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

Basic Study on Eﬀect of Repetitive Perfect Tracking Control on
Iron Loss of IPMSM
Yuhiro Inagaki∗
Osamu Shimizu∗
Hiroshi Fujimoto∗
Yoshiki Yasuda∗∗

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

Student Member
MemberMemberMember

Permanent magnet synchronous motor (PMSM) has wide range of usage opportunities due to its many advantages.
In order to achieve eﬃcient driving of PMSM, a number of high eﬃciency control methods have been developed. However, there are few researches that discuss the relationship between tracking characteristics to the current command anddriving loss. In this paper, we propose to apply repetitive perfect tracking control (RPTC), which is the current control
with high tracking performance to the current command, to the iron loss reduction. RPTC suppresses the harmonic
currents, which are periodic disturbances, and prevents the increase of iron loss caused by them. Simulation will conﬁrm that the presence or absence of harmonic currents aﬀects the iron loss reduction. Experiments will also show that
RPTC reduces the iron loss of IPMSM.
Keywords: IPMSM, iron loss, harmonic current suppression, repetitive perfect tracking control

1.

suppress periodic disturbances such as harmonics, RPTC has
been proposed in the past. RPTC is a repetitive control basedon perfect tracking control (PTC) (5) (6) (7) .
In this paper, the eﬀect of suppressing the harmonic currents on the iron loss is veriﬁed by comparing the iron loss
results of dq axis current control of IPMSM with PI controller
only and RPTC. The veriﬁcation is conducted by simulationand experiment.

Introduction

PMSM has many advantages such as high output powerand controllability, and is widely used in the industrial ﬁeld.
Particularly, interior permanent magnet synchronous motor(IPMSM) is often used for electric vehicles due to its large
torque. For further eﬀective utilization of energy, it is necessary to drive PMSMs with high eﬃciency through control.
Analyzing the causes of driving loss and taking measures
are important in order to achieve the high eﬃcient motor driving. The loss of PMSM is mainly classiﬁed into the copper
loss and the iron loss. The copper loss occurs in the motor
windings and the iron loss occurs in the electromagnetic steel
sheet. Typical examples of high eﬃcient current control are
id = 0 control and maximum torque per ampere (MTPA) control (1) . These methodologies reduce copper loss. However,these are not necessarily the most eﬃcient control methods
because they do not take the iron loss into account. Somemethods which determine the dq axis current command sothat the sum of copper loss and iron loss is minimized havealso been developed (2) (3) . However, the tracking performanceto current command often depends on the general current PI
controller, and there are few studies which examine the eﬀectof changing the current controller on the eﬃciency.
IPMSM contains various harmonics components of the
motor current (4) and these harmonics deteriorate the tracking
performance to the current command. In addition, the tracking error caused by the harmonics current increases the ironloss and reduces the overall driving eﬃciency. In order to2.

Iron loss of PMSM

2.1 Causes of iron loss
The iron loss occurs in theiron core of PMSM and is broadly classiﬁed into the hysteresis loss and the eddy current loss (8) . The IPMSM core contains various harmonic components of magnetic ﬂux density,and these components increase the iron loss caused by minor
loops of the hysteresis loop (9) (10) .
In the simulation, suppressed and not suppressed motor
harmonic currents are set, and the eﬀect of magnitude of harmonics on the iron loss is checked from the magnitude of thehysteresis and eddy current loss.
The experiment will conﬁrm the possibility that the ironloss can be suppressed by the current controller improvement.
2.2 Simulation
The change in iron loss caused by
the harmonic current presence is veriﬁed by simulation.
JMAG-Designer (JSOL) is used for the electromagnetic ﬁeld
analysis and the iron loss analysis, and the direct linkage function of JMAG-Designer and MATLAB & Simulink(MathWorks) is used for the simulation of PI current control.
The simulation model which has the same stator and rotor dimensions as the IPMSM used in the experiment is shown in
Fig. 1. This is a 1/6 partial model for the simulation motor
size, and the analysis is performed for 60 degrees. This result
is converted to a full model using periodicity to reduce the∗

The University of Tokyo
5-1-5, Kashiwanoha, Kashiwa, Chiba, 227-8561 Japan∗∗DAIKIN INDUSTRIES, Ltd.
1-1-1, Nishi-Hitotsuya, Settsu, Osaka 566-8585, Japan© 2021 The Institute of Electrical Engineers of Japan.

296

Fig. 1. IPMSM model on simulation (1/6 model (60◦ )).
Fig. 2. The amplitude of simulated motor phase current.

Table 1. Parameters of simulation IPMSM.
ParameterValued axis inductance Ld
q axis inductance Lq
Resistance R
Permanent magnet ﬂux linkage Ke
Number of pole pairs P

1.03 mH
1.31 mH
85 mΩ
31.2 mWb
6

Table 2. The result of simulation.
without harmonics

3.764 W
1.289 W
5.053 W

3.746 W
1.271 W
5.017 W

Since the dq axis current has coupling terms, the decoupling control is widely used. In order to eliminate the coupling terms, the dq voltages are calculated as follows:
vd = v′d − ωe Lq iq ,
vq = v′q + ωe (Ld id + Ke ).

(3)
(4)

State variables are deﬁned as the dq axis current and inputs
are deﬁned as the decoupled dq axis voltage. The IPMSM
continuous time equation of state is expressed as follows:

The iron loss is simulated using only JMAG. In order tomake the simulation time realistic, the inverter model is notconsidered in the simulation of iron loss, and the phase current values in steady state are used to avoid calculating theloss in transient state.
The rotation speed is set as 1000 rpm. The amplitude ofeach component of phase currents including harmonics set
under this condition is shown in Fig. 2. The trend of ironloss caused by the presence of harmonics is investigated bycomparing this motor current and ideal current source.
The results of iron loss with and without harmonics areshown in Table 2. The iron loss values in this table are averaged over one period of the electric angle. From this result,
it is conﬁrmed that the removal of harmonic currents reduces
the hysteresis loss and the eddy current loss, and the total
iron loss is reduced by about 0.71% compared to that when
the harmonic currents are included. Therefore, RPTC which
suppresses the harmonic currents, is expected to be eﬀective
in reducing the iron loss.

ẋ(t) = Ac x(t) + Bc u(t), y(t) = Cc x(t) + Dc u(t),
where

[

]

id (t)
iq (t)(5)[]v′d (t)x(t) =, u(t) =,v′q (t)


 1
 R

 L
 − L

0 


d
d



Ac = 
 , Bc =  0
R 


0
−

L

 q



 1 0 







Cc =  0 1  , Dc = O.

(6)

0 
 ,
1 

Lq

(7)

3.2 PTC
The block diagram of PTC on the dq axis
is shown in Fig. 4. The subscripts d and q indicate thateach coeﬃcient and variable discussed is calculated on that
axis. The PTC system has feedforward controller and feedback controller (C2 [z]). The feedforward controller is the stable inverse system and achieve zero error to the target value
at the sampling points (5) . For n order plant model, it is usually necessary to switch the control input n times. In the case

Overview of RPTC

3.1 Plant model of IPMSM

with harmonics

dq coordinate system is shown in Fig. 3. The meaning ofeach parameter is given in Table 3. In this study, perfect
tracking controller and PI controller are designed on dq coordinate. The dq axis voltage equation is expressed as follows:
[
] [
][
] [
]
vd
R + sLd −ωe Lq
id
0
=
+
.
vq
ωe Ld
R + sLq
iq
ωe Ke
(2)

calculation scale. 35A360 (JSOL) is used for the material ofelectromagnetic steel sheet, and N39UH (Shin-Etsu Chemical) is used for the material of magnet.
The parameters of this model are listed in Table 1. The
values of the residual ﬂux density and coercivity of magnets
in the simulation are adjusted so that the permanent magnet
ﬂux linkage Ke matches that of the experimental motor.
First, the dq axis current of JMAG motor model is controlled by the PI controller including PWM inverter model ofSimulink model. Phase currents and typical order harmonics(n = 5, 7, 11, 13) in steady state are recorded. As simulation
conditions, q axis current is set to 10 A and d axis current is
set to −0.892 A according to MTPA as follows (11) :
√
Ke2
Ke
−
+ i2q re f .
(1)
id re f =2(Lq − Ld )4(Lq − Ld )2

3.

Loss
Hysteresis loss
Eddy current loss
Total iron loss

The IPMSM model on297

1
Ld s+R

+

vd

+

id
(xd [k + 1])

ωeLq

+

−
u0 [k] +B −1(1 − z −1A)

+

+

vq

1
Lq s+R

Ke

+

+

P

−

1
JM s+BMTωmPz −(N d−1)

Meaning

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

dq axis voltage
dq axis current
dq axis inductanceResistance
Permanent magnet ﬂux linkage
Number of pole pairs
Inertia
Viscous friction coeﬃcient
Mechanical angular velocity
Electric angular velocity+
y0d [k]id ref

−

rf

+
+

Ke

Decoupling control

+

+

Ld

ωe
ωe

Decoupling control

u0q [k]+Dq

Lquq [k] Coupling term−1
H +Lq s+R++

Zero-Order Holder

Nd =
iq

+

+

+
y0q [k]−Switch 1

Td
.
Ts

(11)

For the implementation, a low pass ﬁlter with no phase delay is applied to the compensation signal to remove the sensor
noise. This low pass ﬁlter is expressed as follows:

C2q [z]Samplerz −1Cq++ward or feedback is performed (6) (7) . Switch 1 is turned on torecord errors in the memory during one period of the periodic
disturbance after reaching the steady state. After recording,switch 1 is turned oﬀ and switch 2 is turned on to compensate
the tracking error caused by the disturbance as a feedforward
controller. Feedforward compensation has the advantage of
not exacerbating inter harmonics. However, in this experiment, feedback compensation is used due to factors such asmodeling errors. For feedback compensation, switch 1 andswitch 2 are always ON.
The size of memory is deﬁned as Nd . The memory size Nd
is expressed as equation (11) using one tracking error period
T d and control period T s .

id

+

1
Ld s+R+ud [k] Coupling term

iq refz −(N d−2)

Fig. 6. Block diagram of PSG including low pass ﬁlter.

S

H

r

z −2

Zero-Order Holder(xdd [k + 1])

Bq−1(1 − z −1Aq )Switch 1

Low-pass filter

C2d[z]−(xdq [k + 1])

S−
Sampler

Switch 2

Sampleru0d [k]Bd−1(1 − z −1Ad)

+

+

Q

ParameterDd

id /iqFig. 5. Block diagram of RPTC (dq axis current).

Table 3. Parameters of IPMSM and those meanings.

++
y0 [k]

+

1
Ls+R

z −1

Fig. 3. Block diagram of IPMSM.

++
Zero-Order Holder

ωeKez −1Cd+z −1CSwitch 2

iq

+

H

C2[z]

D

ωeLd

u[k]

+

Ld − Lq−

Coupling term

Decoupling

idref /iqref

SFig. 4. Block diagram of PTC (dq axis current).

r f [k] =z + γ + z−1
r[k],
γ+2

(12)

of the IPMSM, the plant is a one-order model and controlled
by the single rate controller.
For designing the feedfoward controller, the IPMSM discrete time equation of state is shown in equation (8). Thisequation is obtained by discretizing equation (5) by the zeroorder hold (ZOH). The control input period (T u ) is used fordiscretization. From equation (8), the stable inverse modeland nominal output are expressed as equation (9) and (10).

where r f [k] is ﬁlter output,r[k] is PSG output and γ is a design parameter of the ﬁlter. In this experiment, γ is set to 2,
and the cutoﬀ frequency of this ﬁlter is 1.8 kHz. The block
diagram of the PSG including the low pass ﬁlter is shown in
Fig. 6.
3.4 Feedback controller design
The current PI controller C PI (s) is designed by pole zero cancellation method
and expressed as follows:x[k + 1] = Ax[k] + Bu[k], y[k] = Cx[k] + Du[k]. (8)Ls + R.
(13)
τsConsidering equation (5), the transfer function from thereference current id, q re f to the PMSM dq current id, q is derived as follows:
id, q
1
.
(14)
=
id, q re f
τs + 1

u0 [k] = B−1 (1 − z−1 A)xd [k + 1],(9)y0 [k] = z−1Cxd [k + 1] + Du0 [k].

(10)

C PI (s) =In reality, there are disturbances and modeling errors.
However, these are suppressed by a feedback controller
C2 [z], and the tracking performance is basically improved by
the feedforward controller. C2 [z] is discretized PI controller.
3.3 RPTC
The block diagram of RPTC on the dq
axis is shown in Fig. 5. The decoupling control and coupling
term are the same as in Fig. 4. The RPTC system has periodic
signal generator (PSG) by switch 1 and switch 2. By turning
on or oﬀ two switches, the error compensation like feedfor-

In this study, T s is set as 0.1 ms and τ is set as 10T s = 1 ms.
C2 [z] is obtained from C PI (s) by Tustin Transformation. The
period of transform is T s . RPTC has feedforward controller
and C2 [z]. The conventional method has only this feedback
controller.

298

4.

Experiment

4.1 Experimental setup and condtions
The experimental setup is shown in Fig. 7 and the parameters of IPMSM
used in this experiment are shown in Table 4. Since this is asingle rate control, T u and T s are the same value and set to
0.1 ms.
As the drive conditions, the q axis current command is
set to iq re f = 10 A and d axis current command is set to
id re f = −1.86 A according to the MTPA equation (1) andTable 4. The rotation speed is kept to 1000 rpm by the load
servo motor. Thus, the electrical angular frequency is 100 Hz.
The fundamental frequency of the tracking error to be suppressed by RPTC is set as the mechanical angular frequency.
Thus, the memory size Nd is determined as 600 from theequation (11).
In the loss measurement experiment, the sum of iron loss
and mechanical loss is calculated by subtracting the IPMSM
output and copper loss from the input power as follows:

Pir,me = Pir + Pme = Pin − Pout − Pco ,Fig. 7. Experimental setup.
Table 4. Parameters of experimental IPMSM.

(15)

where Pin is input power, Pout is ouput power, Pco is copper loss, Pir is iron loss, Pme is mechanical loss and Pir,me isthe sum of Pir and Pme . The power meter measures the input power and phase currents. The copper loss is calculated
from the phase currents and winding resistance measured inadvance as follows:
Pco = Iu2 rms Ru + Iv2rms Rv + Iw2 rms Rw ,Table 5.

(16)

ParameterValued axis inductance Ld
q axis inductance Lq
Resistance R
Permanent magnet ﬂux linkage Ke
Pole pairs P
Supply voltage Vdc
Carrier frequency F sControl period T sMemory size Nd

0.613 mH
1.21 mH
85.6 mΩ31.2 mWb
6100 V
10 kHz0.1 ms
600

Experimental temperature and phase resistance.
Parameter

Value

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

20.3 ◦ C
86.3 mΩ
88.1 mΩ86.7 mΩ4.30 × 10−3

where Iu rms , Iv rms and Iw rms are phase currents. Ru , Rv and Rw
are resistance of windings.
The output of IPMSM is calculated from the torque measured by the torque meter and the rotation speed measured by
the encoder as follows:
Pout = T m ωm ,(17)

where T m is motor torque.
Power meter is PW3390 (HIOKI) which is able to measureelectric power with an accuracy of 0.1 W. Torque meter is
TMB307/411 (MAGTROL), and this is capable of measuring up to 10 Nm with an accuracy of ±0.1 %.
Measurements of Pir,me are conducted separately by PI control and RPTC for 5 seconds. These procedures as one set
are conducted 12 times. Results of Pir,me are compared on
the same set, and the average of 10 sets, excluding maximum
and minimum diﬀerences of Pir,me , are shown as loss reduction results.
Thermocouple thermometers are attached the bearing andwindings of IPMSM. Since it is diﬃcult to separate the mechanical loss and the iron loss, the bearing temperature is kept
constant in the same set to minimize the variation of mechanical loss. Therefore, it can be considered that the diﬀerence ofPir,me is the amount of reduction in iron loss Pir . In addition,
more accurate copper loss results are calculated by correcting
the winding resistance of each phase as follows (12) :R(T s ) = R(T si ){1 + α(T s − T si )},Fig. 8.

Dq axis currents of PI and RPTC.

Table 6. Current amplitude of dq axis harmonics(average).
Current amplitude

Fundamental

The 6th harmonic

The 12th harmonic

idPI
idRPTC
iqPI
iqRPTC1.86 A
1.86 A
10.0 A
10.0 A632 mA
48.2 mA
555 mA33.3 mA

126 mA34.7 mA
11.7 mA
2.14 mA

Fig. 8 shows the dq axis
4.2 Experimental results
currents of PI control and RPTC. Fig. 9, 10 are FFT resultsof current control of PI control and RPTC. In addition, Table 6 shows the amplitudes of typical harmonic currents. It
is conﬁrmed that RPTC suppresses the 6th and 12th orderharmonics signiﬁcantly.
Table 7 shows the details of power consumption. The values in the table are averages of 10 sets. Assuming that there is(18)

where T s is the winding temperature, T si is the initial winding
temperature and α is the temperature coeﬃcient of copper resistance. The experimental conditions are listed in Table 5.

299

Fig. 9. Comparison of harmonics (d axis current).

Fig. 11. The ratio of loss and error range.

5.

In this paper, the application of RPTC to current control is
proposed to reduce the iron loss of IPMSM. It is pointed outthat harmonic currents increase the iron loss, and described
the method to suppress those harmonics. In RPTC, the harmonic currents of the fundamental frequency are recorded as
errors and compensated by the periodic signal generator tosuppress harmonics and reduce the iron loss. The simulation
results show that the iron loss is reduced when the harmonic
currents are eliminated. The experimental results also showthat RPTC reduces the iron loss by 0.3 %. In Fig. 11, thevariation in output power is large compared to the amount ofmotor loss. This is due to the large variation of the torquemeter relative to the accuracy of power meter and encoder.
For the more detailed analysis of the relationship between
harmonic currents and iron loss, an appropriate processing
method of the torque meter value and a method of separating
the iron loss from mechanical loss are required.
There is a research that shows the possibility of improving eﬃciency when the harmonic current commands are set
to other than 0 A (13) . Veriﬁcation of the change in iron loss
and total loss of PMSM when harmonic currents are actively
controlled is a future work.

Fig. 10. Comparison of harmonics (q axis current).
Table 7. The details of power consumption (average).
Power consumption

PI

RPTC

Input power Pin
Output power Pout
Copper loss Pco
Iron loss + Mechanical loss Pir,me

298.1 W
272.4 W
9.7 W
15.9 W

299.0 W
274.2 W
9.7 W
15.1 W

Table 8. The ratio of loss to input power.
Loss

PI

RPTC

Iron loss + Mechanical loss Pir,me
Copper loss Pco
Total loss

5.4 %
3.3 %
8.6 %

5.1 %
3.2 %
8.3 %

Conclusion

References
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and output power are diﬀerent, the ratio of Pir,me and Pco tothe input power are listed in Table 8 for a fair comparison.
Fig. 11 shows these results and error ranges. RPTC reduces
the sum of iron loss and mechanical loss from 5.4 % to 5.1 %and the total loss from 8.6 % to 8.3 %.
Due to the accuracy of the measuring instruments, it is notpossible to measure the loss with high accuracy. Therefore,
the maximum and minimum values of the results are shownin Fig. 11 as the error range. However, the iron loss reductioneﬀect of RPTC is conﬁrmed by repeating the experiment andaveraging the results.
In the simulation, it is conﬁrmed that the suppression ofharmonic currents reduces the iron loss, and this experimentalso shows that suppression of the 6th and 12th order harmonics on the dq axis reduces the IPMSM loss.

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