PEDS 2007
Phase
Advance Approach to Expand the Speed
Range of Brushless
DC
Motor
BinhMinh Nguyen, Minh C. Ta
Department of Industrial Automation, Faculty of Electrical Engineering
Hanoi University of Technology, 1 Dai Co Viet street, Hanoi, Vietnam
e-mail: ngbminh20021846 a@yahoo.com, minhtc-auto aXmail.hut.edu.vn
Abstract--In order to extend the speed range of a brushless
DC motor drive, it is common to control the phase current to
lead the phase back EMF by an "advance angle". It is based
on the "phase advance approach", which was proposed for
brushless DC motor drive in 1995. This paper completes the
"phase advance approach" theory in several aspects: a) the
stator resistance is taken into account in the analysis and
calculation of phase reference current; b) the situation where
the back EMF is smaller than the applied voltage is also taken
into consideration; and c) an analytical equation to calculate
the advance angle is also derived. The obtained equation are
therefore more general than those in the prior art. Extended
simulation results demonstrate the validity of the approach.
two main disadvantages: it produces more torque ripple and
it's difficult to control at high speed. This paper focuses on
overcoming the second disadvantage. In section 2, the basic
problems of controlling BLDCM in high-speed region
including the "phase advance approach" in literature [1] are
analyzed. The general mathematical derivation of phase
advance approach is developed in details in section 3. In
section 4, the calculation of phase advance angle is
described. Computer simulation results are given in section
5. Finally, the conclusion of this paper is given in section 6.
e,
Index Terms--Trapezoidal-type PM synchronous motors,
Brushless DC Motor Drive, Phase Advance Angle, Wide
Speed-range Operation.
120°
NOMENCLATURE
00
u
e
i
V
E
Ke
Kt
p
R
L
cL)
CO rn
Phase advance angle
Phase applied voltage
Phase back electromagnetic force (BEMF)
Phase current
Amplitude of phase applied voltage
Amplitude of phase back electromotive force
Electromotive force constant
Torque constant
Number of pole pairs
Resistance of phase winding
Inductance of phase winding
Electrical speed
Motor speed
Time constant
I. INTRODUCTION
It is well known that in the trapezoidal-type PM
synchronous motors (often called Brushless DC Motor) the
BEMF wave-form is trapezoidal and the current shape is
rectangular as shown in Fig. 1. Brushless DC Motor
(BLDCM) has many advantages: it requires less
maintenance, operates more quietly than a brushed DC
motors. BLDCM also produces more output power per
frame size than other kinds of motors. Due to the waveforms of BEMFs and currents, the BLDCM has however
1-4244-0645-5/07/$20.00©2007 IEEE
E
l/-o
Fig. 1. Typical phase BEMF and phase current of BLDCM.
II. BASIC PROBLEMS OF CONTROLLING BLDCM
A. Limit of Operation Region
Due to the fact that the BEMF and current of BLDCM is
not sinusoidal, the vector control principle can not be
readily applied. To control a BLDCM, it is common to use
the conventional phase-current-control configuration as
shown in Fig. 2 with two close loops for current and speed
control (R., is the speed regulator and Ri is the current
regulator). With this drive system, BLDCM can operate
with rated torque at any speed under the rated speed.
In many applications such as electric vehicles, tools
machines, etc, wide speed range (over the rated speed) is
required. Above the rated speed, because the power of a
BLDCM is a limit determined value, the torque decreases
inversely proportional to the motor speed. Fig. 3 shows the
typical characteristics of a motor where the bd is often
called the power limit curve, on which the speed is maximal
for a given load torque. Conventional drive system cannot
however operate on the power limit curve due to voltage
saturation. The lower curve bef illustrated in Fig. 3 is the
locus of operating points of conventional drive system
without any additional special control technique. Although
Fig. 3 is general for any kind of motors, the phenomena of
1255
(Om
Fig. 2. Conventional BLDCM drive system with current loop and speed loop.
saturation can be explained for the BLDCM as follows.
PWM technique is used to control the applied voltage of
BLDCM. Operating at rated speed, the power switches are
turned on 120 electrical degrees each half cycle, so the
applied voltage attaints its rated value. This is the maximal
applied voltage for a BLDCM.
The voltage equation for each phase of the motor can be
expressed as
u = Ri+L-+ e
(1)
dt
where u is the phase applied voltage, R is the resistance of
phase winding, L is the inductance of phase winding, i is
the phase current and e is the phase BEMF.
Because the applied voltage is limited, if the motor
current is large enough to satisfy the load torque (for
example T1), the BEMF cannot increase to a higher value.
BEMF is directly proportional to the motor speed, so the
motor speed cannot be developed (C m2 < Co .) as shown by
point e in Fig. 3. Similarly, if we want the motor to operate
at a higher speed (for example cmI ), the output torque of
the motor cannot be developed as required (T2 < T1 point f
in Fig. 3). As the operating points e, f are under the power
limit curve, the total power of the motor cannot be utilized.
Fig. 3. Operating points of BLDCM.
B. Phase Advance Control of BLDCM
In order to expand the speed range of a drive, it is
common to weaken the existing field. For the separately
excited DC motor, as the motor has separate windings for
the flux-producing and torque-producing currents, it is very
easy to control this motor above rated speed by decreasing
the flux in keeping the applied voltage at its rated value.
For a vector-control IM drive, flux-weakening operation
can be realized by decreasing the d-axis (flux-producing)
current. For a vector-control PM drive, the field of PM can
be weakened by introducing a negative d-axis current (Id <
0). For a BLDCM, flux-weakening operation is not a matter
of acting on d-axis current as compared to other of AC
motors, because the vector control is not applicable to this
kind of motor as mentioned previously.
In 1995, C. C. Chan et al. proposed "phase advance
approach" to solve this problem for BLDCM [1]. The main
idea of this approach can be explained as follows. There are
two types of electromotive forces in the phase winding of a
BLDCM. One is called Back electromotive force (BEMF),
e, which is induced by the magnet field of the rotating of
the permanent magnet. Another one is called Transformer
electromotive force (TEMF), L
, which is induced by the
dt
transformer action of the time-varying stator current in the
phase windings. Below the rated speed, BLDCM is
controlled conventionally, the phase current is in phase with
the phase EMF. When operating above the rated speed, the
phase-current leads the phase BEMF. Thus, the TEMF is
utilized to counteract the BEMF. This is equivalent to the
flux-weakening for the DC motor drive.
Chan et al. examined the phase current when the
amplitude of BEMF is higher than the amplitude of phase
applied voltage (E >V) with the assumption that the phase
winding resistance is negligible. But the amplitude of
BEMF can be lower than the amplitude of phase applied
voltage (E < V), and the phase winding resistance is not
always negligible.
There is a corresponding advance angle for every given
reference speed. The locus of these points is the "phase
1256
advance curve". In the paper in 1995, Chan et al. proposed
that the value of "advance angle" is governed by an
approximate linear relationship with the motor speed. In
reality, the "phase advance curve" is not a straight line, but
a complicated curve. In the latter paper in 1998, Chan et al.
proposed an algorithm which is called "Adaptive searching
trajectory" to determine the corresponding advance angle
[2]. This algorithm increases the complicatedness of the
BLDCM drive. Beside the works of Chan et al., other
researches in phase advance approach have been
investigated. J. S. Lawler et al. identified several limitations
of the phase advance approach. The phase advance
approach is especially sensitive to the motor inductance that
must be larger than a threshold value to maintain motor
current within rated value when operating at rated power
and high speed [4]. If the motor inductance is low,
additional cooling will be necessary for the motor and
inverter components and the current rating of the inverter
will have to be increased. J. S. Lawler et al. also proposed
an inverter topology and control scheme which is called
dual-mode inverter control [5]. With this control scheme,
the range of motor inductance is widened.
For the purpose of redounding to the encompassment of
the theory of "phase advance approach", the authors of this
paper examine two cases: E < V and E > V and the phase
winding resistance is taken into account.
Fig. 4. Per-phase equivalent circuit diagram of BLDCM.
f3
III. PRINCIPLE AND MATHEMATICAL DERIVATION OF PHASE
ADVANCE APPROACH
The BLDCM which is used to examine this approach has
three phases, two pairs of poles and wye stator connection.
In the following mathematical analysis, the assumption that
the BEMF is trapezoidal is used. Fig. 4 illustrates the perphase equivalent circuit diagram of BLDCM, Ql and Q2
are the power switchers, D1 and D2 are feedback diodes.
When operating above the rated speed, the conduction
period of QI leads the BEMF by a spatial angle, which is
called the "advance angle" (00). Thus, QI is fired 00
ahead of the instant that the phase BEMF reaches its
positive maximum.
A. Amplitude of Phase BEMF Lower than Amplitude of
Phase Applied Voltage E < V
Starting from the turn-on of Q1, the first half operating
cycle of the equivalent circuit (Fig. 4) is divided into four
intervals as illustrated in Fig. 5. In the interval 1-2, the
phase BEMF is increasing and the phase applied voltage is
positive so that the phase current increases rapidly. In the
interval 2-3, both the phase BEMF and phase applied
voltage reaches their positive amplitudes. Since the phase
applied voltage is larger than the BEMF, the phase current
increases gradually. At the end of the interval 2-3, QI is
turned off. After that, the phase current flows through D2 in
the interval 3-4. Since the phase applied voltage is negative
Fig. 5. Wave form of phase current when E < V.
and the BEMF is positive, the phase current drops rapidly
to zero. In the interval 4-5, since the circuit is opened, the
phase applied voltage equals the BEMF while the phase
current is always zero. Therefore, in both intervals 1-2 and
2-3, the TEMF L- is positive but decreases. In the
dt
interval 3-4, TEMF is negatives. The second half cycle is
analyzed similarly.
The phase current is divided into four stages over the first
half cycle.
a) Stage I, [0 < cot <.0
This stage covers the interval 1-2. The phase applied
voltage and the BEMF are given by
{e= 6Ecouu=V
le
6Ec
t
+17 600 E
(2)
Substituting (2) into (1), the phase current is obtained as
il(t)=A+Bt-Aez
where r -R is the time constant and
R
1257
(3)
6E
AV E + 6EcoL~~~+
R
(4)
7rR' 7rR 00
|
-6Ec
7rR
At the end of this stage, the value of the phase current is
expressed as
I2 =i1K-j=A+B-- AerT
This stage covers the interval 4-5. Because the circuit is
opened, the phase current is always zero. The derivation of
the second half cycle can be obtained similarly.
B. Amplitude of Phase BEMF Higher than Amplitude of
Phase Applied Voltage E > V
(5)
b) Stage II, [ 00 .< ot <.
3]
This stage covers the interval 2-3. The phase applied
voltage and the BEMF are given by
(u =V
(6)
e =E
Substituting (6) into (1), the phase current is obtained as
-
i2(t)=M+12-Me
t-9°J
221/3I
21
(7)
Fig. 6. Wave form of phase current when E > V.
where M = V-E
R
At the end of this stage, the value of the phase current is
expressed as
I3 =2
3co
3
2;r
=M+ 2 -Me
co
(8)
i
.ct <-+2 ,]
3
3
This stage covers the interval 3-4. The phase applied
voltage and the BEMF are given by
c) Stage III, [
(u =-V
{Ue=EV
(9)
e
Substituting (9) into (1), the phase current is obtained as
i3 (t) = P + I3 - Pe
3t 'co
(10)
where P=
R
At the end of this stage, the value of the phase current is
zero, so
_2ir+ 8_2ir
i3 (-+
3
) = P + I3 - Pe
T
=0
Solve
this
equation, the duration of this stage can be found as:
(P + I3)
f ]
<
<fT
d) Stage IV, [ +23.t.
3
Y
(11)
Starting from the turn-on of Q1, the first half operating
cycle of the equivalent circuit (Fig. 4) is divided into nine
intervals as illustrated in Fig. 6. In the interval 1-2, the
negative phase BEMF acts to strengthen the phase applied
voltage so that the phase current increases rapidly. In the
interval 2-3, both the phase applied voltage and the phase
BEMF are positive. Since the applied voltage is still larger
than the phase BEMF, the phase current increases
gradually. In the interval 3-4 and 4-5, the phase BEMF is
larger than the phase applied voltage. Thus, the phase
current decreases gradually. Since the phase TEMF L- is
dt
negative, it acts to counteract the phase BEMF. This
phenomenon is the key point of the approach. At the end of
the interval 4-5, Ql is turned off. After that, the phase
current flows through D2 in the interval 5-6. Since the
phase applied voltage is negative while the BEMF is
positive, the current drop rapidly to zero. Then, in the
interval 6-7 and 7-8, the phase BEMF is larger than the
phase applied voltage, the phase current begins to flows
negatively through DI. When the phase BEMF becomes
smaller than phase applied voltage in the interval 8-9, the
phase current goes back to zero. Since the circuit is opened
in the interval 9-10, the phase applied voltage equals the
phase BEMF and the phase current is always zero. The
second half cycle is analyzed similarly. The phase current
can be divided into six stages over the first half cycle.
a) Stage I, [0 . cot < 00]
This stage covers the intervals 1-2, 2-3 and 3-4. The
phase current can be expressed as
1258
i4(t) = A+Bt-AeT
(12)
where
V-E
At the end of this stage, the value of the phase current is
expressed as
I4 =iK-j=A+B-- Aew)
(13)
b) Stage II, [ 00 < cot < 2
3]
This stage covers the interval 4-5. The phase current can
be expressed as:
-
t-a°J
i2(t)=M+14-Me
(14)
At the end of this stage, the value of the phase current is
expressed as:
_2ir_0o
15 =i(2(
(15)
=M+14-Me
.cot <2+f/3]
3
3
This stage covers the interval 5-6. The phase current can
be expressed as
c) Stage III, [
i3 (t) = P + I5 - Pe
3t 'co
T
(16)
At the end of this stage, the value of the phase current is
zero, so the duration of this stage can be found as
13= (cor P +12
fTR
At the end of this stage, the value of the phase current is
zero, so it's possible to calculate the duration of this stage.
2ff
f) Stage VI, [-+00 +y.< cot <.]
3
This stage covers the interval 9-10. Because the circuit is
opened, the phase current is always zero. The derivation of
the second half cycle can be obtained similarly. All the
factors (A, B, M, P) in this second case are calculated
similarly in the first case.
IV. CALCULATION OF PHASE ADVANCE ANGLE
For each given reference speed there is a corresponding
advance angle. The trajectory of advance angles in function
of speed is a complicated curve which is called "phase
advance curve". Calculating the phase advance
corresponding to the reference speed is an important
problem in the approach of phase advance.
Below the rated speed, the output torque of the motor is
directly proportional to the amplitude of the phase current:
T = Kt.I
(22)
The current flows through the phase winding 120 degrees
each half cycle, so the rms value of the current can be
expressed as
T
Irms
i4 (t)M=M -Me
nt
(21)
B 6Eco
(17)
< ot <-+ 00]
d) Stage IV, [-+fl.c
3
3
This stage covers the interval 6-7. The phase current can
be expressed as
6EcoL
I2 d(cot) = -I
JI2dt
AT0
=
z7
(23)
6
From (22) and (23):
T = Kt Irms
(24)
where
3C
0
T
(18)
At the end of this stage, the value of the phase current is
expressed as
I7=i<42ff+
j=M-Me
e) Stage V, [
+ 00 <cot<<
(19)
+00 +)y]
3
3
This stage covers the intervals 7-8, 8-9. The phase
current can be expressed as
2;T~ ~~(
i5(t) =A +I7 +BB t f--33coco
-Ae
37co co
Kt
=
3
-Kt
(25)
The problem of calculating the advance angle can be
expressed as follows. For a given speed a), and torque
T=
W)m
rated
Trated, find out the phase advance which is
satisfies equation (24). That is with this advance angle, the
current is large enough to satisfy the load torque:
Irms = ('
T
°)=Kt*26
In these expressions, Trated is the rated torque, com the
(20)
mechanical speed, cor -rated the rated mechanical speed of
1259
the electrical speed and m =- =- in our
p 2
case. The rms value of the phase current is a function of c,
00, other motor parameters and the supplied voltage V.
the motor,
c
i3(t)=P+I3-Pe
Table I shows the technical data of the BLDCM which is
used to examine in this paper.
(30)
The expression of the rms value of the phase current is
TABLE I
TECHNICAL DATA OF THE MOTOR
Urated = 12V
Maximum no-load speed
no = 2826 rpm
Rated speed
nrated= 685 rpm
Rated current
lrated= 95A
Rated torque
Trated = 3,75 Nm
Electromotive force constant
Ke 0,0202 V/rad
Torque constant
Kt= 0,0405 N.m/A
Number of pole pairs
p=2
o
irms = IT
I)
The maximum amplitude of BEMF which is correspondent
to the maximum speed is
E=
m max
Ke e
mmax = 0.0202
(2826
~~60
2;T )
_
=5.97(V)<V= 2rated
6(V)
Since the amplitude of BEMF is always smaller than half
of the rated applied voltage, only the first case (E < V) is
examined. The expressions of phase currents contain
exponents, so the Taylor expansion must be utilized.
2
3
~~~x
e =xi+x+-+-+...
(27)
r
L
2
co
I
i3 dt
(31)
2!
3cw
\,c
Phase advance properties
mxu
a
40
r
l0
i2(t)=M+12-Me T
(29)
+
il dt + | i2dt +
60h
rtt-J
= '2 -M
2+#
3 co
3co
For a given reference speed above the rated speed, it is
possible to found an advance angle by using computer to
solve equation (31). The results are summarized in Table II.
Fig. 7 shows the "phase advance curve".
Fig. 8 illustrates the closed loop control system of
BLDCM drive, which is suitable for both constant-power
operation and constant torque operation. When the
reference speed wm is set, the BLDCM starts up. From the
output of the encoder, the speed feedback signal co and
position signal can be obtained. When the speed feedback is
lower than or equal to the rated speed, the BLDCM
operates in the constant torque region and the advance
angle is set to zero. When the speed feedback is above rated
speed, the BLDCM operates in the constant power speed
with a corresponding advance angle. This advance angle is
calculated by using a "lookup table" which is obtained from
table II.
2! 3!
Using the Taylor expansion, the approximate expressions
of phase currents can be expressed as:
-t
-t
I -t2
(28)
il (t) A + Bt Ae Z- =Bt-A+
=
T
-1
=I3
V. COMPUTER SIMULATION RESULTS
Rated applied voltage
3t co
n(rpm)
Fig. 7. Phase advance curve.
(
1260
o
B
Fig. 8. Block diagram of phase advance drive system.
Both the BLDCM drive system in Fig. 8 and the
conventional drive system in Fig. 2 were examined in
using Matlab/Simulink. The simulation was performed
with reference speed n= 2500 rpm and the load torque
T= rated Trated =
685
*3.75=1 .03(Nm) .
If
the
conventional drive system is used, the speed of the
motor can increase up only to 1800 rpm and the motor
torque ripple is considerable as shown in Fig. 9. This
speed is much lower than the reference speed. If the
configuration of Fig. 8 is used, the motor speed can
reach the reference speed of 2500 rpm with less torque
ripple (Fig. 10).
The higher speed, the higher switching frequency of
the power switches which are often MOSFETs. This
TABLE II
REFERENCE SPEED AND CORRESPONDING ADVANCE ANGLE
Mechanical speed
Electrical speed
Advance angle
(rpm)
685
1.25x685 856.25
1.5x685 1027.5
1.75x685 1198.75
2x685 1370
2.25x685= 1541.25
2.5x685= 1712.5
2.75x685= 1883.75
3x685 =2055
3.25x685 = 2226.25
3.5x685 = 2397.5
3.75x685 = 2568.75
2826
(rad/s)
143.46
(degree)
0
179.32
215.2
251.06
286.94
322.78
358.66
394.54
430.40
466.26
502.14
538.00
591.88
1.9
11.75
18.2
27.5
36.1
42.3
45.1
47.9
50.1
51.5
52.5
52.7
phenomenon affects to the torque-producing. So it is
evident that operating at high speed, the BLDCM
produces more torque ripple than in low speed.
Reducing torque ripple, or smoothing the output torque
of BLDCM is also an important problem. It is beyond
the scope of this paper and therefore is not described
here.
Many computer simulations were performed and the
results are shown in Fig. 11 in form of the limitoperating locus with the conventional drive system (the
dot curve) and with the phase advance drive system (the
solid curve). It is clear that the speed range is wider with
phase advance drive system than that of the
conventional drive. For example, if the load torque is T
= 1.03 N.m, with use of the phase advance, the drive
can reach the maximum speed of 2500 rpm, while the
conventional drive can reach the maximal speed of only
1800 rpm.
VI. CONCLUSIONS
Phase advance approach to widen the speed range of
BLDCM is presented. The principle of this approach is
analyzed in more details in this paper compared to other
works by Chan et al. The wave form of phase current is
examined in both cases: E > V and E < V. The stator
resistance is taken into consideration in the analysis and
calculation of phase current and advance angle. The
obtained equations are therefore more general than
those in literature. The calculation of advance angle is
proposed for the first time in this paper. The advance
angle is calculated off-line by computer simulation and
can be then implemented by a "lookup table". Extended
simulation results have shown that the speed range of
the drive can be considerably widened. Although each
motor needs its own "lookup table" or "phase advance
curve", this approach is particularly suitable for mass
productions.
1261
Speed properties
:HH,11
I-------------r-I
Torque propea'ies
I
-------I--
------I---
4
23
16000
o
-nIH
002 004
0
006
008
01
012 014
t(s)
615
018
U
02
----I.
002
0
004
O66
008
01
t(s)
012
014
016
018
062
_
Fig. 9. Speed and output torque of the motor with conventional drive system.
iuuu
I
Speed with phase advance
I
I
I
I---------
I
I
I~~~~~l~~~~r~~~~~~~
I
Torque with phase advance
7
L
6
Gz
1500
-
.
^
.-
.-
.-
.-
--
-
-
--
--
01
012
014
016
018
.-
-i---
4
--
2
EL
,-
-
1000
--
--
*---
--
0
0
-buu
01UU0
002
004
006
008
Fig.
01
t(s)
10.
012
014
016
018
02
0
002
004
006
t(s)
02
Speed and output torque of the motor with phase advance drive system.
I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
4
008
REFERENCES
[1] C. C. Chan, J. Z. Jiang, W. Xia and K. T. Chau, "Novel
wide range speed control of permanent magnet brushless
motor drives", IEEE Trans. on Power Electronics, vol.
10, Sept. 1995, pp. 539 546.
[2] C. C. Chan, W. Xia, J. Z. Jiang, K. T. Chau and M. L.
Zhu, "Permanent magnet brushless drives", IEEE
Industry Application Magazine, Nov./Dec. 1998, pp. 1622.
[3] Bimal K. Bose, Modern Power Electronics and AC
Drives, Prentice Hall PTR, 2002.
[4] J. S. Lawler, J. M. Bailey, J. W. Mc Keever, Ioao Pinto,
"Limitation of the conventional phase advance method
for constant power operation of the brushless DC motor",
Proc. IEEE Southeast Conf, Apr. 2002, pp. 174-180.
-
1.5
I~~~~
.5
n
0
500
1000
1500
n(rpm)
2000
2500
3
0
[5]
Fig. 11. Limit-operating locus of the conventional drive (dot curve)
and the phase advance drive (solid curve).
1262
J. S.
Lawler,
J. M.
Bailey,
J. W.
McKeever,
Joao
Pinto,
"Extending the constant power speed range of the
brushless DC motor through dual-mode inverter control",
IEEE Transactions on Power Electronics, vol. 19, Issue
3, May 2004, pp. 783-793.