Stabilization Method Using Equivalent ResistanceGain Based on V/f Control
for IPMSM with Long Electrical Time ConstantJun-ichi Itoh, Takato Toi, and Koroku Nishizawa
Department of Electrical, Electronics and Information Engineering
Nagaoka University of Technology
Nagaoka, Niigata, Japan
itoh@vos.nagaokaut.ac.jp, t_toi@ stn.nagaokaut.ac.jp, koroku_nishizawa@stn.nagaokaut.ac.jp
Abstract— This paper proposes a novel feedback control
loop based on a damping control in a V/f control in order tostabilize interior permanent magnet synchronous motors(IPMSMs) with a long electrical time constant. A problem of
the conventional damping control is that ignored roots moveto the unstable region due to the conventional damping gain
K1. In addition, the ignored roots are apt to become unstablebecause of its long electrical time constant. Therefore, anovel method is proposed in order to solve this instability
problem. In this paper, first, a boundary condition of stableregion 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 unstable with 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 isstabilized by employing the novel feedback control loop.
Keywords— Damping control; IPMSM; Root locus; V/fcontrolI.

INTRODUCTION

Recently, interior permanent magnet synchronous
motors (IPMSMs) are widely utilized due to their highefficiency and high power density [1–6]. V/f control is
generally employed for applications such as pumps andfans, where high dynamic performance is not demanded,because of its effectiveness and simplicity instead of asensorless field-oriented-control [7–8].
In case that an open-loop V/f control without any
feedback loop is applied for an IPMSM, persistentoscillation occurs in motor speed. Thus, the control
system becomes unstable due to this oscillation.
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, theparameters of the damping control are decided byequations which are acquired from second-order state
equation. In addition, the parameters of the damping
control are calculated by an auto-tuning method. With
this auto-tuning method, the motor parameters are notnecessary in advance. However, due to the influence ofignored roots, this design method still cannot stabilize
some IPMSMs because these roots move to the right inthe s-plane in accordance with the increase of theconventional feedback loop gain K1. In addition, theignored roots are apt to become unstable in the longelectrical time constant. In particular, the electrical time
constant of a high-speed motor becomes long because of
its small winding resistance. Therefore, a high-speed
motor tends to become unstable based on V/f control withthe conventional damping control using active current
feedback.
In this paper, a novel feedback loop is proposed inorder to solve the instability problem. The originality ofthis paper is an additional gain K2 which is the gain in the
proposed feedback loop in order to stabilize the highspeed IPMSMs by increasing the winding resistanceequivalently. The contribution of this paper is that theapplication of the V/f control is enlarged to the longelectrical time constant IPMSM by adding a feedback
loop.
This paper is organized as follows; first, theconventional damping control is introduced. Next, an
auto-tuning method for the conventional damping control
is explained. In addition, the stability of the V/f control isanalyzed in order to clarify the boundary condition of theunstable region with the conventional damping control
method. Next, the stability analysis is conducted withproposed feedback loop. From the experimental results,
the equivalent resistance gain K2 stabilizes the unstable
V/f control system.
II.
A.

V/F CONTROL FOR IPMSM

Damping Control Based on V/f Control

Fig. 1 shows the relation between γδ-frame and dqframe. The d-axis is defined 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, theV/f control is implemented on the γδ-frame. The δ-axis isaligned with the direction of the inverter output voltage,
whereas the γ-axis is defined as the δ-axis delayed by 90
degrees.
Fig. 2 shows the V/f control block diagram with theconventional damping control. The V/f control is basedon the γδ-frame. The constant oscillation occurs in thehigh-speed region due to the resonance between theinertia of the motor and the inductance when the
IPMSMs are driven by the open loop V/f control. Thus,the δ-axis current, which represents the active
component, is utilized in order to stabilize the oscillation.

ω ( Ld id ) + ( Lq iq )2

B. Conventional Design Method for Damping Control
Based on V/f Control
First, in order to analyze the stability of the V/fcontrol, the IPMSM model at steady state is linearized.
The linearized fifth-order state equation is expressed by(1) below.
Here, iγ is the γ-axis current, iδ is the δ-axis current, ωis the motor speed, ω* is the command value of motor
speed, δ is the load angle, x is the output of integrator
from high pass filter (HPF), p is the differential operator,
R is the winding resistance, ψm is the field flux linkage, Vf
is the f/v conversion ratio, K1 is the damping gain, τ is thetime constant of HPF, Pf is the number of pole pairs, andJ is the inertia of the motor. In addition, L0 and L1 aredefined as
Ld + Lq................................................................. (2),L0 =
2Lq − Ld................................................................. (3).
L1 =2The state variables are iγ, iδ, ω, δ, and x. Root locus isobtained from fifth-order state equation.
Table I shows the motor parameters of two IPMSMs.
In this paper, the stability of the two motors is analyzed.
Note that the motor with a rated power of 3.7 kW isdefined as motor A, and the other 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
times and 3.4 times.
Fig. 3 shows the eigenvalue plot of the motor A when
− R ( L0 + L1 cos 2δ 0 ) + 2ω0 L0 L1sin2δ 0L0 2 − L12 Δiγ  
− RL1sin2δ 0 + ω0 L0 2 + L12 − 2ω0 L0 L1 cos 2δ 0  
 Δiδ  
L12 − L0 2
p  Δω  =  2
P
ψ


1
   f m
 Δδ   J sin δ 0 + ψ Ld − Lq iγ 0 sin 2δ 0 + iδ 0 cos 2δ 0 m Δx  
 

0


0


(

(

)

)(

)

(

)

(

)

rm

*

*

Pf

v

f/v conv.

*
1

+

vu*
vv*
vw*

3 /

iu
iv
iw

1
s

-

Damping control

/3

HPF

1

+
-

τ

x

1
s

K1
i
i

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

the damping gain K1 is varied at rated speed and no load.
Note that No. 1-5 represent eigenvalues of fifth orderstate equation. In the conventional design method, thedamping gain K1 is decided to become multiple roots of
No. 2 and No. 3 in order to suppress the overshoot 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, the system is stable.
From fifth-order state equation, the approximatedsecond-order state equation is expressed by (4).

)(−ψ m ( L0 − L1 ) sin δ 0 + 2 L12iδ 0 + 2 L0 L1 iγ 0 sin 2δ 0 + iδ 0 cos 2δ 0

L0 2 − L12

L0 2 − L12()− K1 L0 2 − L12 iγ 0 + 2 L0 L1ω0sin2δ 0 + R ( L0 − L1 cos 2δ 0 )ψ m ( L0 − L1 ) cos δ 0 + 2L12iγ 0 − 2L0 L1 ( iγ 0 cos 2δ 0 − iδ 0 sin 2δ 0 )

L12 − L0 2

L12 − L0 2


Pf 2ψ m 
1
Ld − Lq iγ 0 cos 2δ 0 − iδ 0 sin 2δ 0 
cos δ 0 +
J 
ψm

− K1

)(

(

)

1

(

)

(

)

(

)

)

0
−1
0

τ


2 Ld − Lq
Pf 2ψ m 
Ld − Lqiγ 0iδ 0 sin 2δ 0 +iγ 0 2 cos 2δ 0 − iδ 0 2 cos 2δ 0 
iγ 0 cos δ 0 − iδ 0 sin δ 0 −
J 
ψm
ψm


0
0

v *= 0(−ω0ψ m ( L1 + L0 ) sin δ 0 − 2ω0 L12iδ 0 + 2ω0 L0 L1 iγ 0 sin 2δ 0 + iδ 0 cos 2δ 0
L12 − L0 2

Fig. 1. Relation between γδ-frame and dq-frame.

− K1 L0 2 − L12 iδ 0 + ω0 L0 2 + 2 L0 L1 cos 2δ 0 + L12 + RL1sin2δ 0−ω0ψ m ( L0 + L1 ) cos δ 0 + 2ω0 L12iγ 0 + 2ω0 L0 L1 iγ 0 cos 2δ 0 − iδ 0 sin 2δ 0
L0 2 − L122
K1iδ 0 L sin2δ 0
iδ 0 − 1 2Vf  Δi  
L0 − L12− K1iγ 0   γ    Δiδ   −i − L0 − L1 cos 2δ 0 V f *  Δω  +  γ 0
L12 − L0 2
 Δω
  0   Δδ  0  Δx  1  


K1 
0



1
− 
τ 

(1)

Table I.
Motor parameters of two IPMSMs.
Motor A
Motor B
Rated mechanical power Pm
3.7 kW
3 kW
Rated speed n
1800 r/min
12000 r/min
Rated current In
14 A
17.3 A
Number of pole pairs Pf
3
2
d-axis inductance Ld
6.2 mH
2.04 mH
q-axis inductance Lq
15.3mH
2.24 mH
Winding resistance R
Field flux linkage m
0.27 Vs/rad
0.107 Vs/rad
Inertia moment J
0.037 kgm2
0.0013 kgm2
Electrical time
5.05
38.5
constant %XLd / %R
Electrical time
12.5
42.3
constant %XLq / %R22

3 Pf ψ m 0
2 JLq   Δω  0 
 Δω  *p  =   Δδ  + 1  Δω ............. (4)
Δ
δ
ψ
  
   −1 − K1 m Lq 
It is noted that second-order state equation is derived
under the condition of high-speed region and no load inorder to simplify the analysis of the stability because theinfluence of load is relatively smaller than the influenceof motor speed [11–12].
From (4), characteristic equation is expressed22
ψ
3 Pf ψ m= 0 ........................................ (5).
s2 + K1 m s +
2 JLq
Lq

ωn =

3 Pf ψ m.............................................................. (7)2 JLq

As mentioned before, the damping gain K1 is set inorder to become multiple roots of No. 2 and No. 3.
On the other hand, the cutoff frequency ωc of the HPF
needs to be lower than the natural angular frequency ωnin order to suppress the oscillation of the motor speed.
Fig. 4 shows the eigenvalue plot of the Motor A when
the damping gain K1 is varied under different cutoff
frequency ωc at rated speed and no load. In the figure, theroot of No. 5 is on the real axis regardless of the damping
gain K1 when the cutoff frequency is set as 1/20 of thenatural angular frequency ωn.
From above considerations, the damping gain K1 andthe cutoff frequency ωc are expressed by (8) and (9).
2ωn Lq
K1 =................................................................... (8)

ψm

ωc =

ωn
20

=

1 3 Pf ψ m............................................... (9)20 2 JLq

It is noted that the damping coefficient ζ is set as 1 in
(8).
C. Auto-tuning Method for Parameters of Conventional
Damping Control
In this section, an auto-tuning method for theconventional damping control is explained. In the autotuning, the parameters are identified in order to calculate
the damping gain K1 and cutoff frequency ωc based on
the equations (8) and (9).
The voltage command vδ is expressed by(vδ 2 = ( ωψ m + RIa ) + ω Lq I a2) ............................... (10).
2

It is noted that the equation (10) is derived when id = 0
control is achieved.

Unstable region

800
600

No. 1

K1 : 0 p.u.

400

K1 : 0.15p.u.
K1 : 0p.u.

200

No. 5

No. 2

0

K1 : 0.15p.u.

No. 3

-200

K1 : 0.15p.u.
-400

K1 : 0 p.u.

K1 : 0p.u.

K1 : 0.15p.u.

No. 4

-600
-800
-100

-80

-60

-40

-20

0

20

Real partFig. 3 Loot locus of fifth-order state equation when damping gain K1 isincreased at no-load and rated speed with motor A.
Unstable region

50
40
30

Imaginary part

where s is the Laplace operator.
Then, the damping coefficient ζ and the naturalangular frequency ωn are expressed by (6) and (7).
ψ K
ζ = m 1 ..................................................................... (6)2ωn Lq

No. 2

20

10

10

5

0

0

-10

-5

No. 5

-20

-10

-30

c/

n=1/20

c/

n=1/5

c/

-40
-50
-60

-10

No. 3

-5

0

n=1/10-40-20Real part

020

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

From (10), the field flux linkage ψm is identified at thesteady state by using
2

∧

ψm =

∧

 ∧
vδ 2 −  ω* Lq I a  − R I a.................................. (11).



ω*
∧

Here, vδ is the voltage command, Lq is the identified∧

value of q-axis inductance, R is the identified value ofwinding resistance, Ia is the output current.
Next, the winding resistance R is identified in DC test
at a standstill as (12).

∧R=

VDC ( D − f swTd ).................................................... (12)1.5IuHere, VDC is the DC-link voltage, D is the duty ratio,
fsw is the switching frequency, Td is the dead time, and Iu
is the U-phase current at the steady state.
Next, the q-axis inductance Lq is derived from therelation of the reactive power on the dq-axis and γδ-axis.
The reactive power on the dq-axis is expressed by (13)under id = 0 control.

Qdq = ω Lq I a 2 ............................................................... (13)On the other hand, the reactive power on the γδ-axis isexpressed by (14)Qγδ = vδ iγ ...................................................................(14).
The reactive power on each axis is corresponded.
Therefore, the identified q-axis inductance is expressedas (15) from (13) and (14).
∧vδ iγ
Lq = * 2 ................................................................... (15)ω Ia
The natural angular frequency is identified based on
the hill-climbing method by injecting the sinusoidal wave
into the speed command. The output current becomes the
maximum value at the natural angular frequency.
Therefore, the frequency of the injected sinusoidal wave
is varied in order to search the maximum value of theoutput 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 is accelerated up to 0.5p.u.
because it is impossible to identify the field flux linkage
ψm and the q-axis inductance Lq at standstill as shown in(11) and (15). Next, the q-axis inductance Lq and the field
flux linkage ψm are identified after applying maximum
torque per ampere (MTPA) control based on the hillclimbing method [13]. The motor parameters are notnecessary 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 injectedsinusoidal wave and the magnitude of the output current
based on the hill-climbing method.
Fig. 6 shows the block diagram of the V/f control
during the auto-tuning. From the figure, the motor isaccelerated without the HPF of the damping control.
With this auto-tuning for the conventional damping
control, the motor A, which is stable with theconventional damping control, is stabilized.

START

Calculating Ra in DC test
Identifying

Setting K1 as 0.01p.u.

m2 ∧  ∧vδ −ω* Lq Ia  − RIa


2

∧

ψm =
Accelerating up to
rated speed of 0.5p.u.

No
Injecting sinusoidal
wave to speedcommandIncreasing K1
by 0.01p.u.

Yes
Applying MTPA control
based onhill-climbing method

Identifying n
based on hillclimbing method

Identifying Lq
∧

Lq =

ω*

Stopping injecting
sinusoidal wave tospeed command

vδ iγ

ω*Ia2

Adjusting K1 andc

END
Fig. 5 Flowchart of auto-tuning method
for conventional damping control based on V/f control.

1sFig. 6 Block diagram of V/f control when tuning parameters.
3000
2500
2000
1500
1000
500
0
-500

D. 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, the roots of No.1 and No.4

-1000
-1500
-2000
-2500
-3000
-200

-150

-100

-50

0

50

100

Fig. 7 Loot locus of fifth-order state equation when damping gain K1 isincreased at no-load and rated speed with motor B.

E. Boundary Condition of Conventional Damping
Control
In this section, the unstable condition with only thedamping gain K1 is derived. As mentioned before, theanalysis is conducted at no load condition in order tosimplify the analysis of the stability.
First, the fifth-order state equation is linearized into
fourth-order state equation when the cutoff frequency ismuch smaller than the natural angular frequency of thesecond-order state equation. Then, the characteristic
equation is expressed by (16).
s 4 + a3 s 3 + a2 s 2 + a1 s + a0 = 0 .................................... (16)
where
R
R
+
a3 =
.............................................................. (17),
Ld Lq
2

2

a 2 = ω0 +a1 = K1

2 JLq

ω0 2ψ mLq2

a0 =3Pf ψ m2+

R2...................................... (18),
Ld Lq

2

+

2

3Pf ω 0 ψ m
2 JLq

2

3Pf ψ m R
2 JLd Lq..................................... (19),2....................................................... (20).

Here, ω0 is the steady state value of the motor speed.
From the equations (16)–(20) that the coefficients ofthe fourth-order characteristic equation are positive
regardless of the value of the motor parameters.
Therefore, the necessary condition of the Routh-Hurwitz
stability criterion is satisfied. Thus, the unstable condition
is derived from the Routh table.
Table II shows the Routh table which is acquired fromthe equations (16)–(20). The coefficients of the left end
column are focused in order to evaluate the stability ofthe system. The unstable condition, where thecoefficients b1 and c1 are negative, is expressed as (21)and (22).
2

Ld Ld ω n 2 R 1
ψ m  Ld 
K1
+
+
−
2
2
R
R ω0
Lq ω0
Ld + Lq  R 
Ld 1
3 Pf ψ m<0
2 J Ld + Lq R ω0 2
2

−

(

2

)

........................................................................................ (21)22ψLq  ωn   Ld + Lq  ωn 
m
b 
K1 +
  −  < 0 ....... (22)R
Ld  ω0  
R  ω0 Here, b1' is the left-side term of the inequality equation
(21).
'
1

Table II.
Routh table of fourth order state equation of V/f control with only
damping gain K1.
2

s4
s3
s

2

s1
s

0

2

3 Pf ψ m
R2+
2 JL q
Ld Lq
2
2
ω 2ψ
3 Pf ψ m R
a1 = K 1 0 m +Lq
2 JL qLd2

a2 = ω 0 +a4 = 1RR+
Ld
Lqa a − a 4 a1
b1 = 3 2a3
b a − a 3b2
c1 = 1 1
b1
a0
a3 =

2

a0 =

2

3 ω 0 Pf ψ m
2
JL q

2

0

a0

0

0

0

0

0

100

Real part of No. 1 and 4

move into the right half of the s-plane. Therefore, theseroots make the system unstable.
It is concluded from the above considerations that the
motor B becomes unstable due to the roots of No. 1 andNo. 4 when the damping gain K1 is decided by theconventional design method. It is noted that these roots
are ignored when the fifth order state equation islinearized to second-order state equation.

80
60
40
20
0
-20
-40
-60
-80
-100
0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

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

It is concluded from the inequality equations (21)(22)
and Tables I–II, the left-side term are positive when thedamping gain K1 of 0.15p.u. is designed in the motor A.
On the other hand, b1', which is the left-side term of (21),is negative when the damping gain K1 of 0.05p.u. isemployed in the motor B. Therefore, the system isunstable. These consequences are corresponded to theeigenvalue plots as shown in Fig. 3 and 6.
Fig. 8 shows the real parts of the roots of No. 1 and 4
when the damping gain K1 is varied. The real parts α2,3 of
No. 2 and No. 3 are expressedKψ
α 2,3 = − 1 m ............................................................. (23).
2 LqThen, the real parts α1,4 of No. 1 and No. 4 areexpressed
Kψ
1 
R 1
α1,4 = 1 m −  +  ..................................... (24).

2 Lq
2  Ld Lq 
The real parts of No. 2 and No. 3 decrease
proportionally to the damping gain K1. On the other hand,
the real parts of No. 1 and No. 4 increase proportionallyto the damping gain K1. Therefore, the stable condition islimited depending on the damping gain K1. In addition,
the stable region of the motor B is smaller than that of the
motor A. In particular, the high-speed motor with small
winding resistance is more apt to become unstable asexpressed in (24). It is concluded from above
considerations that there is unstable condition dependingon the roots of No. 1 and No. 4. Therefore, a novel
method is necessary in order to make the roots of No. 1
and No. 4 move into left-half of s-plane.

III.

PROPOSED ADDITIONAL FEEDBACK LOOP

A. Principle of Stabilizing Method with Additional
Feedback Loop
Fig. 9 shows the control diagram of the V/f control
with the novel feedback loop. As shown in Eq. (21), thethird term in the left-side term of the inequality equation
becomes larger when the winding resistance increases.
Furthermore, the negative term, which is the fourth term,
becomes smaller when the winding resistance increases.
In other words, it is possible to stabilize the system when
the term of the winding resistance is increased
equivalently. Therefore, a novel feedback loop is added
to the inverter voltage command vδ*. The feedback loop
consists of an additional gain K2 and the δ-axis current
which is filtered by the HPF.

1sFig. 9. Block diagram of V/f control based on γδ-frame with proposed
feedback loop to output voltage command.
3000

B. Stability Analysis with Additional Feedback Loop
The stability analysis is conducted under the sameprocedure in chapter II, section C. The unstable condition
is derived from the Routh table. It is expressed as
Ld Ld ωn 2 R + K 2 1
ψm
Ld 2
+
+
−
K1
R
R ω0 2Lq ω0 2 ( R + K 2 ) Ld + RLq R

3 Pf ψ m
−
2 J
2

2

Ld 1
R
<0
( R + K 2 ) Ld + RLq R ω0 2........................................................................................... (25),22Lq  ωn   ( R + K 2 ) Ld + RLq  ωn '' ψ mb1 
K1 +    −  <0R
Ld  ω0  R2 ω0 
........................................................................................... (26).
where b1'' is the left-side term of (25).
It is shown in (25) that the additional gain K2 is added
to the numerator of the third term in left-side by adding
new feedback loop. In addition, the denominators of the
fourth and fifth terms are also increased by the additional
gain K2. Therefore, the winding resistance is equivalently
increased by the additional gain K2 in order 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 thedamping gain K1 is decided as multiple roots of No. 2 andNo. 3. The roots of No. 1 and No. 4 move into the left
side of s-plane in accordance with the increase of theadditional gain K2. This stabilization is not achievablewith only damping gain K1.

IV.

EXPERIMENTAL RESULTS

A. Auto-tuning for Conventional Damping Control
In this section, the effectiveness of the auto-tuning
method for the conventional damping control isconfirmed in the experiment with the motor A. It is notedthat the DC test is conducted at standstill. In addition, theother 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

2500
2000
1500
1000
500
0
-500
-1000
-1500
-2000
-2500
-3000
-200

-150

-100

-50

0

50

100

Fig. 10. Root locus when damping gain K2 is increased under rated speed
and no load with motor B.

Fig. 11 Waveforms of DC test.

identified value of the winding resistance converges tothe nominal value in the steady state. In general, winding
resistance of a general-purpose motor is relatively small.
Therefore, the duty ratio is set as 5% in order 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 andthe calculation are conducted in 44 s. The frequency of
the injected sinusoidal wave is varied each 1 Hz in theidentification of the natural angular frequency. 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, thetuning time of the natural angular frequency is relatively
long compared with the other parameters.
Fig. 13 shows the step response before and after the
auto-tuning. In Fig. 13 (a), the damping gain is set as0.01p.u. In addition, the HPF of the damping control is
not utilized. On the other hand, the damping gain K1 is set
as 0.13p.u. and the cutoff frequency is 1/20 of theidentified natural angular frequency. In the figure, themotor 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 the other hand, themotor speed follows the command without an overshoot.
Fig. 14 shows the step responses with the optimal
damping gain and the tuned damping gain. From thefigure, the overshoot with the tuned damping gain isalmost same as the optimal damping gain.
Through the experimental results, the effectiveness of
the auto-tuning method is confirmed.
B. Stabilization with Additional Feedback LoopIn order to confirm the validity of the additional
feedback loop, the experiment is conducted. It is notedthat Motor A which is shown in Table I is used in theexperiment. In addition, inductors of 10 mH areconnected in series in order to evaluate the motor withthe long time constant.
Fig. 15 shows the waveforms of γ, δ-axis current andU-phase current 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 the additional gain K2 is set as
0 at this condition. The diverging oscillation occurs in γ,
δ-axis current after the damping gain K1 is varied 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 theeigenvalue plot is 81 Hz. The experimental value agrees
with the theoretical value with error of 5.5%. The system
becomes unstable due to the influence of the ignored
roots depending on the damping gain K1.
Fig. 16 shows the waveforms of γ, δ-axis current andU-phase current 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 the additional gain K2 even under
the unstable condition with only the damping gain K1. In
addition, the current ripple is also suppressed by theadditional gain K2.
From the experimental results, the effectiveness of theadditional feedback loop is confirmed.
V.

CONCLUSION

In this paper, the designing method for theconventional damping control is explained. In addition,
the auto-tuning method is introduced for the motor whichis stable with the conventional damping control. Then,

Fig. 12 Waveforms of auto-tuning each parameter.

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.

(a) With optimal damping gain (K1 = 0.16p.u.).

(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.

the unstable condition due to the damping gain K1 isderived based on the state equation regarding the V/fcontrol for IPMSMs. From the unstable condition, it isclarified that the 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 theexperiment. In order to solve this instability problem, thenovel 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 unstablecondition with only damping gain K1. In the experiment,
first, the effectiveness of the auto-tuning for theconventional damping control is confirmed. Next, theeffectiveness of K2 is confirmed under 0.9p.u. of rated
speed and 0.7p.u. of rated torque.
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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.

Fig. 16. Waveforms when additional gain K2 is varied
under 0.9p.u. of rated speed and 0.7p.u. of rated torque.