Pseudo vector control – an alternative approach for brushless DC motor drives

Pseudo-vector Control – An Alternative
Approach for Brushless DC Motor Drives


Cao-Minh Ta, Senior Member, IEEE


Abstract- This paper provides an alternative approach called
"Pseudo-Vector Control" (PVC) to reduce torque ripple in
Square-wave PM Motors (BLDC motor). Instead of
conventional square-wave form, the method uses the principle of
vector control to optimally design the wave-form of reference
current in such a way that the torque ripple is minimal. The
currents are however still regulated in phase current control
manner. The proposed PVC has 2 majors advantages: a) the
torque ripple is greatly reduced, and b) the flux weakening for
constant-power high-speed mode can be achieved by injecting a
negative d-axis current into the control system, just like for the
case of PM Synchronous Motors (sine type machines).
Index Terms Brushless DC motor, Trapezoidal type machine,
PM synchronous motor, Sine type machine, Vector control,
Pseudo-vector control, Torque ripple minimization.
I. I
NTRODUCTION

A. Sine-wave PM Motor vs. Square-wave Motor: Vector Control
vs. Phase Current Control
With the recent development of permanent magnet (PM)
materials, PM (brushless) motors have become more and
more popular and find their applications in a wide range of
fields: industry, office use machines, house appliances, space

equipments. The use of permanent magnets has great
advantage in that the created magnetic field is high-density
and without loss, providing a high-efficiency operation of the
whole drive. PM motors can be divided into 2 categories
depending on its current wave-form [1]:
• Sine-wave PM motor, which is also called PM
synchronous motor (PMSM), is fed by sine-wave current
supply;
• Square-wave PM motor that is fed by square-wave
current and it is often called Brushless DC Motor
(BLDCM).
The main difference between two kinds of motors from
the control viewpoint resides on current control techniques
and on the torque quality. The torque produced in a sine-wave
PM motor is smooth as a result of the interaction between the
sinusoidal stator current and the sinusoidal rotor flux. High-
performance PM synchronous motor drives are regulated by
the well-known vector control method, in which motor
currents are controlled in synchronously rotating d-q frame
(Fig. 1). Rotor position at any instant (measured by a high-
resolution position sensor, or estimated by an observer) is the
mandatory requirement in the high-performance vector
control method.
Vector control method cannot however be utilized for
BLDCMs, due to the fact that terminal variables (currents,
voltages) and back-EMFs wave-form are not sinusoidal.
Therefore, the phase current control technique is normally
employed for this kind of motor (Fig. 2). The control
technique is relatively simple because the square-wave
reference currents can be generated in a step manner, every

360/(m*p) electrical degrees, where m is the number of
phases, p is the pole-pair number. Hence, low cost Hall-effect
sensors served as position sensor are enough for the control
purpose. Simplicity in control, BLDCMs suffer however from
a big drawback: because of phase commutation, the torque is
not as smooth as that produced by their counterpart, sine-
wave PM motors.
A BLDCM is, moreover, recognized as having the highest
torque and power capability for a given size and weight due
to its (quasi) square-wave current and trapezoidal form of
back-EMF. In addition to that, a BLDCM also presents the
cost advantage over the sine-wave PM motor, due to its
construction and winding. Therefore, it seems to be evident
that if the torque ripple in BLDCM can be overcome, this
kind of motor would become a very attractive solution for
many industrial and house-appliances applications.
B. Torque ripple in BLDCM and its reduction techniques
Torque ripple is the main concern of the BLDCM because
it limits this kind of motor from many applications. There are
three main sources of torque production (and therefore of the
torque ripple) in BLDCMs: cogging torque, reluctance
torque, and mutual torque [1]. The torque ripple can be
reduced by (a) motor design; (b) control means; (c) or both of
them. If in a BLDCM, either stator slots or rotor magnets are
skewed by one slot pitch, the effect of the first two torque
components is greatly reduced. Therefore, if the waveforms
of phase back-EMFs and the phase currents are perfectly
matched, torque ripple is minimized. However, perfect
matching phase back-EMF and phase current is very difficult,
2011 IEEE International Electric Machines & Drives Conference (IEMDC)

978-1-4577-0061-3/11/$26.00 ©2011 IEEE 1534
considering unbalanced magnetization and/or imperfect
windings [1]. Moreover, due to the finite cut-off frequency of
the current control loop, the transient error of the controlled
currents always occurs, especially in commutation instants,
when the current profile changes drastically and also, the
turn-on and turn-off characteristics of the power devices are
not identical. This is one of the most critical problems in
control of BLDCM drives.
We can find in literature a lot of efforts to reduce the
torque ripple. Ref. [2] - [6] are only some to name. Le-Huy,
Perret and Feuillit [2] analyzed the torque by using Fourier
series and shown that the torque ripple can be reduced by
appropriately injecting selected current harmonics to
eliminate the torque ripple components. Yong Liu, Z. Q. Zhu
and David Howe [5] utilized DTC to reduce torque ripple in
addtion to increasing torque dynamics. Haifeng Lu, Lei
Zhang and Wenlong Qu [6] calculated duty cycles in the
torque controller considering un-ideal back EMFs.
C. Purpose of the present work
The objective of the present work is to provide an
alternative approach to reduce torque ripple in BLDCM drive.
Instead of conventional square-wave form, the method uses
the principle of vector control to optimally design the wave-
form of reference current in such a way that the torque ripple
is minimal. The currents are however still controlled in the
phase current control manner. For this raison, we have called
the proposed approach Pseudo-vector Control.
II. S
YSTEM

C
ONFIGURATION

The configuration of the control system proposed in this
paper is described in Fig. 3 where the part generating current
references is marked by the break-line frame. For the purpose
of comparison, a typical Vector Control scheme for
synchronous motors (sine type machines) is depicted in Fig.
1, followed by a typical Phase Current Control scheme for
brushless DC motors (trapezoidal type machines) in Fig. 2.
The rotor position is normally detected by 3 Hall-effect
sensors in the BLDCM drives (Fig. 2), whereas a position
sensor (such as an encoder) is normally required for the case
of PMSM drives (Fig. 1).
It can be noted that except for the current reference
generation part, the system is nearly the same as of the
conventional control system for BLDC motor (Fig. 2). The
motor is fed by a PWM-controlled MOSFET inverter. As we
can see in Figs. 2 and 3, the currents are controlled by 3
current controllers in stationary frame. By convention, the
superscript
*
refers to the reference variables and quantities.
The reference currents generating part of the proposed
Pseudo-vector Control is consisted of 3 blocks, as shown in
Fig. 3, corresponding to their function that will be described
latter in Sec. III. In order to perform the direct transformation
and the invert one, the rotor position will be needed. The
instantaneous rotor position and speed can be obtained from
the Hall-sensors information using an observer.

a
i
c
i
()
bac
iii=− +
*
c
v
e
θ
+
−
+
+
−
−
*
d
v
*
q
v
*
a
v
*
b
v

d
i
q
i
*
d
i
*
q
i

Fig. 1. Typical block-diagram of Vector Control for PMSM drives.
1535
*
a
i
*
b
i
*
c
i
+
+
+
−
−
−
a
i

c
i
()
bac
iii=− +
*
T
*
a
v
*
b
v
*
c
v
*
I
1
T
k
+
−

Fig. 2. Typical block-diagram of Phase Current Control for BLDCM drives.
INVERTER
BLDC
motor
Load
PWM

Current
Controller
Current
Controller
Current
Controller
*
a
i
*
b
i
*
c
i
+
+
+
−
−
−
a
i
c
i
()
bac
iii=− +
Measured currents
d-q

to
a-b-c
a-b-c
to
d-q
a
e
b
e
c
e
Reference
currents
generation
*
d
i
*
q
i
d
e
Hall-signals
processing
circuit
Hall
sensors
Speed
m
ω

Torque
command
*
T
Position
e
θ
*
a
v
*
b
v
*
c
v
Position
e
θ
+
−
q
e
Look-up
Table

Fig. 3. Block-diagram of the proposed Pseudo-vector Control for BLDCM drives.
III. P
RINCIPLE OF
P

SEUDO
-
VECTOR
C
ONTROL

In the conventional square-wave phase current control
(Fig. 2), for a given torque command, the reference currents
depend only on Hall-sensor signal showing the
communication instants. The torque ripple is therefore
inevitable.
If we can incorporate other information of the drive into
calculation of the reference currents i
a
*
, i
b
*
, i
c
*
, the
performance of the system should be improved. The basic
idea of the proposed “Pseudo-vector control” (or PVC for
short) for BLDCM is to take the back-EMFs into account in
calculation of reference currents (Fig. 3). It can be derived
from the power balance of the control system as follows.

The electromagnetic power of motor is expressed as:
) (

ccbbaae
eieieiP ++=

(1)
where P
e
is the electromagnetic power;
i
a
, i
b
, i
c
is the motor current of the phase a, b, c, respectively;
e
a
, e
b
, e
c
is the back-EMF of the phase a, b, c, respectively.
By neglecting the motor friction loss, (1) can be rewritten:
meccbbaae
TeieieiP
ω
.) ( =++=

(2)
where T
e

is the electromagnetic torque and
ω
m
is the motor
mechanical angular speed.
1536
Therefore, the torque can be expressed as:
(. . .)
aa bb cc
e
m
ie ie ie
T
ω
++
=

(3)
Equation (3) shows the basic equation of the
electromechanical torque, which is a result of the interaction
between the motor currents and the back-EMF produced in
the motor by the permanent magnets (PM). It also explains
the fact that if the currents and EMF do not match each other,
torque ripple occurs. This is therefore interesting to note that:
in the control configuration, if the reference currents are
chosen in the manner that equation (3) is respected, there
would be no torque ripple.
*
***
) (

e
m
ccbbaa
T
eieiei
=
++
ω

(4)
where the * denotes the reference and desired values.
This is now to discuss how to calculate the reference
currents from (4). The back-EMF can easily be estimated
from the motor terminal variables, so they are considered to
be known for a given speed. Let’s work in the synchronously
rotating d-q frame instead of stationary a-b-c frame. The
power equation (1) can be rewritten in d-q frame:
00
(. . . )
3
.( . . . )
2
e aabbcc
dd qq
P ie ie ie
ie ie ie
=++
=++

(5)

where the indexes
d
,
q
,
0
represent the variables in d-axis, q-
axis and zero-sequence, respectively.
For a balance system it is desirable that the zero-sequence
current i
0
is zero. Combining (4) and (5) gives for reference
and desired values:
) (
2
3
.
***
qqddme
eieiT +=
ω

(6)
Equation (6) means that if we can a priori select the
reference current i
d
*
, we can calculate the reference current
i
q

*
, for a given speed and desired torque T
e
*
.
q
ddme
q
e
ieT
i
**
*

3
2
−
=
ω

(7)
As the desired torque contains no torque ripple, the
obtained torque is theoretically free of ripple if the current
controllers work properly to yield currents perfectly equal to
the reference ones.
The back-EMF in d-q frame e
d
, e
q
, e

0
in (5) and (6) are
obtained from 3 phase EMF e
a
, e
b
, e
c
by Park’s
transformation. As previously mentioned, the zero-sequence
current i
0
is forced to be zero in our system, so only e
d
and e
q

need to be calculated using simplified Park’s transformation.
⎥
⎦
⎤
⎢
⎣
⎡
+−
+−−−−
=
)sin()sin()sin(
)cos()cos()cos(
3

2
3
2
e
3
2
ee
3
2
e
3
2
ee
ππ
ππ
θθθ
θθθ
M
(8)
where
e
θ
is the rotor position:
.
ee
t
θω
=
, with ω
e

is the
electrical angular speed (
ω
e
=
ω
m
.p),
ω
m
is the motor
(mechanical) angular speed and p is the pole-pair number.
Having calculated i
d
*
, i
q
*
, the 3 phase reference currents
i
a
*
, i
b
*
, i
c
*
are obtained by inverse Park transformation M
-1

.
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎣
⎡
++−
−−−
−
=
−
)sin()cos(
)sin()cos(
sincos
3
2
e
3
2
e
3
2
e
3
2

e
ee
1
ππ
ππ
θθ
θθ
θθ
M


(9)
It is worth to emphasis again here that by using the
reference currents i
a
*
, i
b
*
, i
c
*
to control the system, there
would be theoretically no torque ripple, because the reference
currents i
a
*
, i
b
*

, i
c
*
was optimally designed taking back-EMFs
into account. The d-q frame is utilized only for calculating
these reference currents, while the phase current control
principle is normally used in a-b-c frame.
IV. S
IMULATION
R
ESULTS

The proposed algorithm has been simulated in
Matlab/Simulink using the motor of which the parameters are
shown in TABLE I.
TABLE I
MOTOR PARAMETER
Nominal voltage 36 [V]
Nominal speed 1500 [rpm]
Phase resistor 0.8 [Ohm]
Phase inductance 2.14 [mH]
Back EMF constant 0.018 [V/rpm]
Nominal current 2 [A]
Nominal torque 0.344 [N.m]
The working condition in simulation is as follows: the
motor is starting up to 1000 rpm without load in the interval
(0 - 0.1) s, then a load is applied at t = 0.15 s when the system
has been already in the steady state. In order to emphasis the
performance of the controlled system, the speed control loop
is omitted in the Figs. 1 - 3 and the currents are controlled by

hysteresis (bang-bang) techniques.
A. Performance of the conventional Phase Current Control
The simulation results of the Phase Current Control for
1537
BLDCM drives is reported in Fig. 4 in the order from top to
bottom: (a) Motor speed, (b) Three phase currents, (c)
Torque, (d) Zoomed current, (e) Zoomed torque. The torque
ripple is clearly detected in the instants of commutation due
to the shape slop of the currents.
(a)
0 0.05 0.1 0.15 0.2 0.25
0
100
200
300
400
500
600
700
800
900
1000
Speed respond
Time [s]
Speed [rpm]
(a)
0 0.05 0.1 0.15 0.2 0.25
0
100
200

300
400
500
600
700
800
900
1000
Speed re spond
Time [s]
Speed [rpm]

(b)
0 0.05 0.1 0.15 0.2 0.25
-6
-4
-2
0
2
4
6
Phase curre nts
Time [s]
Current [A]


Phase A
Phase B
Phase C
(b)

0 0.05 0.1 0.15 0.2 0.25
-6
-4
-2
0
2
4
6
Phase currents
Time [s]
Current [A]


Phase A
Phase B
Phase C

(c)
0 0.05 0.1 0.15 0.2 0.25
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Torque re spond
Time [s]
Torque [N.m]

(c)
0 0.05 0.1 0.15 0.2 0.25
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Torque re spond
Time [s]
Torque [N.m]

(d)
0.4 0.405 0.41 0.415 0.42 0.425
-3
-2
-1
0
1
2
3
t(s)
is
c
(A)

(d)
0.4 0.405 0.41 0.415 0.42 0.425

-3
-2
-1
0
1
2
3
t(s)
is
c
(A)

(e)
0.21 0.215 0.22 0.225 0.23 0.235 0.24
0.4
0.42
0.44
0.46
0.48
0.5
0.52
Torque respond
Time [s]
Torque [N.m]

Fig. 4. Simulation results of conventional Phase Current
Control for BLDCM: (a) Speed, (b) Three phase currents,
(c) Torque, (d) Zoomed phase C current, (e) Zoomed torque.
(e)
0.21 0.215 0.22 0.225 0.23 0.235 0.24

0.4
0.42
0.44
0.46
0.48
0.5
0.52
Torque respond
Time [s]
Torque [N.m]

Fig. 5. Simulation results of the proposed Pseudo-vector
Control for BLDCM: (a) Speed, (b) Three phase currents,
(c) Torque, (d) Zoomed phase C current, (e) Zoomed torque.
1538
B. Performance of the proposed PVC
The most important signals of the proposed control system
are presented in Fig. 5 in the same order as in Fig. 4. As we
can see in Fig. 5 (b), the current are smooth and the torque
ripple due to the 60 degrees interval commutation are
eliminated, as can be seen in Fig. 5 (c).
C. Performance comparison
In order to compare the performance of the conventional
Phase Current Control and of the proposed PVC, let’s have a
closer look to the system with the zoomed current and torque
curves as depicted in Figs. 4 - 5 (d), (e). The peaks at every
60 degree on the current wave form (Fig. 4 (d)) are due to
phase commutation and they correspond to the torque ripple
in Fig. 4 (e). On the other hand, no torque ripple is observed
in the Fig. 5 (e). It is noted that high-frequency noise in the

current and the torque wave forms is the natural consequence
of the hysteresis band in bang-bang current control, but it
does not harm the quality of produced torque.
To further evaluate the performance of the system, the
flux trajectory (ψ
α
- ψ
β
) is examined. Fig. 6 (a) clearly shows
the “cogs”, which have origin from the commutation at every
60 degree in case of Phase Current Control. The sharp change
of current wave-form in commutation instants is reflected in
the flux. Again, the torque produced in the machine can also
be considered as an interaction between rotor and stator flux.
The torque ripple is therefore evident in this case.
In contrast, the flux trajectory of the PVC method in Fig.
6 (b) is nearly round, which can be explained that the current
wave-form of PVC is smooth, as previously shown in Fig. 5
(d). This interesting feature gives another explanation on the
superiority of the proposed PVC over the conventional Phase
Current Control method.
Note that the simulation in Figs. 4 -5 was carried out in
the based-speed region with i
d
*
= 0 in (7). The flux weakening
for constant-power high-speed mode can be achieved by
injecting a negative d-axis current (i
d
*

< 0) into the control
system, just like for the case of PM Synchronous Motors
(sine type machines).
V. C
ONCLUSION

The Pseudo-vector Control (PVC) has been presented in
this paper in order to improve the performance of the
Brushless DC Motor drives. The vector-control principle in d-
q synchronously rotating frame is utilized for the generation
of current references only, and the motor currents are
regulated, as usual for PM trapezoidal type machines, by
individual phase current control in the stationary a-b-c frame.
The performance of the proposed PVC has been tested in
simulation in the Matlab/Simulink environment and the
results have been compared with those of the conventional
phase current control method. The improvement in torque
ripple reduction has been obtained. The PVC can therefore be
a very promising method for applications using BLDCM.
a)
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Flux trajectory
Flux alpha

Flux beta

b)
-0.03 -0.02 -0.01 0 0.01 0.02 0.03
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Flux trajectory
Flux alpha
Flux beta

Fig. 6. The flux trajectory (ψ
α
- ψ
β
): (a) using
conventional Phase Current Control, (b) using PVC.
A
CKNOWLEDGEMENT

A part of this work has been carried out when the
author worked in NSK Co. Ltd, Japan as a Senior
Engineer. The collaboration is very much appreciated.
R
EFERENCES

[1] J. R. Hendershot Jr., T. J. E. Miller, Design of Brushless Permanent-
Magnet Motors, Magna Physics Publications – Oxford Science
Publications, 1994.
[2] H. Le-Huy, R. Perret, and R. Feuillet, “Minimization of torque ripple in
brushless DC motor drive”, IEEE Transactions on Industry
Applications, vol. 22, July/Aug. 1986, pp. 748–755.
[3] Joachim Holtz, “Identification and Compensation of Torque Ripple in
High-Precision Permanent Magnet Motor Drives”, IEEE Transactions
on Industrial Electronics, vol. 43, no.2, April 1996, pp. 309–320.
[4] Thomas M. Jahns and Wen L. Soong, “Pulsating Torque Minimization
Techniques for Permanent Magnet AC Motor Drives - A Review”,
IEEE Tran. on Ind. Electronics, vol. 43, no. 2, April 1996, pp. 321–
330.
[5] Yong Liu, Z. Q. Zhu and David Howe, “Direct Torque Control of
Brushless DC Drives With Reduced Torque Ripple”, IEEE Tran. on
Industry Applications, vol. 41, no. 2, March/April 2005, pp. 599–608.
[6] Haifeng Lu, Lei Zhang and Wenlong Qu, “A New Torque Control
Method for Torque Ripple Minimization of BLDC Motors With Un-
Ideal Back EMF”, IEEE Transactions on Power Electronics, vol. 23,
no. 2, March 2008, pp. 950–958.
1539