized
the expression derived by Halbach:
Motor/Generator
r
The design of a motor/generator using a dipole Halbach
B = B C ln 2
1
0
r
N
r
array is simplicity itself. Furthermore, since there are no
1
iron laminations used in the design the theoretical pre
with r
diction of the properties of such motor / generators is
1 and r2 indicating the inside and outside radius of
the magnet array, respectively, and B
equally simple.
r representing the
remanent field of the permanent magnet material. If M
In order to construct a motor/generator, all that is re
is the number of segments in the magnet,
quired is to insert a single or multi phase winding down
the axis of the dipole field and provide relative motion
2
sin
between the field and the winding s . Relative rotation
M
then generates an EMF in the winding that is linearly
C =
2
N
2
related to the product of the rotation speed and the
amount of flux intercepted by the winding. In the case
M
of a generator, the work performed in sustaining the
relative motion between the windings and the field is
giving CN = 0.90 for M = 8 and CN = 0.97 for M = 16.
transformed directly into ac electrical energy that flows
These equations are derived in reference 2.
out of the machine. Conversely, ac electrical energy
When Equations 1 and 2 are compared with a com
flowing into the winding, if at the proper frequency and
puter code using the correct remanent field of the mag
phase, is transformed into mechanical work that causes
nets, the code results agree within a percent or two of
a relative motion between the windings and the field. At
the analytical result. Arrays constructed using the high
this point it is important to note that only a relative
field material, NdFeB, have measured fields that are
motion between the windings and the field is required;
found to be within a few percent of the predicted values.
electric machines having either the windings or the Hal
bach array in motion could be constructed. There are
Figure 2 is a field plot of a dipole field constructed using
advantages to each type depending upon the application.
an eight segment Halbach array. The permanent mag
nets used for this dipole were ceramic; the field values
measured are uniform to about three percent. The val
ues for the field strength are those measured inside the
Halbach array cylinder; the field decreases somewhat
near the ends of the magnet array. See references for a
detailed discussion and calculation of this decrease.
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
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where A is the area of the winding, i.e., its length multi
plied by its transverse width, and is the rotational ve
locity in radians per second.
For the case of multiple turns per winding, the voltage
becomes
B'
C'
V (t) = KNB sin(wt) 6
0
A'
A
where N is the number of turns and K is a geometric
C
B
constant, close to unity, that accounts for the fact that
the winding now has some physical width that spans an
arc along the circumference of the cylinder upon which
the windings are attached. This constant is readily calcu
lated; it accounts for the relative decrease in magnetic
field since a particular turn is no longer exactly aligned
with the reference direction for the magnetic field. For
such a turn, the correction factor is equal to cos ,
Halbach Magnet Segments
where is the angle between the reference for the mag
Winding Barrier
netic field, usually zero, and the angle at which a par
Litz Wire Windings
ticular turn resides. For a distributed winding with angu
lar spread i.e. its two sides intercept a fraction 2 /2
Figure 3 -- Halbach Array with Three Phase Winding
of the circumference , upon performing the average over
Figure 3 shows a three phase winding inserted into the
this spread the factor K is given by the following expres
Halbach array field. Single phase or higher phase num
sion:
ber machines can also be constructed. This paper con
(
)
centrates on the three phase machine in which the di
K = 2 sin
/ 2 7
pole Halbach array surrounding the winding is rotating.
This configuration has been employed by us in the ap
plication for which it was first proposed, electrome
If = 30, for example, K = 0.989.
chanical batteries flywheel energy storage modules .
In order to calculate the induced EMF, consider first the
simple case of a single turn winding. Assume for the
moment that the winding is stationary and that the
V
BNA
Halbach field, due to the relative motion, is given by
EMF
B(t) = B cos( t) 3
= Speed
0
B = Field
Owing to the linearity of the system the induced EMF is
N = Turns
given exactly by
A = Area of one turn
V (t) = d
4
dt
Figure 4 -- Equivalent Circuit for Electric Machine,
per Phase Basis
where is simply given by the product of the dipole
magnetic field strength and the area, A, intercepted by
Figure 4 is a schematic of the electric machine. It is to
the windings with a small correction, usually of order 10
be interpreted on a per phase basis for machines having
to 20 percent, depending on the relative length of the
multiple phases. Note that this representation contains
windings and the magnets for the fall o of the field
only linear elements, and also note that the value of
strength near the ends of the magnet array.
magnetic field Bo, that is to be used in calculating the
induced voltage, is that of the Halbach array. In this
This expression then reduces to
ironless system, and for all feasible values of the winding
V (t) = B A sin( t)
currents, there is no "back reaction" between the stator
5
0
windings and the inducing magnetic field. There is, of
course, an e ect of the winding inductance on the out
put voltage as well as the usual resistive drop. However,
since the system is ironless, inductances are low, and
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
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with good design, the resistive drops are also low. As will
Advantages
be shown, not only is the power output very high, but
the e ciency is also typically much higher than that of
A fundamental advantage of a machine of this type has
an iron core machine of comparable physical size.
already been mentioned. That is, the fact that the ma
chine can be constructed without the use of magnetic
The equations for torque are also straightforward. Con
material other than the permanent magnets. There is no
sider again the case of a single turn winding and only
need for laminations or back iron. This has two major
one phase. The vector force on a conductor is given by
advantages. First the conventional core loss and eddy
F = 2I(Lx B) 8
current loss in the laminations or back iron does not
exist. The only loss in the machine will be losses in the
where I is the current in the conductor, L is the vector
windings. The second advantage is since there is no back
path along the conductor and B is the vector magnetic
iron or laminations required, the machine is inherently
field. The torque is then
lightweight.
T = r x F 9
The uniform field also results in several important ad
vantages. Since the field is truly uniform, the machine
where r is the radius.
design is no longer constrained by the airgap size. This
o ers the opportunity to solve other system issues. For
For a single turn rectangular winding with longitudinal
example, in the modular EMB application, it allows a
length L, in a dipole field, the magnitude of the peak
vacuum barrier to be placed between the windings and
torque becomes
the Halbach array without appreciable degradation of
T = 2rLIB 10
machine performance.
0
The field uniformity of the dipole Halbach array has
and for the multiple turn case, the peak torque becomes
another, very important result for the extraction of very
T = 2rLIB KN 11
high peak powers over short time scales. Conventional
0
generators employing narrow gaps and iron laminations
where K is the geometric factor as before and N is the
have a problem that is not encountered here. In such
number of turns.
systems, where the drag torque caused by the power is a
function of the gap spacing, there can exist a strong ten
Owing to the complete linearity of this ironless system
dency to drive the rotating system into so called "whirl"
these simple equations form an adequate basis for the
instability. Contrast this with our situation. Since the
design of motor/generator systems employing a dipole
field is uniform, the torque is not a function of the dis
Halbach array. We will later discuss the circuit related
placement of the windings relative to the field the
factors that must be employed in order to calculate the
windings only "know" that they are in a uniform rotating
output power and the e ciency of such generators.
field, origin "unknown" . Thus the potential, from this
Again, as noted, the absence of iron in the magnetic
source, for whirl instability does not exist.
circuit means that only simple air core inductances and
winding resistances must be taken into account to calcu
As mentioned earlier, the major loss mechanism in this
late these quantities.
type of machine is the losses in the windings. These
losses can be minimized by increasing the amount of
copper in the windings. There is a trade o between the
e ciency of the machine and the size of the winding.
2
1
There is also a potential loss due to eddy currents in the
P = V0
Watts/phase 12
R
2
2
conductors; since the field is always present, eddy cur
L
1 + Ro
+
L0
rents will be induced in the conductors due to the rela
R
R
tive motion between the windings and the field. This
L
L
latter loss is easily controlled to low levels, i.e. a few
If the rms output voltage of each of the 3 phase wind
watts, by the use of Litz wire.
ings is V0, the winding inductance is Lo, and its resis
tance is Ro, then the output power of each phase into a
resistive load with resistance RL is given by the equa
tion:
The e ciency is then given by the expression:
=
RL
13
R + R
0
L
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
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Motor Types
Halbach array electric machines could be constructed in
a conventional manner, whereby the Halbach array is
stationary and the windings rotate within the array. As
we have mentioned for use in our modular EMBs, the
Halbach array machine has been constructed "inside
out", with the windings stationary and the magnet array
rotating around the windings. There are advantages to
both types of construction; specific applications drive
the choice.
In a conventional machine configuration, the Halbach
array machine would look physically much like a dc ma
chine; that is, the armature is on the rotor and the field
is stationary. Such a machine could be operated as either
a dc or ac device. If one uses slip rings to bring out the
windings, the machine could be characterized as an ac
machine. However, if now an electronic commutation
circuit is added, the Halbach array machine can be char
acterized as a dc machine. If one desired, one could use
a mechanical commutator as well.
The inherent advantage of this construction technique
is that the inertia of the rotor is dominated solely by the
mass of the windings since there are no laminations re
quired. One can therefor envision the construction of a
very high speed low inertia machine.
If the machine is configured, as we have done, in an
"inside out" geometry, the windings are stationary and
readily accessible. Again, the machine can be character
ized as either an ac or dc machine. The primary advan
tage of the inside out construction is that it is readily
Figure 5 -- Electromechanical Battery
adapted to evacuated systems such as in our EMB since
there is no need for slip rings.
As can be seen in Figure 5, the Halbach array machine is
an integral part of the flywheel construction. The array
Our specific application, the EMB
of magnets is designed into the rotor; the mass of the
As noted earlier, the impetus for the development of
magnets is used advantageously to keep the composite
this new type of electric machine was our application to
material in compression. The advantages listed above are
also utilized in this design. The outer diameter of the
modular electromechanical batteries.5,6,7 The EMB bat
stator windings are typically a centimeter smaller than
tery consists of a high speed flywheel with an integral
the inner diameter of the Halbach array magnets. In this
motor / generator suspended on magnetic bearings and
space, a thin vacuum barrier is placed, still allowing for a
in an evacuated housing. For practical use, a set of power
substantial clearance between the array and the barrier.
electronics is coupled to this module. The flywheel and
This clearance simplifies the design of the bearing/ sus
its motor/generator is a means for energy storage and
pension system, which need not constrain the radial dis
extraction; the power electronics conditions electrical
placements of the rotor assembly to the fraction of a
energy, both for adding energy to and extracting energy
millimeter tolerances that would be required in a con
from the flywheel. Since the goal is to mimic a battery,
ventional iron core machine.
the input/output voltage to the power electronics is
typically dc, although ac based systems can also be con
We conclude this section with a summary of the design
templated, using so called cyclo converters, or the
parameters of one of the modular EMBs that we have
multi phase devices called matrix converters. A sketch
constructed.
of the electromechanical battery is given in Figure 5.
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
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Table 1 --
Table 2 -- Generator Efficiency
Electromechanical Battery Parameters
Versus Output Power
Useful energy
0.6 KW hr
RL (ohms)
Power
Efficiency
100 to 50 speed
(kW)
Maximum speed
84,000 rpm
3.0
23
99.64
Peak power
50 kW
2.5
27
99.57
Open circuit voltage
151 Vrms/phase,
max speed
2.0
34
99.46
Halbach array length
18 cm
1.5
45
99.29
Halbach array outer diameter
10.5 cm
1.0
66
98.93
Halbach array inner diameter
7 cm
Drive Circuitry
Windings turns/phase
6
Although the Halbach array machine can be operated as
Conductor size
1700 strand,
either an ac or dc machine, there are advantages to op
#40 Litz
erate it as a dc machine. The primary advantage is that
the drive circuitry can use a simple 1200 gating wave
Inductance per phase
7.4 microhenries
forms. Normally, in an ac machine, elaborate PWM
techniques are used to generate sinusoidal waveforms in
Resistance per phase
10.8 milliohms
order to minimize harmonics that can cause losses in the
laminations. Since the Halbach array machine does not
have laminations this is no longer of concern. The re
Using these data and equations 12 and 13 we can cal
sulting drive can then be greatly simplified, since in this
culate the power output and the e ciency of this gen
case a simple rectangular wave is very nearly as e cient
erator. We assume that the power from each of the three
in driving the rotor as would be a pure sine wave.
phases is summed independently in calculating the out
put power, and we ignore the eddy current losses in the
Litz wire it is of order one or two Watts . In the table
below the power and e ciency. are calculated as a func
tion of the load resistance the same for each of the
three phases and at full speed 84,000 RPM . As will be
seen at or below the design power the e ciencies are
very high.
Buck Regulator
Electronic Commutator
Halbach Motor
Figure 6 -- Simplified Schematic of Spin-up Circuit
In our specific application, energy is transferred into and
out of the flywheel on a routine basis. In order to add
energy to the flywheel, the power electronics treat the
Halbach array machine as an electronically commutated
dc motor, as the voltage is raised across the motor, the
speed of the motor increases. A schematic of the power
electronics for spin up is given in Figure 6. The elec
tronic commutation is provided by six IGBTs connected
to the three phases of the machine. The dc voltage to
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
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the commutating bridge is controlled by the "buck"
of circuit that creates a true electromechanical battery;
regulator formed by one IGBT, an inductor, and diode.
the flywheel power electronics system mimics a dc bat
The IGBTs in the commutation bridge also serve as an
tery. For those applications where energy compression is
over current protection scheme. The lower IGBT in
important, that is, the energy is added/extracted on a
each leg can be pulse width modulated to control the
much longer time scale than it is extracted/added, there
current in the windings. The switching frequency for
is a cost advantage to separate the spin up and extrac
this modulation is about 20 kHz, while the electronic
tion functions in order to minimize the cost of the
commutation needs to accommodate frequencies up to
power semiconductors.
1400 Hz to achieve 84,000 RPM operation. This pro
tection is not for the windings but for the power semi
Performance
conductors.
Although the flywheel system has not yet been operated
at full speed, the fundamental characteristics of the
Halbach array electric machine have been verified.
AC Operation
Figure 9a is a typical open circuit voltage waveform; the
purity of the sinusoid is given in Figure 9b which shows
that the highest amplitude harmonic is less than a factor
of 300 than the fundamental i.e., more than 50 decibels
lower .
Boost Regulator
Diode Bridge
Halbach Motor
0.447
0.357
Figure 7 -- Simplified Schematic of
0.268
0.178
Energy Extraction Circuit
0.089
A schematic for extraction of power from the flywheel is
0
-.0.089
given in Figure 7. The three phases of the windings are
-0.178
connected to a six pulse diode bridge to convert the ac
-0.268
-0.357
waveforms to dc. Since the speed changes a factor of
-0.447
two while energy is being extracted, the dc voltage will
also decrease by a factor of two. To accommodate this
Figure 9a -- Open Circuit Voltage
change in dc bus voltage, a "boost" regulator is added.
3.16 x 10-1
The boost circuit provides for a constant output voltage
1.00 x 10-1
by changing the IGBT duty cycle.
3.16 x 10-2
1.00 x 10-2
3.16 x 10-3
1.00 x 10-3
3.16 x 10-4
1.00 x 10-4
3.16 x 10-50.0 Hz
100 Hz
Figure 9b -- Open Circuit Voltage Harmonics
Extraction of High Power
Using the flywheel with an integral Halbach array ma
Buck/Boost Regulator Electronic Commutator
Halbach Motor
chine, over 50 kW of power into a load that was not
impedance matched for maximum power in about one
Figure 8 -- Simplified Schematic of
second bursts have been successfully extracted. During
Electromechanical Battery
this high peak power, the machine did not exhibit any
Since IGBTs are normally packaged with an antiparallel
indications of whirl instability. To perform this extrac
diode, the spin up and extraction circuit can be com
tion, the windings were connected to a six pulse diode
bined to one as shown in Figure 8. This circuit works
bridge that was then switched into a resistive load. Fig
best when the energy is added to the flywheel on the
ure 10 shows typical bridge voltage and current during
same time scale as it is extracted. Figure 8 is an example
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
7

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extraction. The peak power that has been extracted to
References
date was 65 kW for about one second.
1 Klaus Halbach; "Design of Permanent Multipole Mag
nets with Oriented Rare Earth Cobalt Material", Nu
clear Instruments and Methods, 169, pp. 1 10, 1980 .
Voltage, 50V/div
2 Klaus Halbach; "Physical and Optical Properties of
Rare Earth Cobalt Magnets," Nuclear Instruments and
Methods, 187, pp. 109 117, 1981 .
3 Klaus Halbach; "Perturbation E ects in Segmented
Rare Earth Cobalt Multipole Magnets," Nuclear In
struments and Methods, 198, pp. 213 215, 1982 .
4 Klaus Halbach; Specialty Magnets, Lawrence Berkeley
Laboratory, LBL 21945, 1986 . AIP 1985 Conference
Current, 98A/div
Proceedings, U.S. Summer School on High Energy Parti
cle Accelerators.
5 Robert F. Post, T. Kenneth Fowler, and Stephen F.
Figure 10 -- Bridge Current and Voltage During Extraction
Post; "A High E ciency Electromechanical Battery,"
Lawrence Livermore National Laboratory, UCRL JC
DC Operation
110861, 1992 .
The flywheel with an integral Halbach array machine
6 Robert F. Post, D. E. Baldwin, Donald A. Bender and
has been routinely operated at 40,000 RPM. During the
T. Kenneth Fowler; Electromechanical Battery Research
spin up, a single phase version of the power electronics
and Development ant the Lawrence Livermore National
as shown in Figure 6 was used. Figure 11 is a plot of
Laboratory, Lawrence Livermore National Laboratory,
speed versus dc bus voltage. The dc bus voltage is always
UCRL JC 113905, 1993 .
slightly higher than the emf to ensure power flow into
7
the motor during spin up. This operation is typical of a
Robert F. Post, Donald A. Bender and Bernard T. Mer
dc motor with constant excitation.
ritt; "Electromechanical Battery Program at the Law
rence Livermore National Laboratory," IECEC, Mon
250
terey, CA, August 1994.
200
150
DC Bus
ltage
o
V 100
Peak EMF
50
00
200
400
600
800
1000
1200
1400
Speed (Hz)
Figure 11 -- Flywheel Speed versus DC Bus Voltage Graph
Publishing History
This work was performed under the auspices of the U.S.
Department of Energy by Lawrence Livermore National
Laboratory under contract No. W 7405 ENG 48.
Reformatted and color illustrations added by Mark
Duncan on July 2009.
Halbach Festschrift Symposium, Berkeley, CA; February 3, 1995
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