Open-loop Control for Permanent Magnet
Synchronous Motor Driven by Square-wave Voltage
and Stabilization Control
Daisuke Sato

Jun-ichi Itoh

Department of Electrical Engineering
Nagaoka University of Technology
Nagaoka, Niigata, Japandsato@stn.nagaokaut.ac.jp

Department of Electrical Engineering
Nagaoka University of Technology
Nagaoka, Niigata, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— In this paper, the open-loop control for permanent
magnet synchronous motor (PMSM) driven by a square-wave
voltage is proposed. The proposed control is able to transit fromthe PWM region to the square-wave voltage region via the overmodulation region seamlessly. When the PMSM is driven by thesquare-wave voltage using the proposed control, the lowfrequency torque vibration except 6th or 12th order harmonics
occurs due to the transition to the square-wave voltage region.
Therefore, a variable band pass filter (BPF) is introduced as the
reduction method of the low-frequency torque vibration. The lowfrequency component in the active current for the stabilization
control is extracted by the filter. As an experimental result, theproposed method reduces the low-frequency vibration which canbe represented for the torque by over 40%.

feedback control in the over modulation region [9]. This control
is able to be applied into both the PWM region and the overmodulation region. However, the operation is not considered in
the square-wave voltage region. Furthermore, the instantaneous
current control in the square-wave voltage region has been
proposed [10]. This control method is also applied to both the
PWM region and the over modulation region. Therefore, theswitchover of the controller is not necessary, although a fieldprogrammable gate array (FPGA) is required in order to apply aspace vector modulation to the inverter. The inverter is required
to switch every 180 degrees in order to generate the square-wave
voltage. Therefore, the FPGA which operates in several MHz is
required. In other words, it is impossible to switch every 180
degrees accuracy only by the single-chip microcomputer which
generates PWM based on triangular-wave comparison method.
Therefore, the error in the inverter output voltage occurs. As aresult, low-frequency torque vibration except 6th or 12th orderharmonics occurs in the square-wave voltage region.

Keywords—Permanent magnet synchronous motor; Squarewave voltage; Open-loop control; Low-frequency torque vibration

I. INTRODUCTION
Recently, adjustable speed drive systems with a permanent
magnet synchronous motor have been actively studied [1-5]. A
voltage source inverter is applied to the drive systems widely,whereas the PMSM is driven by the PWM voltage, the overmodulation voltage or the square-wave voltage. In addition, the
PMSM is typically controlled by a field oriented control.
However, it is difficult to control the PMSM by the typical fieldoriented control in the over modulation region and the squarewave voltage region because the controller becomes instable dueto the harmonic currents [6]. Therefore, the control strategies in
the over modulation region and the square-wave voltage region
have been proposed, respectively [7-8]. The harmoniccomponent is removed from the detected current by a low passfilter in order to implement a current feedback control in the overmodulation region [7]. On the other hand, the torque iscontrolled by adjusting the voltage phase in the square-wave
voltage region [8]. However, the switchover of the controller isnecessary according to each voltage region. Therefore, theimplementation of the controller becomes complex and thestability of the system is difficult to design.
In addition, it is proposed that the harmonic current isestimated from the motor and the inverter parameters used for aposition sensorless control in order to implement the current

This paper proposes a control strategy for the PMSM basedon the open-loop control implemented into the single-chip
microcomputer, the control period of which is constant. The
proposed method is able to transit from the PWM region to thesquare-wave voltage region via the over modulation region
seamlessly. In particular, V/f control implemented with thestabilization control is applied in both the PWM region and theover modulation region [11-12]. In the square-wave voltage
region, only the frequency is controlled. In principle, the torqueincludes the ripple, the frequency of which is six times thefundamental frequency. However, an additional lower orderharmonic component also occurs in the square-wave voltage
region, which fluctuates the torque. Hence, a reduction methodof this low-frequency torque vibration is necessary. Therefore, avariable BPF is introduced. It extracts the low-frequency
component from the active current for the stabilization control.
This method reduces the low-frequency torque vibration without
an additional sensor. First, this paper presents the characteristicof the open-loop control for the PMSM. Then, the operation
verification of the PMSM is simulated in the square-wave
voltage region. Next, the cause of the torque fluctuation isclarified and the low-frequency torque-vibration reduction

q

+

−

δ

+

e

v

−
+

−

i

iδ

ϕ

γ

1
s

iγ

d
Fig. 1
Relationship between d-q coordinate and γ−δ coordinate. iδ isactive current and iγ is reactive current.

method is proposed. Finally, the proposed method is verified by
the simulation and the experiment.

Fig. 2Control block diagram of open-loop control with stabilization
control for PMSM. The stabilization control is nesessary for the PMSM
drive.

II. OPEN-LOOP CONTROL OF PMSM
A. Square-wave Voltage Drive by Open-loop Control
Generally, the direction of the flux in the permanent magnet
is defined as d-axis and the direction of the back electromotive
force is defined as q-axis in the field oriented control of the
PMSM. However, d-q coordinate is not able to be defined in the
open-loop control because the position of the magnetic pole isunknown. Therefore, the direction of the inverter output voltage
is defined as δ-axis, whereas the axis which lags by π/2 rad fromδ-axis can be defined as γ-axis in the open-loop control system.

(a)

PWM region.

Fig. 1 shows the relationship between the d-q coordinate andthe γ−δ coordinate. The δ-axis component of the current vectoriδ expresses the active current because the δ-axis aligns in thesame direction of the output voltage vector. Similarly, the γ-axis
component of the current vector iγ expresses the reactive current.
Fig. 2 shows the control block diagram of the open-loop
control with the stabilization control for the PMSM [11-12]. The
PMSM is driven by the V/f control in the PWM region and theover modulation region. In particular, the relationship between
the fundamental amplitude of the output voltage and themodulation index is nonlinear in the over modulation region.
Hence, the transition control, the modulation index in which iscorrected, is necessary [9, 12]. In addition, the voltage amplitude
is constant and the frequency is controlled only in the squarewave voltage region. Furthermore, in case that the PMSM isdriven by a simple open-loop control, the control system
becomes instable due to the fluctuation of the load angle.
Therefore, the active current iδ is fed back in order to adjust theoutput frequency command ω*. By this method, the control
system becomes stable because the equivalent feedback of theload angle is realized [11].
Fig. 3 shows the triangular-wave comparison method in eachvoltage region. The amplitude of the modulating wave is smaller
than that of the triangular-wave in the PWM region. On the other
hand, the amplitude of the modulating wave becomes larger thanthat of the triangular-wave by the transition control in the overmodulation region and the square-wave voltage region.
Therefore, each voltage waveform is able to be generated by thetriangular-wave comparison method.

(b)

(c)

Over modulation region.

Square-wave voltage region.

Fig. 3
Triangular-wave comparison method in each voltage region.
The each voltage waveform is able to be generated by the triangular-wave
comparison method.

Then, the PMSM is driven by the open-loop control andtransited from the PWM region to the square-wave voltage
region in simulation. The parameters of the PMSM in thesimulation as well as the experiment are shown in Table 1. The
two-level inverter, the carrier frequency of which is 10 kHz isapplied in the simulation.
Fig. 4 shows the simulation results of the PMSM driven by
the open-loop control. The speed is controlled smoothly fromthe PWM region to the square-wave voltage region via the overmodulation region. However, the torque fluctuation increases inparticular speed of the square-wave voltage region.

TABLE I.

PARAMETERS OF THE PMSM

Rated power

Fig. 5 shows the torque harmonics analysis result at several
points of speed in the square-wave voltage region. In principle,
when the PMSM is driven by the square-wave voltage, theharmonic components, the orders of which are the multiples ofsix compared to the fundamental frequency occur in the torque.
However, the lower order harmonic which is not the multiple ofsix also occurs. This significantly increases the torquefluctuation at ω = 0.74 p.u. and ω = 0.96 p.u. Because theexcessive torque fluctuation worsens the speed response, thereduction of the low-frequency torque vibration is necessary.

3 kW

Rated torque8 Nm

Maximum speed

12000 min-1

Rated current

17.3 Arms

Pole number

4d-axis inductance

2.04 mH

q-axis inductance

2.24 mH

Winding resistance

133 mΩ

Linked flux

0.1066 Vs/rad

Moment of inertia

0.0013 kgm2

B. Analysis of Control System
The cause of the increase in the low-frequency torquevibration is investigated. Therefore, the gain characteristics ofthe control system are obtained. In addition, the PMSM
subjected to the analysis is a surface PMSM which the d-axis
inductance equals to the q-axis inductance in order to simplify
the analysis. First, the state equation and the output equation of
the control system are obtained. The voltage of the PMSM on
the γ−δ coordinate system is expressed in (1),dvγ   R + dt L − ω1 L  iγ  sin ϕ    + ωψ m 
 =

d
cosϕ vδ   ω1 L
R + L iδ 
dt 


(1),

where vγ is the γ-axis voltage, vδ is the δ-axis voltage, R is thearmature winding resistance, L is the armature inductance, ψm is
the maximum value of armature linked flux, ω is the angularfrequency of the PMSM, ω1 is the angular frequency of inverter
output and ϕ is the phase difference between the d-q coordinate
and the γ-δ coordinate. Next, the torque is expressed in (2),T = Pf ψ m iq = Pf ψ m (iγ sin ϕ + iδ cos ϕ )(2),Fig. 4
Simulation results of PMSM driven by open-loop control. The
torque fluctuation increases in the square-wave voltage region.

where Pf is the number of the pole pairs and iq is the q-axis
current. In addition, the relationship between the angularfrequency of the PMSM and the torque is expressed in (3).

(3)

Furthermore, the rotational speed of the γ-δ coordinate is sameas that of the d-q coordinate in the steady state. However, therotational speed of two coordinates is varied by changing therotational speed and the stability control in the transient state.
Therefore, the differential of the phase difference is derived asin (4),

0.74 p.u.
0.85 p.u.
0.96 p.u.

Lower order harmonics

Torque [p.u.]Pdω= f T
dt
J

0.15

0.1
6t h order harmonics

0.05
12th order harmonics

0

1000

2000
3000
Frequency [Hz]

4000

5000

Fig. 5Torque harmonics in square-wave voltage region. The lower
order harmonic which is not the multiple of six also occurs.

dϕ = ω1 − ω − K 1iδ
dt

(4),
0

Gain [dB]

where K1 is the gain of the stability control. In addition, the phasedifference ϕ is same as the load angle by the definition of the γδ coordinate.
The open-loop control system is expressed as the fourthorder state equation by (1), (3) and (4). These equations are
nonlinear. Therefore, these equations are linearized in thesteady-state operating points. The state equation and the output
equation after the linearization can be expressed in (5) and (6),respectively. Suffix ‘0’ implies the values at the steady-state
operating points. Then, the gain characteristics of the control
system are obtained by (5) and (6). In addition, in order toevaluate the relationship between the gain characteristics and the
speed of the PMSM, the gain characteristics are obtained atseveral different points of speed.

10

−

R
K 1iγ 0 −L−

2

Pf ψ mJ

cos δ 0

− K1

ψm
L

ψm
L

sin δ 0cos δ 00−1[

ΔT = Pfψ m sin ϕ 0

100
1000
Angular frequency [rad/s](a)Characteristic from Δvγ to ΔΤ.

Gain [dB]

-20

-40

0.67 p.u.
0.79 p.u.
0.85 p.u.
0.90 p.u.
1.00 p.u.

-60
10

100
1000
Angular frequency [rad/s]

(b)

10000

Characteristic from Δvδ to ΔΤ.

Fig. 6
Gain characteristics of open-loop control for PMSM. The
control system has resonance frequency, which equals to the inverter
output frequency.

fluctuation of the γδ-axis voltage, which worsens the torquefluctuation.
III.

REDUCTION METHOD

OF

LOW-FREQUENCY TORQUEVIBRATION
The reduction method of the low-frequency torque vibrationis considered. The low-frequency torque vibration increases
ψm


1

 Δiγ  Lψm
 Δiδ  
ω 0 sin δ 0



0
+

L
2
  Δω  
Pf ψ miγ 0 cos δ 0 − iδ 0 sin δ 0   Δϕ   0

J

0
0

−

L

ω 0 cos δ 0(

Pfψ m cos ϕ 0

10000

0

The γ-axis voltage and the δ-axis voltage are fluctuated dueto the transition to the square-wave voltage. In the square-wave
voltage region, the control response becomes slow because thevoltage changes only twice a period. Furthermore, the error of
the voltage occurs because it is impossible to switch every 180
degrees by the single-chip microcomputer which generate the
PWM based on triangular-wave comparison method. Therefore,
the load angle easily fluctuates with the inverter output
frequency which results in the fluctuation of the γδ-axis voltage.
Consequently, the low-frequency torque vibration occurs andthe δ-axis current which is fed back to the stabilization control
also fluctuates. The fluctuation of the δ-axis current leads to theω 0 − K 1 iδ 00.67 p.u.
0.79 p.u.
0.85 p.u.
0.90 p.u.
1.00 p.u.

-40

-60

Fig. 6 shows the gain characteristics of the open-loop control
for the PMSM. In the control system, these are three gain
characteristics because the state equation has three input
variables. However, Δω1 is zero at the steady-state operating
points. Therefore, only the gain characteristics of Δvγ and Δvδ areverified. From Fig. 6, it is understood that the control system has
resonance frequency. The resonance frequency equals to theinverter output frequency. Hence, in case that Δvγ andΔvδ contain the resonance frequency component whose
frequency is similar to the inverter output frequency, the lowfrequency torque vibration increases.


R
−

L
Δiγ    − ω0d Δiδ =dt  Δω   P 2ψ   f m sin δ0 Δϕ   J
0


-20

)

 Δiγ  Δi
0 Pfψ m iγ 0 cos ϕ 0 − iδ 0 sin ϕ 0  δ  Δω  
 Δϕ 

(

)]

0
1
L
0
0


iδ 0 
  Δv γ − iγ 0   Δvδ 
0  Δω1 

1 

(5)

(6)

because the γ-axis voltage and the δ-axis voltage are fluctuatedby the transition to the square-wave voltage. Therefore, it isnecessary to suppress the effect of the fluctuation of the δ-axis
current for the drive system.
Fig. 7 shows the control block diagram of the open-loop
control with the low-frequency torque vibration reduction
control in the square-wave voltage region. The proposed control
consists of the band pass filter (BPF). The resonance frequency
component in the δ-axis current is extracted by the BPF. Then,
it is fed back to adjust the inverter output frequency command.
In addition, the mid-band frequency of the BPF is changed by
the inverter output frequency command (variable BPF) because
the resonance frequency of the control system changes
according to the speed, i.e. the inverter output frequency. The
variable BPF consists of the biquad filter. Furthermore, theproposed control is not necessary in the PWM region and theover modulation region. Therefore, the filter gain K2 is zero inthese regions.

0

v

+

−
+

f/V
conv.

HPF
Variable
BPF

v

Transition
control
+
+
-

K1

+

K2

uvw

+

−

Sv
Sw

1
s

i

Low-frequency torquevibration reduction control

−

Su

i

uvw

iu
iv
iw

Fig. 7
Control block diagram of open-loop control with low-frequency
torque vibration reduction control in square-wave voltage region. The
reduction control is consist of the valiable BPF.

Fig. 8 shows the block diagram of the variable biquad filter.
The coefficients are varied by the inverter output frequency. In
addition the biquad filter is expressed by (7),y n = b0 u n + b1 z −1 u n + b 2 z −2 u n − a1 z −1 y n − a 2 z −2 y n(7),where un is the input signal, yn is the output signal and a1, a2, b0,
b1 and b2 are the coefficients. If the BPF consists of the biquadfilter, the coefficients are expressed by (8) ~ (12),−2 cos ω c
1+ α

(8),

1−α
1+ α

(9),

α
1+ α

(10),

fc is the center frequency, fs is the sampling frequency and Q is
Q factor. In this case, only a1, a2 and b0 is calculated due to b1 =0 and b2 = -b0. In addition, the coefficients are changed by thecenter frequency.

b1 = 0

(11),

α
= −b0
1+ α

(12),

Fig. 9 shows the coefficient characteristics of the BPF on thebiquad filter. The sampling frequency is 10 kHz and Q factor is
0.7. The all coefficients are proportional to the center frequency,
i.e., the coefficients are expressed by the linear equation.
Therefore, the memory usage of the controller is not large.

2πf c
fs

(13),

a1 =

a2 =

b0 =

b2 = −
whereωc =Fig. 8Block diagram of variable biquad filter. The coefficients arevaried by the inverter output frequency.

α=sin ω c2Q

(14).

Fig. 10 shows the simulation result of the PMSM drive by
the open-loop control with the low-frequency torque vibration
reduction control. The simulation condition is same as Fig. 3. It
is confirmed that the fluctuation of the torque is reduced by the
proposed method in the square-wave voltage region.

a1

-1.65
-1.7
-1.75
250

300
350
Center frequency [Hz]

(a)

400

a1

0.8
a2

0.78
0.76
0.74
0.72
0.7

250

300
350
Center frequency [Hz]

(b)

400

a2

Fig. 10
Simulation result of PMSM drive by open-loop control with
low-frequency torque vibration reduction control. The fluctuation of thetorque is reduced by the reduction method.

0.15

b0

0.14
0.13

0.74 p.u.
0.85 p.u.
0.96 p.u.

0.14

0.12

0.12

0.1
250

300
350
Center frequency [Hz]

(c)

400

b0

Torque [p.u.]

0.11

0.1
0.08

Lower order harmonics
6th order harmonics

0.06
0.04

12t h order harmonics

0.02

Fig.9
Coefficient characteristics of band pass filter on biquad filter. The
all coefficients are proportional to the center frequency.

Fig. 11 shows the torque harmonics when the low-frequency
torque vibration reduction control is applied. The six-ordercomponent and 12-order component do not change compared toFig. 4. On the other hand, the lower order component is reduced.
Therefore, the stabilization of the torque is can be achieved.

0

1000

2000
3000
Frequency [Hz]

4000

5000

Fig. 11Torque harmonics with low-frequency torque vibration
reduction control. The lower order component is reduced compared to Fig.
4.

Fig. 12 shows the comparison of the low-frequency torquevibration. The low-frequency torque vibration decreases byapproximately 70% at the speed when the torque fluctuation
becomes large (ω = 0.74 p.u. and ω = 0.96 p.u.). However, thelow-frequency torque vibration is not removed completely
because of the delay of the control response with the squarewave voltage as explained above.
IV. EXPERIMENTAL RESULTIn this section, the proposed control is verification by theexperiment. The control period is 50 μs, whereas the variable B
PF is consist of two-order IIR filter. In addition, the parameterof the PMSM in the experiment is shown in Table 1.

Fig. 12Comparison of low-frequency torque vibration. The lowfrequency torque vibration decreases by approximately 70% at the speed
when the torque fluctuation becomes large (0.74 p.u. and 0.96 p.u.).

Fig. 13 shows the waveform of the line-to-line voltage andthe phase current. The motor speed is 1.0 p.u. From Fig. 13, it isconfirmed that the inverter is driven by the proposed control
method in the square-wave voltage region.

Fig. 14 shows the waveform of the speed and the q-axis
current when the PMSM is accelerated. In this experiment, it isdifficult to measure the torque waveform directly. Therefore, theq-axis current is evaluated instead of the torque. In addition, thepole position is detected by the position sensor only in order tocalculate the q-axis current. The controller does not use theposition sensor. Furthermore, the acceleration time is 6.67 s
from ω = 0.6 p.u. to 1.0 p.u. because the torque fluctuation is
likely to have a negative effect on the drive system when the
motor is accelerated slowly. From Fig. 9(a), it is understood thatthe fluctuation of the q-axis current increases at several points ofspeed in the square-wave voltage region. On the other hand, inFig. 9(b), the fluctuation of the q-axis current is reduced by the
proposed control.
Fig. 15 shows the waveform of the speed and the q-axis
current at ωm = 0.943 p.u. The fluctuation of the q-axis current
also occurs in the steady state as with the accelerating. Moreover,the fluctuation component of the q-axis current is 0.325 p.u.
without the low-frequency torque vibration reduction control. If
the proposed control is applied, the fluctuation component is
improved to 0.172 p.u.

Fig.13
Waveform of line-to-line voltage and phase current in the squarewave voltage region. The motor speed is 1.0 p.u.
Square-wave voltage region

Speed 0.2 p.u./div
m = 0.6 p.u.

m = 0.909 p.u.

m = 0.847 p.u.

Fig. 16 shows the comparison of lower order harmonic of the
q-axis current. The speed to compare are 0.848 p.u., 0.908 p.u.
and 0.943 p.u. At any compared speed, the fluctuation of q-axis
current is large in Fig. 9. From Fig. 10, the lower order harmoniccomponent of the q-axis current decrease by over 40% at any
compared speed.

q-axis current 0.2 p.u./div

V. CONCLUSION
In this paper, the control strategy for the PMSM based on the
open-loop control in the single-chip microcomputer is proposed.
The proposed method is able to transit from the PWM region to
the square-wave voltage region via the over modulation region
seamlessly. In addition, the low-frequency torque vibration
except 6th or 12th order harmonics occurs because the voltage
is fluctuated by the transition to the square-wave voltage.
Therefore, the low-frequency torque vibration reduction control
using the variable BPF is proposed. As an experimental result,the low-frequency vibration decreases by over 40%.

m = 0.943 p.u.

1 s/div(a)Without low-frequency torque vibration reduction control. The
fluctuation of the q-axis current increases at several points of speed in
the square-wave voltage region.

(b)With low-frequency torque vibration reduction control. The fluctuationof the q-axis current is reduced by the proposed control.

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[1]

[2]

[3]

[4]

[5]

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Fig. 14
Waveform of motor speed and q-axis current. The q-axis current
is evaluated instead of the torque.
[6]

[7]

A. M. Khambadkone and J. Holtz, “Compensated Synchronous PI Current
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[8]H. Nakai, H. Ohtani, E. Satoh, and Y. Inaguma, “Development andTesting of the Torque Control for the Permanent-Magnet Synchronous
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Speed 0.2 p.u./div

q-axis current 0.2 p.u./div

5 ms/div(a)

Without low-frequency torque vibration reduction control.
Speed 0.2 p.u./div

q-axis current 0.2 p.u./div

5 ms/div(b)
Fig. 15

With low-frequency torque vibration reduction control.
Waveform of motor speed and q-axis current at ωm = 0.943 p.u.

Fig. 16
Comparison of low-frequency q-axis current. The lower orderharmonic component of the q-axis current decrease by over 40% at any
compared speed.