電気学会論文誌●（●●●●●●●部門誌）
IEEJ Transactions on ●●●●●●●●●●●●●●●
Vol.●● No.● pp.●-●●
DOI: ●.●●/ieejeiss.●●.●

Paper

Stabilization Method for IPMSM with Long Electrical Time ConstantUsing Equivalent Resistance Gain Based on V/f Control
Jun-ichi Itoh＊a), Senior member,

Takato Toi＊, Member,

Koroku Nishizawa＊, Student member(Manuscript received Jan. 00, 20XX, revised May 00, 20XX)
This paper proposes a novel feedback control loop based on a damping control in a V/f control in order to stabilize interior
permanent magnet synchronous motors (IPMSMs) with a long electrical time constant. A problem of the conventional damping
control is that ignored roots move to the unstable region due to the conventional damping gain K1. In addition, the ignored roots
are apt to become unstable because of its long electrical time constant. Therefore, a novel method is proposed in order to solve thisinstability problem. In this paper, first, a boundary condition of stable region is derived based on state equation. Then, a novel
current feedback loop of the current is added to an output voltage command. As experimental results, the motor becomes unstablewith the conventional damping control under a rated speed of 0.9p.u. and a rated torque of 0.7p.u. Under the common operation
condition, the motor is stabilized by employing the novel feedback control loop.
Keywords : Damping control, IPMSM, Root locus, V/f control

1.

stabilize the high speed IPMSMs by increasing the winding
resistance equivalently. The contribution of this paper is that theapplication of the V/f control is enlarged to the long electrical time
constant IPMSM by adding a feedback loop.
This paper is organized as follows; first, the conventional
damping control is introduced. Next, an auto-tuning method for theconventional damping control is explained. In addition, the stabilityof the V/f control is analyzed in order to clarify the boundary
condition of the unstable region with the conventional damping
control method. Next, the stability analysis is conducted withproposed feedback loop. From the experimental results, theequivalent resistance gain K2 stabilizes the unstable V/f control
system.

Introduction

Recently, interior permanent magnet synchronous motors(IPMSMs) are widely utilized due to their high efficiency and highpower density(1)-(6). V/f control is generally employed for
applications such as pumps and fans, where high dynamic
performance is not demanded, because of its effectiveness andsimplicity instead of a sensorless field-oriented-control(7)-(8).
In case that an open-loop V/f control without any feedback loop
is applied for an IPMSM, persistent oscillation occurs in motor
speed. Thus, the control system becomes unstable due to thisoscillation. Therefore, the employment of the feedback loop of anactive current is proposed as one of stabilizing methods(9). By
suppressing the oscillation with this damping control, stable
operation is achieved. In this method, the parameters of thedamping control are decided by equations which are acquired fromsecond-order state equation. In addition, the parameters of thedamping control are calculated by an auto-tuning method. With thisauto-tuning method, the motor parameters are not necessary inadvance. However, due to the influence of ignored roots, this design
method still cannot stabilize some IPMSMs because these roots
move to the right in the s-plane in accordance with the increase of
the conventional feedback loop gain K1. In addition, the ignored
roots are apt to become unstable in the long electrical time constant.
In particular, the electrical time constant of a high-speed motor
becomes long because of its small winding resistance. Therefore, ahigh-speed motor tends to become unstable based on V/f control
with the conventional damping control using active current
feedback.
In this paper, a novel feedback loop is proposed in order to solve
the instability problem. The originality of this paper is an additional
gain K2 which is the gain in the proposed feedback loop in order to

2. V/f Control for IPMSM
2.1 Damping Control Based on V/f Control
Fig. 1shows the relation between -frame and dq-frame. The d-axis isdefined as the direction of the flux vector of the permanent magnet.
The q-axis is defined as the electro motive force vector. On the other
hand, the V/f control is implemented on the -frame. The -axis isaligned with the direction of the inverter output voltage, whereasw  Ld id    Lq iq 
2





RIa

wym

v
Ia
i

b

q
f

i
d

a) Correspondence to: Jun-ichi Itoh. E-mail: itoh@vos.nagaokaut.ac.jp＊
Nagaoka University of Technology.
1603-1, Kamitomioka-machi, Nagaoka, Niigata, Japan 940-2188© 200● The Institute of Electrical Engineers of Japan.

q2Fig. 1. Relation between -frame and dq-frame.
1Stabilization Method for IPMSM using Equivalent Resistance Gain Based on V/f Control (Jun-ichi Itoh et al.)the -axis is defined as the -axis delayed by 90 degrees.
Fig. 2 shows the V/f control block diagram with the conventional
damping control. The V/f control is based on the -frame. The
constant oscillation occurs in the high-speed region due to theresonance between the inertia of the motor and the inductance whenthe IPMSMs are driven by the open loop V/f control. Thus, the axis current, which represents the active component, is utilized inorder to stabilize the oscillation.
2.2
Conventional Design Method for Damping Control
Based on V/f Control
First, in order to analyze the stability of
the V/f control, the IPMSM model at steady state is linearized. The
linearized fifth-order state equation, which is expressed by (1)below, is derived from the following 4 equations: γδ-axis voltage
equations (2), relational expression between torque and speed (3),relational expression between load angle and speed (4), andrelational expression between input and output of HPF applied in
damping control (5).
 i 
 v   R  p  L  L cos 2q   w L sin2q pL sin2q  w  L  L cos 2q    
R  p  L  L cos 2q   w L sin2q  i v   pL sin2q  w  L  L cos 2q ....... (2)
0

1

1

1

1 1

0

1

1

1

0

0

v* = 0

wrm*

Pf2
J





d

q





1



Ld  Lq2.................................................................... (5)

L1 













L0  L1



iu
iv
iw

i
i

Lq  Ld
2

Motor A
3.7 kW
1800 r/min
14 A
3
6.2 mH15.3mH
0.69 W
0.27 Vs/rad
0.037 kgm2

Motor B
3 kW
12000 r/min
17.3 A2
2.04 mH
2.24 mH
0.133 W
0.107 Vs/rad
0.0013 kgm2

5.05

38.5

12.5

42.3

. ....................................................................... (7)y m  L0  L1  sin q 0  2 L12 i 0  2 L0 L1 i 0 sin 2q 0  i 0 cos 2q 0
L0 2  L12

 K1 L0 2  L12 i 0  2 L0 L1w0 sin2q 0  R  L0  L1 cos 2q 0 

y m  L0  L1  cosq 0  2 L12 i 0  2 L0 L1  i 0 cos 2q 0  i 0 sin 2q 0 L  L0L12  L0 2

2
1

2

Pf 2y m 1
L  Lq i 0 cos 2q 0  i 0 sin 2q 0 cos  0 
J 
ym d

 K1
1















2

L12  L0 2
2 Ld  Lq
Pf 2y m 
Ld  Lqi 0 i 0 sin 2q 0 i 0 2 cos 2q 0  i 0 2 cos 2q 0 i 0 cosq 0  i 0 sin q 0 
J 
ym
ym







0
0

2



0
1
0



w0y m  L1  L0  sin q 0  2w0 L12 i 0  2w0 L0 L1 i 0 sin 2q 0  i 0 cos 2q 0



1
s

K1

L0 2  L12

w0y m  L0  L1  cosq 0  2w0 L1 i 0  2w0 L0 L1 i 0 cos 2q 0  i 0 sin 2q 0
2x K1 L0 2  L12 i 0  w0 L0 2  2 L0 L1 cos 2q0  L12  RL1sin2q 0



2



The state variables are i, i, w, q, and x. Root locus is obtained
from fifth-order state equation. Note that the variables with
subscription 0 mean the variables at the operating point.
Table 1 shows the motor parameters of two IPMSMs. In thispaper, the stability of the two motors is analyzed. Note that the
motor with a rated power of 3.7 kW is defined as Motor A, and theother is Motor B(10). In addition, electrical time constants %XLd / %R
and %XLq / %R of Motor B are longer than that of Motor A by 7.6, ..................................................................... (6)-Rated mechanical power Pm
Rated speed wn
Rated current In
Number of pole pairs Pf
d-axis inductance Ld
q-axis inductance Lq
Winding resistance R
Field flux linkage ym
Inertia moment J
Electrical time
constant %XLd / %R
Electrical time
constant %XLq / %R
 R  L0  L1 cos 2q 0   2w0 L0 L1sin2q 0
L0 2  L12 i  
 RL1sin2q 0  w0 L0 2  L12  2w0 L0 L1 cos 2q 0   i  
L12  L0 2
p  w    2
P
y

1


   f m q   J sin q 0  y Ld  Lq i 0 sin 2q 0  i 0 cos 2q 0 m x  
 
0



0


3f/

1
s

Table 1. Motor parameters of two IPMSMs.

where, i is the -axis current, i is the -axis current, w is the motor
speed, w* is the command value of motor speed, q is the load angle,
x is the output of integrator from high pass filter (HPF), p is thedifferential operator, R is the winding resistance, ym is the field flux
linkage, Vf is the f/v conversion ratio, K1 is the damping gain,  isthe time constant of HPF, Pf is the number of pole pairs, and J is theinertia of the motor. Note that the definition of per unit value of K1
is a product of rated speed and rated current (rad/sA). Furthermore,
following approximations are used in the derivation for (1):1) product of more than two perturbation (Δ) terms is 0, and2) load angle perturbations are small; sin Δq = Δq, cos Δq = 0,
when Δq is small.
In addition, L0 and L1 are defined as

L0 

vu*
vv*
vw*

Fig. 2. Block diagram of V/f control based on -framewith conventional damping control.

pq  w*  K1  i  x   w .................................................... (4)px   i  x 

/3f

HPF

1

+

1 1



w 1*

Damping control

y i sinq  i cosq    L  L i cosq  i sinq i sinq  i cosq  ...... (3)
m

*

-

1

1

v

f/v conv.
+

 sinq  wy m cosq 

pw 

w

Pf

*


K1i 0 L sin2q 0
i 0  1 2
VfL0  L12  i   K1i 0     i   i 0  L0  L1 cos 2q 0 V f 22*  w   
L1  L0
 w




0   q  0  x  
1

 


K1 
0




1
 
 

(1)

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Stabilization Method for IPMSM using Equivalent Resistance Gain Based on V/f Control (Jun-ichi Itoh et al.)times and 3.4 times.
Fig. 3 shows the eigenvalue plot of the Motor A when thedamping gain K1 is varied at rated speed and no load. Note that No.
1-5 represent eigenvalues of fifth order state equation. In theconventional design method, the damping gain K1 is decided to
become multiple roots of No. 2 and No. 3 in order to suppress theovershoot of the motor speed. As shown in Fig. 3, the value of
0.15p.u. is the optimum value for the damping gain K1. In addition,
all roots are located in the left half of the s-plane; therefore, thesystem is stable.
From fifth-order state equation, the approximated second-orderstate equation is derived with the application of the following
approximations, and expressed by (8):1) under no-load condition; q0 = 0, and iδ0 = iγ0 = 0,
2) under high-speed condition; ω0L >> R, and ω0 >> K1iδ0,3) mechanical time constant is sufficiently larger thanelectrical time constant; pΔiγ = pΔiδ = 0,
4) root of HPF has small influence on the stability; Δx = 0.

Imaginary part

600

Lq

ym

K1 : 0.15p.u.

No. 4

-80

-60

-40

Real part

-20

0

20

Unstable region

40

Imaginary part

30

No. 2

20

10

10

5

0

0

-10

-5

No. 5

-20

-10

-40
-50
-60

-10

No. 3

wc/wn=1/20
wc/wn=1/10
wc/wn=1/5

-30

-40

-20

0

-5

0

20

Real partFig. 4. Loot locus of fifth-order state equation under different
cutoff frequency wc at no-load and rated speed.

2

wc 

wn
20



3 Pf y m. .............................................. (13)
2 JLq

1
20

It is noted that the damping coefficient  is set as 1 in (12).
2.3
Auto-tuning
Methodfor
ParametersofConventional Damping Control
In this section, an autotuning method for the conventional damping control is explained.
In the auto-tuning, the parameters are identified in order to calculate
the damping gain K1 and cutoff frequency wc based on the equations(12) and (13).
The voltage command v is expressed by

3 Pf y m. ............................................................. (11)2 JLq

2wn LqK1 : 0 p.u.

K1 : 0p.u.

50

As mentioned before, the damping gain K1 is set in order tobecome multiple roots of No. 2 and No. 3.
On the other hand, the cutoff frequency wc of the HPF needs tobe lower than the natural angular frequency wn in order to suppress
the oscillation of the motor speed.
Fig. 4 shows the eigenvalue plot of the Motor A when thedamping gain K1 is varied under different cutoff frequency wc at
rated speed and no load. In the figure, the root of No. 5 is on thereal axis regardless of the damping gain K1 when the cutoff
frequency is set as 1/20 of the natural angular frequency wn.
From above considerations, the damping gain K1 and the cutoff
frequency wc are expressed by (12) and (13).

K1 

K1 : 0.15p.u.

No. 3

-200

Fig. 3. Loot locus of fifth-order state equation when damping gain
K1 is increased at no-load and rated speed with Motor A.

y m K1. ................................................................... (10)2wn Lq

wn 

0

-800
-100

where s is the complex variable in Laplace transform.
Then, the damping coefficient  and the natural angularfrequency wn are expressed by (10) and (11).

K1 : 0p.u.

No. 5

No. 2

-600

3 Pf y m
 0, .......................................... (9)
2 JLq
2

s

K1 : 0.15p.u.

200

K1 : 0.15p.u.

It is noted that second-order state equation is derived under thecondition of high-speed region and no load in order to simplify the
analysis of the stability because the influence of load is relatively
smaller than the influence of motor speed(11)-(12).
From (8), characteristic equation is expressed

ym

No. 1

K1 : 0 p.u.

400

-400

2
2

3 Pf y m 0
2 JLq   w  0  * w  p        w . ...................... (8)y m     1 
   
1  K1Lq s 2  K1

Unstable region800

v 2  wy m  RI a   w Lq I a2

 . .................................... (14)2

It is noted that the equation (14) is derived when id = 0 control isachieved. In addition, Ia = iq under id = 0 control.
From (14), the field flux linkage ym is identified at the steadystate by using
2



ym 





v 2   w * Lq I a   R I aw*. ................................... (15). ................................................................. (12)Here, v is the voltage command, Lq is the identified value of

q-axis inductance, R is the identified value of winding resistance,
Ia is the output current.
3

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Stabilization Method for IPMSM using Equivalent Resistance Gain Based on V/f Control (Jun-ichi Itoh et al.)Next, the winding resistance R is identified in DC test at astandstill as (16).
VDC  D  f swTd 
1.5I u. .................................................... (16)

Calculating Ra in DC test

where D is the duty factor of the u-phase upper arm, and Td is thedead-time.
The reactive power on the dq-axis is expressed by (17) under id= 0 control.

2



ym 

Qdq  w Lq I a . ................................................................ (17)2Applying MTPA control
based onhill-climbing method

The reactive power on each axis is corresponded. Therefore, theidentified q-axis inductance is expressed as (19) from (17) and (18).

Identifying Lq

v i

w Ia
*

Identifying wn
based on hillclimbing method2. ................................................................... (19)



Lq 

The natural angular frequency is identified based on the hillclimbing method by injecting the sinusoidal wave into the speedcommand. The output current becomes the maximum value at thenatural angular frequency. Therefore, the frequency of the injectedsinusoidal wave is varied in order to search the maximum value ofthe output current.
Fig. 5 shows the flowchart of the auto-tuning. First, the winding
resistance R is identified in DC test at standstill. Then, the motor isaccelerated up to 0.5p.u. because it is impossible to identify the field
flux linkage ym and the q-axis inductance Lq at standstill as shownin (15) and (19). Next, the q-axis inductance Lq and the field flux
linkage ym are identified after applying maximum torque per
ampere (MTPA) control based on the hill-climbing method(13). The
motor parameters are not necessary in the MTPA control because
the operation point is searched by the relation of the output current.
In addition, the natural angular frequency is identified by the
relation between the frequency of the injected sinusoidal wave andthe magnitude of the output current based on the hill-climbing
method.
Fig. 6 shows the block diagram of the V/f control during the autotuning. From the figure, the motor is accelerated without the HPFof the damping control.
With this auto-tuning for the conventional damping control, theMotor A, which is stable with the conventional damping control, isstabilized.
2.4
Unstable Condition with Conventional Damping
Control
Fig. 7 shows the eigenvalue plot of the Motor B when
the damping gain K1 is varied at the same condition of rated speed
and no load. As shown in Fig. 7, the roots of No. 2 and No. 3 are
multiple roots when the damping gain K1 is 0.05p.u. However, theroots of No.1 and No.4 move into the right half of the s-plane.
Therefore, these roots make the system unstable.
It is concluded from the above considerations that the Motor B
becomes unstable due to the roots of No. 1 and No. 4 when thedamping gain K1 is decided by the conventional design method. It
is noted that these roots are ignored when the fifth order state
equation is linearized to second-order state equation.

Stopping injecting
sinusoidal wave tospeed commandv i

w* Ia2Adjusting K1 and wc

END

Fig. 5. Flowchart of auto-tuning method
for conventional damping control based on V/f control.
w+*v* = 0
v* +

Vf

+

/3f

vu**
vv
vw*

3f/

iu
iv
iw

-

v

Asinwat

wn
I

2
a

Ia

Ia2



w*

Injecting sinusoidal
wave to speedcommandIncreasing K1
by 0.01p.u.

YesQ  v i . ..................................................................... (18)   v 2 w* Lq Ia   RIa

No

Accelerating up to
rated speed of 0.5p.u.

On the other hand, the reactive power on the -axis is expressedby (18)Lq 

Identifying ym

Setting K1 as 0.01p.u.

wa

Ia



R

START

v
MTPA control based onhill climbing method
+
1
w1
s

Identifying wn based onhill climbing method

K1
Damping control
without HPFi
i

Fig. 6. Block diagram of V/f control when tuning parameters.
Unstable region
3000
2500
2000

No. 1

K1 : 0 p.u.

K1 : 0.05p.u.

Imaginary part

1500
1000

No. 2

No. 5

500

K1 : 0 p.u.

0
-500
-1000

K1 : 0.05p.u.

No. 3

K1 : 0.05 p.u.

K1 : 0 p.u.

-1500

K1 : 0 p.u.

-2000

K1 : 0.05p.u.

No. 4

-2500

-3000
-200

-150

-100

-50

Real part

0

50

100

Fig. 7. Loot locus of fifth-order state equation when damping gain
K1 is increased at no-load and rated speed with Motor B.
4

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Stabilization Method for IPMSM using Equivalent Resistance Gain Based on V/f Control (Jun-ichi Itoh et al.)Table 2. Routh table of fourth order state equation of V/f control
with only damping gain K1.

2.5
Boundary Condition of Conventional Damping
Control
In this section, the unstable condition with only thedamping gain K1 is derived. As mentioned before, the analysis isconducted at no load condition in order to simplify the analysis of
the stability.
First, the fifth-order state equation is linearized into fourth-orderstate equation when the cutoff frequency is much smaller than thenatural angular frequency of the second-order state equation. Then,the characteristic equation is expressed by (20).

3 Pf y m
R2
2 JL qLd Lq
2a2  w 0 

s4

a4  1

s3

R
R

Ld
Lq
a a  a 4 a1
b1  3 2a3
b a  a 3b2
c1  1 1
b1
a0

s2
s1
s0

2

a3 

a1  K 1

w 0 2y m
Lq

2

3 Pf y m
2 JL q2

2

3 w 0 Pf y m
2
JL q
2

a0 

R
Ld

2

2

0

a0

0

0

0

0

0

s 4  a3 s 3  a2 s 2  a1s  a0  0, ....................................... (20)

where

R R
 , ............................................................... (21)
Ld Lq

a2  w0 2 
a1  K1
a0 

Real part of No. 1 and 4

a3 

100

3Pf 2y m 2
2 JLq

w0 2y mLq



3Pf 2w0 2y m 2
2 JLq



R2
, ......................................... (22)
Ld Lq

3Pf 2y m 2 R
2 JLd Lq

, ......................................... (23)8060Stable region of motor A

40

Stable regionof motor B

20
0
-20
-40
-60

Motor A
Motor B

-80
-100

. ......................................................... (24)

0

2
2
Ld 1
3 Pf y m
 0.
2 J Ld  Lq R w0 2



wrm

*

0.2

0.25

0.3

0.35

0.4

Pf

HPF

Additional gain



vu*
vv*
vw*

i 3f/

iu
iv
iw

1
s

K1
K2

................... (25)/3fv* = 0**v* + vf/v conv.
-

w1*

+ w1
-

i

Fig. 9. Block diagram of V/f control based on -frame withproposed feedback loop to output voltage command.

22
Lq  wn  y Ld  Lq  wn 
b1'  m K1    0. ............. (26)
Ld  w0  
R  w0  RThen, the real parts 1,4 of No. 1 and No. 4 are expressed1,4 

Here, b1' is the left-side term of the inequality equation (25).
It is concluded from the inequality equations (25), (26) andTables 1–2, the left-side term are positive when the damping gain
K1 of 0.15p.u. is designed in the Motor A. On the other hand, b1',
which is the left-side term of (26), is negative when the damping
gain K1 of 0.05p.u. is employed in the Motor B. Therefore, thesystem is unstable. These consequences are corresponded to theeigenvalue plots as shown in Figs. 3 and 6.
Fig. 8 shows the real parts of the roots of No. 1 and 4 when thedamping gain K1 is varied. The real parts 2,3 of No. 2 and No. 3
are expressed 2,3  

0.15

Fig. 8. Real part of No.1 and 4 when damping gain K1 isincreased under rated speed and no load.

2



0.1

Damping gain K1 [p.u.]

Here, w0 is the steady state value of the motor speed.
From the equations (20)–(24) that the coefficients of the fourthorder characteristic equation are positive regardless of the value ofthe motor parameters. Therefore, the necessary condition of the
Routh-Hurwitz stability criterion is satisfied. Thus, the unstablecondition is derived from the Routh table.
Table 2 shows the Routh table which is acquired from theequations (20)–(24). The coefficients of the left end column arefocused in order to evaluate the stability of the system. The unstablecondition, where the coefficients b1 and c1 are negative, is expressedas (25) and (26).

Ld Ld wn 2 R 1
y m  Ld 



K1
2
2
R
R w0
Lq w0
Ld  Lq  R 

0.05

K1y m R  1
1 
 

 . ......................................... (28)
2 Lq
2  Ld Lq 

The real parts of No. 2 and No. 3 decrease proportionally to thedamping gain K1. On the other hand, the real parts of No. 1 and No.
4 increase proportionally to the damping gain K1. Therefore, thestable condition is limited depending on the damping gain K1. In
addition, the stable region of the Motor B is smaller than that of theMotor A. In particular, the high-speed motor with small winding
resistance is more apt to become unstable as expressed in (28). It is
concluded from above considerations that there is unstablecondition depending on the roots of No. 1 and No. 4. Therefore, anovel method is necessary in order to make the roots of No. 1 andNo. 4 move into left-half of s-plane.

K1y m. .............................................................. (27)2 Lq

5IEEJ Trans. ●●, Vol.●●, No.●, ●●●Stabilization Method for IPMSM using Equivalent Resistance Gain Based on V/f Control (Jun-ichi Itoh et al.)3.

Proposed Additional Feedback Loop

Unstable region
3000

3.1
Principle of Stabilizing Method with Additional
Feedback Loop
Fig. 9 shows the control diagram of the V/fcontrol with the novel feedback loop. As shown in Eq. (25), thethird term in the left-side term of the inequality equation becomeslarger when the winding resistance increases. Furthermore, thenegative term, which is the fourth term, becomes smaller when thewinding resistance increases. In other words, it is possible tostabilize the system when the term of the winding resistance isincreased equivalently. Therefore, a novel feedback loop is added
to the inverter voltage command v*. The feedback loop consists ofan additional gain K2 and the -axis current which is filtered by the
HPF. Note that the definition of per unit value of K2 is a quotient ofrated voltage divided by rated current (Ω).
3.2
Stability Analysis with Additional Feedback Loop
The stability analysis is conducted under the same procedure in
section 2.4. The unstable condition is derived from the Routh table.
It is expressed as
Ld Ld wn 2 R  K 2 1
ym
Ld 2



K
2
2
R
R w0
Lq w0
 R  K 2  Ld  RLq R 1
3 Pf y m
2 J
2



2

2500

Imaginary part

1500

K2 : 0.2p.u.

K2 : 0 p.u.

1000

No. 3

500

No. 5

0

No. 2

-500
-1000
-1500

K2 : 0 p.u.

K2 : 0.2p.u.
No. 4

-2000
-2500
-3000
-200

-150

-100

-50

0

50

100

Real partFig. 10. Root locus when damping gain K2 is increased underrated speed and no load with Motor B.

Identified value of Ra [0.5 W/div] 0.759 W
0.693 W

(29)

Nominal value of Ra [0.5 W/div]

0

Ld 1
R
 0,
 R  K 2  Ld  RLq R w02

U-phase current [5 A/div]
6.6 A

2
2
y
Lq  wn    R  K 2  Ld  RLq  wn 
m
b 
K1 

 

  0. (30)
Ld  w0  R2
 w0 
 R


0

''
1

where b1'' is the left-side term of (29).
It is shown in (29) that the additional gain K2 is added to thenumerator of the third term in left-side by adding new feedback
loop. In addition, the denominators of the fourth and fifth terms arealso increased by the additional gain K2. Therefore, the winding
resistance is equivalently increased by the additional gain K2 inorder to stabilize the system.
Fig. 10 shows the eigenvalue plot when K2 is increased at the
rated speed and no load. It is noted that the damping gain K1 isdecided as multiple roots of No. 2 and No. 3. The roots of No. 1 andNo. 4 move into the left side of s-plane in accordance with theincrease of the additional gain K2. This stabilization is notachievable with only damping gain K1.

4.

No. 1

2000

20 ms/div

DC test is started.

Fig. 11 Waveforms of DC test.

Identified value of Lq [20 mH/div]
0

Identified value of ym [0.2 Vs/div]

12.2 mH
0.255 Vs

00

Identified value of wn [0.1p.u./div]
41 rad/s (0.0726p.u.)
Damping gain K1 [0.1p.u./div]
0.01p.u.

0.133p.u.

0

Experimental Results

Estimation is started.

10 s/div

Fig. 12 Waveforms of auto-tuning each parameter.

4.1
Auto-tuning for Conventional Damping Control
In this section, the effectiveness of the auto-tuning method forthe conventional damping control is confirmed in the experimentwith the Motor A. It is noted that the DC test is conducted atstandstill. In addition, even though the dead-time has not been
considered in the analysis, the dead-time influence appears as thelarger winding resistance in the experiment. Therefore, in theexperiment, the system might become stable even in the unstableregion in the root locus analysis result. In order to ignore this stableregion difference between the experimental result and analysis
result, the other parameters are identified under the condition of a
rated speed of 0.5p.u. and a rated torque of 0.27p.u.
Fig. 11 shows the waveform of the DC test. The identified valueof the winding resistance converges to the nominal value in thesteady state. In general, winding resistance of a general-purpose

motor is relatively small. Therefore, the duty ratio is set as 5% inorder to prevent overcurrent in the test. From the figure, the winding
resistance is identified with an error of 9.6%.
Fig. 12 shows each identified parameter and calculated damping
gain K1. From the figure, the identification and the calculation are
conducted in 44 s. The frequency of the injected sinusoidal wave isvaried each 1 Hz in the identification of the natural angularfrequency. In addition, one period of the sinusoidal wave needs to
be injected at least. Thus, the injecting time of the sinusoidal wave
on each frequency is set as 1 s. Therefore, the tuning time of thenatural angular frequency is relatively long compared with the otherparameters.

6

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Stabilization Method for IPMSM using Equivalent Resistance Gain Based on V/f Control (Jun-ichi Itoh et al.)Fig. 13 shows the step response before and after the auto-tuning.
In Fig. 13 (a), the damping gain is set as 0.01p.u. In addition, the
HPF of the damping control is not utilized. On the other hand, thedamping gain K1 is set as 0.13p.u. and the cutoff frequency is 1/20of the identified natural angular frequency. In the figure, the motor
speed oscillates when the step speed command is employed before
applying the auto-tuning. In addition, the steady-state error remains
in the motor speed because the HPF is not implemented. On theother hand, the motor speed follows the command without anovershoot.
Fig. 14 shows the step responses with the optimal damping gain
and the tuned damping gain. From the figure, the overshoot withthe tuned damping gain is almost same as the optimal damping gain.
Through the experimental results, the effectiveness of the autotuning method is confirmed.
4.2
Stabilization with Additional Feedback LoopIn
order to confirm the validity of the additional feedback loop, theexperiment is conducted. It is noted that Motor A which is shownin Table 1 is used in the experiment. In addition, inductors of 10
mH are connected in series in order to evaluate the motor with thelong time constant.
Fig. 15 shows the waveforms of , -axis current and U-phasecurrent at 0.9p.u. of rated speed and 0.7p.u. of rated torque when
the damping gain K1 is varied from 0.1p.u. to 0.2p.u. Note that theadditional gain K2 is set as 0 at this condition. The diverging
oscillation occurs in , -axis current after the damping gain K1 isvaried to 0.2p.u. Then, the motor is stopped due to the overcurrent
detection. Each frequency of the oscillation is 81 Hz and 78 Hz,
whereas, the frequency which is obtained from the eigenvalue plotis 81 Hz. The experimental value agrees with the theoretical value
with error of 5.5%. As a conclusion, the system with the motor, of
which electrical time constant is long, easily becomes unstable onlywith the conventional damping gain K1 due to the influence of theignored roots depending on K1.
Fig. 16 shows the waveforms of , -axis current and U-phasecurrent at 0.9p.u. of rated speed and 0.7p.u. of rated torque when
the additional gain K2 is varied. The system is stabilized by theadditional gain K2 even under the unstable condition with only thedamping gain K1. In addition, the current ripple is also suppressedby the additional gain K2.
From the experimental results, the effectiveness of the additional
feedback loop is confirmed.

5.

Speed command[0.01p.u./div]
0.01p.u.

Speed command[0.01p.u./div]
0.01p.u.
0.9p.u.

0.9p.u.

Motor speed[0.01p.u./div]

Motor speed [0.01p.u./div]
U-phase current [0.5p.u./div]

00U-phase current
400 ms/div[0.5p.u./div]

400 ms/div(a) Original damping gain
without HPF.

(b) Tuned damping gain
with HPF.

Fig. 13 Waveforms of step response before and after auto-tuning
under 0.9p.u. of rated speed and 0.8p.u. of rated torque.
0.92p.u.

Speed command [0.01p.u./div]
Max: 0.914p.u.
0.91p.u.

0.01p.u.
0.9p.u.

Motor speed [0.01p.u./div]
400 ms/div(a) With optimal damping gain (K1 = 0.16p.u.).
0.92p.u.

Speed command [0.01p.u./div]
Max: 0.913p.u.
0.91p.u.

0.01p.u.
0.9p.u.

Motor speed [0.01p.u./div]
400 ms/div(b) With tuned damping gain (K1 = 0.13p.u.).
Fig. 14 Waveforms of step response with different damping gain
K1 under 0.9p.u. of rated speed and 0.8p.u. of rated torque.

Conclusion

-axis current[1p.u./div]

In this paper, the designing method for the conventional damping
control is explained. In addition, the auto-tuning method is
introduced for the motor which is stable with the conventional
damping control. Then, the unstable condition due to the damping
gain K1 is derived based on the state equation regarding the V/fcontrol for IPMSMs. From the unstable condition, it is clarified thatthe high-speed motor, which the electrical time constant is long, is
apt to be unstable. The validity of the analysis is confirmed by the
eigenvalue plot and the experiment. In order to solve this instability
problem, the novel feedback loop is added to the inverter voltage
command in order to increase the winding resistance. As a result,the stable condition is achieved by the additional equivalent
resistance gain K2 even under the unstable condition with only
damping gain K1. In the experiment, first, the effectiveness of the

12.3 ms00

-axis current[1p.u./div]
0 U-phase current[20 A/div]

0

0

0

-axis current[1p.u./div]

-axis current
12.8 ms [1p.u./div]

U-phase current[20 A/div]
Overcurrent is detected.

Over current is detected.
K1 = 0.1p.u.

1 s/div
K1 = 0.2p.u.

20 ms/div

Fig. 15. Operation waveform without feedback loop K2 when
damping gain K1 is varied from 0.1p.u. to 0.2p.u. under 0.9p.u. of
rated speed and 0.7p.u. of rated torque.
7

IEEJ Trans. ●●, Vol.●●, No.●, ●●●Stabilization Method for IPMSM using Equivalent Resistance Gain Based on V/f Control (Jun-ichi Itoh et al.)K1 = 0.2p.u.,
K2 = 0.1p.u.

K1 = 0.2p.u.,
K2 = 0.2p.u.

(8)

K1 = 0.2p.u.,
K2 = 0.3p.u.

(9)

-axis current [0.5p.u./div]

(10)

0

-axis current [0.5p.u./div]

(11)
(12)

0

(13)

U-phase current [20 A/div]0

Jun-ichi Itoh

(Senior member) received his M.S. and Ph.D. degree
in electrical and electronic systems engineering from
Nagaoka University of Technology, Niigata, Japan in
1996, 2000, respectively. From 1996 to 2004, he waswith Fuji Electric Corporate Research andDevelopment Ltd., Tokyo, Japan. He was with
Nagaoka University of Technology, Niigata, Japan as
an associate professor. Since 2017, he has been aprofessor. His research interests are matrix converters, dc/dc converters,
power factor correction techniques, energy storage system and adjustablespeed drive systems. He received IEEJ Academic Promotion Award (IEEJ
Technical Development Award) in 2007. In addition, he also received Isao
Takahashi Power Electronics Award in IPEC-Sapporo 2010 from IEEJ, 58th
OHM Technology Award from The Foundation for Electrical Science andEngineering, November, 2011, Intelligent Cosmos Award from Intelligent
Cosmos Foundation for the Promotion of Science, May, 2012, and Third
prize award from Energy Conversion Congress and Exposition-Asia, June,
2013. Prizes for Science and Technology (Development Category) from theCommendation for Science and Technology by the Minister of Education,
Culture, Sports, Science and Technology, April 2017. He is a senior memberof the Institute of Electrical Engineers of Japan, the Society of Automotive
Engineers of Japan and the IEEE.

2 s/div
-axis current[0.2p.u./div]

12.3 ms
012.5 ms

11.9 ms

0

0

0

0

0

0

-axis current[0.2p.u./div]0U-phase current[20 A/div]0

10 ms/div

10 ms/div

G. L. Kruger, A. J. Grobler, S. R. Holm : "Improved non-ideality
compensation for the V/f controlled permanent magnet synchronous motor,"in Conf. Proc. Industrial Technology, pp. 331-336, 2013.
J. Itoh, N. Nomura, H. Ohsawa: "A comparison between V/f control andposition-sensorless vector control for the permanent magnet synchronous
motor," in Conf. Proc. Power Conversion Conf., Vol. 3, pp. 1310-1315, 2002.
D. Sato, J. Itoh: "Open-loop control for permanent magnet synchronous
motor driven by square-wave voltage and stabilization control," in Conf. Proc.
Energy Conversion Congress and Exposition, EC-0319, 2016.
Motion Control, INTECH, pp. 439-458, 2010.
Z. Tang, X. Li, S. Dusmez, B. Akin : "A New V/f Sensorless MTPA Control
for IPMSM Drives," IEEE Trans. on Power Electron., Vol. 31, No. 6, pp.
4400-4415, 2016.
J. Itoh, T. Toi, M. Kato : "Maximum Torque per Ampere Control Using Hill
Climbing Method Without Motor Parameters Based on V/f Control," in Conf.
Proc. European Conference on Power Electronics and Applications, DS3dTopic40283, 2016.

10 ms/div

Fig. 16. Waveforms when additional gain K2 is varied
under 0.9p.u. of rated speed and 0.7p.u. of rated torque.
auto-tuning for the conventional damping control is confirmed.
Next, the effectiveness of K2 is confirmed under 0.9p.u. of rated
speed and 0.7p.u. of rated torque.

Takato Toi(Member) received his B.S. and M.S. degrees inelectrical, electronics and information engineeringfrom Nagaoka University of Technology, Niigata,Japan in 2016 and 2018, respectively. Since 2018, hehas been with Mitsubishi Electric Corporation, Japan.
He is a member of IEEJ and IEEE. His researchinterests include PM motor drives.

Koroku Nishizawa(Student member) received his B.S. degree inelectrical, electronics and information engineeringfrom Nagaoka University of Technology, Niigata,Japan in 2015. And he has been in 5-year integrated
doctoral course of Nagaoka University of Technology.
He is the student member of IEEJ and IEEE. His
research interests include PM motor drives andmodulation method of power converters.

References
(1)

(2)

(3)

(4)

(5)

(6)

(7)

K. Kondo, S. Doki : "Position Estimation System for PMSM Position
Sensorless Control in Inverter Overmodulation Drive," IEEJ J. Ind. Appl.,
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R. Takahashi, K. Ohishi, Y. Yokokura, H. Haga, T. Hiwatari : "Stationary
Reference Frame Position Sensorless Control Based on Stator Flux Linkage
and Sinusoidal Current Tracking Controller for IPMSM," IEEJ J. Ind. Appl.,
Vol.6, No.3, pp.181-191, 2017.
A. Shinohara, Y. Inoue, S. Morimoto, M. Sanada : "Correction Method ofReference Flux for Maximum Torque per Ampere Control in Direct-TorqueControlled IPMSM Drives," IEEJ J. Ind. Appl., Vol.6, No.1, pp.12-18, 2017.
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for PMSMs with a Real-time Simulator Installed Behavior Motor Model,"IEEJ J. Ind. Appl., Vol.5, No.2, pp.167-173, 2016.
X. Ji, T. Noguchi : "Online q-axis Inductance Identification of IPM
Synchronous Motor Based on Relationship between Its Parameter Mismatchand Current," IEEJ J. Ind. Appl., Vol.4, No.6, pp.730-731, 2015.
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8

IEEJ Trans. ●●, Vol.●●, No.●, ●●●