Torque Ripple Reduction Methodof Permanent Magnet Synchronous Motor
by Current Sensor Gain Unbalance Correction
Jun-ichi Itoh, Yuuki Ikarashi, Koroku Nishizawa and Hiroshi Tamura
Department of Electrical Engineering
Nagaoka University of Technology
Nagaoka, Niigata, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— This paper presents a new algorithm, which corrects
gain unbalances of current sensors, applied in a conventional twophase current detection for adjustable speed drive system of
IPMSM. The gain unbalance of current sensors is corrected byutilizing the second harmonic components which is extracted byDiscrete Fourier Transform (hereafter DFT) from q-axis voltage
command. Besides, the correction process is performed insynchronization with the process of DFT. The validity of the
proposed method is confirmed by experimental results. The
torque ripples of the second harmonic components caused by thegain unbalance of current sensors is reduced by 94%.

I.

INTRODUCTION

Recently, motor drive systems using power electronics
technologies which realize saving energy are rapidly becoming
widespread due to increasing global environmental regulations.
Particularly, a permanent magnet synchronous motor (hereafter
PMSM) has a high efficiency and a small size compared withan induction motor. Thus, the PMSM is applied to the driving
source of electric vehicle [1-3]. In order to further improve theefficiency and a torque control performance of the motor drive
systems using the PMSMs, it is important to reduce torqueripple. The torque ripple is caused by spatial harmonics of apermanent magnet flux of the PMSM, the dead time of a voltage
source inverter, and current sensor detection errors. Especially,
the current detection is important to improve the performance.
The current sensor detection errors can be generally classified
into two categories, an offset error and a gain error.
Many on-line compensation methods for the current sensor
detection errors have been researched [4-6]. Ref. [4] utilized an
integral output of the d-axis current to compensate for thecurrent sensor detection errors. However, the compensation
performance heavily depends on an accuracy of a feed-forward
back EMF in the current controller. Ref. [5, 6] utilizes thevoltage command generated by a current controller tocompensate for the current sensor detection errors. However,these compensation performances deteriorate with increasing
the motor parameter errors. These compensation methods
require the motor parameters, such as a winding inductance ora back EMF, for reducing the torque ripples. Then, anotheralgorithm that can compensate for the current sensor detection
errors without using the motor parameters has been also
reported by authors [7-9]. These method can compensate thecurrent sensor offset errors by using DC voltage components
included in phase voltage commands. However, thesecompensation method cannot correct the gain unbalance of thecurrent sensors.
This paper presents a new correction method for the gain
unbalance of the current sensors at a two-phase current
detection system for driving Interior PMSM (hereafter IPMSM)
without using the motor parameters. Firstly, the proposedcorrection method extract the second harmonic components,
which is caused by the unbalance of the current sensors gain
and included in the q-axis voltage command generated by thecurrent controller, by DFT. Then, by compensating V-phasegain so as to match U-phase gain, the gain unbalance of thecurrent sensors is corrected.
In this paper, Section II describes the proposed algorithms
for correcting the gain unbalance of the current sensors. Section
III discusses the effectiveness of the proposed method withreference to experimental results.
II.

PROPOSED CORRECTION METHOD OF GAINUNBALANCE

A. Torque ripples resulting from the detection errors of thecurrent sensors
Fig. 1 shows the conventional adjustable speed drive system
of IPMSM with two-phase current detection.

cs

cs

Detected dq-axis currents i d and i q are calculated fromcs
cs
the phase currents i u and i v which are detected at the U and
V-phase current sensors.
2
icsucos (θ+ π)
3] [ ics]v
2
sin (θ+ π) ics ics
3
u v

2

ics 2 cos θ cos (θ 3 π)
[ dcs ] = [23iq-sin θ sin (θ π)3Auto Current Regulator
Kpd
id*



iq*

++-

+
Kid +
s
Kiq
s +
+
Kpq

vd*
++

true

2

cs
true
iv ≈Gcs
v Im cos (θ π+ϕ) +Ivoffset

U- and V-phasecurrent detection
systemcs

cs

where G u is a detection gain of U-phase current sensor, G v is a
detection gain of V-phase current sensor, ϕ is a phase differencecs
between a pole position θ and the detected motor current, I uoffset
cs
is a detected offset current in current sensor of U-phase, I voffset istrue
a detected offset current in current sensor of V-phase, I m is atrue amplitude of fundamental current.
From (3), it is found that the detected dq-axis currents
contain dc-components, fundamental frequency components
and second harmonic components. These components can beseparated by using addition theorem.
cs
Gcs
u sin (3 π+ϕ) +Gv sin (3 πϕ)





2

2

cs
Gcs
u cos (3 π+ϕ) +Gv cos (3 πϕ)
2

2

]

cs
Ics
uoffset sin (θ 3 π) Ivoffset sin θ

2
√3



[ cs

Iuoffset cos (θ 3 π) Ics
voffset cos θ

Itrue
m
√3

2

cs

cs

Gvcs

q

P

PS

PMSM

2

Load

]

[ cs cs]2(Gu Gv ) cos (2θ 3 π+ϕ)T  Pn keiq  Pn Ld  Lq id iq 3v*diniv*[ d* ] = [ * ] +
vq
vqini

Kid
s



) (i*d icsd)


Kiq * cs
(Kpq +) (iq iq )s[]

Kid cs2) {Iuoffset sin (θ π) Ics
voffset sin θ}
s
3
+ [
]
K
2
iq
√3 (K + ) {Ics cos (θ π) Ics cos θ}pquoffsetvoffset
s
3
2



where Pn is number of pole pairs, ke is the back EMF constant
Ld is the d-axis inductance and Lq is the q-axis inductance.
When the d-axis current is controlled to zero, the torque of
PMSM is proportional to the q-axis current. Then, from (4) and(5), it is confirmed that torque contains the ripples offundamental frequency components caused by the offset error(Kpd +



3

Kid
2
2
cs
(Kpd + ) {(1Gcsu ) sin ( π+ϕ) +(1Gv ) sin ( πϕ)}v*dini Itruems33≈[ * ]+
[
]
Kiq
2
2
vqini √3cs))
(Kpq + ) {(1Gcs
cos
(
π+ϕ)
+(1G
cos
(
πϕ)}
u
v
s
3
3





2

i*d Itrue sin (3 π+ϕ) + sin (3 πϕ)]
[ *] = m [
2
2
√3
iq
 cos ( π+ϕ) + cos ( πϕ)

2(Gu Gv ) sin (2θ 3 π+ϕ)--

B. Relationship between the current sensor detection errors
and the dq-axis voltage commands
When the detected dq-axis currents expressed as (4) are*regulated so as to match the dq-axis current commands i d and*i q expressed as (6) by conventional PI controller, dq-axis**
voltage commands v d and v q can be expressed as (7).

Besides, the equation of the PMSM’s torque is given by (5).


iwcal

Gucs

of the current sensors and the ripples of the second harmoniccomponents caused by the gain error of the current sensors.
Therefore, the correction for gain errors of the current sensors
is effective to reduce the torque ripples.

2



dq

ivcs

Fig.1. The conventional adjustable speed drive system of
IPMSM with two-phase current detection.

2
2
cos (θ π) cos (θ+ π)
33 ] ∙
22
sin (θ π) sin (θ+ π)
3
3

true
cs
Gcs
u Im cos(θ+ϕ) +Iuoffset2
cs true
Gv Im cos (θ 3 π+ϕ) +Ics
voffset2cs truecs true
cs
cs
)

G
I
cos(θ+
ϕ

G
I
cos
(θ
π+
ϕ
)

I

I
v m
uoffset voffset ]
[ u m
3



w

ke



ics Itrue[ dcs ] ≈ m [
iq
√3


iucs

uvw

iqcp



3

ics 2 cos θ[ dcs ] ≈ [iq3sin θ

iwtruewLq

idcp

cs truecs
ics
u ≈Gu Im cos(θ+ϕ) +Iuoffset

Voltage
SourceInverter
iutrue iv

wLd

Equation (1) can be expressed as (3) by defining thedetected dq-axis currents as (2).
{ cs

uvw

vw*

-

where θ is a motor’s electrical angle.



v u*
vv*

vq*

+
-

dq

(Kpd +

K



Itrue

+m [
√3

2

cs
(Kpd + id) (Gcsu Gv ) sin (2θ π+ϕ)

(Kpq+

s
Kiq
s

3
2

cs
) (Gcsu Gv ) cos (2θ π+ϕ)

]



3

where v*dini is an initial value of the d-axis voltage command,v*qini is an initial value of the q-axis voltage command, Kpd is ad-axis proportion gain, Kid is a d-axis integral gain, Kpq is a qaxis proportion gain, Kiq is a d-axis integral gain and s is the
Laplace operator.

3rd term of right hand side of (7) is the fundamental
frequency components caused by the current sensor offset error.
4th term of right hand side of (7) is the second components
caused by the current sensor gain error.
C. Correction algorithms for the gain unbalance of thecurrent sensor
Fig. 2 shows the two-phase current detection system for
driving IPMSM and the control block diagram of the proposed
method for the gain unbalance correction. This is a generalcontrol system of IPMSM where the proposed correction
method for the gain unbalances of the current sensors added.
Current sensors are installed in U-phase and V-phase.
In the proposed method, by using the second harmoniccomponents included in the q-axis voltage command value,
expressed as 4th term of right hand side of (7), the gain
unbalance of the current sensors are corrected. In particular, atfirst, it extracts the second harmonic components from the qaxis voltage command by using DFT. The DFT is expressed asfollows.
N 1

Aq k   aq n e



j

2kn
N



Here, Δθ is determined as follows.
q 

Aq 2 





2

q

N



Aq 2  a ne j 2 nq

q

n0



q

n 0

 2 
1
 q 



 2 
1
 q  a n  cos2nq   j   a n  sin 2nq 
q

q

n 0

ext

vqext2 h 

2NA  cosq
2

q2

q2
2



q



 2  1
 \  2  1 q    q   aq n cos2nq      aq n sin 2nq  n0
  n0


 


2




 \  2  1
 1  n0q   aq n sin 2nq  
* cos tan  2 
  \  q  1 a n cos2nq  

q
n0



ext

The extracted second harmonic components v q2h from theq-axis voltage command v*q is equal to 4th term of right handside of (7). Therefore, (14) is obtained.
 2  q  1 a ncos2nq   j sin 2nq 

Finally, the second harmonic component v q2h included in
the d-axis voltage command v*q can be extracted by substituting(12) to (13).

where N is a number of sampling in one cycle of electrical angleof the motor, k is a harmonic order and aq(n) is a sampled value.





 2  q 1



n 0



N is expressed by (9). By substituting k = 2 and (9) to (8),(10) can be calculated. And from this (10), second harmoniccomponents can be extracted from the q-axis voltage command
value.

m  0,1,2, 

Then, (12) is obtained by expanding (10) by using an
Euler’s formula.



n0

2
4k  mvqext2 h 

K iq  cs2 Gu  Gvcs cos 2q      
I mtru e  K p q 
s 
3


3




1







Auto Current Regulator
Kpd
id*
iq*

++-

+
Kid +
s
Kiq
s +
+
Kpq

vd*
++

vq*

+
-

dq

vu*
vv*

uvw

vw*

Voltage
SourceInverter

-

wLd

true

iutrue iv

Gain unbalancecompensator

wLqvqext2 htodc

Eq.(16)

idcp
iqcp

ke

w

Eq.(13)

Eq.(15)



Eq.(19)cs
I um

Amplitude
detector

U- and V-phasecurrent detection
system
iucs

uvw
dq

ivcp
iwcal

--

iwtruevqext2 h

Gvcp

ivcs

q

P

Fig.2. The control block diagram of the proposed method.

Gucs

Gvcs

PS

PMSM

Load

By dividing both sides of (14) by cos 2q      , (15) is

2
3

Table I. Experimental parameters for two-phase-current
detection system.



obtained.
vqext2 h2cos 2q     
3





1
3

SymbolPnke
Ld
Lqiuoerr
Ivoerr

K iq  cs Gu  Gvcs  vqext2 h to d c 
I mtru e  K p q 
s 






ext

cs

From (15), it is found that when v q2htodc is positive, G u is
cs
ext
cs
larger than G v . And when v q2htodc is negative, G u is smallercs
than G v . Based on these relationships, the correction gain ofext
V-phase sensor is increased when v q2htodc is positive and theextcorrection gain of V-phase sensor is decreased when v q2htodc is
negative, as shown in (16).
Gvcp  Gvocpld  Kig  vqext2h to d c csGuMeaning
Number of pole pairs
Back EMF constantd-axis inductance (typical)
q-axis inductance (typical)
U-phase current sensor offset error
V- phase current sensor offset error
U-phase current sensor gain

cs

Gv
Kig
Δθ



Value
3
0.284 [Vs/rad]4.3 [mH]
10.2 [mH]
0 [A]
0 [A]
1.2

V-phase current sensor gain

0.9

Integral gain for gain error correction
Predetermined amount of motor
electric angle change

0.001
π/8

Correction start*cpv

q-axis voltage command vq [10 V/div]

cp
vold

where G is a correction gain of V-phase sensor, G is aprevious value of correction gain of V-phase sensor and Kig is
an integral gain for gain error correction.
The detection gain of V-phase can be matched that of Uphase by multiplying the detection gain of V-phase sensor by
the correction gain of V-phase sensor as shown in (17).
Gvcp Gvcs  Gucs Extracted second harmonic components vqext2 h [10 mV/div]0

4 mVp-p
15 mVp-p

66% reduction

cp

Also, the detected current i u from U-phase current sensor canbe expressed as (18) by applying the current sensor offset error
compensation method in advance. Then, the phase difference
ϕ can be calculated by (19).
iucs  Gucs I mtru e cosq     I ucso ffset  Iˆucso ffset
 Gucs I mtru e cosq   



I

tru e
um



cosq   

 i cs   cos1  ucs   q 
 Iu m 



cs
um

where I is the amplitude of the fundamental current detectedin U-phase current sensor and Iˆucso ffset is the estimated offset
current value of U-phase current sensor.
III.

EXPERIMENTAL RESULTS AND SIMULATION RESULTS

Table I shows the experimental conditions. The
experimental system for the two–phase-current detection
consisted of dc-link voltage of 282 V, a general three-phase VSI,an IPMSM with rated power of 5.5 kW. This section indicatesthe effectiveness of the proposed method for reducing torqueripple of the second harmonic components caused by the gain
unbalance of the current sensors.
Fig.3 shows the experimental waveforms of the q-axis*
voltage command v q and the extracted second harmonicext*
components v q2h , expressed as (13), from v q . The extractedext
second harmonic components v q2h is used for the gain

[2 s/div]

Fig.3. Experimental waveforms of q-axis voltage commandand the extracted second harmonic components from q-axis
voltage command.
unbalance correction as (15). Also, this components decrease if
the gain unbalance of the current sensors is corrected. As aext
result, the amplitude of v q2h is reduced by applying the
proposed method.
Fig.4 shows the experimental waveforms of gain of U-phasecurrent sensor, the corrected gain of V-phase current sensor
expressed as (17) and the calculated torque when the proposed
method is applied. Here, it is assumed that the gain errors of thecurrent sensors are within 20% in order to confirm theeffectiveness of the proposed method. U-phase and V-phasegain errors of the current sensors set to 1.2 and 0.9, respectively.
V-phase gain is corrected with 25% relative error so as to match
U-phase gain by applying the proposed method. Also, it isconfirmed that the torque ripple is reduced by approximately
50 % by correcting V-phase gain. The gain correction error iscaused by low resolution of DFT.
Fig.5 shows the expanded experimental waveforms of thecalculated torque. (a) is the waveforms of the calculated torquewithout the proposed method and (b) is that with the proposed
method. Torque ripple of the second harmonic components
caused by the gain unbalance of the current sensors isdrastically reduced by applying the proposed method.

Fig.6 shows the harmonic analysis results of the torque. It is
confirmed that the ripple of second harmonic components
caused by the gain unbalance of the current sensors is reducedby 94% by applying the proposed method. Here, remaining
torque ripples of 6th-order components and 12th-ordercomponents are resulting from the spatial harmonics ofpermanent magnet flux and the dead time of inverter.

Detection gain of U-phase sensor Gucs [0.25 -/div]
Corrected gain of V-phase sensor Gvcp Gvcs [0.25 -/div]
Calculated torque [1.75 Nm/div]

CONCLUSION

This paper presents a new algorithm which can correct thegain unbalance of the current sensors in the conventional twophase current detection system for driving IPMSM by using thesecond harmonic components included in the q-axis voltage
command value. From experimental results, it is confirmed thatthe torque ripple is reduced by 50% by correcting the gain
unbalance of the current sensors, in particular, by matching Vphase gain and U-phase gain. From harmonic analysis results,
the torque ripple of second harmonic components caused by thegain unbalance of current sensors is reduced by 94%.

[2]

[3]

[4]

[5]

[6]

[7]
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50% reduction

0[2 s/div]0

Fig.4. Experimental waveforms of the gains of U and Vphase current sensors and the calculated torque when the
proposed method is applied.

REFERENCES[1]

0.875 Nmp-p

1.75 Nmp-p

Calculated torque[1.75 Nm/div]

Calculated torque[1.75 Nm/div][5 ms/div][5 ms/div]00(a) Without the proposed method.

(b) With the proposed method.

Fig.5. Expanded experimental waveforms ofthe calculated torque.

8.0
7.0

Amplitude [Nm]IV.

Correction start

Without proposed method

6.0With proposed method

5.0※ Fundamental frequency components :750 rpm * Pn /60 = 37.5 Hz

4.0
3.0
2.0

94% reduction

1.0

0.0
0

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15

Harmonic order※ [-]Fig.6. Harmonic analysis results of torque ripples.