Stabilization Method of Current Regulator forElectric Vehicle Motor Drive Systems under Motor
Parameter Mismatch Conditions
Masakazu Kato

Jun-ichi Itoh

Department of Electrical engineering
Nagaoka University of Technology, NUT
Niigata, Japan
katom@stn.nagaokaut.ac.jp

Department of Electrical engineering
Nagaoka University of Technology, NUT
Niigata, Japan
itoh@vos.nagaokaut.ac.jp

Abstract— In this paper, the stability of a current regulator for
high-speed motor drives in electric vehicle is analyzed using a
motor parameter mismatch model between the current regulator
and actual motor parameters. As a result, a fundamental
decoupling control which adds the cross terms between d-axis andq-axis to the voltage commands causes instability when the naturalangular frequency of a current regulator is low under the motor
parameter mismatch. The conventional method requires a lowpass filter (LPF) that has an electrical time constant in order toachieve compensation for the d-q axis coupling components.
Therefore, a high carrier frequency is required in order toimplement the LPF on the controller. However, it is difficult toimplement the LPF because the carrier frequency is limited by the
total efficiency of the motor drive systems. Therefore, this paper
also proposes a stabilization method for the current regulator
based on an equivalent resistance gain in order to overcome thisinstability problem. One of the features in the proposed method is
no use of the LPF. In addition, the proposed method reduces thecurrent overshoot by 1.1 p.u. compared with the fundamental
decoupling control.

(it is often same as a sampling frequency) over a synchronous
frequency becomes less than ten in the high-speed motor drive
systems [11]. Therefore, the LPF cannot be implemented on the
ACR. Moreover, a natural angular frequency of ACR at least 10
times than that of the carrier frequency should be designed inorder to achieve the stable operation of the control system. In
consequence, the ratio of the synchronous frequency over thenatural angular frequency of ACR becomes less than 1. In
addition, the ACR of high-speed drive system generates theovershoot due to the low natural angular frequency. As a result,the output current becomes unstable and the inverter is trippedby an over current in the IPMSM drive systems. In case of the
IM drive systems, this overshoot causes a torque fluctuation
because the output torque is product of the torque current andthe secondary flux which is generated by the d-axis current
excitation current.

Keywords—Interior permanent magnet synchronous motor,
Induction motor, High-speed drive, Current control,I.

INTRODUCTION

In recent years, an interior permanent magnet synchronous
motor (IPMSM) is actively researched in order to achieve thehigh efficiency in the motor drive system for electric vehicles(EV) [1]. On the other hand, a high performance induction motor(IM) is often a good choice for a low/medium-cost EV [2]. In
addition, the EVs require a high-speed motor drive system inorder to achieve downsizing and high output power density [35]. In the high speed drive region, a current regulator (ACR)
becomes unstable which is caused by incomplete decoupling
control for d- and q-axis cross terms due to detection delay,
control delay and motor parameter errors [6]. Thereby, thedynamic decoupling control methods have been examined inorder to achieve a robust ACR [7-9]. This method requires afilter which has an electrical time constant to achieve the inverse
model of the motor on the decoupling controller.
However, a carrier frequency of an inverter is limited due tothe efficiency [10]. As a result, the ratio of a carrier frequency

This paper proposes a stabilization method for the ACR for
IPMSM and IM drive systems based on an equivalent resistancegain in order to overcome the instability. The proposed method
does not require the LPF which has electrical time constant. The
stability of decoupling control with the parameter mismatch isanalyzed using the model which considers a motor parametermismatch between the current regulator and an actual motor
parameter. According to the stability analyses result, theequivalent resistance gain is designed for the drive systems. The
proposed method is robust for motor parameters variations incomparison with the current regulator without the parameter
error compensation.This paper is organized as follows; firstly
the stability of decoupling control with the parameter mismatchfor IPMSM and IM drive systems are analyzed. Secondary the
proposed equivalent resistance gain is designed based on theeach stability analysis. In addition, upper limit of the gain isdesigned consider the detection delay. Thirdly, the simulation
results of IPMSM and IM drive systems are shown in order toconfirm the effect of the equivalent resistance gain for thestabilization. Finally, the experimental results of the IPMSM
drive system are shown in order to verify the availability of the
proposed method.

II.

ANALYSIS OF CURRENT CONTROL SYSTEM FOR IPMSM

A. Transfer Function of Fundamental Decoupling Control
Fig. 1 illustrates a block diagram of current control systemwith a fundamental decoupling control for an IPMSM. Note thatthe d-axis is defined as the direction of the flux vector by thepermanent magnet and the direction of the q-axis is defined as
the electromotive force vector. In addition, in this paper, thestability is analyzed in the continuous system shown in Fig.1
under an assumption that the discretization error can be ignoredby the discretization error compensation method [11]. The
IPMSM has cross terms between d- and q-axis. The current
controller could be regard the cross terms as a disturbance if the
motor were modeled as RL load. In order to eliminate the crossterms, a fundamental decoupling control adds the cross terms to
the voltage commands of the d- and q-axis as shown in Fig.1. In
the ACR has no parameter errors, the open-loop transfer
functions between current command and output current become
simple integral elements. Note that the proportional gain andintegral gain is designed to be a first order lag response.
However, the motor parameters are varied according totemperature variation and magnetic saturation. Hence, thecontroller has mismatch between the ACR and the actual motor
parameters. Furthermore, the mismatch results in the decoupling
control error. The open-loop transfer functions which haveparameter errors represented as (4) and (5). Note that in thispaper, the parameters of ACR are defined as the product of theparameter error coefficients and the motor parameters.
O
id _ K dq

Gs  

GiqO _ K dq s  kˆ pd s  kˆidskˆ pq s  kˆiq

Pd s (4)

Pq s s

(5)

Controller
d-axis ACR

id*+Rˆ  K R R , Lˆd  K Ld Ld , Lˆq  K Lq Lq

(7)

 
   L s  R  kˆ s  kˆP s  R  sL L s  R  kˆ s  kˆ    s
Pd s  

Lq s 2  R  kˆ pq s  kˆiq
R  sLd  Lq s 2  R  kˆ pq s  kˆiq   dq  qd s

(8)

2

d

q

pd

id

pd

id

2

q

d



dq

(9)

qd



(10)

 qd  re Ld 1  K Ld (11) dq  re Lq 1  K Lq

R is the armature winding resistance, Ld and Lq are the d- and qaxis components of the armature self-inductance, re is the
motor speed in electric angler frequency, c is the naturalangular frequency of the ACR, kpd and kpq are proportional gainsv d +

vd +

s

-

+

-

1
R  Ld s re L̂d

iq*

+
-

id

 re L q re L̂qdq axis coupling
component

decoupling
controlv q++kˆ pq s  kˆiqs

q-axis ACR+

 re Ld

vq + -

1R  Lq siqre mreˆ mFig. 1. Block diagram of current control system with fundamental decoupling
control for IPMSM. Note that the stability is analyzed in the continuous system
under an assumption that the discretization error can be ignored by thediscretization error compensation method [11].

of ACR, kid and kiq are integration gains of ACR, KR is the error
coefficient of armature winding resistance in the ACR. KLd andKLq are the error coefficients of the armature self- inductances.
In addition, in case of the error coefficient is equal to 1, the ACR
has no parameter errors. Thus, the open-loop transfer functions
which has parameter errors could not maintain the simple
integral characteristic due to the decoupling control error dq andqd.
The crossed loop transfer function of Fig.1 is represented as
the following equation.

id   Gd s   Fdq s  id * 
i    F s  G s    * q
 q   qd
 iq (12)b4 s 3  b3 s 2  b2 s  b1
s  a 4 s 3  a3 s 2  a 2 s  a1

(13)

c3 s 2  c2 ss  a4 s 3  a3 s 2  a2 s  a1

(14)

d 4 s 3  d 3 s 2  d 2 s  d1
s 4  a4 s 3  a3 s 2  a2 s  a1

(15)

e3 s 2  e2 ss 4  a4 s 3  a3 s 2  a2 s  a1

(16)

where,
Gd s  

4

Fdq s  where,kˆ pd  c K Ld Ld , kˆ pq  c K Lq Lq , kˆid  kˆiq  c K R R (6)IPMSMkˆ pd s  kˆid

Gq s  

Fqd s  

a4 

4



 

Ld R  kˆ pq  Lq R  kˆ pd





Ld Lq

,



R  kˆ pd R  kˆ pq  kˆiq Ld  kˆid Lq   dq  qd (17)
a3 
,
Ld Lq
a2 

b4 
b2 







kˆid R  kˆ pq  kˆiq R  kˆ pd

Ld, b3 

id iq

1

Ld Lq

kˆ pd

, a  kˆ kˆ



Ld Lq



kˆ pd R  kˆ pq  kˆid Lq



Ld Lq

kˆ pd kˆiq  kˆid R  kˆ pq
Ld Lq

 , b  kˆ kˆ

,

id iq

1

Ld Lq

(18)

kˆ pq  dq
Ld Lq

d4 
d2 

e3 

kˆ pq
Lq

kˆiq  dq, c2 

, d3 

(19)

Ld Lq





kˆ pq R  kˆ pd  kˆiq Ld



Ld Lq

kˆ pq kˆid  kˆiq R  kˆ pd
Ld Lq

kˆ pd re Ld 1  K Ld 
Ld Lq

,

 , d  kˆ kˆ
id

1, e2 

(20)

iq

Ld Lq

kˆidre Ld 1  K Ld 
(21)
Ld Lq

c / reR2

 re Ld Lqc  RK R  K Ld  RK R  K Lq  re   re Lq re Ld
2

Table 1. Parameters of test PMSM.

(22)



2

   K Ld
1   c   re  1  K Ld

12000 r/min

Pole number

4

Maximum torque

8 Nm

Rated current

24.5 Ad-axis inductance Ld

2.04 mH

q-axis inductance Lq

2.24 mH

Linked flux m

0.1066 Vs/rad

cFig. 2 shows the unstable condition of current control systemwith the parameter errors. Note that the parameter errors are
small when the horizontal and vertical axis values in Fig.1 areclose to the origin, and large when the values are far from theorigin. According to the unstable condition (23), in case that theparameter error coefficients KLd and KLq are placed on the rightside of each curved lines determined by the ratio of thesynchronous frequency over the natural angular frequency of
ACR c/ re, the current control system is unstable. For example,
when the error coefficients are placed on Point C (KLd = 0.7, KLq= 2.0) and the ratio c/re is 0.5, the system is stable. On theother hand, when the error coefficients are placed on Point B(KLd = 0.6, KLq = 2.0), the system becomes unstable.
Table 1 shows the motor parameters that are used in thestability analysis. Fig. 3 shows the placement and tracking ofroots of the current control system with the parameter mismatch.
Fig. 3 (a) shows the root loci under the condition that the error
coefficients are placed on Point D (KLd = 0.8, KLq = 2.0), andImaginary part(23)

0.133 500 rad/s(This value is determined to be half of the motor speed
re. Moreover, the condition c / re = 0.5 is comparablewith the conditions re = 15000r/min, Pf = 12 and c =4700rad/s.)

200
150

No.1

100
50

No.4

No.3

0
-50

-100
-150
-200
-1600
-1600

No.2
-1400

-1200
-1200

(a)

200
150
150
200

Imaginary part

1

3 kW

Maximum speed

ACR natural frequency

Then, the first, second and third term on the left hand side are
reduced when re becomes large. Therefore, the left side fourthterm becomes dominant when the ACR has the error of armature
self- inductance in high-speed region. In order to simplify thisequation, it is assumed reL >> R and KR = 1, then the unstablecondition is given by (23).

K Lq 

Rated power

Winding resistance R 
 K Ld K Lq  c   1  K Ld  1  K Lq  0
  re c / re c / reFig. 2. Unstable condition of current control system with parametermismatches. When the motor parameter mismatches condition is placedon the right side of each line c / re, the system is unstable.

B. Unstable condition of Current Control System
From (13), the coefficients of the characteristic equation a1,a2and a4 are positive value irrespective of the motor parameters,
the parameter errors and the natural angular frequency of ACR.
However, the coefficient a3 becomes negative due to the motor
parameter errors. As a result, this current control system
becomes unstable with the parameter errors. The unstablecondition is obtained by (22).
2c / re= 0.50= 0.75 = 1.00
55 = 0.25
Instability condition
4.5
1
Unstable
StableKLq 
2regionregion c _ ACR  KLd
44


1 

 re  1  KLd
Stable Unstable
3.5
regionregion33 Stable Unstableregion2.5 region
Unstable
Stableregionregion
A
D
C
B
22
1.5
11
1
0.8
0.6
0.4
0.2
1
0.9 0.8
0.7 0.6of d-axis
0.5 inductance
0.4 K0.3
0.2 0.1
Error coefficient
Ld

Error coefficient of q-axis inductance KLq

c3 

-1000

-800
-800
-600
Real part

-400
-400

-200

00

Placement of four roots.

C (KLd = 0.7, KLq = 2.0, c/ re = 0.5) 〇A (KLd = 0.5, KLq = 2.0, c/ re = 0.5)
D (KLd = 0.8, KLq = 2.0, c/ re = 0.5)
B (KLd = 0.6, KLq = 2.0, c/ re = 0.5)
A

B

C

D

No.1

100
100

50
50

No.3

00
-50
-50

Unstable
regionStableregion

-100
-100

-150
-150
-200
-200
-150
-150

No.2

-100
-100

-50
-50

00

50
50

100
100

Real part(b)

Roots locus of No. 1-3.

Fig. 3. Placement and tracking of roots. The roots move into the right halfplane when the error coefficients KLd and KLq increase. Each curved line
determined by c/re shown in Fig.2 corresponds with imaginary axis incomplex place.

c/ re = 0.5. The crossed loop transfer function (11) gives the

KLd = 0.6, KLq = 2.0

30

Gain [dB]2020

KLd = 0.5, KLq = 2.0

KLd = 0.7, KLq = 2.0
KLd = 0.8, KLq = 2.0

10

00

-3 dB

-10

KLd = 1.0, KLq = 1.0

-20
-20

500 rad/s(KLd = 1.0, KLq = 1.0)

-30
-40
-40
360 10360

825 rad/s(KLd = 0.7, KLq = 2.0)

100

1000

10000

KLd = 0.6, KLq = 2.0

315

270
270

Phase [deg]Fig. 4 shows the closed-loop frequency response of thetransfer functions Gq(s) and Fqd(s). From the frequency responseof the Gq(s), since the error coefficients KLd and KLq increase, thebandwidth of current regulator is widened. However, themaximum gain value is increased with the increase of the error
coefficients. Moreover, the current control system becomes
unstable when the error coefficients KLd = 0.6 and KLq = 2.0.
From the frequency response of the Fqd(s), the maximum gain
value is also increased with the increase of the error coefficients.
In particular, the maximum gain is 18.3 dB at the operation point
D shown in Fig. 3(b) in spite of the stable condition. Thereby,even if the d-axis current command is zero, an excessive current
flows through the motor by the q-axis current command. The
excessive current is the cause of an inverter trip due to an
overcurrent.

re = 1000 rad/s, c = 500 rad/s, c/ re = 0.5

40
40

225
180
180

KLd = 0.5, KLq = 2.0

135
90
90

KLd = 0.7, KLq = 2.045

KLd = 0.8, KLq = 2.000-45

KLd = 1.0, KLq = 1.0

-90
-90
10
10

10000
10k

100
1000
1k
100
Frequency [rad/s](a) Closed-loop frequency response of Gq at c/re=0.5.

re = 1000 rad/s, c = 500 rad/s, c/ re = 0.54040KLd = 0.6, KLq = 2.0

KLd = 0.5, KLq = 2.030

KLd = 0.7, KLq = 2.0
Mp = 18.3 dB

20
20

Gain [dB]root loci of the current control system. This system has four roots,
because the system is fourth -order system. The roots No. 1-3
are located nearest to the imaginary axis. Therefore, the stabilityof the system is discussed based on the roots No. 1-3. Fig. 3 (b)shows the roots No. 1-3 locus with the variations in KLd and KLq.
The roots move into the right half plane when the error
coefficients KLd and KLq increase. In particular, when thecondition of error coefficients changes from Point C to B, theroots move into the right half plane. According to the above
results, each curved line is determined by c/re shown in Fig.2
corresponds with the imaginary axis in complex place.

1000

KLd = 0.8, KLq = 2.0

-10

-20
-20

-30

ANALYSIS OF CURRENT CONTROL SYSTEM FOR IM

A. Characteristic Equation of Current control System
Fig. 5 illustrates a block diagram of current control systemwith the fundamental decoupling control for an IM which isdriven by a field-oriented control. Here, the back EMF isassumed to be canceled by the feedforward compensation term.
From Fig. 5 and (12), the characteristic equation of the current
control system for IM is expressed by (24). In this case, the daxis PI gains kpd1 and kid1 are set to the same value as the q-axis
PI gains kpq1 and kiq1 respectively because each plant of the PI
controller is same. Then, PI gains of d-, q-axis are defined as kp1
and ki1.

L s  R  kˆ s  kˆ 

2

2

1

1

p1

i1

  re L1 1  K  K L1   s 2  0
2

2

(24)

-40
-40
270
10
270

100

1000

KLd = 0.6, KLq = 2.0

225

180
180

Phase [deg]

III.

10000

KLd = 0.5, KLq = 2.0

135
90
90

KLd = 0.7, KLq = 2.0

4500

KLd = 0.8, KLq = 2.0

-45
-90
-90

-135

-180
-180
10
10

10000
10k

100
1000
1k
100
Frequency [rad/s](b) Closed-loop frequency response of Fqd at c/re=0.5.
Fig. 4. Closed-loop frequency response of the transfer functions Gq(s) andFqd(s) based on Fig.1. The maximum gain of Fqd(s) is 18.3 dB at theoperation point D shown in Fig. 3(b) in spite of the stable condition.

2

Controller
d-axis ACRvd1 +kˆ s  kˆs-*where,

id1 +Mˆ 2ˆ  K    1 
Lˆ Lˆ
1

pd1

-

id 1

2

 reˆLˆ1(26)

k p1  c K L1 L1 , ki1  c K R1 R1(27)

R1 is the primary winding resistance, R2 is the secondary winding
resistance, L1 is the primary self-inductance, L2 is the selfinductance, M is the mutual inductance  is the leakage
coefficient, and Kl1 , Kl2, KM are error coefficients.

vd1 +
+

reˆLˆ1
decoupling
control(25)

ˆ , Lˆ  K l  K M
ˆ ,Mˆ K MLˆ1  K l1l1  K M M
2
l2 2
M
M

Induction Motor

iq1*

+
-

kˆ pq1s  kˆiq1v q++svq + -

1
R1 L1s

id1

reL1
dq axis coupling
componentreL1
1
R  Lq s

iq1

q-axis ACR

Fig. 5. Block diagram of current control system with fundamental
decoupling control for IM. Here, the back EMF is assumed to be canceledby the feedforward compensation term.

The first term on the left hand side of (24) is the characteristic
equation when the current control system has no parameter
errors. On the other hand, the second term on the left hand sideof (24) is the error term of the decoupling control.

B. Stability Analysis using Approximation
Equation (24) shows the 4th order of the state equation thatis complicated to evaluate the stability. In order to simplify thisequation, it is assumed that there is a sufficient distance between
No.3, No.4 and No.1, No2 by the large parameter errors, then(24) can be approximated as the 2nd order state equation as (28).
2

a

idq

a

pdq

2

re



2

2

a



(28)

2

Fig. 7 shows a comparison between the 4th order model
based on Fig.5 and the approximate model roots locus. Note that
Fig. 7 shows the roots No. 1, 2 locus shown in Fig. 6. From Fig.
7, the roots of the approximate model is similar to the 4th ordermodel when the error coefficient is large. Thus, theapproximated model (28) is valid for the modeling of the l4 thorder model of current control system for IM with the output theparameter error. In order to introduce the damping factor , thevariables of (28) compared with the characteristic equation of
the second order system is given by (29).
2Then, the damping factor  and the natural angular frequency nis expressed as (30).
 

n 





2

2

34.5 mH

No.3

No.1

No.4

No.2

K  2.5

-200
-200
-400
-400

-600
-600
-600
-600

-500
-500

-400
-400

-300
-300
Real part50
50

K  Kl1  Kl2  KM 1

30
30
20
20

2k i1L1  R1  k p1  re  2 L1 1  K K L1 
2

2

(31)

2

PROPOSED METHOD FOR STABILIZATION OF SYSTEM

A. Esuivalent Registance Gain for Stabilization of Current
Control System
Fig. 8 shows the block diagram of the current regulator withthe equivalent resistances kr. In order to increase the armature

00

-100
-100

Approximate modelNo.1 shown in Fig.6

10
10

K  2.5

00

-10
-10

No.2 shown in Fig6

-20
-20
-30
-30
-40
-40

Approximate model

-50
-50
-160
-160

-120
-120

-80
-80
Real part

00

-40
-40

Fig. 7 Comparison between the 4th order model and the approximate
model roots locus. Note that Fig. 7 shows the roots No. 1, 2 locus shownin Fig. 6.
Controller
id*+

PI

-

Motor
+
-

vd* +

-

vd +
+

1
R  Ld s

idkr

Coupling

Fundamental
Decoupling
Control

iq*

ki1

-200
-200

Fig. 6. Roots locus with variations in Kl1, Kl2 and KM. Note that the error
coefficients are assumed to same value. Each of the parameter error
coefficient is varied K = 2.5 from each of the no parameter error values.

2

According to (30),  and n become small when the error
coefficient is large. Hence, the system becomes unstable.
IV.

1.24 mH

Mutual inductance M

K  2.5(30)

2k i1L1  R1  k p1    re  L1 1  K K L1 
2

2

Secondary leakage inductance l2

Equivalent resistanceR1  k p1
2

0

(29)

s 2  2 n s   n  00.423 1.24 mH

200
200

40
40

 2 Ra  K pdq K idq s  K idq  0

Secondary resistance R2
Primary leakage inductance l1

400
400

2

La

4
0.414 K  Kl1  Kl2  KM 1

Imaginary part

2L K  R  K    L 1  K  s

Pole number
Primary resistance R1

600
600

Imaginary part

Table 2 shows the motor parameters of the IM that are usedin the stability analysis. Fig. 6 shows the roots locus based on(24) with the variations in Kl1, Kl2 and KM. In addition, the ratioc/ re is 0.5. Note that the error coefficients Kl1, Kl2 and KM are
assumed to same value K. Each of the parameter error coefficientis varied K = 2.5 from each of the no parameter error values. In
order to stabilize the system, the roots should be located in thearea negatively distant from the imaginary axis. The roots No.3
and No.4 move to the left side from the imaginary axis when theerror coefficients increase. In contrast, the roots No.1 and No.2
approach the imaginary axis when the error coefficients increase.
Therefore, the stability of the system is discussed based on theroots No.1, and No.2.

Table 2. Parameters of test IM.

+-

component

+

PI

*
vq+
+

-

vq + -

1R  Lq siqkr
Equivalent resistanceFig. 8. Block diagram of current regulator with equivalent resistances. The
armature winding resistance becomes equivalent to the sum of R and kr.
The unstable phenomenon of current regulator is preventable by kr in thehigh speed region.

winding resistance R, the product of detection current and kr issubstracted from the inverter voltage command v*. In
consequence, the armature winding resistance becomes
equivalent to the sum of R and kr. In the IPMSM, depending on
the increase of the armature winding resistance, the first, second

B. Equivalent Registance Gain Design for IPMSM
Fig. 9 shows the transition of the root locus with theproposed method when the gain kr_PM is gradually changed. At
the equivalent resistance gain kr_PM = 0, the poles are placed on
the right half plane. By contrast, when the equivalent resistancegain kr increases, the poles move into left half plane. In order todefine the lower limit of kr_PM,it is necessary to solve thecondition in which the coefficient a3 becomes positive value.
The lower limit is derived from (22) in consideration ofequivalent resistance. Then, the lower limited is given by (32).
k r _ PM 2c / re = 0.25, KLd = 0.75, KLq = 2.5500
kr_PM increase
400
300
200
kr_PM = 0
kr_PM = 0
100
0
-100
-200
-300 Conventional method
Proposed method-400(With equivalent resistance gain kr_PM)-500-1400 -1200 -1000 -800 -600 -400 -200
0 200
Real partFig. 9. Root locus with proposed method. The poles of conventional
method are placed on the right half plane. By contrast, when theequivalent resistance gain kr increases, the poles move into left handplace.

(32)

R

+
-

kˆ pq s  kˆiq
s







K 2  K R RLd  Lq   Ld Lq K Ld K Lq  re 1  K Ld 1  K Lq  (34)2

Fig. 10 shows the block diagram of q-axis current control
system with a detection delay. Table 3 shows the Routh table ofthe current control system with the detection delay shown in
Figure 5. Note that the dead time is approximated by first-order
Pade approximation. The value of kr_PM is limited by thedetection delay because 1 becomes negative due to the increaseof kr. In order to derive this equation, it is assumed Td Tf  0.
Then, the upper limit of kr_PM is obtained by (35).
2 Lq Td  2T f (35)

Td Td  4T f 

C. Equivalent Resistance Gain Design for IM
Fig. 11 shows the transition of the root locus with theproposed method when the gain kr_IM is gradually changed. At
the equivalent resistance gain kr_IM = 0, the poles are placed on
close to the imaginary axis. On the other hand, when theequivalent resistance gain kr increases, the poles move the left
side from the imaginary axis. In order to derive the kr_IM, it isnecessary to solve (30) for R1. The kr_IM is derived from (30) in
consideration of equivalent resistance, as given by2

 K 3  K 3  4K 4

(36)

2

where,

K1  2 c La 2  Ra  K pdq   2  1K 2  Ra  K pdq   2  1
2



(37)

2

 



  re La 1  K La   2c La K R R    1
2

2

2

2

2

(37)

Note that the upper limit of kr_IM is the same as (35).

iq

DisturbancereLd id  Kd id 

Delay

iq

(33)

esTd

1
1 TfsFig. 10. Block diagram of q-axis current control system of proposed
method with a detection delay. The detection delay of the control
system is approximated by dead time and time constant of a first-orderlag system.
Table 3. Routh table of the current control system with the detection
delay shown in Fig. 5. Note that the dead time is approximated by firstorder Pade approximation.
s4
s3
s2
s1
s

0





Td
T 
T

2  Ra Tf  d   Lq  d K pq  krq 0  Kiq
2
2
2

Td
Td 
T

3  RaTf  Lq Tf  
1  Ra  krq  K pq  d Kiq
0
2
2
2

32 41
30
1 
2 
0

1  LqTf

3   
1  1 1 3 2
1
0

50
50
40
40
30
30
20
20
10
10
00
-10
-10
-20
-20
-30
-30
-40
-40
-50
-50
-160
-160

Imaginary part

K1  re K Ld Ld  K Lq Lq

k r _ IM 

+
-

kr

where,

k r _ PM 

Motor
v
q
1
+
Ra  sLq-

Controller
iq*2

 K 1  K1  4 K 2

Imaginary partand third term on the left hand side of (2) are increased. In
addition, in the IM, the first term on the left hand side of (24) isincreased. Therefore, the unstable phenomenon is preventableby additional gain kr in the high speed region. Note that the
proposed method does not require the LPF which has electricaltime constant. Therefore, the proposed method can be applied tothe high-speed drive systems.

3

0

0

0

0

K  Kl1  Kl 2  KM  2.5, c / re  0.5
Approximate model

4th order model

kr = 0.5 p.u.

-140

-120
-120

kr =0

-100

-80
-80
-60
Real part

-40
-40

-20

00

Fig. 11 Root locus with proposed method when the gain kr_IM is
gradually changed. When the equivalent resistance gain kr increases,
the poles move to the left side from the imaginary axis.

Fig. 13 shows the configuration of IPMSM drive system. In
order to confirm the effectiveness of the proposed method, theexperiments are demonstrated with a motor - generator set
shown in Fig. 13. The rated power of this motor - generator set
is 3.0 kW. The test motor is IPMSM shown in Table 1.
Fig. 14 shows an output d-q axis current response with andwithout the proposed method. Figure 14 (a) shows the result of
the conventional method under the conditions Kd = 0.7, Kq = 2,
c / re = 0.5 without the proposed method, (b) shows the resultwith the conventional under the conditions Kd = 0.7, Kq = 2, c /
re = 0.5. In Fig. 14 (a), a output current overshoot is 60%.
Furthermore, the inverter is tripped by an over current. This
overshoot current is due to gain characteristic of Fqd(s) shown in
Fig.4 (b). In order to avoid the inverter trip, the output torque islimited by the magnitude of the overshoot current. On the other
hand, the proposed method suppresses the overshoot because thethe unstable phenomenon is preventable by additional gain kr.
Moreover, the designed response is achieved with the proposed
method. As a result, it is confirmed that the proposed method isrobust for motor parameters variations.
Fig. 15 shows output current response with the speed control
in a field-weakening control region. In Fig. 15 (a), an output
current becomes unstable and the inverter is tripped by an overcurrent. On the other hand, in the proposed method, the motor is

20 ms

q-axis current [p.u.]With proposed method (kr_PM=0.5 p.u.)

Without proposed method

With proposed method

Output current [p.u.]d-axis current [p.u.]

B. Experimental Results
According to Fig.12, the stabilization method of IPMSM ismore effective than that of the IM. Therefore, this paper
evaluates the stability of the proposed method for IPMSM by theexperiments.

With proposed method

20 ms

Without proposed method
20 ms

With proposed method(a) IPMSM under the conditions Kd = 0.7, Kq = 2.0, c / re = 0.5 withand without proposed method an output current overshoot of the system
without proposed method is 230%. In controast, the proposed method
reduces the d-, q-axis current overshoots by 230%

q-axis current [p.u.]

A. Simulation Results
Fig. 12 shows an output d-q axis current response with andwithout the proposed method. Fig. 12 (a) shows the results of thestep response for the IPMSM under the conditions Kd = 0.7, Kq= 2.0, c / re = 0.5, (b) shows the results of the step response
for the IM under the conditions Kl1=Kl2=KM=2.5, c / re = 0.5.
IN Fig.12 (a), an output current overshoot of the system without
the proposed method is 230%. In addition, the peak value of theoutput current is 3.3 times large as rated current of test motor. In
this case, in order to prevent the motor and drive circuit such as
inverter from breakdown, the drive circuit is tripped. In controast,
the proposed method reduces the d-, q-axis current overshootsby 230% because of the proposed equivalent resistance gain incomparison with the conventional method. Hence, it isconfirmed that the proposed method achieves no overshoot of
the output current in the simulations. In Fig.12 (b), the system
without the proposed method generates a d-axis current
overshoot of 105%. This overshoot causes a torque fluctuation
because the output torque of IM is product of the q-axis current(torque current) and the secondary flux which is generated by
the d-axis current (excitation current). On the other hand, theproposed method suppress the overshoot to 30%. Furthermore,
the response time correspond to the deign value. Hence, it isconfirmed that the proposed method achieves less overshoot of
the output current than the conventional method in thesimulations.

d-axis current [p.u.]

SIMULATION AND EXPERIMENTAL RESULTS

Output current [p.u.]

V.

With proposed method

Without proposed method
100 msWith proposed method

Without proposed method
100 msWith proposed method
Without proposed method
100 ms(b) IM under the conditions Kl1=Kl2=KM=2.5, c / re = 0.5 with andwithout proposed method. The system without proposed method
generates a d-axis current overshoot of 50%.
Fig. 12. Output current response with/without proposed method.
200 V / 50 Hz

Rectifier2-levelinverter

Test
motor
Coupling

DCsupply2-levelinverter

Load
motor

Fig. 13. Configuration of PMSM drive system. The rated power of thismotor - generator set is 3.0 kW. The test motor is IPMSM shown in
Table 1.

q-axis current reference iq* [1.0 p.u./div]
0.5
0.1 0

q-axis current iq [1.0 p.u./div]

accelerated to the field-weakening control region without thecurrent overshoot. As a result, the stability is confirmed under
the instability region of the fundamental decoupling control
method with parameter mismatch.

0d-axis current id [1.0 p.u./div]

1.60Output current Ia [1.0 p.u./div]I  i 2 i 2 
 a
d
q 


[5 ms/div]
The inverter is tripped by an over current protection.

(a) Without proposed method at the condition C shown in
Fig.2.
(c=500 rad/s, re/c=0.5, Kd=0.7, Kq=2.0, kr=0)
q-axis current reference iq* [1.0 p.u./div]
0.5
0.1 0
0

q-axis current iq [1.0 p.u./div]
2 ms
d-axis current id [1.0 p.u./div]

Output current Ia [1.0 p.u./div]

VI.

This paper presents a stabilization method of the current
regulator for high speed motor drives under motor parametermismatch condition without the LPF which has electrical time
constant. In order to increase the armature winding resistance R
for stabilization of the motor drive systems, the product of thedetection current and the equivalent resistances is substracted
from the inverter voltage command.The proposed method has
no output current overshoot under the instability region of thefundamental decoupling control method. From simulation result,the proposed method suppress the overshoot of IM to 30%.
Furthermore, the response time correspond to the deign value.
From experimental result, the propsed method suppress theoutput current overshoot of IPMSM by 1.1 p.u.. Moreover, thedesigned response is achieved with the proposed method. As aresult, it is confirmed that the proposed method is robust for
motor parameters variations.

0.5
0.1 0

REFERENCES[5 ms/div](b) With proposed method at the condition C shown in Fig.2.
(c=500 rad/s, re/c=0.5, Kd=0.7, Kq=2.0, kr=0.5)
Fig. 6. Output current response with/without proposed method. As
a result, the stability is confirmed under the instability region of thefundamental decoupling control method with parametermismatches.
The inverter is tripped by an over current protection.

Output current Ia [1.0 p.u./div]
0
0

0
0.6

d-axis current id [1.0 p.u./div]

q-axis current iq [1.0 p.u./div]
Motor speed [0.4p.u./div][50 ms/div]

0.10(a) Conventional method at the condition C shown in Fig.2.
(c=1000 rad/s, Kd=0.7, Kq=2.0, kr=0)
Output current Ia [1.0 p.u./div]00d-axis current id [1.0 p.u./div]
q-axis current iq [1.0 p.u./div]

0
0.6
0.1
0

Motor speed [0.4p.u./div]

c / re =0.66[50 ms/div](b) Proposed method at the condition C shown in Fig.2.
(c=1000 rad/s, Kd=0.7, Kq=2.0, kr=0.5)
Fig. 7. Output current response with the speed control in a fieldweakening control region. In the proposed method, the motor isaccelerated to the field-weakening control region.

CONCLUSIONS[1]

D. Sato, J. Itoh: "Total Loss Comparison of Inverter Circuit Topologies
with Interior Permanent Magnet Synchronous Motor Drive System", 5thIEEE Annual International Energy Conversion Congress and Exhibition,
Vol. , No. 5-5-4, pp. 537-543 (2013)[2] Xi Zhang, “Sensorless Induction Motor Drive Using Indirect Vector
Controller and Sliding-Mode Observer for Electric Vehicles,” IEEE
Transactions on Vehicular Technology, Vol. 62, No. 7, pp. 3010 - 3018 ,sept 2013[3] A. M. EL-Refaie, J. P. Alexander, S. Galioto, P. B. Reddy, K. K. Huh, P.
Bock, and X. Shen, “Advanced High-Power-Density Interior Permanent
Magnet Motor for Traction Applications,” IEEE Transactions On Industry
Applications, Vol. 50, No. 5, pp. 3235-3248 , Sep./Oct., 2014[4] E. Sulaiman, T. Kosaka, N. Matui, “High Power Density Design of 6Slot-8-Pole Hybrid Excitation Flux Switching Machine for HybridElectric Vehicles,” IEEE Trans. On Magnetics, Vol. 47, No. 10, pp. 44534456, Oct., 2011[5] D. Matsuhashi, K. Matsuo, T. Okitsu, T. Ashikaga, T. Mizuno,“Comparison Study of Various Motors for EVs and the Potentiality of a
Ferrite Magnet Motor,” IEEJ Journal of Industry Applications, Vol. 4, No.
4, pp. 503-511 , July 2015[6] D. W. Novotny, and T. A. Lipo, “Vector control and dynamics of ac
drives,” Oxford Science Publications, 1996
[7] J. Jung, and K. Nam, “A Dynamic Decoupling Control Scheme for HighSpeed Operation of Induction Motors,” IEEE Trans. On Industrial
Electronics, Vol. 46, No. 1, pp. 100-110, Feb. 1999
[8] H. Zhu, X. Xiao and Y. Li, “PI Type Dynamic Decoupling Control
Scheme for PMSM High Speed Operation,” Applied Power Electronics
Conference and Exposition (APEC) 2010, pp. 1736-1739, Feb. 2010[9] Yukihiro Yoshida, Kenji Nakamura, Osamu Ichinokura, “QuickResponse Technique for Simplified Position Sensorless Vector Control in
Permanent Magnet Synchronous Motors”, IEEJ Journal of Industry
Applications, Vol. 4, No. 5, pp. 582-588 , Sep. 2015[10] Yukihiro Yoshida, Kenji Nakamura, Osamu Ichinokura, “Evaluation of
Influence of Carrier Harmonics of SPM Motor Based on Reluctance
Network Analysis”, IEEJ Journal of Industry Applications, Vol. 3, No. 4,
pp. 304-309 , July 2014[11] B. H. Bae, and S. K. Sul, “A Compensation Method for Time Delay of
Full-Digital Synchronous Frame Current Regulator of PWM AC Drives,”
IEEE Transactions On Industry Applications, Vol. 39, No. 3, pp. 802-810 ,May/June 2003