IEEE Transactions on Energy Conversion, Vol. 13, No. 3, September 1998
257
Commutation Torque Ripple Minimization for Permanent Magnet
Synchronous Machines with Hall Effect Position Feedback
Todd D. Batzel and Kwang Y. Lee, Senior Member, IEEE
Department of Electrical Engineering
The Pennsylvania State University
University Park, PA 16802
Abstract: A permanent magnet synchronous motor (PMSM) with
sinusoidal flux distribution is commonly coininutated using discrete
rotor position feedback from Hall sensors. A commonly used stator
current excitation strategy used in such a system is a six-step
current waveform. Application of sinusoidal current waveforms is
shown to produce smooth torque in the PMSM. This paper shows
how a pseudo-sensorless rotor position estimator may be used with
Hall sensors to provide sinusoidal current excitation in place of sixstep currents to reduce the torque ripple associated with the sixstep strategy. Performance evaluation of the rotor position
estimator in a PMSM drive is provided through simulation.
Keywords: torque ripple, permanent magnet syncl~ronousmotor,
sensorless commutation, Hall effect sensor
I. INTRODUCTION
In recent years, the permanent magnet synchronous motor
(PMSM) has become a popular choice for applications such
as machine tool drives, computer peripherals, robotics, and
electric propulsion. Much of this popularity is due to the
PMSM’s reduced maintenance (no brushes), superior power
density and efficiency, and low rotor inertia [I].
In a PMSM, the ability to separately control the stator
current and angle allows the choice of operating re,’mimes,
such as maximum torque per unit current and maximum
power. In addition, separate current and angle control
permits flux weakening (for operation above rated speed)
and phase advancing (for reluctance torque augmentation)
[2]. In order to individually control stator current and angle
and thus fully exploit the characteristics of the PMSM, high
resolution rotor angle information is required - usually from
a device such as a resolver or encoder. These high resolution
position sensors, which are usually attached to the rotor, add
length to the machine, raise the system cost, increase rotor
inertia, and require additional cabling.
PE-959A-EC-0-05-1997 A paper recommended and approved by the
IEEE Electric Machinery Committee of the IEEE Power Engineering
Energy
Society for publication in the IEEE Transactions on
Conversion. Manuscript submitted January 2,1997; made available for
printing May 23, 1997.
One of the most commonly used methods for angular
position sensing is the Hall effect sensor [3]. These coarse
shaft position sensing devices are mounted on the magnetic
axis of each stator phase winding, thus providing an
aggregate resolution of 60 electrical degrees. Figs. 1 and 2
show how the discrete signals generated by Hall effect
sensors may be used to decode position data as well as to
generate a six-step stator excitation. Operation with six-step
excitation tends to increase torque ripple [4], and due to the
discrete nature of the feedback, does not easily allow some
advanced PMSM operating regimes to be used [ 5 ] .
Many applications may benefit from the use of sinusoidal
excitation yet, due to various constraints, do not permit the
use of an external position sensing device other than the
built-in Hall sensors. Thus, the designer is faced with a
choice of six-step stator excitation using the Hall Sensors, or
using one of the many sensorless techniques that have been
developed for the PMSM [6-101. The six-step approach
suffers from torque ripple, and many of the sensorless
techniques suffer performance limitations at low speed.
In this paper, a method for obtaining high resolution shaft
position information from coarse Hall sensor data, stator
voltage, and stator current measurements is proposed. This
method provides the reliability of the Hall sensors to
commutate the machine at very low speeds, and the high
resolution angle information needed for sinusoidal
excitation. This pseudo-sensorless method is shown through
simulation to reduce the torque ripple associated with sixstep stator currents commonly used in conjunction with Hall
effect position sensors.
11. PMSM MACHINE MODEL
A two pole, 3 phase salient pole machine is shown in Fig.
1. For such a machine, the flux linkage and voltage
equations are given in the stationary reference frame [ 111.
The flux linkages of the PMSM machine in the stationary
reference frame may be represented by
where v,i, 1, and R are recpective voltage. current- flux
linkage. and resistance for phase a,b. or c. In addition,
0885-8969/98/$10.00 0 1997 IEEE
258
7
hm cos(8)
where the flux terms are defined in (2), and di is the
increment in the phase current. The torque may then be
derived from the coenergy expression
2a
awc
where L v are the self and mutual inductances for each phase,
h,is the flux due to the permanent magnet of the rotor, and
8 is the rotor angle. The self and mutual inductances for the
PMSM may be written as shown in (3), where the terms LI,
La,, and L, represent leakage, nominal, and position
dependent inductance terms, respectively:
Lll(8) =Ltl+La,-LsCOS(28)
(e) = Ll+ La,
-
L, cos(28 + -1
-
Luv
2
-
(3)
2n
L, cos(28 - -)
3
T = -hm[sin(8) i,+sin(8--j
o
1
0
1
1
1500 to 2100
210O to 270°
0
1
0
0
270° to 330°
1 0
90 9 0 1500
+-)ic].
3
2n
3
2n
j b +sin(O+-)ic].
3
(7)
Next, we may write the generated back emf expression as
[
1
(A) expressions, the
1
Wc = -[ ~ u i+;~ 2 z i-+i ~33i3+[
Luiaib + ~ u i+ ~~ z ii iC]
b~
2
(6)
2n
2n .
r
- 30°to 30'
30°t0900
(5)
Using (9,and assuming a non-salient rotor, the torque for
the machine may now be written as
3
L12(0)=L21(0)=
Using (2) and (3) to obtain the flux
coenergy function can be found to be
+hm[cos(8)ia+cos(8--)ib+cos(8
3
2n
3
2n
~ , , ( e )= L~+L,,- L ~ C O S --I
(~B
L22
T(i~,ib,i~,O)=
-.a0
0
0
d axis
/
1
h cos(0)
hnrcos(8 --)
=$xnl
3
27t
cos(e + -)
3
o
\ qaxis
Fig. 1. PMSM analytical model.
It is apparent from these equations for the PMSM
machine, that the rotor angle (e> is a function of the motor
voltage, current, and flux linkages, as well as the machine
parameters - winding resistance, inductance, and the
permanent magnet nux linkage (hI,).
A. PMSM Torque Generation
An expression for the torque generated by the PMSM may
be obtained by using the coenergy of the electromagnetic
system [8]. The coenergy is defined as
1
sin(@)
2n
= -0L Slll(0 --)
3
2n
sin(8 +-)
3 -
(8)
where e, represents the back emf of each phase and 0 1s the
rotor speed in electrical radians per second. Using (8), we
inay rewiite (7) in terms of the back emf as
T-e,i,I
a
27c
r
I
o
e&-.
o
P Ienia I ebib
2 O I ~ Urn
I
ecic ,,
Om
(9)
where a,,,
is the mechanical shaft speed and P is the number
of poles for the machine.
From (8) and (9), it can be seen that any mismatch
between the back emf waveform and the corresponding
phase currents will result in a torque ripple. Thus, for a
sinusoidally wound PMSM, torque ripple may be minimized
by utilizing sinusoidal phase cuirents. In the analysis of the
PMSM, it is generally assumed that the stator windings may
be approximated as: sinusoidally distributed windings. Since
most machines are designed so that the windings produce a
relatively good approximation of a sinusoidally distributed
ax gap mmf , this appears to be a justified assumption.
The phase currents to minimue corque ripple are
2n
2n
i, = I,sin(8) ; ib = I,sin(B --) 3 ;ic = /,sin(B tT) , (10)
where 1, is the amplitude of the phase currents. The
resulting torque expression for the currents described by (10)
is
259
amplifier. The output of the amplifier represents the voltage
required to force the phase current to its reference value.
-
This shows that for the sinusoidal phase currents with a
constant amplitude given in (lo), the resultant torque is
constant and independent of shaft position.
7
............ L.. ......................._
"0
0 4
0 2
0.6
0.8
0.6
0.8
1
time (s)
111. PMSM COMMUTATION
Two techniques are commonly used for the commutation
of PMSM machines - six-step and sinusoidal. A high
resolution position sensor such as a resolver or encoder
usually replaces Hall sensor feedback when sinusoidal
commutation is used. Brushless motors exhibit back emf
characteristics which may be either sinusoidal or trapezoidal,
with each phase being 120" apart.
.... ........... i....
0.4
0.2
time
.................
......
(5)
.......
0
0
0.4
0.2
tin- e ( s )
Fig. 2a. Hall sensors used for six-step cumnt generation.
8'
A . Six-Step Commutation
0 4
0 2
Six-step commutation is widely used for commutation of
PMSM machines. For six-step commutation, Hall effect
sensors are mounted as an integral part of the motor and
aligned with the motor back emf at the factory.
With Hall sensor feedback, an average 90" torque angle
may be maintained by forcing it to remain between 60" and
120". When the torque angle has fallen to 60", the drive
electronics directs the current in the stator such that the
torque angle is increased to 120°, and keeps it there for the
next 60" of rotation. Clearly, for six-step commutation, the
forced stator currents will not match the sinusoidal back emf
characteristic of the PMSM, and torque ripple' will result.
The effect of this torque ripple tends to be a slight kick at the
commutation points, which may be detrimental to high
performance positioning
and
velocity
regulation
applications.
Several simulations of the PMSM with Hall sensor
feedback and six-step currents were performed, with the
results shown in Figs. 2 and 3. The results shown in Fig. 2
depict an acceleration from zero speed, while Fig. 3 shows
low speed operation with a high load torque. The torque
ripple generated by the six-step stator currents is shown
clearly in both simulations.
The overall system diagram of the PMSM model, current
controller, and commutation logic used for the computer
simulations is shown in Fig. 4. The input to the simulation
is the current command (Icmd),which the commutation logic
uses with the rotor position feedback to produce the optimum
phase current references. The rotor position feedback
consists of simulated Hall effect signals for the six-step
simulations, and high resolution angle for the sinusoidal
excitation simulations. The current controller compares the
actual phase current measurements to the reference currents
and operates on the error to generate the input to the
0 6
0 8
time ( s )
L ...........
.;
!
...........................
0.2
0
0 4
0 6
0.8
i .;.-.. Y
0 2
..
0.2
0 4
0 6
0.8
time (s)
I
2
0
\.............
.......
260
h
u1
6 1
I
2
1
I
1
2,
I
2 7
2 75
2 8
2 85
2 0
2 05
3
0
-1,
a'
.
.
.
06
0 8
s
1
I
I
-1
.......... 1 ..........i..
/
i
0 4
02
,
I
1
0 2
0 4
0 6
I
I
I
,
I
0 8
1
I
I
2.75
2.8
2 85
time
I
2.9
2.95
3
(5)
Fig. 3b. Torque, velocity, and actual position for six-step current.
i
1............
3
dz
I
I
Rotor Position (Selectable - Hall Sensor or High Resolution)
Fig. 4. PMSM current controller block diagram.
-0
IV. ROTOR POSITION ESTIMATOR
The block diagram of a high resolution rotor position
estimator developed is shown in Fig. 6. Inputs to the
position estimator are the PMSM stator currents, voltages,
and the Hall sensors. The output is a high resolution
estimate of the rotor position.
The flux linkages may be estimated by rearranging (1) and
integrating:
j.....
0 4
0 2
0 6
0 8
tim e (s)
glo
.....
- ........... ............
1 ..I
0-, 0
0
I
I
0.2
0.4
.......
I
time (s)
0 6
,
0.8
Fig. 5b. Torque, m o r angle, aud shaft speed for sinusoidal current.
B . Sinusoidal Commutation
Sinusoidal commutation is used when very smooth torque
response is desired. Typically, this is when a high degree of
velocity regulation or position accuracy must be maintained
while the motor is operating under heavy load torque at low
speed.
A much higher resolution rotor position sensor is
required for sinusoidal commutation if smooth rotation at
low speeds is desired. In this way, it is possible to maintain
a constant torque angle very accurately, resulting in very
smooth low speed rotatioil and negligible torque ripple.
The results of a simulation of a sinusoidally-fed PMSM
are shown in Fig. 5. The current controller shown in Fig. 4
was used for the simulations with the assumption that exact
rotor position is available. As expected, there is no torque
ripple.
i
............ ..........................
Iab<-
v
h,,,,
Flux
E m ma for
z-+
Coarse
?
Eqtmiator
tFl" x
Fig. 6 Block diagram of rotor position estimator.
h,,, = j (v - iR) ,
(12)
where
v, and i are the three-phase flux linkage estimate,
voltage. and current vectors defined in (1).
This flux linkage estimate may then be used to generate a
coarse current estimate by using (2) and the decoded Hall
sensor data. The current estimate error vector, I,,, , is then
calculated from the difference between measuied stator
currents and the current estimate vector.
A position error, Ae, which represents a measure of the
error between the coarse Hall sensor position data and the
actual rotor position, may then be formed. Since the flux
linkage is a function of the stator currents and the rotor
position, the scalar A8 may be calculated from the
permanent magnet flux linkage vector, the inductance
~
263
matrix, the current estimation error vector, as shown in (13).
The flux linkage estimate is assumed to be correct when
calculating A0 ,
A0
=[[$r[$IT[$]
T
[AX-LAi],
rotor position estimator tracks the actual position well, the
exception being at very low speeds. The tracking error at
low speeds may be attributed to the lack of a developed back
emf under these conditions. Recall from (12) that the back
emf is integrated in the estimation of the flux vectors.
8 ,
I
where L is the inductance matrix defined in (2) and A
denotes changes in variables. A non-salient rotor is assumed
in (13). The rotor position estimate is then calculated from
6 =0h,,+A8.
(14)
Finally, the position estimate is used to calculate the error in
the estimated flux linkages. This flux correction loop is
necessary to remove any accumulated errors due to the
integration process, measurement errors, and uncertainty in
the initial conditions of the flux linkages.
The flux correction process is performed by utilizing the
high resolution position estimate and the flux estimate to
obtain a high resolution current estimate I* via (2). Another
current estimation error I*err is then formed from the
difference between the measured current and this high
resolution current estimate. A measure of the flux estimation
error may then be formed in a manner similar to the
formation of (13). In (13), the flux estimate was assumed to
be correct (AM).To calculate the flux correction, however,
the position estimate is assumed to be correct (A0=0). Thus,
the flux correction is calculated as
ai
Ai=-AI*
ai
The rotor position estimator shown in Fig. 6 was used in a
series of system simulations. The PMSM was subjected to
step changes in the current command and load torque.
t i m e (s)
Fig. 8 Response of rotor angle estimator to step load change
Fig. 8 displays the rotor position estimation performance
due to a step change in the load torque. For this test, a step
load torque is applied to reverse the acceleration of the rotor
at 0.5 seconds. It is seen from Fig. 8 that the step load
change does not have an effect on the position estimate.
V. SYSTEM SIMULATION
In order to generate sinusoidal stator currents (and thus
reduce torque ripple) without the use of a high resolution
rotor position transducer, the rotor position estimator shown
in Fig. 6 is used. A block diagram of the system used to run
the simulations is shown in Fig. 9. The motor parameters
used in the simulations are
R = 1.91!2, L,, = 9.55 mH,
L, = 0, K r =.332
*
-
vA
, 6 poles.
I
7 ,
Lit”
c (6)
Fig. 7. Actual and estimated position for step cunent reversal at 0.5 sec.
Fig. 7 shows a comparison of the estimated angular
position and the actual rotor position for a step change in the
current command at 0.5 seconds. In this test, the rotor is
initially at rest, and is accelerated through a constant torque
command in the positive direction. At 0.5 seconds, the
torque command is reversed. It is clear in Fig. 7 [hat the
I
I’
Fig. 9. Block diagram of pseudo-sensorless system.
Fig. 10 shows a comparisoii of the resulting torque during
an instant current reversal at 0.5 sec. for both the simulated
meudo-sensorless svstem and an ideal system for which the
rotor position is known exactly. The data of Fig. 10 shows
262
that the torque for the pseudo-sensorless system is very close
to the ideal system, with slight deviations when the system
crosses through zero-speed. The torque overshoot at 0.5
sec. in Fig. 10 is due to small overshoots in the phase
currents.
0.5
0.55
0.6
0.65
0.7
0.75
0 8
0.85
0.9
lime (n)
40
I
I
I
I
I
I
I
I
I
I
I
I
,
I
055
06
065
07
075
08
085
I
--0 5
0.5
0.55
0.6
0.65
0.7
0 75
0.8
0 85
[2]P. Pillay and R. Krishnan, “Application characteristics of permanent
magnet synchronous and brushless DC moton for servo diives,” IEEE
Trans. on Ind. Appl., vo1.27,no.5, pp. 986-996, September-October 1991.
[3] B. Drafts, “Selecting Hall effect devices for BLDC motor coinmutation,”
Power Conversion & Intelligent Motion, vol. 22, no. 5, pp. 55-61. May
1996.
[4] T.M. Jaluis, “Torque production in pennanent-magnet synchronoqs motor
drives with rectangular current excitation,” IEEE Trans. on Ind. Appl.,
vol. 20, pp. 803-813, April 19S4.
[SI S. Vlaliu,“New bmsliless AC servo drive uses isolated gate bipolar power
transistors and a CPU to obtain high dynaniic performance with
exceptionally high reliability ‘and efficiency,” Proc. Ind. Appl. Sociery
Annual Meeting, pp. 685-690,1990.
[6] T. Liu and C. Clieng, “Adaptive control for a sensorless bmshless
permanent-magnet synchronous motor drive,” IEEE Trans. on Aerospace
and Electronic Sys. vol. 30, no. 3, pp. 900-909, July 1994.
[7] S. Matsui, “Sensorless operation of blushless DC motor drives,” IECON
Proceedings vol. 2, pp. 739-744, 1993.
[SI A. Consoli, “Sensorless operation of bmshless DC motor drives,” IEEE
Trans. on /ndustrial Electronics vol. 41. no. 1,Pp. 91-95, Feb. 1994.
[9] R. Wu, G.R. Slernon, “A pennanent magnet motor drive without a shaft
sensor,”IEEE T r m . ~on
. lnd. Appl. vol. 27, no. 5 , pp. 1005-1011, 1991.
[10]J.S. Kin and S.K. SUI, “New aproach for high peifonnance PMSM drives
without rotational position sensois,” lEEE Applied Morion Conf. And
&yo, pp. 381-386. 1995.
[ll]P.C. Krause and 0. Wasynzuk, Electromechanical Motion Devices, New
York, McGraw-Hill. 1989.
OS
09
time (sec )
Fig. lob. Velocity and angular position for pseudo-sensorless simulation.
Todd D. Batzel received his B.S. degree in Electrical
Engineering from the Pennsylvania State University, State
College, PA, in 1984, and his M.S. degree in Electrical
Engineering from the University of Pittsburgh, Pittsburgh,
PA, in 1989. He is presently a Research Assistant with the
Applied Research Laboratory of the Pennsylvania State
University and a Ph.D. student in Electrical Engineering at
the Pennsylvania State University. His research interests
include machine controls; electric drives, and power
electronics.
VI. CONCLUSIONS
In this paper, a pseudo-sensorless rotor position estimator
for a PMSM has been implemented toward the goal of
reducing the torque ripple for a sinusoidally wound PMSM.
Simulations revealed that the pseudo-sensorless system was
able to track the rotor position extremely well. This allowed
the use of sinusoidal stator currents, which was found
through simulation to significantly reduce the magnitude of
torque ripple. The proposed system combines sensorless
technology with coarse Hall sensor feedback in an innovative
way to achieve torque ripple reduction. Such a system may
be useful in many motion control applications - particularly
where high precision is extremely important but the addition
of an external position sensor is not permissible.
VII. REFERENCES
.
“Permanent Magnet Machines” in Nasar S.A. (Ed.)
Handbook of Elecrrical Machines, McGraw-Hill, New York, 1987.
[l] K.J. Binns,
Kwang Y. Lee received his B.S. degree in Electrical
Engineering from Seoul National University, Korea, in 1964,
his M.S. degree in Electrical Engineering from North
Dakota State University, Fargo, ND, in 1968, and his Ph.D.
degree in Systems Science from Michigan State Univei-sity,
East Lansing, MI, in 1971. He has been on the faculties of
Michigan State, Oregon State, University of Houston, and
the Pennsylvania State University, where he is currently a
Professor of Electrical Engineering. He is Director of Power
Systems Control Laboratory, and a CO-Director of Intelligent
Distributed Controls Research Laboratory at Penn State €€is
interests include control systems, artificial intelligence,
neural networks, fuzzy logic control, genetic algorithms, and
their applications to power plants and power systems control,
operation and planing. He is a Senior Member of IEEE,
active in Power Engineering Society and Control Systems
Society.