FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
BLDC Ripple Torque Reduction via Modified
Sinusoidal PWM
Shucheng Wang
Abstract — This paper introduces a BLDC fan motor driver
system using modified sinusoidal PWM (SPWM). The driver
board is assembled in the motor to minimize system size.
Based on the signal from the hall position sensor on the board,
the controller generates the SPWM to the power module to
drive the BLDC motor. Because it's SPWM instead of square
PWM, there is no pulsation torque. The simulation and
experimental results are presented to verify the stability of the
driver system.
I. INTRODUCTION
The permanent magnet (PM) motor has many
advantages. Compared to DC motors, they require
lower maintenance due to the elimination of the
mechanical commutator and they have a high-power
density, making them ideal for high torque-toweight-ratio applications. Compared to induction
machines, they have lower inertia allowing faster
dynamic response to reference commands. They are
more efficient due to the permanent magnets, which
result in virtually zero rotor losses.[1]
There are two kinds of PM motors, the permanent
magnet synchronous motor (PMSM) and the
brushless DC (BLDC) motor. They both have a
permanent magnet on the rotor and require
alternating stator currents to produce developed
torque. The difference between the two is that the
PMSM and the BLDC have sinusoidal and
trapezoidal back-EMFs, respectively. The BLDC
motor has the advantage of being a simple machine
with higher power density, simple discrete position
sensors, and simple control compared to a
sinusoidal machine[2]. However, its disadvantage is
the pulsating torque problem[3]. In theory, these two
machines can be driven by either sinusoidal or
rectangle state current. Practically, the PMSM
requires sinusoidal stator currents to produce
constant torque. For a BLDC motor, due to finite
phase induction, the sum of the commutating
currents is never constant and this is the reason for
the generation of pulsation torque. The increment of
current in one phase as a result of the other twophase commutation can sometimes be fatal.
An improved implementation of direct torque
control (DTC) to a permanent-magnet, brushless
DC (BLDC) drive is introduced in reference [4].
The commutation torque ripple is minimized by
combining the conventional two-phase switching
mode with a controllable three-phase switching
mode during periods when the phase currents are
being commutated.
A strategy for reducing commutation torque ripple
in a position sensorless BLDC motor drive is
proposed in reference [5]. The proposed method
directly measures the commutation interval from the
motor terminal voltage waveforms and does not
require a current sensor and current control loop.
However, when the duty cycle of the PWM is high,
the compensation effect is not good, since the duty
for commutation is limited to the maximal PWM
duty. To compensate the duty of commutation, the
PWM cycle is modified to synchronize the end of
the point of commutation. This complex
computation requires advanced MCU/DSP to
achieve.
A low-cost method to minimize the pulsating
torque is introduced in this paper. In this method,
the sinusoidal terminal voltage is employed. This
induces a sinusoidal current with harmonics because
of non-sinusoidal back-EMF. The harmonics and
optimal angle for the voltage PWM calculation has
been deduced in this paper. The torque ripple
comparison between the proposed method and
traditional six-step control for BLDC is shown by
simulation. Compared to others, this method doesn’t
require a current-sampling circuit, complex
computation, or precision encoding. However, this
method can’t be used where good dynamic
performance is required because there is no current
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
control loop or precision encoding.
The testing was conducted on a fan motor used on
an air-conditioner that requires high efficiency,
quiet running, and low cost. The advantage of this
method is to meet the requirement. The BLDC
motor has been more and more popular for the outdoor fan application field because of low cost and
high efficiency.
K e ( −θ r ) = −K e (θ r )
where
and
dλe (θ r )
dθ r
K e (θ r ) ≡
ωr ≡
dθ r
dt
(6)
is the back-EMF coefficient
is the rotor electrical speed (rad/s).
Figure 1a shows the ideal
Ke(θ r )
of the BLDC motor.
II. BLDC MOTOR D,Q MODELING
Because the BLDC motor is fed with a sinusoidal
voltage, the harmonics can be analyzed in a d,q
model[6][7]. Since there is different back-EMF
between the BLDC motor and the PMSM, the d,q
model of BLDC is unlike that of the PMSM. The
harmonics of the back-EMF are analyzed in this
section. The harmonics of the current and the
harmonic torque induced by the harmonics in
current and back-EMF are also presented. Finally,
the optimal input voltage angle is introduced.
A. Modeling of Back-EMF
The BLDC motor is made with trapezoidal backEMF and the model is different from the PMSM.
The BLDC motor is generally assumed to have
three balanced phases connected in a Υ
configuration; the N and the P pole of the PM is
symmetrical in electrical position; and each pole
pair is symmetrical in mechanical position. Based
on the previous assumption, the flux linkage
characters can be summarized as:
λam = λ (θ r )

 λbm = λ (θ r − α )
 λ = λ (θ + α )
r
 cm
(1)
λ (θ r + π ) = −λ (θ r )
(2)
λ ( −θ r ) = λ (θ r )
(3)
where λam , λbm , and λcm are the phase a, b, and c
flux-linkage due to permanent magnet, respectively;
θ r is the electrical position of rotor; and α ≡ 2π / 3 .
The back-EMF character of each phase can be
derived from Equations 1, 2, and 3 as:
ea = ωr K e (θ r )

 eb = ωr K e (θ r − α )
 e = ω K (θ + α )
r e r
 c
(4)
K e (θ r + π ) = −K e (θ r )
(5)
Fig. 1. Back-EMF in the a b c frame and d,q frame.
The voltage equations for the PM motor in the
synchronous reference frame (SRF) are:
r 
v ds
rs + pLs
 r =
v qs   ωr Ls
ea 
r 
− ωr Ls  i ds
 
  r  + T (θ r )eb 
rs + pLs  i qs 
 
ec 
(7)
where Ls and rs are the stator equivalent inductance
and resistance, respectively, and T (θ r ) is the
transformation matrix from three-phase to SFR.
The definition of
T (θ r ) ≡
2  cos θ r

3 − sinθ r
T (θ r )
is:
cos(θ r − α )
cos(θ r + α ) 

− sin(θ r − α ) − sin(θ r − 2α )
(8)
Substituting Equations 8 and 4 in Equations 7,
the new voltage equation of BLDC motor is:
r 
v ds
r + pLs
 r =s
v qs   ωr Ls
where
K ed (θ r ) =
r 
K ed (θ r )
− ωr Ls  i ds

 r  + ωr K (θ )
rs + pLs  i qs 
 eq r 
 
(9)
2 2
∑ cos(θ r − iα )K e (θ r − iα )
3 i =0
(10)
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
K eq (θ r ) = −
2 2
∑ sin(θ r − iα )K e (θ r − iα )
3 i =0
(11)
K e (θ r ) =
∑ − ai
i =1,3,5,7,9...
sin(iθ r )
(17)
Considering Equations 5 and 6, the characters of
K ed (θ r ) and K eq (θ r ) can be obtained as:
Table I shows the harmonic relationship between
the K e (θ r ) in three-phase and K ed (θ r ) & K eq (θ r ) in
K ed (θ r + α / 2) = K ed (θ r )

K eq (θ r + α / 2) = K eq (θ r )

K ed ( −θ r ) = K ed (θ r )
K ( −θ ) = −K (θ )
r
eq r
 eq
SFR.
Fig. 1(b) shows the
A0 , A6 ,
and
B6
can be calculated as:
(12)
K ed (θ r ) and K eq (θ r ) of
the ideal
 A0 = a1

 A6 = a7 − a5
B = −a − a
7
5
 6
back-EMF calculation of each point. The result is
coincident to Equation 12.
(18)
TABLE I
HARMONICS RELATIONSHIP
K ed (θ r )
K eq (θ r )
− a1 sin(θ r )
− a3 sin(3θ r )
0
0
a1
0
− a5 sin(5θ r )
− a5 sin(6θ r )
− a5 cos(6θ r )
− a7 sin(7θ r )
− a9 sin(9θ r )
− a7 sin(6θ r )
0
a7 cos(6θ r )
0
− a11 sin(11θ r )
− a13 sin(13θ r )
− a11 sin(12θ r )
− a13 sin(12θ r )
− a11 cos(12θ r )
a13 cos(12θ r )
K e (θ r )
B. Modeling of the Current and the Torque
As for the sixth harmonic in Equation 9, the pLs
are dominant and the ωr Ls , −ωr Ls , and rs can be
ignored to simplify the analysis process. The sixth
harmonic of the current in SRF can be expressed as:
Fig. 2. Ked and Keq harmonics amplitude.
Based on Equation 12 the Ked(Θr) and Keq(Θr)
can be written as:
K ed (θ r ) =
∑ Bi sin( iθ r
i = 6,12,18...
K eq (θ r ) =
∑ Ai cos( iθ r
i = 0,6,12,18...
(13)
)
)
(14)
Figure 2 shows the harmonics of the Ked(Θr) and
Keq(Θr) of the ideal trapezoidal back-EMF
waveform. In Figure 2, the sixth harmonic is
dominant. If the twelfth and higher harmonics are
ignored, the back-EMF function can be written as:
K ed (θ r ) = B6 sin 6θ r
(15)
K eq (θ r ) = A0 + A6 cos 6θ r
(16)
Based on Equations 5 and 6, the back-EMF
coefficient can be expressed in Fourier form as:
r
i ds
(6) =
−B6 cos 6θ r
6ωr Ls
(19)
r
i qs
(6) =
A6 sin 6θ r
6ωr Ls
(20)
The DC value of the current in the SRF can be
expressed as:
r
r
i ds


ωr Ls   v ds
 rs
1
 r (0)  =




r
2
2
2
rs  v qs − ωr Ke 
i qs(0)  rs + ωr Ls − ωr Ls


(21)
The torque expression is:
Te =
(
3P r
r
i qs K eq + i ds
K ed
4
)
(22)
Ignoring the higher harmonics, the detailed
torque expression can be expressed as:
Te =

AA

3P 
i qs0 A0 +  6 0 + i ds0B6  sin6θr + A0i qs0 cos6θr 

4 
6
L
ω
 r s


(23)
= Te0 + Te6 sin(6θr + γ )
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
where δ is the angle between the voltage vector and
back-EMF vector. Figure 4 shows the stable voltage
relationship. In MTPA control, the terminal voltage
is:
where Te0 = 3P i qs 0 A0 ,
4
Te 6 =
2
 A6 A0

2 2


 6ω L + i ds 0B6  + A0 i qs 0
 r s

3P
4

, and
6 A0 i qs 0ωr Ls

.

A
A
+
6
i
B
ω
L
ds 0 6 r s 
 6 0
γ = arctan
The sixth harmonic component ratio is:
η 6 ≡ Te 6
(24)
Te0
For a typical BLDC motor with the load torque
linear to running frequency, the sixth harmonic
torque ratio is shown in Figure 3. When the
operation frequency is higher than 6Hz, the sixth
harmonic torque ratio is stable at A0/A6=5%. When
it’s lower than 1Hz, the ratio is about 50%.
Vdc

ua = − 2 MI sin(θ r + δ )

Vdc

MI sin(θ r − α + δ )
u b = −
2

u = − Vdc MI sin(θ + α + δ )
r
 c
2
(27)
where Vdc is the DC link voltage and MI is the
modulation index ( MI , defined as the peak phase
voltage divided by half of the DC link voltage).
qr
ωr
r
r
v qs
=
r si q
s
r
Ke
r
ω
+
K eq
ωr
δ
vs
dr
r
r
v ds
Fig. 3. The sixth harmonic torque ratio in a typical motor.
=
−ω
L
i qs
s
r
Fig. 4. The voltage vector for MTPA
C. Optimal Angle of the Input Voltage
When the motor works at the mode of maximum
torque per ampere (MTPA), the copper losses are
the smallest. It can be achieved by adjusting the
input voltage of the BLDC. Normally, BLDC motor
permanent magnet is surface mounted and there is
no reluctant torque in the developed torque. Thus,
the MTPA can be achieved by irds=0. Considering
the DC value of the BLDC model, the stable voltage
equation can be written as:
r
r
v ds
= −ωr Ls i qs

 r
r
v qs = rs i qs + ωr K eq

r
ω r Ls i qs
r
 rs i qs
+ ω r K eq




A. Simulation Model
The BLDC motor model has been presented in
reference [8].The dynamic system equations are:

Ls


Ls

L
 s
di a
= uan − rs i a − ea
dt
di b
= u bn − rs i b − eb
dt
di c
= u cn − rs i c − ec
dt
(28)
(25)
where uan, ubn, ucn, ia, ib, ic, eb, eb, and ec are the
phase voltages, currents, and back-EMF of phase a,
b, and c, respectively. The developed torque and
mechanical equation are:
(26)
 P  e i + eb i b + ec i c
Te =   a a
ωr
2
The optimal angle of the voltage vector is:
δ = arctan
III. SYSTEM SIMULATION
(29)
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
 2  dω r
Jm 
= Te − TL

 P  dt
(30)
where TL , P , and J m are load torque, pole number,
and rotor inertia.
Considering ia+ib+ic=0, the following can be can
be derived from Equation 28:
uan + u bn + ucn = ea + eb + ec
(31)
The neutral point voltage is:
un =
1
(ua + ub + uc − ea − eb − ec )
3
(32)
Combining Equations 28 and 30, the system
operation status equations, can be expressed as:
 di a
Ls dt = ua − rs i a

Ls di b = ub − rs i b
 dt
 di
Ls c = uc − rs i c
dt

  2  dω r
 J m  P  dt = Te
  
− ea − u n
− eb − un
− ec − u n
(33)
− TL
B. SPWM Method
In this system, the carrier waveform is changed for
lower loss. Normally, the SPWM is generalized as in
Figure 5. The sinusoid modulating signals are
expressed in Equation 27. The three phases
modulating signals are shown in Figure 5(a). By
comparing the modulating signal and the carrier signal
in Figure 5(b), the PWM is shown in Figure 5(c).
Fig. 6. Modified modulation of a three-phase sinusoidal waveform.
To save the loss, a modified SPWM is used, as
shown Figure 6. The maximum MI value can be
increased from 1 to 1.154 (= 2 / 3 ). The modified
voltages are expressed as:
u a' = u a − min(u a , u b , u c )
 '
(34)
u b = u b − min(u a , u b , u c )
'
u = u − min(u , u , u )
c
a
b
c
 c
C. System Configuration
Figure 7 shows the schematic of the simulation.
The hall sensor generates the absolute position
( θ n ) at time ( t n ) to the speed calculator. In the
speed calculator, the low pass filter (LPF) generates
estimated rotor speed by the input of the transient
speed. The estimated rotor position is adjusted by
adding the incremental angle ( ∆θ r ) in ( t − t n ) to θ n .
Finally, the three phases of PWM are generated by
the inverter. The inverter mechanism is based on
Equations 29 and 33, and the motor model uses the
Equation 32. There is no current loop and voltage
loop in the inverter block.
Fig. 5. General three-phase PWM.
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
Vdc
θ n − θ n −1
Fig. 7. Simulation schematic.
D. Simulation Results and Comparison
Figure 8 shows simulation results from the start.
When the speed is low, the current is not stable due
to the low resolution position sensor. The torque
and current become stable after the MI increases to
the maximal value. Figure 9 shows the stable
segment of the simulation. There are 6n harmonics
in the developed torque due to the non-sinusoidal
back-EMF. The currents are synchronized with the
back-EMF by fine tuning the MTPA angle.
Figure 10 shows the simulation result for the
traditional six-step control result. The simulation
uses the same motor parameters and load characters.
There are six developed torque pulses in a electrical
cycle. To compare the torque ripple, the torque
pulse has been minimized by optimizing the
commutation angle.
In Figure 9 and 10, the operation frequencies are
almost the same. That means these two kinds of
methods work at the same load point. In Figure 10,
the magnitude of the torque ripple is 0.15Nm. In
Figure 9, this value is 0.08Nm, which is half the
traditional control result in Figure 10.
Fig. 8. Modified SPWM control simulation results from 0 to 0.6s.
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
Fig. 9. Modified SPWM control simulation results from 5(s) to 5.2(s).
Fig. 10. Six-step control simulation results from 5(s) to 5.2(s).
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
IV.EXPERIMENT RESULTS
A. Experimental Setup
The system is configured in Figure 11. The
system includes the BLDC motor, hall sensor,
controller, and Fairchild Smart Power Module
(SPM®), and can be assembled in the motor. In
this experiment, the selected SPM is the
FSB50450, a 500V three-phase FRFET® inverter
that includes a high-voltage integrated circuit
(HVIC). The controller is the TB6551, a
sinusoidal PWM controller for motor control. The
three hall sensors are EW632’s.
The motor parameters are shown in Table II.
Figure 12 shows the back-EMF of this BLDC fan
motor. It’s a typical 100w BLDC and the fifth and
seventh harmonic back-EMF are about 5% and
0.5%, respectively.
Fig .11. System configuration.
TABLE II
PARAMETER OF THE EXPERIMENTAL MOTOR
Pole Number P
8
Stator Resistance rs
33
Ω
Equivalent Self-Inductance L s
0.1
H
Maxim Back-EMF Coefficient
Terminal Voltage Angle
Inertia
Jm
K e max
0.18
δ
V·s/rad
0.08
rad
0.00012
Kg·m2
Fig. 12. One-phase back-EMF of the fan motor.
Figure 13 shows the air-conditioner outdoor
unit in which the tested fan is assembled. Figure
14 shows the tested fan motor in detail, while
Figure 14(a) shows the fan assembled in the air-
conditioner. The case of the outdoor unit has been
dissembled. Figure 14(b) shows the fan separately,
while Figure 14(c) and (d) show the built-in PCB
board and the fan motor, respectively. The back
case of the motor is the heat sink of the SPM after
assembly. The SPM has been marked. SPM
simplifies the built-in PCB design because of
high-density integration and good thermal design.
The motor winding has concentrated full-pitch
distribution. The hall sensor is positioned on the
bottom side of the PCB to detect the direction of
the magnetic field.
B. Experimental Results
Figure 15 shows the starting waveform.
Channel 1 and channel 3 are the PWM and the
current of phase a, respectively. Figure 16 shows
the stable working waveform. Channels 1-4 are
the PWM in phase a, PWM in phase b, current in
phase a, and PWM in phase b, respectively.
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
There is no current spike and this shows the
motor operation is stable and that the position
calculation works well.
and dynamic performance are not required.
Fig. 13. The tested fan assembled in the air-conditioner.
Fig .15. Operation waveform from start.
(a)
(b)
(c)
(d)
Fig. 14. The fan motor for test.
Fig .16 Operation waveform when the motor is stable.
V. CONCLUSION
There is a 6n harmonic torque instead of
pulsating torque in the motor developed torque
when the BLDC motor is driven by sinusoidal
PWM. In the simulation, compared to the
traditional six-step control, this method can
reduce ripple about 50%. The test result shows
this method can be used when the speed precision
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[1]
[2]
[3]
Fernando Rodriguez and Ali Emadi, “A Novel Digital Control
Technique for Brushless DC Motor Drives,” IEEE Trans. Ind.
Electron., vol. 54, no. 3, pp. 2365-2373, Oct. 2007.
P. Pillay and R. Krishnan, “Application Characteristics of Permanent
Magnet Synchronous and Brushless DC Motors for Servo Drives,”
IEEE Trans. Ind. Appl., vol. 27, no. 5, pp. 986–996, Sep./Oct. 1991.
Bimal K.Bose, Modern Power Electronics and AC Drivers. Prentice
Hall, 2002.
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FAIRCHILD SEMICONDUCTOR POWER SEMINAR 2008 - 2009
[4]
[5]
[6]
[7]
[8]
Y. Liu, Z. Q. Zhu, and D. Howe, “Direct Torque Control of
Brushless DC Drives with Reduced Torque Ripple,” IEEE Trans.
Ind. Appl., vol. 41, no. 2,pp. 599–608, Mar./Apr. 2005.
D. K. Kim, K. W. Lee, and B.-I. Kwon, “Commutation torque ripple
reduction in a position sensorless brushless DC motor drive,” IEEE
Trans.Power Electron., vol. 21, no. 6, pp. 1762–1768, Nov. 2006.
H.M.Ryu,” Synchronous Reference Frame d-q Modeling of Interior
Permanent Magnet Synchronous Motor (IPMSM),” unpublished.
C. W. Lu, “Torque Controller for Brushless DC Motors,” IEEE
Trans. Ind. Electron., vol. 46, no. 2, pp. 471–473, Apr. 1999.
P. Pillay and R. Krishnan, “Modeling of Permanent Magnet Motor
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Nov. 1988.
Shucheng Wang was born in Jinzhou, China in
1979. He received a B.S. degree from Haerbin
University of Science & Technology, Haerbin,
China in 2001, and an M.S. degree in electronic
engineering from Huazhong University of
Science & Technology, Wuhan, China in 2004.
Mr. Wang works for Fairchild Semiconductor in
Shenzhen, China, where he is a technical and
marketing engineer for the application support
center team. His research interest is motor
control.
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