Note : Les descriptions sont présentées dans la langue officielle dans laquelle elles ont été soumises.
CA 02769084 2012-02-27
r
METHOD AND APPARATUS FOR HIGH VELOCITY RIPPLE SU RESSION OF BRUSHLESS
i
DC MOTORS HAVING LIMITED DRIVE/AMPLIFIER BANDW,,rI H
BACKGROUND OF THE INVENTION
I. Field of the Invention
[0001] The present invention is directed to brushless DC motors, and more
particularly to a
method and apparatus for ripple suppression in brushless DC motors having
limited
drive/amplifier bandwidth when operated at high velocity.
2. Description of the Related Art
[0002] Brushless DC (BLDC) motors, or electronically commutated motors (ECMs,
EC motors)
are electric motors that are powered by direct-current (DC) electricity via
electronic commutation
systems. BLDC motors exhibit linear current-to-torque and frequency-to-speed
relationships and
are commonly used as servo drives for precision motion control in numerous
applications
ranging from silicon wafer manufacturing, medical, robotics and automation
industries to military
applications.
[0003] BLDC motors comprise a rotor having a plurality of permanent magnets, a
stator with
electromagnetic coil armature windings, and a commutator for continually
switching the phase of
the current in the armature windings to induce motion in the rotor. More
particularly, an
electronic controller causes the commutator to apply excitation current to the
armature windings
in a specific order in order to rotate the magnetic field generated by the
windings thereby
causing the rotor magnets to be pulled into alignment with the moving magnetic
fields and
thereby drive rotation of the rotor.
[0004] Prior art relating generally to the construction of BLDG motors and
controllers
therefor, includes: US 6,144,132; US 7,715,698; US 7,852,025; US 7,906,930; US
2010/0109458; US 2010/01487101; US 2010/0176756; US 2010/0181947; US
2010/0270957;
US 2010/0314962; US 2011/0031916; US 201110074325; US 201110133679. The
following
patent documents are directed to methods of controlling speed (velocity) of
BLDG motors: US
4,720,663; US 4,855,652; US 5,563,980; US 5,757,152; US 6,049,187; US
6,313,601; US
6,822,419; US 6,828,748; US 6,922,038; US 7,375,488; US 7,994,744; US
2005/0035732; US
2009/0128078; US 2010/0001670; US 2010/0134055; US 2010/0171453.
[0005] In order to control the speed of rotor rotation, the electronic
controller requires
information relating to the rotor's orientation/position (relative to the
armature windings). The
following patent documents are directed to methods of commutating and
controlling BLDG
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CA 02769084 2012-02-27
motors relying on feedback of the sensed rotor position: US 2010/0117572; US
2010/0134059;
US 2010/0237818; US 2010/0264862; US 2010/0270960; US 2011/00.43146; US
2011/0176229.
MOW Some electronic controllers use Hall Effect sensors or rotary encoders
to directly
measure the rotor position. Other controllers measure the back EMF in the
windings to infer the
rotor position. The following patent documents disclose methods of commutating
and controlling
BLDG motors relying on feedback of sensed back electromotive force (back EMF):
US
5,057,753; US 5,231,338; US 2011/0074327; US 2011/0115423; US 2011/0121770; US
2011/0156622; US 2011/0210688. Additional background patent literature
relevant to
commutating BLDG motors using feedback includes: US 7,893,638; US 7,969,108;
US
2010/0052584; US 2010/0090633; US 2010/0237813; US 2010/0237814; US
2010/0315027;
US 201110006712; US 201110025237; US 201110043144; US 201110084639; US
201110148336; US 201110202941; US 201110205662.
[0007] Typically, the excitation current generated by the commutator has
insufficient power to
be fed directly to the coils, and must be amplified to an appropriate power
level by a
driver/amplifier. The conventional driver/amplifier of a BLOC motor produces
sinusoidal
armature current waveforms to the armature windings for smooth motor
operation. However, in
practice, the actual magneto-motive form generated by a non-ideal motor is not
perfectly
sinusoidally distributed, and can therefore result in 'torque ripple'. It is
known that suppressing
torque ripple in the motor drive of a servo system can significantly improve
system performance
by reducing speed fluctuations (see Park, S. J., Park, H. W., Lee, M. H. and
Harashima, F.: A
new approach for minimum-torque-ripple maximum-efficiency control of BLDG
motor, IEEE
Trans. on Industrial Electronics 47(1), 109-114 (2000); and Aghili, F.,
Buehler, M. and
Hollerbach, J. M.: Experimental characterization and quadratic programming-
based control of
brushless-motors, IEEE Trans. on Control Systems Technology 11(1),139-146
(2003)
[hereinafterAgnili et al., 2003].
[0008] One known solution to reducing torque ripple in commercial high-
performance electric
motors is to increase the number of motor poles. However, such motors tend to
be expensive
and bulky due to the construction and assembly of multiple coil windings.
10009] Other control approaches for accurate torque control in electric
motors, and their
underlying models, are set forth in the patent literature. For example, the
following patent
documents are directed to various methods and systems for stabilizing or
reducing torque ripple
in synchronous electric motors, particularly BLDC motors: US 4,511,827; US
4,525,657; US
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4,546,294; US 4,658,190; US 4,912,379; ; US 5,191,269; US 5,569,989; US
5,625,264; US
5,672,944; US 6,437,526; US 6,737,771; US 6,859,001; US 7,166,984; US
7,859,209; US
7,952,308; US 201110175556.
[0010] Additional prior art approaches for providing accurate torque
production in electric
motors are set forth in the non-patent literature.
[0011] For example, Le-Huy, H., Perret, R. and Feuillet, R.:Minimization of
torque ripple in
brushless dc motor drives, IEEE Trans. Industry Applications 22(4), 748-755
(1986), and Favre,
E., Cardoletti, L. and Jufer, M.: 1993, Permanent-magnet synchronous motors: A
comprehensive approach to cogging torque suppression, IEEE Trans. Industry
Applications
29(6), 1141-1149, describe a method of reducing the torque-ripple harmonics
for brushless
motors by using several current waveforms.
[0012] Ha and Kang: Explicit characterization of all feedback linearizing
controllers for a
general type of brushless dc motor, IEEE Trans. Automatic Control 39(3), 673-
6771994 (1994)
characterize, in an explicit form, the class of feedback controllers that
produce ripple-free torque
in brushless motors.
[0013] Newman, W. S. and Patel, J. J.: Experiments in torque control af the
AdeptOne
robot, Sacramento, California, pp. 1867-1872 (1991) discuss the use of a 2-D
lookup table and
a multivariate function to determine the phase currents of a variable-
reluctance motor with
respect to position and torque set points.
[0014] Optimal torque control schemes for reducing torque ripples and
minimizing copper
losses in BLOC motors have been proposed (see Hung, Y and Ding, Z.: Design of
currents to
reduce torque ripple In brushless permanent magnet motors, IEEE Proc. Pt. B
140(4) (1993)
[hereinafter Hung-Ding 19931; Aghili, F., Buehler, M. and Hollerbach, J. M.:
Optimal commutation
laws in the frequency domain for PM synchronous direct-drive motors, IEEE
Transactions on
Power Electronics 15(6), 1056-1064(2000) [hereinafter Aghili et al., 20001;
Park, S. J., Park, H.
W., Lee, M. H. and Harashima, F.: A new approach for minimum-torque-ripple
maximum-
efficiency control of BLDG motor, IEEE Trans. on Industrial Electronics 47(1),
109-114 (2000);
as well as Aghili et al., 2003, above..
[0015] Wang, J., Liu, H., Zhu, Y., Cui, B. and Duan, H.: 2006, A new
minimum torque-ripple
and sensoriess control scheme of bldc motors based on rbf networks, IEEE Int.
Conf. on Power
Electronics and Motion Control, Shanghai, China, pp. 1-4 (2006), proposes a
method for
minimizing the torque ripples generated by non-ideal current waveforms in a
BLDG motor
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CA 02769084 2012-02-27
having no position sensors, by adjusting actual phase currents.
[0016] Similarly, the electrical rotor position can be estimated using
winding inductance, and
the stationary reference frame stator flux linkages and currents can be used
for a sensorless
torque control method using d-axis current, as set forth in Ozturk, S. and
Toliyat, H. A.:
Sensoriess direct torque and indirect flux control of brush/ass do motor with
non-sinusoidal
back-EMF, IEEE Annual Conf. on Industrial Electronics IECON, Orlando, FL, pp.
1373-1378
(2008).
[0017] Lu, H., Zhang, L. and Qu, W., A new torque control method for torque
ripple
minimization of BLDC motors with un-ideal back EMF, IEEE Trans. on Power
Electronics 23(2),
950-958 (2008), sets forth a torque control method to attenuate torque ripple
of BLDG motors
with non-ideal back electromotive force (EMF) waveforms, wherein the influence
of finite dc bus
supply voltage is considered in the commutation period.
[0018] A low cost BLDG drive system is set forth in Feipeng, X., Tiecai, L.
and Pinghua, T.: A
low cost drive strategy for BLDG motor with low torque ripples, IEEE Int.
Cont. on Industrial
Electronics and Applications, Singapore, pp. 2499-2502 (2008), wherein only a
current sensor
and proportional-integral-derivative controller (PID controller) are used to
minimize the
pulsating torque.
[0019] In the prior art set forth above, it is assumed that the phase
currents can be
controlled accurately and instantaneously and that they may therefore be
treated as control
inputs, such that the waveforms of the motor phase currents may be adequately
pre-shaped so
that the generated torque is equal to the requested torque. However, at high
rotor velocity the
commutator generates high frequency control signals that the finite bandwidth
motor dynamics
of the driver/amplifier may not be able to respond sufficiently quickly. Thus,
complete
compensation for the position nonlinearity of the motor torque cannot be
achieved in the
presence of amplifier dynamics, with the result that pulsation torque
therefore appears at high
motor velocities. As discussed above, torque ripple can significantly
deteriorate the performance
of the servo control system and even lead to instability if the ripple
frequency is close to the
modal frequency of the closed-loop system.
[0020] In Aghili et al., 2000, above, an optimal commutation scheme is set
forth based on
Fourier coefficients in BLDC motors. This was followed by Aghili, F.: Adaptive
reshaping of
excitation currents for accurate torque control of brush/ass motors, IEEE
Trans. on Control
System Technologies 16(2), 356-364 (2008), which developed a self-tuning
adaptive version of
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CA 02769084 2012-02-27
the commutation law that estimates the Fourier coefficients of the waveform
associated with the
motor's electromotive force based on measurements of motor phase voltage and
angle.
[0021] Other prior art has addressed the application of Fourier analysis to
BDLC motor
control. For example, US 6,380,658 discloses a method and apparatus for torque
ripple
reduction in a sinusoidally excited brushless permanent magnet motor for
automotive
applications (for electric power steering, as an alternative to hydraulic
power steering). In
essence, the physical components of the BLDC motor are designed to a
predetermined
geometry for reducing torque ripple when the motor is sinusoidally excited
(i.e. driven by a
sinusoidal current supplied to its armature coils). More specifically, the
method of the '658 patent
applies an elementary Fourier analysis to determine one specific dimension of
the rotor of a
given shape which minimizes the fifth harmonic component of the magnetic flux
in the air gap
between the stator and the rotor, when the motor is driven by a sinusoidal
current. The fifth
harmonic is identified as the lowest harmonic having adverse influence on
torque ripple and
therefore the one that should be eliminated to the extent possible.
[0022] US 7,629,764 discloses a method of controlling a high-speed servomotor,
such as a
BLDC motor, operating under control of a PWM (pulse-width modulation)
controller, to attain
optimal performance and stability margins across an operational range
encompassing the entire
torque versus speed curve of the motor. According to the '764 patent, the
torque versus speed
curve for the motor is divided into operating regions, and control parameters
are calculated for
= each region. Fourier analysis (transforms) of various feedback signals
received from the
operating motor (e.g., the electric current flowing through the motor and its
actual speed) is
carried out in real time to produce fundamental and harmonic components of the
feedback
signals, which components are then used to produce an output voltage command
for controlling
the motor.
[0023] Additional patent literature is directed to methods and systems for
controlling and/or
operating electric motors using Fourier transforms of signals, including: FR
2,825,203; JP
2001238,484; JP 2007143,237; US 4,7440,41; US 5,280,222; US 5,455,498; US
5,844,388.
The foregoing patent references disclose various methods for controlling or
commutating
electric motors, in particular BLDC motors, using the results of Fourier
transforms of various
periodic signals, feedback or otherwise.
[0024] Additional non-patent literature relevant to this disclosure
includes:
= Murai, Y, Kawase, Y, Ohashi, K., Nagatake, K. and Okuyama, K.: 1989,
Torque ripple
CA 02769084 2012-02-27
improvement for brushless dc miniature motors, Industry Applications, IEEE
Transactions on 25(3),441-450;
= Deleduse, C. and Grenier, D.: 1998, A measurement method of the exact
variations
of the self and mutual inductances of a buried permanent magnet synchronous
motor
and its application to the reduction of torque ripples, 5th International
Workshop on
Advanced Motion Control, Coimbra, pp. 191-197;
= Wallace, R. S. and Taylor, D. G.: 1991, Low-torque-ripple switched
reluctance motors
for direct-drive robotics, IEEE Trans. Robotics &Automation 7(6), 733-742;
= Filicori, E, Blanco, C. G. 1. and Tonielli, A.: 1993, Modeling and
control strategies for
a variable reluctance direct-drive motor, IEEE Trans. Industrial Electronics
40(1),
105115;
= Matsui, N., Makin , T. and Satoh, H.: 1993, Autocompensation of torque
ripple of
direct drive motor by torque observer, IEEE Trans. on Industry Application
29(1), 187-
194;
= Taylor, D. G: 1994, Nonlinear control of electric machines: An overview,
IEEE
Control Systems Magazine 14(6), 41-51;
= Kang, J.-K. and Sui, S.-K: 1999, New direct torque contra of induction
motor for
minimum torque ripple and constant switching frequency, IEEE Trans. on
Industry
Applications 35(5)1076-1082.;
= French, G. and Acamley, P.: 1996, Direct torque control of permanent
magnet drives,
IEEE Trans. on Industry Applications 32(5), 1080-1088;
= Kang, J.-K. and Sul, S.-K.: 1999, New direct torque control of induction
motor for
minimum torque ripple and constant switching frequency, IEEE Trans. on
Industry
Applications 35(5)1076-1082;
= Xu, Z. and Rahman, M. F: 2004, A variable structure torque and flux
controller for a
DTC IPM synchronous motor drive, IEEE 35th Annual Power Electronics
Specialists
Conference, PESC04., pp. 445-450, Vol. 1).
SUMMARY OF THE INVENTION
[0025] According to the invention a method and apparatus are provided for
ripple suppression
of brushless DC motors at any given velocity irrespective of the limited
bandwidth of the
6
driver/amplifier supplying the excitation currents to the armature. In a
preferred
embodiment, Fourier coefficients of the current waveform are calculated as a
function
of rotor velocity by taking into account the driver/amplifier's finite
bandwidth dynamics
in order to eliminate pulsation torque. Unlike Aghili et al., 2000, above, the
inventive
commutation scheme updates the Fourier coefficients of the current waveform
based
on the desired velocity so that the torque ripple and velocity fluctuation are
eliminated
for a given velocity while power losses are simultaneously minimized. More
particularly, since the control signal is a periodic function (i.e. a
waveform), it can be
approximated by a truncated (finite) Fourier series. For a given velocity,
Fourier
coefficients of the series approximating the waveform (control signal) are
calculated
as a function of the rotor velocity and the amplifier dynamics, to generate a
waveform
that results in no torque or velocity pulsations. When changing the motor
speed, the
coefficients are updated (recalculated) based on the new desired velocity (and
amplifier dynamics), resulting in generation of an updated waveform that
results in no
torque or velocity pulsations at the new motor speed.
[0025a] According to the present invention, there is provided a method of
driving a
load via a brushless DC motor, comprising:
receiving a desired motor velocity wd,;
receiving actual velocity w and rotor position angle e from said load;
generating armature phase drive currents i*k, for a desired torque Td from the
desired motor velocity wd, actual velocity w and rotor position angle 0 using
a
modified commutation law;
amplifying the armature phase drive currents via a driver/amplifier; and
applying the amplified armature phase drive currents ik to an armature of said
brushless DC motor for inducing rotation of said rotor and thereby rotating
said load;
wherein the modified commutation law comprises calculating compensated Fourier
coefficients c' of a truncated (finite) Fourier series approximating the
armature
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phase drive currents for said desired motor velocity wd as a function of the
actual
velocity w and transfer function of said driver/amplifier, wherein said
modified
commutation law further comprises:
a) computing fixed Fourier coefficients c of said Fourier series at zero
velocity;
b) computing said compensated Fourier coefficients c' at said desired motor
velocity wd;
c) computing the armature phase drive currents i*k by modulating the desired
torque Td with said corresponding commutation functions uk(e) according to
i*kd'0)=1-d Uk(C)' Vk=1, . , p; and
u(0) = Elrit=-N Cnejncle;
and
d) returning to b).
[0025b] According to another aspect of the invention, there is provided a
controller
for controlling rotation of a load by a brushless DC motor having a stator and
a rotor,
comprising:
a motion controller for receiving a desired motor velocity wd,, actual
velocity
w and rotor position angle U from said load;
a commutator for generating armature phase drive currents i*k, for a desired
torque Td from the desired motor velocity wd, actual velocity w and rotor
position
angle 0;
7a
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a driver/amplifier for amplifying the armature phase drive currents and
applying the amplified armature phase drive currents ik to an armature of said
stator
for inducing rotation of said rotor and thereby rotating said load;
wherein said motion controller and commutator generate said armature
phase drive currents i*k according to a modified commutation law comprising
calculating compensated Fourier coefficients c' of a truncated (finite)
Fourier series
approximating the armature phase drive currents for said desired motor
velocity wd
as a function of the actual velocity w and transfer function of said
driver/amplifier,
wherein said modified commutation law further comprises:
a) computing fixed Fourier coefficients c of said Fourier series at zero
velocity;
b) computing said compensated Fourier coefficients c' at said desired motor
velocity wd,;
c) computing the armature phase drive currents i*k by modulating the desired
torque Td with said corresponding commutation functions uk(e) according to
i*hd
k(Td'0)=Td Uk(0)' Vk=1, . . . , p; and
u(0) = cneinge;
and
d) returning to b).
[0025c] According to yet another aspect of the invention, there is provided a
brushless DC motor, comprising:
a rotor for rotating a mechanical load;
a stator having armature coils;
7b
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controller for receiving a desired motor velocity wd, actual velocity w and
rotor
position angle e from said load;
a commutator for generating armature phase drive currents i*k, for a desired
torque Td from the desired motor velocity wd, actual velocity w and rotor
position
angle 8;
a driver/amplifier for amplifying the armature phase drive currents and
applying the amplified armature phase drive currents ik to the armature coils
of said
stator for inducing rotation of said rotor and thereby rotating said load;
wherein said controller and commutator generate said armature phase drive
currents i*k according to a modified commutation law comprising calculating
compensated Fourier coefficients c' of a truncated (finite) Fourier series
approximating the armature phase drive currents for said desired motor
velocity wci
as a function of the actual velocity w and transfer function of said
driver/amplifier,
wherein said modified commutation law further comprises:
a) computing fixed Fourier coefficients c of said Fourier series at zero
velocity;
b) computing said compensated Fourier coefficients c' at said desired motor
velocity wd.
C) computing the armature phase drive currents i*k by modulating the desired
torque Td with said corresponding commutation functions uk(8) according to
1*k(1d'8)=1k(0)' Vk=1, . , p; and
u(e)= nn=-N Cneincle ;
and
d) returning to b).
7c
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[0026] The above aspects can be attained by a method and apparatus of driving
a
load via a brushless DC motor, the method comprising: receiving a desired
motor
velocity wd,; receiving actual velocity w and rotor position angle 0 from said
load;
generating armature phase drive currents i*k, for a desired torque Td from the
desired
motor velocity cud,, actual velocity w and rotor position angle 0 using a
modified
commutation law; amplifying the armature phase drive currents via a
driver/amplifier;
and applying the amplified armature phase drive currents i*k to an armature of
said
brushless DC motor for rotating said load; wherein the modified commutation
law
comprises calculating compensated Fourier coefficients c' of a truncated
(finite)
Fourier series approximating the armature phase drive currents for said
desired motor
velocity wd; as a function of the actual velocity w and transfer function of
said
driver/amplifier.
[0027] These together with other aspects and advantages which will be
subsequently
apparent, reside in the details of construction and operation as more fully
hereinafter
described and claimed, reference being had to the accompanying drawings
forming
a part hereof, wherein like numerals refer to like parts throughout.
BRIEF DESCRIPTION OF THE DRAWINGS
[0028] Figure 1 is a block diagram of a brushless DC motor according to a
preferred
embodiment of the invention.
[0029] Figure 2 is a graph showing the motor phase waveform for the brushless
DC
motor in
7d
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CA 02769084 2012-02-27
Figure 1.
[0030] Figure 3 is a flowchart showing a method for ripple suppression in
brushless DC motors
having limited drive/amplifier bandwidth when operated at high velocity,
according to a preferred
embodiment.
[0031] Figure 4 is graph showing waveforms of the commutation function
corresponding to
different rotor velocities for the brushless DC motor in Figure 1.
[0032] Figures 5A and 5B are bode plots of the driver/amplifier and the torque
transfer function
for the brushless DC motor in Figure 1.
[0033] Figures 6A and 6B are step response curves for the closed-loop PI
(proportional-
integral) velocity controller of Figure 1.
[0034] Figures 7A and 7B are step response curves of a velocity controller for
a conventional
commutation system.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0035] Turning to Figure 1, a BLDC motor 100 is illustrated, comprising a
rotor 110 having a
plurality of permanent magnets (not shown) for driving a mechanical load 160,
a stator 120 with
electromagnetic coil armature windings, and a controller for controlling
rotation of the rotor 110,
including a commutator 130 for generating drive current (ri,i r3) which is
then amplified via
driver/amplifier 140 (4, 12, /3) and applied to the stator 120 in a specific
order, and a motion
controller 150 for pre-shaping the current waveform on the basis of an input
signal representing
a desired motor velocity wd. The actual velocity w and rotor position angle 0
are fed back from
the mechanical load 160 driven by the rotor 110.
[00361 In order to better understand the scientific principles behind the
invention, the theory
relating to modeling and control of motor torque is discussed below in terms
of Fourier series,
followed by a discussion of how the commutation law is modified at high
velocity for ripple
compensation and derivation of the torque transfer function, taking into
account the dynamics of
the driver/amplifier 140.
[0037] Consider the BLDC motor 100 with p phases (in Figure 1, p = 3), and
assume that there
is negligible cross-coupling between the phase torques and no reluctance
torque. Then, the
torque developed by a single phase is a function of the phase current ik and
the rotor position
angle 19, as follows:
rk(i*,9)¨ik MO) k=1,---,p (1)
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[0038] Where y(0) is the position nonlinearity, or torque shape function,
associated with the kth
phase. The motor torque z is the superposition of all phase torque
contributions,
(2)
i=-==1
[0039] The torque control problem requires solving equation (2) in terms of
current, ik(6),Td ), as
a function of motor position, given a desired motor torque rd. For any scalar
torque set point,
equation (2) permits infinitely many (position dependent) phase current wave
forms. Since the
continuous mechanical power output of an electrical motor is limited primarily
by heat generated
from internal copper losses, the freedom in the phase current solutions may be
used to
minimize power losses,
PIONS C (3)
where i = col (ih... ip) is the vector of phase currents.
[0040] In rotary electric motors, the torque shape function is a periodic
function. Since
successive phase windings are shifted by 27./p, the following relationship
exists,
yk(e9)= y(q614- 2,r(k ¨1)), V k =1,¨ , p (4)
where q is the number of motor poles. The electronic commutator 130 commands
the phase
currents rk, through
rk(r d , 0) = Td k(0), Vk 1,...,p= (5)
where uk (9)) is the commutation shape function associated with the kth phase.
The individual
phase control signals can be expressed based on the periodic commutation
function, 40) which
is also a periodic function, i.e.,
u,(0) = u(q9+ 22z(k ¨1)),
[0041] Since both the commutation shape function u(6) and the torque shape
function y(0) are
periodic functions with position periodicity of 27z/q, they can be
approximated effectively via the
truncated complex Fourier series
9
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u(8)= (6)
n--N
30) d mejmq (7)
rn=-N
where j and N can be chosen arbitrary large, but 2Nlp must be an integer.
Since both
are real valued functions, their negative Fourier coefficients are the
conjugate of their real ones,
= and d= c7õ . Furthermore, since the magnetic force is a conservative
field for linear
magnetic systems, the average torque over a period must be zero, and thus cc,
= 0.
[0042] The motor model and its control can be described by the vectors c,d E
CM of the Fourier
coefficients of u(19) and y(0) , respectively, by
c = cogct , c2 , -= = , cN), (8)
d = , d2 ,= - = ,dy). (9)
[0043] In the following, coefficient c may be determined for a given torque
spectrum vector d
so that the motor torque r becomes ripple free, i.e. independent of the motor
angle 8. It may
be assumed that the driver/amplifier 140 delivers the demanded current
instantaneously, i.e.,
ik =1: for k =1,- -,p. In this case, after substituting equations (4)-(1) into
(2), we arrive at
j(n+mXg0+21r(4-1))
r = rcndme . (10)
n=-Ntn=-
nr-0 [two
[0044] This expression can be simplified by noting that the first summation
vanishes when
1= rn+n is not a multiple of p, i.e.,
2x(k-1)
p if 1= p, 2p, 3p,===
P = (11)
k=1 0 otherwise.
[0045] Defining p := pq , the torque expression (10) can be written in the
following compact
form
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N L(N4-mypi
r'rc1PE E d,acne-iqmoeicne (12)
In¨ r n-L(-N+mYpi
,no
[0046] The expression of the torque in (12) can be divided into two parts: The
position
dependent torque, rrip(0,rd), and the position independent torque, rim (rd).
That is
r1tm(rd )+ rrõ(6,1-0, (13)
in which
ri&,(rd)" park (14)
2Nlp
rnp(0,re)= re EArc, e-fPw , (15)
1-2Nlp
1*0
where k, are the Fourier coefficients of the motor torque, and can be
calculated by
ff 1 < ¨N
Ic,¨ (16)
P E endpi, otherwise.
n=p1-N
[0047] The term ko in (14) is the constant part of the circular convolution of
14(9) and y(0).
This, in turn, is equal to twice the real part of the inner product of the
vectors c and d,
ko = 2pRe <c,d > . (17)
[0048] A ripple-free torque implies that all coefficients lc, except ko are
zero and ko I so that
r . That is, the spectrum of the current excitation, c, must be calculated
so that
= if 1 n= 0
0 otherwise
[0049] This problem has infinitely many solutions. In this case, it is
possible to minimize the
power dissipation by noting that the average of dissipated power per unit
command torque over
one period, assuming constant speed, is
1
Pion Ili(0112dt
[0050] By changing the integral variable from time Ito 0, where d0= codt and
col' = 27rIq , we
11
CA 02769084 2012-02-27
have
a P fdq
Plost oc (0)th 9, (18)
2ir
[0051] where rd a 1. By virtue of Parseval's theorem, the power loss per unit
commanded
torque, i.e. ra =1, is
Pi. cc Pile12 (19)
[0052] A person of skill in the art will understand that minimizing power loss
is tantamount to
minimizing the Euclidean norm of the commutation spectrum vector 114
[0053] Where the spectrum of the excitation current c e CN represents the set
of unknown
variables then, according to (16), in order to minimize power loss, we must
solve
mm Icli2
(20)
subject to: Ac+ ¨ = = 0, (21)
c A eo1.(1 0 ¨ e It2N/P4.1 (2 Ar
¨+1)x N
where . .1" , and matrices A, B EC P
can be constructed
from the torque spectrum vector. For example, for a three phase motor (p = 3),
the A and B
matrices are given as
(12 413 j4
d2 d, 0 /it d2--=d4 dN3
d5 d4 d3 d2 d1 = = = 67N-7 ;IN-6
. . .
(22)
dNl d,_2 4_3 44 dõ_5 === d1 0
0 0 dN 4_1 d,_2 = = = d4 d3
= = = =
0 0 0 0 0 = = = 0 d,
12
CA 02769084 2012-02-27
= d2 d3 d4 d, ===
d, d, d6 d7 di, = = = 0
= dg d9 dii === 0
= = - ' = = =
d,_2 0 0 0 0 0 . (23)
0 0 0 0 0 = = = 0
. . . . . . .
' = " " = = = - = = = =
0 0 0 0 0 = = = 0
[0054] By separating real and imaginary parts, equation (21) can be rewritten
as
Re(A+ B) ¨Im(A¨B) Re(c)] [61
(24)
Im(A + B) Re(A¨ B) Irn(c) 0
, _
Q(d)
[0055] In general, for motors with more than two phases (i.e. p> 2 ), there
are fewer equations
than unknowns in (24). Therefore, a unique solution is not expected. The
pseudo-inverse offers
the minimum-norm solution, i.e. minimum 114 , which is consistent with the
minimum power
losses. Thus
c [' fINi(2+ [5], (25)
0
where V" represents the pseudo-inverse of matrix Q and IN is the Nx1V identity
matrix.
[0056] Having explained the theory relating to modeling and control of motor
torque in terms of
Fourier series, the following explains how the commutation law is modified at
high velocity,
according to the present invention, for ripple compensation arid derivation of
the torque transfer
function taking into account the dynamics of the driver/amplifier 140.
R*57] First, with respect to ripple compensation, since motor phase currents
are determined
based on sinusoidal functions of the motor angle, high motor velocities result
in a high drive
frequency that makes it difficult for the commutator 130 to track the
reference current input.
Therefore, design of ripple-free commutation at high velocities necessitates
taking the dynamics
of the driver/amplifier 140 into account.
[0058]
[0059] In practice, the range of motor velocities in which torque-ripple
compensation (without
velocity compensation) becomes problematic depends on three factors: I)
bandwidth (BW) of
13
CA 02769084 2012-02-27
the driver/amplifier (in rad/s); ii) the number of motor poles q; and iii) the
maximum harmonics
of the motor back-emf waveform n. Specifically, the motor velocity w (in rpm)
should be less
than (30 * BVV)/(pi *q * n). For example, for BW=100 red/a, q=2, and n=10,
velocity induced
torque ripple becomes problematic at 47.8 rpm and higherWith h(t) defined as
the impulse
response of the driver/amplifier 140, the actual and dictated phase currents
are no longer
identical, rather they are related by
4(t)¨ h(t¨ 4) d4- = 1,= = =,p. (26)
[0060] After substituting (5) and (1) into (26), the total motor torque can be
expressed as
k-1 k-1
P N m(ge(t)4-2x ¨) N
r(rd,9)=(E dme P ) X rd(4-)1c,,e P
k=1 n--N
N (n+N)lp
= E cc pi, elPle frri(4-)e-i'*-4.)h(t ¨ 4") dc)
1-(n-N)/p
n.t0
(27)
[0061] Equation (27) is obtained by using (11) and assuming a constant
velocity, i.e.,
= co(t¨C). The integral term in the right-hand-side of (27) can be written as
the
convolution integral, ra(t)*Cigwrah(t), where function e3gmr"12(t) can be
interpreted as the
impulse response of a virtual system associated with the nth harmonics. Then,
the
corresponding steady-state response to the step torque input response is given
by rdll(jgan2),
where H(s) is the Laplace transform of function of h(t), i.e., the amplifier's
transfer functions.
Now, define coefficients
cõ' = cp,H(jqna)) Vn=1,--=,N (28)
and the corresponding vector c' = co/(c. = =,c'h.) is related to vector c by
c' = D(a))c (29)
where
D(a))=diag(H (jqco), f (j2qa)),= = = ,H(jNqa))).
[0062] A person of skill in the art will understand that the angular velocity
variable a) in (28) and
(29) should not be confused with the frequency. Since H(¨jqnci.))= H(jqna)),
the new
coefficients satisfy
c=c Vn=1,===,N
[0063] Therefore, en are the compensated Fourier coefficients of a commutation
law that in the
presence of actuator dynamics yields the same steady-state torque profile as
the commutation
14
CA 02769084 2012-02-27
law set forth above. This means that all commutations that yield ripple-free
torques at constant
velocity a) must satisfy the constraint equation (21) with c being replaced by
c'. Therefore, the
Fourier coefficients of the commutation law that at rotor velocity co yields
ripple-free torque must
satisfy:
AD(w)c + BD(o))e - = 0,
or equivalently
Ac'+Bc'-0. (30)
[0064] Furthermore, as discussed below, power dissipation in the presence of
amplifier
dynamics is proportional to 102. Taking into account the relationship between
actual and
dictated phase currents set forth in (26), the average power dissipation is
1 ,v
Pioss cc E lim Ec h(t - Od02 dt
k-1 -Om
P = T rr (cnej" A le-ign"h(v)dv)2dt
= lim (cr eign")2 dt (31)
T.-z n
(MW) it follows from (18)- (19), above, that
/joss cc Plicir (32)
[0066] Therefore, in view of (30) and (32), one can conclude that the problem
of finding a
coefficient c' that minimizes power dissipation and yields ripple-free torque
at particular motor
velocity co can be similarly formulated as set forth in (20) if c is replace
by c'. Having
determined c', the spectrum of actual commutation, c, can be obtained from the
linear
relationship set forth in (29) through matrix inversion.
[0067] Next, in order to derive the torque transfer function in view of
amplifier dynamics, the
position independent part of the generated torque is
rlin(r P Eadn 1rd ()e-A-0-oh(t¨ (14"
n¨N
n*0
- g(t) * r (t) (33)
where * denotes the convolution integral and g(t) is the impulse function of
the system,
g(t) = 2p I a õ1 cos(q cont Za õ)h(t), (34)
n=1
with a,, = cndr, _
[0068] Transforming function (34) into the Laplace domain, the system torque
transfer function
CA 02769084 2012-02-27
= becomes
G(s) = pEaH(s + jqnco) + jqnco)),
(35)
n-1
where
G(s)- r11(8) .
d(s)
= [0069] Returning to the BLDC motor 100 of Figure 1, the Fourier
coefficients of the
commutation law may be calculated experimentally as a function of the desired
velocity,
according to the method set forth above. First, assume that the three-phase
motor has two
poles and drives mechanical load 160 (including the rotor 110) with inertia
0.05 kgm 2 and
viscous friction 4 Nm.strad, and that the load is driven by commutator 130
under control of a
Proportional Integral (P1) controller 150 within a velocity feedback loop,
where the controller
gains are set to
Kp =1.1 Nin.slrad and K, =18 Nmlrad,
so that the controller 150 achieves a well-damped behavior. A time varying
desired velocity 6)4i
is input to the controller 150, which changes from 2 rrliFs to 20 radls and
then to Wracks in
three 2-second intervals. The relatively low bandwidth driver/amplifier 140 is
characterized by
the the following transfer function
60
H(s)
s 60
[0070] The motor torque-angle profile (motor phase waveform) is shown in
Figure 2, and the
corresponding complex Fourier coefficients, are listed in Table 1, which shows
simulation results
for the commutator 130 operating according to the conventional commutation law
and according
to the modified commutation law of the present invention, in terms of harmonic
content of the
torque shape function and the commutation shape functions:
Table 1 (Fourier Coefficients)
Harmonics Torque Function Conventional Commutation Modified Commutation
1 -0.0e78 - 0.9977i -0.0105 -
0.1603i 0.0964 - 0.1673i
3 -0.0405 - 0.2107i -0.0051 -
0.0305i 0.0558 - 0.04071
-0.0063 - 0.0216i 0.0055 + 0.0098i -0.0272 + 0.0283i
7 0.0300 + 0.0315i -0.0026 -
0.0062i 0.0261 - 0.0162i
9 0.0365 + 0.0288i 0,0017 +
0.00181 -0.0093 + 0.0123i
11 0.0602 + 0.0173i -0.0025 +
0.0012i -0.0113 - 0.0174i
16
CA 02769084 2012-02-27
13 -0.0137 + 0.01041 0.0102 + 0.00181 -
0.0056 + 0.09021
_
15 0.0073 + 0.0027i 0.0030 + 0.00011 0.0021 + 0.02971
. 17 -0.0019 + 0.0017i -0.0031 - 0.00001 -
0.0027 - 0.0347i
-
19 -0.0039 + 0.00191 -0.0013 - 0.00031
0.0030 - 0.01691
21 -0.0005 - 0.0011i 0.0008 + 0.00111 -0.0142 + 0.0098i
23 -0.0017 - 0.00091 - 0.0004 - 0.00061
0.0099 + 0.00611
100711 Based on the detailed discussion and experimental results set forth
above, the method
according an aspect of the present invention for generating a ripple-free
desired torque pa to a .
load (i.e. combination of rota- 110 and mechanical load 160), is shown in the
flowchart of Figure
3, and comprises receiving a desired motor velocity wd (step 300), receiving
the actual velocity
wand rotor position angle 0 from the mechanical load 160 (step 310),
generating armature
phase drive currents rk from the desired motor velocity wd, actual velocity w
and rotor position
angle 0 using a modified commutation law (step 320), amplifying the armature
phase drive
currents via driver/amplifier 140 (step 340) and applying the resulting
currents ik to armature 120
(step 350) for rotating the rotor 110 and load 160, wherein the modified
commutation law (step
.320) comprises computing the fixed Fourier coefficients of the commutation
function at zero
velocity according to equation (25) c' = [IN "N1Q+ [ C (step 320a);
computing the
0 '
compensated Fourier coefficients at any desired velocity according to equation
(29)
e = D-1(m)ei (step 320b); computing the phase currents by modulating the
desired torque with
the corresponding commutation functions according to eqquations (5) and (6)
N
rdui(0), Vk=1,===,p and u(8)= E ,eingo, ( step 320c); and
returning
to step 320b.
[0072] The spectrums of (i) the conventional commutation scheme (i.e. without
taking the
frequency response of the amplifiers into account) based on the fixed Fourier
coefficients of
equation (25) ii) the modified commutation scheme based on velocity dependent
Fourier
coefficients (29), for cod =20 radls , are given in the third and fourth
columns of Table 1,
respectively.
[00731 The waveforms of the commutator corresponding to various rotor
velocities for one cycle
are illustrated in Figure 4.
10074] The bode plots of the torque transfer function obtained from (35) along
with the amplifier
17
CA 02769084 2012-02-27
transfer function are illustrated in Figure 5.
[0075] Trajectories of the motor torque and velocity obtained from the step-
input response of
the closed-loop system with the conventional and modified commutation
functions are illustrated
in Figures 6A, 6B and 7A, 7B, respectively.
[0076] It is apparent from the foregoing that the conventional commutation law
is able to
eliminate pulsation torque only at low velocity whereas the modified
commutation method
according to the present invention does so for any velocity.
[0077] In conclusion, an apparatus and method are set forth for electronically
controlled
commutation based on Fourier coefficients for BLDC motors operating at high
velocity while the
driver/amplifier bandwidth is limited. As discussed above, the excitation
currents are preshaped
based on not only rotor angle but also velocity in such a way that the motor
always generates
the requested (desired) torque while minimizing power losses. Unlike prior art
commutation
schemes, perfect ripple cancelation is effected at every velocity making the
commutation
method of the present invention particularly suitable for velocity servomotor
systems, (e.g.,
flywheel of a spacecraft). The simulations set forth above demonstrate that
the performance of a
velocity servomotor system is compromised by velocity if the conventional
prior art commutation
scheme is used whereas the commutation scheme according to the present
invention
significantly reduces torque ripple and velocity fluctuation.
[0078] The many features and advantages of the invention are apparent from the
detailed
specification and, thus, it is intended by the appended claims to cover all
such features and
advantages of the invention that fall within the true spirit and scope of the
invention. Further,
since numerous modifications and changes will readily occur to those skilled
in the art, it is not
desired to limit the invention to the exact construction and operation
illustrated and described,
and accordingly all suitable modifications and equivalents may be resorted to,
falling within the
scope of the invention.
18
