IEEJ International Workshop on Sensing, Actuation, Motion Control, and Optimization

Eﬀect of Harmonic Current Control of IPMSMs by Repetitive Perfect
Tracking Control on Torque Ripple and Iron Loss
Yuhiro Inagaki∗
Osamu Shimizu∗
Hiroshi Fujimoto∗
Yoshiki Yasuda∗∗

Student Member,
Member,Senior Member,
Member,Masahiro Mae∗
Sakahisa Nagai∗
Takayuki Miyajima∗∗
Akio Yamagiwa∗∗

Student Member
MemberMemberSenior Member

Interior permanent magnet synchronous motors (IPMSMs) have broad applicability in the industry owing to theiradvantages of fast response and high torque. However, harmonic currents caused by the spatial harmonics of the magnetic ﬂux owing to the motor structure increase iron loss and torque ripple. Suppressing harmonic currents by current
control is thought to improve the drive eﬃciency of IPMSMs because it removes the cause of the increase in iron loss.
However, this is not a control method aimed at suppressing torque ripple, including cogging torque. Many current
control methods to suppress torque ripple add compensation signals to the current command, and these signals are
expected to increase iron loss. In this study, we experimentally veriﬁed the eﬀects of harmonic current suppression
control and torque ripple suppression control on iron loss and torque ripple, and conﬁrmed the trade-oﬀ relationship.
The current controller in the experiment is based on repetitive perfect tracking control (RPTC).
Keywords: IPMSM, iron loss, harmonic current suppression, torque ripple suppression, repetitive perfect tracking control

1.

in the case of harmonic currents.
To solve the problems of harmonic current and torque ripple, our research group has proposed repetitive perfect tracking control (RPTC) (6) (7) that focuses on their periodicity synchronized with the rotation. RPTC is based on perfect tracking control (PTC) (8) .
In this paper, a current controller that can change the degree of harmonic current suppression and torque ripple suppression is implemented using RPTC. The eﬀects of harmonic current suppression control and torque ripple suppression control on iron loss and torque ripple were veriﬁed bythis current controller. By suppressing the harmonic current
through RPTC, the cause of iron loss increase can be removed, and iron loss can be reduced without using a complicated iron loss model. In addition, since the degree of
torque ripple suppression can be changed with simple parameters, system users can adjust the degree of eﬃciency improvement and torque ripple suppression according to theirrequirements.

Introduction

Permanent Magnet Synchronous Motors (PMSMs) have awide range of applications in the industry because of theirhigh torque and fast control response. Among PMSMs,
IPMSMs can obtain higher torque because they can utilize reluctance torque. However, harmonic current owing to spatial
harmonics of the magnetic ﬂux caused by the motor structurecause degradation of various characteristics. For example,
the harmonic current increases the copper loss caused by theresistance of the windings and the iron loss caused in the electromagnetic steel sheet, decreasing the operating eﬃciency.
In addition, the harmonic current also increases torque ripple, which causes vibration and noise in the equipment.
Many current controllers to improve the eﬃciency considering harmonic current have been proposed. However, somemethods do not consider iron loss (1) , and even those that consider the iron loss have problems with the accuracy of theiron loss model (2) (3) . Since the iron loss is diﬃcult to measure
directly, modeling is problematic, and there are many issues
to be considered, such as simplifying the model into a form
that is easy to handle in the control system while maintaining
accuracy, and the existence of parameters that are diﬃcult tomeasure in advance.
Many torque ripple suppression controllers use a compensation signal for torque ripple added to the current command(4) (5). However, the compensation signal added in this way isthought to have the same problem of increasing iron loss as

2.

Modeling of IPMSM and Characteristics ofIron Loss and Torque Ripple

2.1 Plant Model of the IPMSM
The IPMSM modelon the dq-axis system is shown in Fig. 1. In this study, current controllers were designed on the dq-axis. The diﬀerent
parameters are listed in Table 1. The dq-axis voltage equation
is expressed as follows:
[
] [
][
] [
]
vd
R + sLd −ωe Lq
id
0
=
+
.(1)
vq
ωe Ld
R + sLq
iq
ωe Ke

∗

The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba, 277-8561, Japan∗∗DAIKIN INDUSTRIES, Ltd.
1-1-1, Nishi-Hitotsuya, Settsu, Osaka 566-8585, Japan

In general, the dq-axis voltage is decoupled as follows:
vd = v′d − ωe Lq iq , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·(2)© 2022 The Institute of Electrical Engineers of Japan.

379

1
Ld s+R

+

vd

+

id

Motor

Load

𝑇𝑚𝑒𝑡𝑒𝑟ωeLq
Ld − Lq
ωeLd
+

vq

−

1
Lq s+R

iq

Ke

+

+

P

−

TM 0

1
JM s+BM

ωm

P

𝐾𝑠

ωe

𝑇𝑀

Ke

Load Side

Table 1. Parameters of the IPMSM and their interpretation.
Meaning

vd , vq
id , iq
Ld , Lq
R
Ke
P
JM
BM
ωm
ωe

dq-axis voltage
dq-axis current
dq-axis inductanceResistance of windings
Permanent-magnet ﬂux linkage
Number of pole pairs
Motor inertia
Viscous friction coeﬃcient
Mechanical angular velocity
Electric angular velocity

𝑇𝐿

Fig. 2. Schematic diagram of the two-inertia system.

Fig. 1. Block diagram of the IPMSM model.

Parameter

𝜔𝐿

𝜔𝑀

TL
+

+

θs
TmeterωL

1
JL s+BL

Ks

1
s

−
+

TM − Tev+

1
JM s+BMωM

Motor Side

Fig. 3. Block diagram of the two-inertia system.

vq = v′q + ωe (Ld id + Ke ). · · · · · · · · · · · · · · · · · · · · · · · · ·(3)loss is calculated for each harmonics order, and the sum ofthese is used as the iron loss as follows (11)∼(13) :∑Pir =Pir, n ,· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (9)When the state variable is deﬁned as the dq-axis current and the input as the decoupled dq-axis voltage, thecontinuous-time state and output equations of the IPMSM are
expressed as follows:nẋ(t) = Ac x(t) + Bc u(t), · · · · · · · · · · · · · · · · · · · · · · · · · · (4)y(t) = Cc x(t) + Dc u(t),· · · · · · · · · · · · · · · · · · · · · · · · · · (5)where

where n is harmonics order and Pir is the iron loss. From
Eq. (9), it is expected that suppressing the harmonic current
and reducing the harmonic components of the magnetic ﬂux
density reduces the iron loss.
2.3 Torque Ripple Measurement
Various eﬀorts
have been made in the measurement method in the study of
torque ripple suppression. Torque meters, which are often
used for torque measurement, measure the motor torque by
multiplying the shaft torsion angle caused by the torque ofthe driving side and load side motors by the spring coeﬃcient. In this study, the torque ripple is estimated by assuming that the system consisting of motors and a torque meter isa two-inertia system. The schematic diagram and block diagram of a typical two-inertia system are shown in Fig. 2 andFig. 3.
To control and evaluate the torque ripple, it is necessary toestimate the torque T M in Fig. 3 and obtain its frequency component. Using encoder position data θ M and measured valueof the torque meter T meter , motor torque T M is estimated asfollows (14) :

[

]
[ ′
]
vd (t)
id (t), u(t) =, · · · · · · · · · · · · · · · (6)v′q (t)iq (t)


 R
 1

 − Ld
 Ld 0 
0 




,



A =
,B =


− LRq  c  0 L1q  c  0][· · · · (7)





 Cc = 1 0 , Dc = O.

0 1x(t) =

2.2 Harmonic Components in General Iron Loss
Model
Many eﬃciency improvement control methods
considering iron loss use iron loss models based on Steinmetz iron loss formula or Bertotti iron loss formula (3) (9) . Asan example, the basic form of Bertotti’s equation is shown as
follows (10) :ρir ≃ C0 B2max fm +π2 σd2(Bmax fm )2 + C1 (Bmax fm )1.5 ,(8)6δ

T M = T meter + T ev = T meter + J M θ̈ M + BM θ̇ M .· · · · · (10)where ρir is the iron loss per unit mass, C0 and C1 are parameters determined by material properties, σ is the electrical conductivity, d is the thickness of the lamination, δ is thedensity of the material, Bmax is the maximum magnetic ﬂux
density, and fm is the magnetization frequency. From thismodel, it can be seen that the magnetic ﬂux density of thehigher frequency components increases the iron loss.
In addition, when calculating the iron loss owing to harmonics, an approximate calculation is sometimes used. Iron

In this research, the central diﬀerence was used to calculate the angular velocity and angular acceleration, assumingthat the RPTC can use the one sample ahead data because it
records the encoder position information in memory.
3.

Design of Current Controller

3.1 Design of the feedback controller
In general,
the PI controller is used for current control. A typical current
feedback control system is shown in Fig. 4. The PI controller

380Decoupling

idref /iqref+

Coupling term
+

−
+ u

CP I [z]H−+

Decouplingd(t)(xd [k + 1])

id /iq

−
+

1
Ls+R

Coupling term

idref /iqref++−
u0 [k] +B −1(1 − z −1A)

Holder

Switch 2

+

z −1C

+
y0 [k]

Samplerz −(Nd−1)

Fig. 4. General current PI controller.

+
PWM Hold1
Ls+R

id /iq+S−
Sampler

+
+

+

H

C2[z]

D

S

u[k]

+

Switch 1

z −1
10Fig. 7. Block diagram of RPTC with switches.
5axis is shown in Fig. 6. Subscripts d and q indicate that eachcoeﬃcient and variable are calculated on its axis. The control
system by PTC has a stable inverse system as feed-forward
controller and PI controller (C2 [z]) as feedback controller.
In the n th-order plant model, it is necessary to switch thecontrol input n times. However, the dq-axis plant model of
IPMSM shown in Fig. 6 is a ﬁrst-order model, and single-rate
implementation is suﬃcient.
To design the feed-forward controller of PTC, the state
equation (Eq. (4)) and the output equation of (Eq. (5)) are
discretized by the PWM hold (15) . PWM hold is the strict holdmethod based on the pulse voltage composed of the inverter
supply voltage ±E [V] and 0 [V], instead of assuming thatthe inverter can output an arbitrary voltage. The control input
u[k] is the inverter ON time, and PWM hold is more suitable than the zero-order hold for instantaneous value control (16) (17) .The discrete time state and output equation of the
IPMSM based on the PWM hold are expressed as follows:

0

-5

-10
50

100

150

200

250

300

350

400

450

500

Fig. 5. Sensitivity function of the PI controller and therepetitive control based on the internal model principle.
+

z −1Cd+y0d [k]+id refC2d[z]

u0d [k]++PWM Hold

Decoupling control

S

iq

+

Lq

Ke
ωe

S
ωe

iq refx[k + 1] = Ax[k] + Bu[k], · · · · · · · · · · · · · · · · · · · · · · (13)y[k] = Cx[k] + Du[k]. · · · · · · · · · · · · · · · · · · · · · · (14)Ke

Sampler

S

Ld+

Sampler

id

id

+

1
Ld s+R+ud [k] Coupling termH−

(xdd [k + 1])

S
SamplerDd
Bd−1(1 − z −1Ad)

−

Ld

+

+

Sampler

Lq
Decoupling control(xdq [k + 1])

u0q [k]Bq−1(1 − z −1Aq )+Dq

uq [k] Coupling term−1
H +Lq s+R++PWM HoldA, B, C, and D are calculated using Eq. (15). T u is thecontrol input period. E is the DC power supply voltage of a3-phase inverter.

iq

C2q [z]A = eAc Tu , B = eAc Tu /2 Bc E, C = Cc , D = Dc .· · · · (15)Samplerz −1Cq+++y0q [k]−S

From Eq. (13), and Eq. (14), the stable inverse model, andthe nominal output of PTC are expressed as Eq. (16), and Eq.
(17).

Fig. 6. Block diagram of the PTC (dq-axis current).

C PI (s) designed by pole-zero cancellation is as follows:u0 [k] = B−1 (1 − z−1 A)xd [k + 1], · · · · · · · · · · · · · · · (16)y0 [k] = z−1Cxd [k + 1] + Du0 [k]. · · · · · · · · · · · · · · · (17)

Ls + RC PI (s) =.· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (11)τs

The inverse system of PTC improves the tracking characteristic to the command value, and the tracking errors owing
to modeling errors and external disturbances are suppressedby the feedback control of the PI controller (C2 [z]).
3.3 RPTC (6)3.3.1 Harmonic Current Suppression RPTC (HCS
RPTC) (7)This section describes HCS RPTC which suppresses the harmonic currents. The block diagram of HCS
RPTC is shown in Fig. 7. The decoupling control and coupling terms are the same as those in Fig. 6. The RPTC system
has a periodic signal generator (PSG). The PSG records the
tracking error caused by the harmonic current as the periodic
error in the memory and uses it for compensation. The internal structure of the PSG is shown in detail in Fig. 8.
As shown in the block diagram in Fig. 7 and Fig. 8, HCS

τ is often set to approximately 10 times the control period.
C PI [z] is the discretized C PI (s). From Eq. (4) and Eq. (5),the transfer function from the current command id, q re f to theactual current id, q is as follows:id, q
id, q re f=

1.· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (12)τs + 1

Fig. 5 shows the sensitivity function from the output end
disturbance (d(t) in Fig. 4) to the output. This ﬁgure shows
the sensitivity functions of PI controller and repetitive control based on the internal model principle. It can be seen that
it is diﬃcult to suppress the harmonic current and extend thecontrol bandwidth with PI control.
3.2 PTC (8)The block diagram of the PTC on the dq-

381

rf

r

1
m

Q
Low-pass
filterz −(Nd−1)
Switch 2

Decoupling++

Coupling term

i∗q
iqref

Switch 1

averaging+(xd [k + 1])

+
+

−

−
u0 [k] +B (1 − z A)
−1

−1

Fig. 8. Block diagram of the PSG.

Switch 4

Switch 2z −(Nd−1)

1
Ls+R

iq

C2[z]

D
+

z −1C

H

+
PWM Hold

+

z −1

+

u[k]

+

+

y0 [k]S−

Sampler++

Switch 1

1
Kt

Tˆr

KT RS

z −1

[0]

[2]

[1]
[2]

z −(Nd−1)

[1]

d[i]

+
+

Switch 3

z −1

[0]

[3]

Fig. 10. Block diagram of TRS RPTC (Proposed
method).

[n-1][Nd -2][Nd -1]

Memory(size : Nd )Rotary Encoder(Resolution : R enc )Fig. 9. PSG for handling speed variations.

RPTC is equipped with two switches. At ﬁrst, only switch
1 is turned ON to start recording the error, and when therecording is complete, switch 1 is turned OFF and switch 2
is turned ON. By turning on switch 2, RPTC can suppress
the harmonic current like feed-forward control. In contrast to
repetitive controllers based on the internal model principle,
the feed-forward-like compensation signal of RPTC has theadvantage of not exacerbating the inter-order harmonics.
To design the PSG, we explain the memory settings at ﬁrst.
The memory size is deﬁned as Nd , and expressed by Eq. (18)using the cyclical tracking error period T d and the control period T s .
Nd =Fig. 11. Experimental setup.

The operation procedures of these switches and the operation
ﬂow of the TRS RPTC are shown as follows.
( 1 ) At ﬁrst, all switches are left OFF, and tracking to thecurrent command is started.
( 2 ) Turn on the switch 1 ON both dq-axis and start
recording the current tracking errors.
( 3 ) After recording is completed for m cycles, theswitch 1 is turned OFF and the switch 2 is turned
ON. This operation starts the harmonic suppression oneach dq-axis through the low-pass ﬁlter Q.
( 4 ) After reaching the steady-state, turn ON the switch
3 on the q-axis to start recording the position data of
encoder. No new operation is applied to the d-axis,
and the d-axis harmonic current suppression is continued.
( 5 ) The position data of encoder is recorded in thememory for p periods and averaged. The torque ripple
T r is estimated from that data and the measured valueof the torque meter. The estimated data is divided by
the torque coeﬃcient Kt and converted into the compensation signal to be given to the current command.
( 6 ) After the compensation signal is generated, theswitch 3 is turned OFF and the switch 4 is turned ON.
The compensation signal is added to the current command through the low-pass ﬁlter Q and a parameter
KT RS (0 ≦ KT RS ≦ 1) set by users.
The torque ripple is estimated from the angular velocity
and angular acceleration calculated from the encoder, the inertia and friction coeﬃcient obtained from the system identiﬁcation, and the torque meter measurement value as shownin Eq. (10). This estimated value also includes the cogging
torque.
The torque constant Kt is obtained as follows, taking thereluctance torque into account:

Td.· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (18)Ts

In addition, to remove asynchronous components and noisefrom the compensation signal, the errors of m periods arerecorded and averaged, then passed through a low-pass ﬁlter Q before the compensation signal is added to the current
command. The low-pass ﬁlter Q takes the following form to
avoid the phase delay (18) .
r f [k] = Qr[k] =z + γ + z−1
r[k],· · · · · · · · · · · · · · · · (19)γ+2where r f [k] is the ﬁlter output, r[k] is the PSG output, and γ
is the design parameter of the ﬁlter.
Assuming that the harmonic current is inﬂuenced by thespatial harmonics of the magnetic ﬂux, the appearance of thecompensation signal is considered to be highly dependent on
the rotor position. Using this characteristic, RPTC links thememory and the encoder position data as shown in Fig. 9.
This makes it possible to handle speed variation (19) .
3.3.2 Torque Ripple Suppression RPTC (TRS RPTC)(14)
In this section, Torque Ripple Suppression RPTC of
IPMSM is described. The block diagram of TRS RPTC isshown in Fig. 10. Assuming that the magnet torque is suﬃciently larger than the reluctance torque, this block diagram
controller is applied to the q-axis current. On the d-axis current, HCS RPTC is used to suppress the harmonic current
continuously. This controller has four switches on the q-axis.

Kt = P{Ke + (Ld − Lq )Id }.· · · · · · · · · · · · · · · · · · · · · · (20)To extract only the ripple component T̂ r from the estimated

382

0.2

0.2

0.2

0.18

0.18

0.18

0.16

0.16

0.16

0.14

0.14

0.14

0.12

0.12

0.12

0.1

0.1

0.1

0.08

0.08

0.08

0.06

0.06

0.06

0.04

0.04

0.04

0.02

0.02

0
-100

0

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400

0
-100

0.02

0

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400(b) HCS RPTC (KT RS = 0)(a) PI controller

0
-100

0

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400(c) TRS RPTC (KT RS = 1)

Fig. 12. Magniﬁed view of the motor torque DFT results focusing on the 6th order component.
Table 2. Parameters of the experimental IPMSM andexperimental condition.
ParameterValued-axis inductance Ld
q-axis inductance Lq
Resistance R
Permanent magnet ﬂux linkage Ke
Pole pairs P
Supply voltage Vdc (E)Carrier frequency F sControl period T s
Motor inertia J M
Viscous friction coeﬃcient BM

0.613 mH
1.21 mH
85.6 mΩ31.2 mWb
6100 V
10 kHz0.1 ms3.11 × 10−3 kgm2
3.21 × 10−4 Nm · s/rad

Table 3. Experimental temperature and phase resistance.
Value
20.3 ◦ C
86.3 mΩ
88.1 mΩ86.7 mΩ3.92 × 10−3

Pco = Iu2 rms Ru + Iv2rms Rv + Iw2 rms Rw , · · · · · · · · · · · · (24)where Pin is the input power, Pout is the output power, Pco isthe copper loss, Pir is the iron loss, and Pme is the mechanical loss. Iu rms , Iv rms and Iw rms are the phase currents. Ru ,Rv and Rw are the resistances of the windings. The winding
resistance used in the copper loss calculation is temperaturecorrected as follows (21) :torque, the zero-order component is removed by subtracting
the average of one period of the memory from the estimatedtorque when it is used as the compensation signal.
4.

Parameter
Initial winding temperature T S T i
U phase resistance Ru (T S T i )V phase resistance Rv (T S T i )W phase resistance Rw (T S T i )Temperature coeﬃcient α

R(T S T ) = R(T S T i ){1 + α(T S T − T S T i )},· · · · · · · · · · (25)Experiment Using Motor Bench

4.1 Experimental Setup
Fig. 11 shows the motor
bench used in the experiment. The power meter PW3390(HIOKI) can measure in units of 0.01W, and the torque meter
TMB307/411 (MAGTROL) can measure up to 10 Nm withan accuracy of ±0.15%. The bandwidth of this torque meteris 5 kHz. The parameters of IPMSM and experimental conditions are listed in Table 2. The rotation speed condition was
set to 1000 rpm (electric angular frequency:100 Hz), and theq-axis current command given to the IPMSM was set to 10A,
and the d-axis current command was set to −1.86 A based on
the MTPA conditional formula (20) .
√
Ke2
Ke
−
+ i2q re f .· · · · · (21)id re f =2(Lq − Ld )4(Lq − Ld )2

where T S T is the winding temperature, T S T i is the initial
winding temperature and α is the temperature coeﬃcient of
the copper resistance (20.3 ◦ C). The parameters related to resistance are as shown in Table 3.
4.2 Experimental Result
The iron loss and thetorque ripple of pole-zero cancellation PI controller, HCS
RPTC (KT RS = 0) and TRS RPTC were measured. All measurements are averaged over 1 second, and the average of 10
times for each controller is used. The relationship between
the degree of torque ripple suppression and the iron and total
loss was investigated by changing the coeﬃcient KT RS , which
adjusts the magnitude of the compensation signal, to 0, 0.25,
0.5, 0.75, and 1.
At ﬁrst, the discrete Fourier transform (DFT) results of
torque are shown in Fig. 12, which are magniﬁed graphs focusing on the 6th order torque ripple. Furthermore, the ratio of the 6th order torque ripple to the average torque measured by the torque meter are shown in Fig. 13. The resultsin Fig. 13 are the average of 10 times for each case (PI, HCS,KT RS = 0.25, 0.5, 0.75, 1), and the results in Fig. 12 arerepresentative examples. Fig. 14 shows the ratio of iron loss
and total loss to the input power. TRS suppresses all integer
multiple components of the PSG memory period. However,
we focus on the 6th order component, which is particularly
signiﬁcant.
Fig. 13 shows that 18 % of the torque ripple amplitudecaused in the case of HCS RPTC was reduced in the caseof KT RS = 1. As for the iron loss, it was conﬁrmed that 0.2 %of the iron loss caused by PI control was reduced by HCS
RPTC. In addition, Fig. 14 shows that the iron loss in HCS

All memory sizes Nd used in RPTC were determined to be
600 with the basic period to be suppressed set to the rotation frequency. The average number of current errors m was
10 and the average number of position data for torque ripple
measurement p is 150. The torque ripple measurement also
averaged the encoder position data 150 times. The parameterγ of low-pass ﬁlter Q was set to 2, and the cutoﬀ frequency
was 1.8 kHz. The parameter τ of PI controller was set to
10T s = 1 ms. Tustin transformation was used for discretization, and the transformation period was T s .
The iron loss was measured by subtracting the output andother losses from the input power to the IPMSM as follows:
Pir = Pin − Pout − Pco − Pme , · · · · · · · · · · · · · · · · · (22)Pout = T meter ωm , · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (23)

383

(4)

Ratio of the 6th order torque ripple
to torque meter measured value [%]

4.2
3.8

(5)

3.4
3
2.6

(6)
2.2
1.8

(7)

1.4
0

0.25

0.5

0.75

1

Fig. 13. Ratio of the 6th order torque ripple to average
torque for each torque ripple compensation signal coeﬃcient KT RS .

(8)

(9)

7

2.45

Ratio of total loss to input power [%]Ratio of iron loss to input power [%]

2.4
2.35
2.3
2.25
2.2
2.15
2.1

6.9

6.8

6.7

( 10 )
6.6( 11 )

6.5

2.05
2
0

0.25

0.5

0.75

1

6.4

(a) Iron loss

0

0.25

0.5

0.75

1

(b) Total loss

( 12 )

Fig. 14. Ratio of iron and total loss to average input
power for each torque ripple compensation signal coefﬁcient KT RS .

( 13 )

RPTC was increased by 8.9 % in KT RS = 1 case. The eﬃciency in HCS RPTC was 93.4 %, while that in KT RS =1 casewas 93.2 %.
As the coeﬃcient KT RS multiplied by the torque ripplecompensation signal was increased, the iron loss increased,
and the eﬃciency decreased. In addition, the torque ripple tended to be generally reduced. However, the case of
KT RS = 0.75 deviated from the tendency. It is assumed that
this is because the compensation signal caused the modelingerror on both the dq-axes owing to the coupling term that
could not be completely decoupled.
5.

( 14 )( 15 )( 16 )( 17 )

Conclusion

In this paper, we focused on the harmonic current of
IPMSMs and conﬁrmed the trade-oﬀ relationship between
the iron loss and the torque ripple when the harmonic current
was suppressed by HCS RPTC and when the torque ripple
was suppressed by TRS RPTC. Users of this system can easily adjust the balance by looking at these torque ripple andloss characteristics.

( 18 )( 19 )( 20 )

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( 21 )

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