2014
International Symposium on Power Electronics,
Electrical Drives, Automation and Motion

Off-Line Parameter Identification ofInterior Permanent Magnet Motor by Searching
Minimum Point of Current Norm Characteristics
Xiang Ji

Toshihiko Noguchi

Department of Environment and Energy Systems,
Graduate School of Science and Technology,
Shizuoka University
Hamamatsu, Shizuoka 432-8561, Japan
f5345014@ipc.shizuoka.ac.jp

Department of Electrical and Electronics Engineering,
Graduate School of Engineering,
Shizuoka University
Hamamatsu, Shizuoka 432-8561, Japan
ttnogut@ipc.shizuoka.ac.jp

Abstract—This paper describes a novel parameter
identification technique of an interior permanent magnet (IPM)
motor. The controller of the IPM motor requires its parametersin the decoupled current control loops. Off-line parametertuning is indispensable to achieve initial setup and starting up of
the motor control. The proposed technique can tune the every
motor parameter by changing the current controller structure
and by seeking the minimum point of the current norm on thebasis of a hill-climbing algorithm. The paper presents somesimulation results of the identification of the d-axis inductance, qaxis inductance, magnetic flux linkage, and winding resistance,
followed by the theoretical analysis. In the end of the paper, theexperimental test results of the q-axis inductance and winding
resistance identification are demonstrated to confirm thefeasibility of the proposed technique.

of the motor parameters must be identified individually in as asimple manner as possible.
This paper proposes a novel technique to achieve the offline identification of the IPM motor parameters, which
requires only the motor current norm information. The
proposed technique has several steps to complete theidentification of the all parameters, where the current
controller structure is changed every time the identification
step proceeds, and the minimum operation point of the current
norm is sought by tuning the specific motor parameter until
the parameter mismatch is cancelled out. The search for thecurrent norm minimum point is achieved by an ordinary hillclimbing algorithm with respect to the parameter mismatch.
In the following sections, some theoretical analyses aredeveloped, followed by the simulation results, andexperimental test results are also presented to show thefeasibility of the proposed technique.

Keywords— identification; parameter; IPMSM; hill-climbing;current norm; minimum point; current controller.

I.

INTRODUCTIONII.

In recent years, interior permanent magnet (IPM) motors
are widely used in a variety of industry, home appliance, andautomotive applications, owing to its high-efficiency andhigh-power-density features. The IPM motor is usuallycontrolled with a field-orientation technique (vector control),and requires a current controller on the synchronous rotatingreference frame (dq-reference-frame) for the instantaneous
torque and magnetic flux control. The current control on thedq-reference-frame mainly consists of the coordinatetransformation, the PI regulation, and the decoupling
compensation, which is based on the mathematical model ofthe motor. Therefore, it is indispensable not only to detect themagnetic-pole-position and the motor currents but also toknow the motor parameters accurately. Identification of the
motor parameters is important to control the motor properly instarting up of the control as well as the running operation, andthe off-line parameter identification is particularly required forthe initial controller setup and starting up. There are fourparameters to be off-line identified in the motor model, i.e.,the d-axis inductance Ld, the q-axis inductance Lq, themagnetic flux linkage ψ, and the winding resistance R. Each978-1-4799-4749-2/14/$31.00 ©2014 IEEE

OFF-LINE PARAMETER IDENTIFICATION USING
CURRENT NORM CHARACTERISTICS

In the standard control system of the IPM motor, the fieldoriented control (the vector control) technique is often used tocontrol the instantaneous torque and the magnetic flux. The
field-orientation is achieved by controlling the two-axis
armature current components, i.e., the d-axis current and the qaxis current on the synchronously rotating dq-reference-frame.
Figure 1 shows the whole system block diagram including thedecoupling current controller and the IPM motor. In the figure,the three-to-two-phase coordinate transformation and therotational coordinate transformation are omitted forconvenience because the magnetic-pole-position is accurately
acquired by the mechanical position sensor such as a rotary
encoder or a resolver. As can be seen in the block diagram,several motor parameters are required to compose the two-axis
current control loops and to control the two-axis currents
independently.
The motor parameters can be simply
calculated by using the voltage and current information on thebasis of the inverse motor model. However, detecting the real
voltage is not pragmatic because no voltage sensors are

282

1
R+sLd

ωL̂q

ωLq

ωL̂d

ωLd

id

1
R+sLq

iq∗

d-axis and q-axis currents (A)

id∗

iq

ψ

1
R + sLd

ωLq

ωL̂d

ωLd

0

id∗

P

+
-

+
+

iq

iq∗

+

-

PI

-

80

1
R

4.6

Fig. 5. Simulation result of motor current norm characteristic withrespect to Lq mismatch.
Symbol
R
P
ζ
p
ω
ψ
Lq
Ld
K Pd

iq

ψ

4.7

-0.008 0.000
0.008
L q m ism atch (H )

id

1
R

T heoretical C h aracteristic
S im ulatio n C haracteristic

4.8

4.5

ω

ωLq

ωL̂q

60

time (ms)Fig. 2. Structure of current control loops on Lq identification.

-

40

20

Fig. 4. Current step responses of P and PI regulators.

ψ

+

0.1

id

1
R + sLqiq∗

0.2

C urrent norm (A )ωL̂q

Idorder(PI)
Id(PI)
Iqorder(PI)
Iq(PI)
Idorder(P)
Id(P)

0.3

ω

Fig. 1. Decoupling current control system of IPM motor.

id∗

0.4

ω

Fig. 3. Simplified current control loops on Lq identification.

installed in the standard controller. Even though the voltage
commands generated by the current regulators are used insteadof the real voltages, there must be errors in the voltage
commands because of the dead time and the voltage drop of
the power transistors in the inverter. On the other hand, theaccurate current detection is inherently available in thecontroller by employing some Hall-effect current transducers.
It is the simplest and pragmatic way, therefore, to utilize only
the current information without any complicated mathematical
calculation in the motor parameter identification. In addition,
it is necessary to avoid the undesirable mutual interferencesamong the parameters in each parameter identification process.

TABLE I PARAMETERS USED IN SIMULATION.
Description
Value
Winding resistance
0.48 Ω
Rated output power
1.5 kW
Damping coefficient
0.0002 Ns/r
Number of poles
4
Rotation speed
50 r/sMagnetic flux linkage
0.0674 Wb
q-axis inductance
12.0 mHd-axis inductance
7.3 mHP Gain of d-axis regulator
1.0 V/A

the q-axis current regulator remains the PI form to suppress
the interferences from the d-axis and the back electromotive
force (e.m.f.). No matter how much parameter mismatchexists between the real d-axis inductance Ld and the set valueL̂d in the controller, the steady state error of the q-axis current
is perfectly eliminated because the PI regulator is employed in
the q-axis current loop, i.e., iq* = iq is guaranteed in the steadystate. Therefore, the modified current controller in Fig. 2 canbe simplified as shown in Fig. 3 in the steady state. The q-axis
PI regulator has the following transfer function:

In order to achieve this goal, the proposed techniqueintroduces modifications of the current control loop structure,and focuses on the motor current norm characteristics withrespect to the parameter mismatches.

⎛
1 ⎞⎟
.
GPIq ( s ) = K Pq ⎜1 +⎜sτ Iq ⎟⎠⎝

(1)

In the above transfer function, the P gain KPq and the integral
time constant τIq are designed as follows:

A. Identification of Lq
In the case of the Lq identification, the structure of thecurrent control loops is changed as shown in Fig. 2, where thed-axis PI regulator is replaced with a low-gain P regulator tolet the current norm have a large variation. On the contrary,

K PI = ωc Lq and τ q =

283

Lq
R

,

(2)

Inmagnetic (A)

0.08
0.06

Theoretical Characteristic
Simulation Characteristic

0.04
0.02
0.00
0.065 0.066 0.067 0.068 0.069 0.070
Magnetic Flux Linkage (Wb)Fig. 9. Simulation result of motor current norm characteristic with respectto ψ mismatch.

Fig. 6. Simulation result of proposed Lq identification technique.
+

∗
d

i

-

1
R

PI

+

-

P

+
+
++

-

Lq
Ld
L̂d

1
R

K Pd

iq
ψ

ψ̂

ω

id

ωLd

ωL̂d

iq∗

Symbol
ψ
ψˆ

id

iq∗

1
R

iq

ψ

ψ̂

ω

ω

Fig. 8. Simplified current control loops on ψ and Ld identification whenid* = 0.

Figure 6 shows another simulation result of the Lq
identification with the interferences from the d-axis and theback e.m.f. It is observed from the result that the current norm
reaches the minimum value when the set value of the q-axis
inductance also reach the real value of the motor. The result
proves robustness of the proposed identification techniqueagainst the other parameter mismatches of Ld and ψ in thesteady state.

where ωc is a designed crossover frequency of the q-axis
current control loop. Figure 4 shows an example of thecurrent step responses of the P-regulator-based d-axis and the
PI-regulator-based q-axis current control loops. As can beseen in the figure, the d-axis current control loop has a large
steady state error after the q-axis current command is given to
the system.
From the simplified current controller shown in Fig. 3, thefollowing relationship is mathematically derived:*q

iq = i ,id =B. Identification of ψ
In a similar manner, ψ and Ld can be identified with thesame approach. However, some minor modification isrequired to identify these parameters separately. After Lq is
identified as described in the previous section, the current loop
structure is changed to that shown in Fig. 7. Figure 7 shows
the structure of the current control loops changed for the ψand Ld identifications, where the d-axis employs the PI
regulator but the q-axis current regulator is replaced with a
low-gain P regulator. Therefore, the steady state error in thed-axis current control loop is perfectly eliminated, but the qaxis current involves some amount of steady state error due tothe back e.m.f. and the Ld mismatch. In order to carry out the(3)

i*q ωc ( Lˆq − Lq )K Pd + R

in = id2 + iq2 .

, and

1.0 V/Avariable and parameter values iq* , ωc, KPd, and R. Figure 5
shows a simulation result of the current norm characteristicwith respect to the Lq mismatch. The simulation condition islisted in TABLE I. This current norm characteristic is
obtained by gradually changing the set value of Lq in thecontroller. As illustrated in the figure, the current norm has aparabolic function characteristic whose minimum point can beobserved when the Lq mismatch becomes zero. In otherwords, the real value of the q-axis inductance can be estimatedby searching the minimum point of the current norm
characteristic. This search operation is carried out by meansof a hill-climbing algorithm.

Fig. 7. Structure of current control loops on ψ and Ld identification.

1
R

P Gain of q-axis regulator

becomes minimum when Lˆq = Lq regardless of any otherω

id∗

TABLE II PARAMETERS USED IN SIMULATION.
Description
Value
Magnetic flux linkage
0.0674 Wb
Initial set value of magnetic flux linkage
0 Wb
q-axis inductance
12.0 mHd-axis inductance
7.3 mH
Initial set value of d-axis inductance
0 mH

(4)
(5)

In the above equations, in is the motor current norm. It isfound that in is a parabolic function of the Lq mismatch andthat it takes the minimum value when the Lq mismatch iscompletely canceled out. It should be also noted that in always

284

+

id∗

+

iq∗

PI

1
R

ωL̂d

ωLd

-

-

P

+

+

-

1
R

equal to KPq and the d-axis current command is changed to thevalue equal to iq* , the resultant d-axis current comes to havethe same value as the second term of (7). This implies that thesecond term of (7) can be found with no knowledge of thewinding resistance. By memorizing the simulated second term
value and by subtracting the memorized value from (7), themost important part can be extracted from (7) as follows:

id

iq

iq′ =

Fig. 10. Simplified current control loops on Ld identification.

Current norm (A)in = id2 + iq′2 =

4.7

4.5
-0.0080.000
0.008
L d mismatch (H)Fig. 11. Simulation result of motor current norm characteristic withrespect to Ld mismatch.

In the first identification step of ψ, the d-axis current
command i*d = 0 is given to the controller. Since the d-axis
current control loop employs the PI regulator, id = 0 isachieved with no steady state error. In addition, a crossdecoupling term through ωLd and ωL̂d from the d-axis to theq-axis can be ignored because id = 0 is guaranteed. In thiscase, only the parameter mismatch of ψ affects the steady state
error of the q-axis current. Taking the above discussion into
account, the current controller structure can be simplified asshown in Fig. 8, where the back e.m.f. compensation block isvirtually added to the q-axis current control loop in thecontroller for the purpose of the ψ identification.
Mathematical expressions of id and iq can be derived
from Fig. 8 as presented by (6) and (7):ωc (ψˆ − ψ )K Pq + R

iq* K Pq
K Pq + R.

ωc (ψˆ − ψ )K Pq + R

.

(9)

iq =

id* K Pd ω c ( Lˆ d − Ld ),( K Pq + R )( K Pd + R )

(10)

id =

i d* K Pd , andK Pd + R(11)

in = id2 + iq2 .

(6)

+

(8)

C. Identification of Ld
After identification of Lq and ψ, the identification of Ld iscarried out in the similar manner as described in the previous
sections. This identification is still based on the system
configuration shown in Fig. 7. The only difference between
the case of the ψ identification is the operating condition of
the system, i.e., i*q = 0 and i*d ≠ 0 are given to the system. In
the case, the current controller is simplified as shown in Fig.
10, and its mathematical expressions in terms of the d-axis andq-axis currents can be derived from the block diagram as
follows:

separate identification of the two parameters, the identification
process is divided into the following two steps.

iq =

.

Figure 9 shows a simulation result of the current norm
characteristic when the ψ mismatch is varied by graduallychanging the set value in the controller, and the simulation
conditions are listed in TABLE II. As expressed by (9), thecurrent norm characteristic is linear as a function of the ψ
mismatch. The current norm in has the minimum value whenthe ψ mismatch becomes zero, where the minimum point of incan be sought by means of the hill-climbing algorithm, also.

4.6

id = id* = 0 , andK Pq + R

By using (8), the motor current norm is expressed as

Theoretical Characteristic
Simulation Characteristic

4.8

ωc (ψˆ − ψ )(12)Figure 11 shows a simulation result of the current norm inwith respect to the Ld mismatch. As can be seen in the figure,the current norm curve has a parabolic characteristic as afunction of the Ld mismatch. It should be noted that inbecomes minimum only at the point of Lˆd = Ld as the Ldmismatch is varied by changing the set value in the controller;
hence, it is possible to identify Ld by adopting the similar hillclimbing algorithm as described previously. In addition, theminimum point of in can be observed only when Lˆd = Ld , nomatter how much the winding resistance R varies. This meansthat the proposed method is basically robust to the temperature
variation of the windings and the motor cable length.

(7)

The second term of (7) affects the identification of ψ, so thispart must be eliminated from the calculation without using thewinding resistance of the motor. This can be done by using
the d-axis current control loop as a simulator of the second
term in (7). As mentioned previously, d-axis current control
loop no longer has any interference from the q-axis because Lq
is already identified and Lˆq = Lq . Therefore, if the d-axis
current regulator is changed to a P regulator whose gain is

285

1.2

1.63

0.8

In (A)R (Ohm)Simulation Characteristic
Expermental Characteristic

Ridentification
Rreal
Raverage

1.0

0.6

True Value
1.62

0.4

Speed Unstable
0.2

200

400
time (ms)

0.02

600

0.03

0.04

Lq (H)

Fig. 14. Motor current norm characteristics calculated by simulation andtraced by proposed identification technique with respect to Lq mismatch.

Fig. 12. Experimental result of R identification.

Fig. 13. Experimental setup.
TABLE III OPERATION PARAMETERS OF EXPERIMENTAL TEST.
SymbolDescription
Value
R
Winding resistance
0.48 Ω
P
Rated output power
1.5 kWζ
Damping coefficient
0.0002 Ns/rp
Number of poles
6ω
Rotation speed
5.33 r/sψMagnetic flux linkage
0.0674 Wb
ψˆInitial set value of Magnetic flux linkage
0 Wb
Lqq-axis inductance
24.5 mHL̂q

Range of q-axis inductance

16.0~36.0 mH

Ldd-axis inductance

13.0 mH

L̂d

Initial set value of d-axis inductance

10.0 mH

Lq (H)

0.0234KP

Gain of P regulator

1.0 V/A

Gain of PI regulator

0.131 V/AτI

Time constant of PI regulator

0.051 s∗qq-axis current command

1.6 A

0.0225
500

1000
t (s)

1500

Fig. 16. Experimental result of Lq identification by using hill-climbing
algorithm.

Figure 12 shows an example of the experimental test resultof the R identification. The black line is the winding
resistance value calculated by (13), and the red line is its
average value. As can be seen in the figure, the average valueof R converges to the true value that has been measured by theparameter measurement test. The identified average value is0.485 Ω, while the true value is 0.48 Ω, where theidentification error is approximately 1 %. .
III.

EXPERIMENTAL SETUP AND TEST RESULTS

Experimental tests were conducted with the experimental
setup shown in Fig. 13, where the two same IPM motors weremechanically connected through the torque meter. One motor
is controlled by the current controller as shown in Fig. 1, andthe other is used merely as a generator with load resistors.
The operation parameters of the experimental setup are listed
in TABLE III, whose most parameters are the same as thoseused in the simulations. Attention must be paid to the initial

From (11), R is simply calculated by the following
expression:
K Pd ( i d* − i d ) .
id

Lq

0.0228

D. Identification of R
As described in the above discussion, the proposed
identification technique of the IPM motor parameters does not
require any information of the winding resistance R, and isbasically robust to the variation of R; thus, it is not necessary
to identify R inherently. The R identification, however, can be
optionally realized by using the identification algorithms
described so far.

R=

0.02310

K PI

i

Fig. 15. Simulation result of Lq identification by using hill-climbing
algorithm.

(13)

286

parameter values set in the controller, i.e., ψ and Ld
intentionally have the initial errors. These errors make itpossible to verify the robustness of the proposed identification
technique. Prior to the experimental tests of the proposed
technique, the relationship between the motor current norm
and the Lq mismatch was confirmed, where the set value in thecontroller L̂q had the tuning range from 16.0 to 36.0 mH as
indicated in TABLE III.

motor parameters can be achieved because the minimum pointof the current norm is found only when the parametermismatch is cancelled out between the controller and the real
motor.
The proposed identification technique is also
theoretically proven by the mathematical model of the IPM
motor drive.
Several simulations have been conducted to check the
motor current norm characteristics and the identification
process of the motor parameters. In addition, the experimental
test of the Lq identification has been conducted to confirm theperformance of the proposed technique as an example. The
resultant error of the Lq identification is approximately 4.5 %.
In the end of the paper, the reasons of this identification error
has been discussed. Furthermore, the winding resistanceidentification technique has been also discussed, which was an
optional function of the proposed identification technique
because the technique basically requires no knowledge of theresistance.

Figure 14 shows the motor current norm characteristics
obtained by the simulation and the experimental test, where
the black and the red lines are the simulation and theexperimental results, respectively. As can be seen in the graph,
the two curves have the minimum points at the same values of
Lq. However, the experimental test was broken off in therange of the high Lq region, and the parabolic characteristic
was lost due to the unstable rotation of the test motor.
As the parabolic characteristic was confirmed in the above
preliminary test, the simulation adopting the hill-climbing
algorithm was conducted to check the dynamic behavior of the
Lq identification. Figure 15 shows the simulation result. It isfound that the set value in the controller converges to the true
value within the identification error of 0.3 %.

Identification of ψ and Ld will be also examined throughthe experimental tests in the future work to confirm total
feasibility of the proposed technique.
In addition,
improvement of the identification accuracy will be consideredon the basis of the discussion suggested in this work.

Figure 16 shows the experimental result of the proposed
technique. The dynamic convergence behavior of the set
value of Lq has a slight overshoot with a damping, and thefinal converged value is 23.4 mH. The true value of Lq is
24.5mH, so the identification error is 4.5 % in the test. From
this result, the proposed technique is capable to identify Lq
stably within the error of approximately ± 5 %. It isconsidered that the identification error is caused by thefollowing possible reasons:

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1) ripples of the motor current norm, and
2) undesirable fluctuation of the low-rotation-speed, which
has an impact to the current norm.
To overcome these problems, the following countermeasures
can be effective solutions:
1) reduction of ripple effects using averaging scheme of
the motor current norm, and2) testing in higher speed region to reduce the fluctuatedspeed effect or adding the speed control loop to the
proposed identification system to make the speed
control more stable.
IV.

CONCULUSION

The off-line identification technique of the IPM motor
parameters has been proposed in this paper. The most uniquefeature of the technique is capability to identify the parameters
by using only the motor current norm information androbustness to the winding resistance variation.
The
identification proceeds step by step with changing thestructure of the current controller and with seeking theminimum point of the current norm on the basis of the wellknown hill-climbing algorithm. The identification of the287