On-line Parameter Identification of IPM Motor Using
Instantaneous Reactive Power for
Robust Maximum Torque Per Ampere Control
Toshihiko Noguchi

Yuki Kumakiri

Department of Electrical and Electronics Engineering
Graduate School of Engineering, Shizuoka University
3-5-1 Johoku, Naka-ku, Hamamatsu, Shizuoka 432-8561,
Japan
ttnogut@ipc.shizuoka.ac.jp

Department of Electrical and Electronics Engineering,
Graduate School of Engineering, Shizuoka University
3-5-1 Johoku, Naka-ku, Hamamatsu, Shizuoka 432-8561,
Japan
f0330108@ipc.shizuoka.ac.jp

Abstract—The reluctance torque of an IPM motor can beeffectively utilized by applying the maximum torque per ampere(MTPA) control algorithm, which improves the efficiency and thepower density of the motor. However, the motor is unable tooperate accurately on the MTPA curve due to the parametermismatch caused by magnetic saturation and/or temperature
variation. This paper proposes the d-axis inductance, the q-axis
inductance, and the magnetic flux linkage identification strategy
by using the instantaneous reactive power of the IPM motor. The
key feature of the technique is robustness against the winding
resistance variation. Several experimental tests have been
conducted to verify the feasibility of the proposed technique, andit has been confirmed that every parameter identification can beachieved within 5 % error.

This paper proposes an on-line parameter identification
strategy of the inductances and the magnetic flux linkage of
the IPMSM. The output torque characteristic is improved by
the modified MTPA curve using the identified parameters.
The identification must be performed with robustness to theother parameters such as a winding resistance. The algorithm,
which can identify the motor parameters individually, is basedon the separate frequency component injection technique. The
algorithm does not have parameter sensitivity to the armature
resistance. In the following sections, the variation of the
MTPA curve caused by the magnetic saturation is examined,
and the several experimental test results, i.e, the on-line
identification of the parameters, the output torquecharacteristic and the parameter sensitivity are presented.

Keywords—maximum torque control, magnetic saturation,
parameter identification, instantaneous reactive power

II. IMPACT OF MAGNETIC SATURATION
A relational expression of the d-axis current id and the qaxis current iq which maximizes the torque per unit current canbe given by

I. INTRODUCTION
An interior permanent magnet synchronous motor(IPMSM) has several advantageous points such as highefficiency and high power density. In recent years, in order toutilize these features, the range of the applications expands toindustrial, home appliance, automotive sectors and more.
Moreover, from the view point of the improvement of thecontrol performance, vector control is generally applied toobtain high response of the motor torque. Particularly, it isstrongly desired to obtain the quick torque response for thetraction control of the hybrid vehicles and the electric vehicles[1]-[3]. Under the vector control, the IPMSM can deliver the
reluctance torque as well as the magnet torque by optimizing
the motor current phase. Therefore, the maximum torque per
ampere (MTPA) control which can deliver the torque with theminimum current norm is applied to the IPMSM drive. Due tothe decrease of the inductance and/or the variation of themagnetic flux linkage; however, the operating point deviatesfrom the optimal value[4]-[7]. As a result, enlargement of thecopper loss occurs, resulting in the deterioration of theefficiency; hence, it is necessary to correct the operating pointaccording to the operating condition on the basis of the on-line
parameter identification of the motor.

978-1-4799-7799-4/15/$31.00 ©2015 IEEEd-axis and q-axis inductances Ld , Lq (mH)

id =

ψf
2( L q − L d )−ψf2
4( Lq − Ld )2

+ iq 2 .

40

id = 0 A

35

id = -2 A

30

iq = 2 A25

iq = 4 A

Lq

20
15
10
5
0

Ld
0

1

2

3

4

d-axis and q-axis currents id , iq (A)Fig. 1. Magnetic saturation characteristics.

793

(1)

5

6

The torque can be efficiently delivered by controlling thecurrents to satisfy this equation. If the parameters in (1) arevaried by the magnetic saturation, the actual operating point
turns away the optimum operating condition. As a result, thecopper loss increases as the current amplitude increases; thus,
the loss increase degrades the efficiency characteristic.

TABLE I.

Figure 1 shows the magnetic saturation characteristics ofthe test motor. Although the effect of the cross saturation canbe slightly seen, the d-axis inductance and the q-axis
inductance monotonically decrease as the corresponding
currents increase, respectively.
The q-axis inductancedecreases by 30~35 %, and the d-axis inductance decreases by
10 % from their nominal values. This paper discusses a robust
identification strategy of these parameters.

RATED VALUES AND NOMINAL PARAMETERS OFTEST MOTOR
Number of poles

8

Rated power (W)1000

Rated speed (r/min)2000

Rated current (A)3.70

Armature resistance (Ω)

1.10

Number of magnetic flux linkage (Wb)

0.174

d-axis inductance (mH)11.0

q-axis inductance (mH)

25.0

III. VARIATION OF MTPA CURVE
Table I shows the rated values and the nominal parametersof the test IPM motor. The experimental system has been set
up as shown in Fig. 2. The test motor is directly coupled with
the AC servo motor (load machine) through the torque sensor.
A DSP (TMS320C6713, TI) gives the gate signals to theinverter, and controls the test motor. The AC servo motor iscontrolled in the speed control mode, whereas the inverter
controls the output torque of the test motor. The common
vector control is carried out by using the rotor position which
is detected by the rotary encoder. The inverter carrier
frequency is 10 kHz, and the dead time is 4 μs.

θFig. 2. Experimental system.
8

Torque T (Nm)

The measurement of the output torque has been conducted
to search the true MTPA curve. The possible combination of
the d-axis current and the q-axis current has been checked out
according to the following equations:(2)

1( L d − L q ) I a 2 sin 2β } .
2Ia = 3.0 A

5

Ia = 1.0 A

Ia = 2.0 A

4
3
2
1

Figure 3 shows the result of the above output torque test.
The output torque has been measured at 1000 r/min. The
current is controlled as expressed in (2) where the current
amplitude Ia is maintained constantly and the current phase β
is advanced. In Fig. 3, the plots are the measured values, andthe solid lines are the torque curves calculated by (3) using thenominal values.
T = p{ψ f I a cos β +Ia = 4.0 A

6

0
-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

d-axis current id (A)Fig. 3. Torque characteristic.
6
5

(3)

4

Here, p is the pole pair number. Figure 4 shows the MTPA
curves. The d-axis current and the q-axis current given by (2)that maximize the output torque are plotted. The solid line is
the MTPA curve given by Eq. (1) when the nominal values areused for calculation.

3
2

Measured valueEq.(1) using nominal values

q-axis current iq (A)id = − I a sin β , and iq = I a cos β .

Ia = 5.0 A

71

According to Fig. 3, significant errors between themeasured values and the calculated values are observed. The
errors expand as the current norm increases due to themagnetic saturation.

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0

d-axis current id (A)Fig. 4. Maximum torque per ampere control curves.

In Fig. 4, the true optimum operating points do notcoincide with the MTPA curves calculated by (1) with thenominal values.
Although the deviation is not very
remarkable when the mismatch of the inductance is small, theerror between the true and the calculated MTPA curves isnotably enlarged as the magnetic saturation becomesremarkable. As mentioned above, the MTPA control isdetrimentally affected by the magnetic saturation. Therefore,794the operating points must be correctly modified according tothe operating situation.
IV. PROPOSED IDENTIFICATION STRATEGY
The parameter identification method have been
proposed[7]-[11]. In this paper, the strategy using the
instantaneous reactive power is focused on. The instantaneous
reactive power Q is calculated by a vector product of the twophase currents and voltages on the stator reference frame andthe synchronously rotating reference frame asQ=

3 ⎡iα ⎤ ⎡vα ⎤ 3⎢ ⎥ × ⎢ ⎥ = (iα v β − i β vα )2 ⎣⎢i β ⎦⎥ ⎣⎢v β ⎦⎥ 2

3 ⎡i d ⎤ ⎡ v d ⎤ 3= ⎢ ⎥ × ⎢ ⎥ = (i d v q − i q v d )2 ⎢⎣ i q ⎥⎦ ⎢⎣ v q ⎥⎦ 2Q̂Ψˆ f , Lˆd , Lˆq

Fig. 5. Identification system.
.

(4)

As shown in the identification system depicted in Fig. 5,
the mathematical model (8) and the magnetic flux linkage are
dynamically modified on the basis of the following
identification algorithm:

Then the command voltages in the controller are used forvd and vq. The voltage equation of the IPM motor in the steadystate is given by (5). The instantaneous reactive power is also
obtained as (6) by substituting (5) into (4);⎡v d ⎤ ⎡ Ra
⎢ ⎥=⎢⎢⎣ vq ⎥⎦ ⎣ωLd
Q=

− ωLq ⎤ ⎡id ⎤ ⎡ 0 ⎤⎥ , and⎥⎢ ⎥ + ⎢Ra ⎦ ⎢⎣iq ⎥⎦ ⎢⎣ωψ f ⎥⎦3
ω ( L d i d 2 + Lq i q 2 + ψ f i d ) .
2ψˆ f = γψ ∫

dt ,

(9)

After the above identification process completes, themismatch of ψf in the third term of (7) becomes zero. The
mathematical model Q̂Ld is given by(6)

3Qˆ Ld = ω ( 2 Lˆ d i d + ψ f )i dh cos ω h t .
2(10)

Ld can be identified by using the following algorithm in thesimilar manner:

In order to identify Ld and Ψf, the high-frequency signal
whose angular frequency is ωh and amplitude is idh is injectedto the d-axis current control loop. This can be achieved bysuperimposing the high-frequency current command onto thed-axis current command. In the case, (6) is rewritten as
follows:

Lˆ d = γ Ld ∫QLd − Qˆ Ld
dt ,2ω id idh cos ω h t(11)where QLd is the reference model and γLd is the identification
gain.

(7)

As shown in (7), Q has a ωh component and a 2ωhcomponent as well as the DC component. If ωh component isseparated from other components with a band-pass filter(BPF), the third term that does not include Lq can be extracted.
It is preferable to set idh as small as possible. Therefore, theforth term is very small, and this means that it is impossible touse the 2ωh component for the d-axis inductance identification.
In addition, the mathematical model Q̂ψ which has only the
information of ψˆ f is obtained as (8) if the d-axis current is
controlled to be zero;3Qˆψ = ω ψˆ f id h cos ω ht .
2

ω idh cos ω h t

where γΨ is the identification gain. ψˆ f is updated until theerror between the reference model and the mathematicalmodel converges to zero.

(5)

According to (4) and (6), the armature winding resistance
Ra is not included at all in both equations. Therefore, theinstantaneous reactive power is never affected by the variationof Ra caused by the temperature change[12].

3LQ = ω{( Ld id 2 + Lq iq 2 + ψ f id ) + d idh 2
22
Ld+ ( 2 Ld id + ψ f )idh cos ωht +idh 2 cos 2ωht}.
2Qψ − Qˆψ

Similarly to the case of ψf, L̂d converges to the true value
by integrating the error between the reference model and themathematical model. The denominator of (9) and (11) has
multiple zero cross points because it includes cosωht. In order
to avoid the division by the zero, the parameter identification
is carried out by using the root mean squared value as shownin Fig. 5.
Since the mismatch of Ld and ψf have been canceled out by
the above technique, Lq can be identified by using the DC
component of (6) which is filtered out of (7) by using the LPF.
The modified equation is given by
Q − Qˆ
Lˆq = γ Lq ∫dt ,
ω iq 2(8)(12)

where Q is the reference model, Q̂ is the mathematical modeland γLq is the identification gain. According to the above

795

Instantaneous reactive
power Q (Var)

Q
^

Q

0.20
0.15
0.10
0.05
0.00
-0.05

Time 5 (ms/div)

0.20

40

0.19

35

0.18
0.17
0.16

Measured value
500 r/min
1000 r/min
1500 r/min

0.15
0.14

Q
^

Q

12
10
8
6
4
2
0
-2

Time 10 (ms/div)

Fig. 8. d-axis inductance identification characteristic.

d-axis inductance Ld (mH)

Magnetic flux linkage ψf (Wb)Fig. 6. Magnetic flux linkage identification characteristic.

0.13
0.12

60
50
40
30
20
10
0
-10

d-axis inductance
Ld (mH)

Instantaneous reactive
power Q (Var)
Identified magnetic
flux linkage ψf (Wb)

70
60
50
40
30
20
10
0
-10

30

Measured value (iq = 2 A)
Measured value (iq = 4 A)

25

Identified value (iq = 2 A)
Identified value (iq = 4 A)

20
15
10
5

0

1

2

3

q-axis current iq (A)

4

5

6

0
0

1

2

3

4

5

6

5

6

d-axis current id (A)Fig. 7. Experimental results of magnetic flux linkage(a) 500 r/min

identification.
40

technique, each parameter can be identified independently by
the high-frequency signal superimposed onto the d-axis
current command.
Additionally, the identification time
becomes constant regardless of the rotating speed by setting
the identification gain as an inverse function of the injected
frequency[13]-[15]. The MTPA curve is corrected by theparameters identified by the method described above.
Assuming that the proposed method is applied for the motor
mounted on the vehicle, the filter time constant of the current
commands is very large, i.e, the LPF with an approximately
500-ms time constant is inserted in the current control system;
thus, the current varies slowly. Therefore, the proposed
identification strategy must identify parameters in less than
500 ms.

d-axis inductance Ld (mH)35
Measured value (iq = 2 A)

30

Measured value (iq = 4 A)

25

Identified value (iq = 2 A)
Identified value (iq = 4 A)

20
15
10
5
0
0

1

2

3

4

d-axis current id (A)(b) 1500 r/min

V. EXPERIMENTAL RESULTS

Fig. 9. Experimental results of d-axis inductance identification.

A. Magnetic Flux Linkage Identification
Figure 6 shows the dynamic identification characteristic of
the magnetic flux linkage based on (9) at 1000 r/min. The
current command is set to the following values: id*=0 A andiq*=3 A. In (8), the high frequency signal condition is set tothe following values: idh=0.3 A and ωh=2000π rad/s. Figure 7
shows the experimental result of the magnetic flux linkage
identification when the rotating speed is controlled at theconstant value and the q-axis current is varied.

identified value of the magnetic flux linkage converges to thetrue value in the test, where the identification time is 50 ms.
The identification is completed within 1/10 of the time
constant of the current command. In Fig. 7, the solid line
shows the measured values of the magnetic flux linkage which
is 0.174 Wb. It can be seen that the proposed method canidentify the magnetic flux linkage within 5 % error regardlessof the rotating speed and the q-axis current.

As shown in Fig. 6, the mathematical model stably
converges to the reference model. It was confirmed that the

796

20

30

Identified value (id = 0 A)

Nominal value (id = -2 A)
Identified value (id = -1 A)
Identified value (id = -2 A)

25

18

Lq (mH)

Nominal value (id = 0 A)

q-axis inductance

35

16
14
12
10

20

q-axis voltage vq (V)q-axis inductance Lq (mH)

40

15
10
5

19.4
19.3
19.2
19.1
19.0
18.9
18.8

nominal Ra+20% Ra variedTime 5 (ms/div)

0
0

1

2

3

4

5

6

Fig. 11. Impact of winding resestance variation on q-axis

q-axis current iq (A)inductance identification characteristic.

(a) 500 r/min

40
35

35

Measured value (id = 0 A)

30

Identified value (id = 0 A)

q-axis inductance Lq (mH)q-axis inductance Lq (mH)40
Measured value (id = -2 A)
Identified value (id = -1 A)

25

Identified value (id = -2 A)

20
15

Nominal value (id = -1 A)

25

20 % increase (id = -1 A)

20 % increase (id = 0 A)

20
15
10
5

10

0

5
0

Nominal value (id = 0 A)

30

0

1

2

3

4

5

6

q-axis current iq (A)
0

1

2

3

4

5

6

Fig. 12. Parameter sensitivity of winding resistance to q-axis

q-axis current iq (A)inductance identification.

(b) 1500 r/min
Fig. 10. Experimental results of q-axis inductance identification.

id

B. d-Axis Inductance Identification
The d-axis inductance identification is carried out after themismatch of the magnetic flux linkage has been eliminated.
Figure 8 shows the dynamic convergence characteristic of thed-axis inductance identification which is based on (11). The
rotating speed is 1000 r/min, and the current commands are set
at id=-3A and iq=3A. The injected high-frequency signal is thesame as the case of the magnetic flux linkage identification.
The experimental results of the d-axis inductance at 500 r/min
and 1500 r/min are shown in Figs. 9 (a) and 9(b). The d-axis
inductance saturation characteristic is also depicted in Fig.9.

iq

*

*

+

+
−

vd

*

vq

*

−

vα
vβ

*

*

θ

id

iα

iq

iβ

Fig. 13. Adaptive MTPA control system.

experimental results based on (12) and the measured
saturation characteristics of the q-axis inductance at 500 r/min
and 1500 r/min, respectively. Although the error between theidentified and the measured values is observed in the low qaxis current range, the both characteristics agree very well as
the q-axis current increases. The reason of the error is theminute mismatch of the magnetic flux linkage and the d-axis
inductance. The q-axis inductance identification algorithm(12) is discussed on the assumption of no mismatches of themagnetic flux linkage and the d-axis inductance, but from
Figs. 7 and 9, the small identification error is observed.
Therefore, in the low instantaneous reactive power range, theThe convergence time is 70 ms as demonstrated in Fig. 8.
In comparison of the identified and the measured saturation
characteristics, the figure shows very good agreement between
them regardless of the rotating speed and the current
amplitude.
C. q-Axis Inductance Identification
The q-axis inductance is identified after the above
mentioned identification processes, which is in the case of noparameter mismatches in the magnetic flux linkage and the daxis inductance.
Figures 10 (a) and 10(b) show the797

impact of these errors is large enough to affect detrimentallythe q-axis inductance identification.

8
7
6

q-axis current iq [A]

D. Verification of Parameter Sensitivity
In order to verify the impact of the armature resistancevariation caused by the temperature rise, the resisters areintentionally inserted between the inverter and the IPMSM.
Assuming the situation where the armature resistance varies
by +20 % of the nominal value, the inserted resistance value is
0.2 Ω. Figure 11 shows the q-axis inductance identification
characteristics when the external resistance is inserted. The
current command is set at id=0 A and iq=5 A. The comparisonof the q-axis saturation characteristics is shown in Fig. 12.
The rotating speed is 1000 r/min, and the d-axis current is
controlled to be constant; id=0 A and id=-1 A in the test.

5
4
Measured Value
Nominal Value
Identified Value

3
2
1
0
-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

d-axis current id [A]Fig. 14. Comparison of MTPA curves.
1.07

Torque/current amplitude T/Ia (Nm/A)

As shown in Fig.11, even if the armature resistanceincreases by 20 %, the identified parameter is not affected by
the resistance variation at all.
The identification
characteristics when the external resistance is inserted agreewith those of the nominal value as shown in Fig.12. This
experimental result verifies that the proposed method has nosensitivity to the armature resistance.
E. Adaptive Modification of MTPA Curve
The MTPA curve given by (1) is modified by theparameters identified by the above mentioned technique asshown in Fig. 13. Figure 14 shows the three MTPA curves,
i.e., the MTPA curve calculated by (1) using the nominal
parameters, the MTPA curve sought by the experimental
measurement and the MTPA curve recalculated by (1) using
the identified parameters by the proposed technique. In Fig.
14, the measured value is identical to the plots in Fig. 4, andthe plots of the nominal values are obtained by substituting thenominal parameters to (1). The plots of the identified values
are also given by modifying (1) by using the identified
parameters. Figures 15 (a) and 15(b) show the MTPA
characteristics at 500 r/min and 1500 r/min. These arecalculated by dividing the motor torque by the current
amplitude. Similarly to Fig. 14, the measured values (the true
MTPA curve), the calculated values by substituting thenominal values into (1), and the modified MTPA values byusing identified parameters are plotted.

1.06
1.05
1.04
1.03
1.02

Measured value
Nominal value
Identified value

1.01
1.00
0.99
0.98
0

1

2

3

4

5

6

7

8

5

6

7

8

q-axis current iq (A)(a) 500 r/min
Torque/current amplitude T/Ia (Nm/A)

1.07
1.06
1.05
1.04
1.03
1.02

Measured Value
Nominal Value
Identified Value

1.01
1.00
0.99
0.98
0

1

2

3

4

q-axis current iq (A)(b) 1500 r/min

As can be seen in Fig. 14, the MTPA control points using
the nominal values digress from the optimum points. The
MTPA curve is turned to watch the measured curve (the true
MTPA curve). The motor torque per unit current amplitude is
improved by using the proposed method as shown in Fig. 15.
Particularly the effectiveness of the proposed method isremarkable as the q-axis current increases. The MTPA
characteristic agrees well with the measured characteristic
regardless of the rotating speed. It is found that the test motor
can be controlled to deliver the maximum torque per unit
current amplitude by applying the proposed method.

Fig. 15. Comparison of MTPA characteristics.

confirmed that the proposed method can identify the magnetic
flux linkage, the d-axis inductance and the q-axis inductanceby using the instantaneous reactive power which has nosensitivity to the armature winding resistance.
The
identification time of these parameters are within 500 ms, andthe identification errors are within 5 %. It is substantiated thatthe proposed method is not affected to the temperature
variation. The deviated operating points which are calculatedby the MTPA formulas can be properly modified by using theidentified parameters; hence, the motor is always controlled to
track on the true MTPA curve. It was confirmed that the
motor torque per unit current norm can be maximized.

VI. CONCLUSION
In this paper, a robust MTPA control strategy against the
motor parameter variations caused by the magnetic saturationand the temperature change has been proposed. It has been

798

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