PULSATING TORQUE IN BRUSHLESS DC MOTORS
Ronald De Four
Department of Electrical & Computer Engineering, The University of the West Indies
St. Augustine, Trinidad
rdefour@eng.uwi.tt
Emily Ramoutar
University of Trinidad & Tobago, Point Lisas Campus, Trinidad
eramoutar@tstt.net.tt
Juliet Romeo
Department of Electrical & Computer Engineering, The University of the West Indies
St. Augustine, Trinidad
jnromeo@hotmail.com
Brian Copeland
Department of Electrical & Computer Engineering, The University of the West Indies
St. Augustine, Trinidad
bcopeland@eng.uwi.tt
Abstract
Brushless dc motors are increasingly being employed in many high performance applications
due to the simplicity of their control. However, in many of these applications, constant
ripple-free torque is an essential requirement. In an effort to broaden the range of
applications for these motors, pulsating torque minimization techniques must be employed.
This can only be performed effectively with the development of an electromagnetic torque
equation that incorporates the factors responsible for the production of the pulsating torque.
In this paper, vector analysis was applied to the brushless dc motor to develop the
© Copyright 2007 by Ronald De Four
All rights reserved. No part of this publication may be reproduced in any material form without
the written permission of the Copyright Owner, except in accordance with the provisions of the
Copyright Act 1997 (Act No. 8 of 1997) or under the terms of a License duly authorized and issued
by the Copyright Owner.
electromagnetic torque equation. It was observed that the electromagnetic torque varies with
the magnitude of the phase currents and the rotor position for motor operation between
commutations and during commutation. Hence, any attempt of constant torque production
must include both phase current and rotor position.
Keywords: Pulsating; torque; bldcm.
1. Introduction
Three-phase permanent magnet brushless dc motors (bldcm) are characterized by trapezoidal
phase back emfs having crest widths of 120 electrical degrees. These motors are increasingly
being employed in many high performance applications due to the simplicity of their control.
However, in many of these applications, constant ripple-free or non-pulsating torque is an
essential requirement.
Pulsating torque can be divided into two components, cogging torque and ripple torque. These
two pulsating torque components are defined as follows [1-2]:
(1) Cogging torque is the pulsating torque component produced by the variation of the
air-gap permeance or reluctance of the stator teeth and slots above the magnets as the
rotor rotates. No stator excitation is involved in cogging torque production.
(2) Ripple torque is the pulsating torque component generated by the interaction of stator
current magnetomotive force (MMF) and rotor MMF. This torque component can
take two forms:
(a) Alignment torque, which results from the interaction of the stator current MMF
with the rotor magnetic flux distribution. This is the dominant torque producing
mechanism in most permanent magnet motors.
(b) Reluctance torque, which results from the interaction of the stator current MMF
2
with the angular variation in the rotor magnetic reluctance.
This paper examines the production and nature of the pulsating alignment electromagnetic torque
developed by brushless dc motors, when ideal trapezoidal back emfs are being generated. The
pulsating alignment electromagnetic torque is computed between commutations, when two phase
windings are energized and during commutation of a phase winding, when three phase windings
are energized. The electromagnetic torque equation traditionally utilized for permanent magnet
brushless dc motors was derived by equating the electrical power absorbed by the motor to the
mechanical power produced. A different approach would be adopted in this paper. The Lorentz
force equation applied to charged particles forming a current would be utilized to develop the
electromagnetic torque developed by the permanent magnet brushless dc motor.
2. Literature Review
Many authors have addressed the issue of torque production and minimization of pulsating
torque in permanent magnet brushless dc motors [1-5]. Pulsating torque minimization techniques
during commutation were examined in [6-8]. The electromagnetic torque equation for modeling
and torque control of the permanent magnet brushless dc motor was obtained by equating the
electrical power absorbed by the motor to the mechanical power produced [1], [4-6] and [9-11],
which produces
Te =
1
(ea i a + eb ib + ec ic )
ω
(1)
where, T e is the electromagnetic torque developed, ω is the rotor speed, ea , eb and ec are the
phase back emfs and i a , i b and ic are the corresponding phase currents.
3
Hanselman in [4], Carlson et al. in [6], Pillay and Krishnan in [10], and Ying and Ertugrul in [11]
employed the electromagnetic torque Eq. (1) and stated that under trapezoidal back emf,
rectangular excitation currents are required for the production of constant electromagnetic
torque. Pillay and Krishnan in [10], and Ying and Ertugrul in [11] developed a, b, c, phase
variable models of the brushless dc motor and utilized hysteresis and pwm current controllers in
an attempt to produce constant current and hence constant torque motor operation. The steadystate simulation results produced by model of Ying and Ertugrul in [11] resulted in current and
torque pulsations when two phase windings were energized, with the current controller disabled.
However, the rotor speed-time plot was not presented. Al-Badi and Gastli in [9] employed Eq.
(1) in their Matlab and Pspice dynamic models of the brushless dc motor, and produced nonconstant current plots when two phase windings were energized under steady-state conditions,
but did not include electromagnetic torque and rotor speed plots.
The brushless dc motor models developed by Pillay and Krishnan in [10], and Carlson et al. in
[6] utilized Eq. (1) and the simulations produced significant torque pulsation during
commutation of a phase current.
It can be concluded from the literature that alignment electromagnetic torque pulsations are
produced in brushless dc motors between commutations, when two phase windings are energized
and during commutation, when three phase windings are energized. A knowledge of the sources
of these alignment electromagnetic torque pulsations would be useful to a pulsating torque
minimization exercise.
It was stated that rectangular excitation currents are required for the production of constant
electromagnetic torque. This implies that the electromagnetic torque developed by the motor is
only proportional to the phase current. When the relationship between the electrical power
absorbed and the mechanical power produced by the motor Eq. (1) is accompanied with the
4
above requirement for constant electromagnetic torque, it suggests that the back emf to rotor
speed ratios in Eq. (1) are all constants and equal. However, Berendsen et al. in [8] stated square
currents are required to control the torque in brushless dc motors and the amplitude of the
currents are proportional to the desired torque. They also added that two active stator phases are
energized so that the stator field is in advance of the rotor field, and the angle between them
varying between π/3 and 2π/3, with an average angle of π/2. In addition, Jahns in [3] reported
that the net shaft torque was contributed by the interaction of the permanent magnet flux density
(
r r
and the instantaneous phase currents due to the integration of the Lorentz force density J × B
)
along each of the individual coil sides. However, the net stator current vector was reported to be,
on average, in space quadrature with the rotor flux vector, thereby producing maximum average
torque/ampere.
These two latter contributions suggests that the alignment electromagnetic torque developed by
the motor varies with rotor position in addition to the phase currents, and the rectangular current
requirement for constant torque is necessary but not sufficient.
In an attempt to supply constant current to the brushless dc motor, current controllers were
utilized in [3], [6], [10] and [11]. However, instead of producing constant current, these current
controllers produced pulsating currents within a selected band, thereby restricting the pulsating
electromagnetic torque developed by the motor within a band.
The objective of this paper is to develop alignment electromagnetic torque equations for the
brushless dc motor between commutations and during commutation, which are not based on the
relationship between the electrical power absorbed and the mechanical power produced by the
motor.
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3. BLDCM Electromagnetic Torque Production
3.1. Energization sequence
Three-phase brushless dc motors are operated by energizing two of its three phase windings at a
time. The two windings being energized are dependent on the position of the rotor. Given the
energization sequence of the phase windings for anti-clockwise operation of the brushless dc
motor as ac, bc, ba, ca, cb, ab and ac again in that sequence, where, a, b and c in this winding
pair arrangement represents phase windings aa', bb' and cc' respectively. Each pair of phase
winding is energized for 60 electrical degrees, starting with rotor position at θ = 0°. This
energization sequence applied to one cycle of operation to a two-pole three-phase star connected
brushless dc motor is shown in Fig. 1. The ends of each phase winding are labeled and the
primed end of each winding are connected together to form the star point n.
Fig. 1 Energization Sequence of Two-Pole, Three-Phase BLDCM
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From Fig. 1, when the rotor position is in the range 0° ≤ θ ≤ 60° , phase winding pair ac is
energized. As the rotor rotates and its position changes to satisfy one of the six rotor position
ranges in (a) to (f) of Fig. 1, the winding pair for that rotor position range is energized.
3.2. Torque production by vector method
A cross section of a two-pole three-phase brushless dc motor is shown in Fig. 2 and only the
center conductors of phase winding a, designated by a and a' and hereafter called winding aa' are
shown for the determination of the torque developed due to current flowing in that winding.
The current i a in the a and a' sides of phase winding aa' is under the influence of the uniformly
r
distributed rotor flux density B m . Applying the Lorentz force equation to the charged particles
forming the current i a flowing through the a and a' sides, each of length l meters yields
(
r r
r
F = ia l × Bm
)
(2)
r
r
r
where l is the displacement vector in the direction of current i a . When B m and l are
orthogonal, then
r
r
F = Bm ia l j
(3)
r
r
r
where j is the unit vector perpendicular to the plane contained by B m and l .
7
Fig. 2 Cross Section of Two-Pole Three-Phase BLDCM Showing Winding aa' Only
The two forces of magnitude F acting on both sides of the winding aa' and separated by the
perpendicular distance d p as shown in Fig. 2, develops electromagnetic torque on the winding
given by
r
r
T e1 = B m i a l d Sin δ1 r
(4)
r
where d is the diameter of winding aa', δ1 = 90 − θ and, r is a unit vector.
With winding aa' containing N turns and area Aaa ' = l d , φm = B m Aaa ' and λ m = N φ m , the
electromagnetic torque developed by winding aa' in Eq. (4) can be written as
r
r
T eaa 's = λ m i a Sin δ1 r
(5)
where λ m is the peak flux linkage due to the permanent magnet rotor.
8
This electromagnetic torque acts on the stator causing rotation in a clockwise direction, provided
the rotor magnet is prevented from moving and the stator is free to move. If however the stator is
held fixed and the rotor is free to move, the rotor would experience a torque of same magnitude
but opposite in direction to that applied to the stator and given by
r
r
Sin
k
=
λ
δ
i
T eaa 'r
m a
1
(6)
r
r
where k is a unit vector opposite to that of r .
But Eq. (6) can be rewritten in the form
r
r
r
T eaa 'r = λ m × i a
(7)
r
where i a is the magnetic current vector along the positive magnetic axis of phase winding aa',
whose magnitude is equal to that of the scalar current i a flowing in the winding [12-13].
δ1
r
r
r
which is a function of the rotor position, is the angle between the vectors λ m and i a and k is
r
r
the unit vector perpendicular to the plane containing vectors λ m and i a .
Eq. (7) presents a very powerful statement relating to the mechanism of electromagnetic torque
production in electrical machines. It states that electromagnetic torque is developed by the cross
r
product of peak flux linkage vector and current vector. Since current vector i a is fixed on the
r
stator and peak flux linkage vector λ m is fixed on the rotor, the electromagnetic torque
developed by the motor that causes rotor movement, decreases the angle δ1 between these two
vectors. Since the electromagnetic torque varies with phase winding current ia and Sin δ1 , which
varies with rotor position, then a pulsating electromagnetic torque is developed by the brushless
dc motor, even if the phase winding current ia is kept constant. However, the developed
electromagnetic torque can be made constant by varying the phase current to compensate for the
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variations in Sin δ1 with changes in rotor position. Hence, a current controller that varies the
phase current by 1/ Sin δ1 about a reference value would ensure constant torque operation of the
brushless dc motor.
3.3. Torque production due to two winding energization
Since brushless dc motors are operated by energizing two of its three phase windings, the
electromagnetic torque developed for rotor position range 0° ≤ θ ≤ 60° when phase winding pair
ac is energized is obtained using the vector diagram of Fig. 3.
r
Phase current i a flowing through winding aa' establishes magnetic current vector i a along the
positive magnetic axis of winding aa', while, phase current ic flowing through winding cc'
r
establishes magnetic current vector i c along the negative magnetic axis of winding cc' as shown
r
in Fig. 3. The angles between the peak flux linkage vector due to the rotor magnet λ m and
r
r
current vectors i a and i c are given by δ1 and δ2 respectively. Since phase windings aa' and
cc' are series connected, then, scalar currents
i a = ic ,
(8)
r
r
ia = ic .
(9)
and magnetic vector current magnitudes
10
Fig. 3 Space Vectors of BLDCM for Energization of Phase Winding Pair ac
r
r
The two phase magnetic vector currents i a and i c are added vectorially to produce resultant
r
r
stationary stator current vector i ac that makes an angle δ with peak flux linkage vector λ m due
to the rotor magnet. The mechanism of electromagnetic torque production when winding pair ac
r
is energized can now be seen by the interaction of flux linkage vector λ m and resultant
r
stationary current vector i ac . As electromagnetic torque is developed and the rotor turns in an
anti-clockwise direction, the torque angle δ decreases from 120 electrical degrees at rotor
position θ = 0° , to 60 electrical degrees at θ = 60° . Hence, for energization of phase winding
pair ac, the electromagnetic torque developed by the brushless dc motor is given by
r
r
r
r
r
r
r
T e = λ m × i ac = (λ m × i a ) + (λ m × i c )
(10)
T e = λ m i ac Sin(δ ) = (λ m i a Sin(δ1)) + (λ m i c Sin(δ2 ))
(11)
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where, δ1 = (90 − θ) , δ2 = (150 − θ) and δ = (120 − θ)
Therefore, as the rotor develops torque and rotates in an anti-clockwise direction, the angles δ ,
δ1 and δ2 decreases as rotor position θ increases, hence the electromagnetic torque developed
by the brushless dc motor for this rotor position range 0° ≤ θ ≤ 60° is dependent on both stator
current and the changing angle between the flux linkage and current vectors. Hence this machine
is characterized as a pulsating torque machine even if current controllers are introduced to ensure
constant stator current during the energization of a pair of phase windings, and the developed
electromagnetic torque is maximum at θ = 30° or δ = 90° .
3.4. Torque production during commutation
If winding pair ac remain energized up to the point where the rotor position is θ = 120° , then
Sin(δ ) = 0 and no electromagnetic torque would be developed by the motor. Hence, at rotor
position θ = 60° , which corresponds to δ = 60° , phase winding aa' is commutated and winding
pair bc is connected to the supply for rotor position range 60° ≤ θ ≤ 120° as shown in Fig. 1(b).
Phase winding aa' would take some time to commutate its current ia , and hence during
commutation of winding aa', all three phase windings would be conducting. The electromagnetic
torque developed by the machine during commutation of winding aa' is obtained using the vector
diagram of Fig. 4.
The three phase currents i a , i b and ic in Fig. 4 are changing since i a is being commutated and
r r
i b is growing from zero. These phase currents establishes magnetic current vectors i a , i b and
r
i c along the positive magnetic axes of windings aa' and bb' and along the negative magnetic
axis of winding cc' respectively. In addition, the angles δ1 , δ2 and δ3 between current vectors
12
r
r r
r
i a , i b and i c respectively and the position of the peak flux linkage vector λ m are all changing
with rotor position θ during commutation.
Fig. 4 Space Vectors During Commutation of aa' and Energization of Winding Pair bc
The electromagnetic torque developed by the motor during commutation is given by
r
r
r
r
r
r
r
T eCom = (λ m × i a ) + (λ m × i b ) + (λ m × i c )
(12)
T eCom = (λ m i a Sin(δ1)) + (λ m ib Sin(δ2 ) + (λ m i c Sin(δ3)))
(13)
where, δ1 = (90 − θ) , δ2 = (210 − θ) and δ3 = (150 − θ) .
The electromagnetic torque developed during commutation of phase current i a as shown in Eq.
(13) is a function of the varying phase currents i a , i b and ic and the varying angles δ1 , δ2 and
δ3 and hence is non constant and difficult to predict and control.
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3.5. Torque between commutations for constant current operation
The electromagnetic torque developed by the brushless dc motor over a cycle of motor operation,
ignoring the developed torque during commutation, for constant stator current operation is
obtained using the given energization sequence of Fig. 1, and the fact that the three phase
windings aa' , bb' and cc' are displaced from each other by 120 electrical degrees. Fig. 5 shows
one cycle of electromagnetic torque developed by the motor for constant current operation of I
amps, ignoring the commutation torque. The peak to peak torque ripple is seen to be 0.134 λ m I .
Fig. 5 Electromagnetic Torque for Constant Current Operation
4. Conclusion
The application of vector analysis to three-phase brushless dc motors has been instrumental in
the development of a simple, but powerful expression for the alignment electromagnetic torque
developed by the motor. In comparison to the conventional electromagnet torque equation
produced by equating the electrical power absorbed to the mechanical power produced by the
motor, the vector cross product of peak flux linkage vector due to the rotor magnet and the
resultant stationary stator current vector electromagnetic torque equation, exhibits the
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mechanism of torque production in the motor. It shows that the electromagnetic torque
developed is pulsating in nature between commutations and during commutation, with a peak to
peak ripple value of 0.134 λ m I between commutations, under constant current operation. Hence
a current controller, whose output depends on the rotor position, which varies the phase current
by 1/ Sinδ about a reference value would ensure constant torque operation of the brushless dc
motor between commutations.
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