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

V/f Control for Switched Reluctance Motor
Hirotaka Kato＊a) Student Member,
Takahiro Kumagai＊ Member
Jun-ichi Itoh＊ Senior Member,
Keisuke Kusaka＊ Member
Masakazu Kato＊＊ Member(Manuscript received Jan. 00, 20XX, revised May 00, 20XX)
This paper proposes a novel V/f control method for switched reluctance motor (SRM). The proposed method does not require
rotor position information. In addition, the wide operating range including magnetic saturation region of the SRM and hightorque/ampere ratio are achieved without complicated pre-measurement or finite element analysis (FEA). This paper proposed thefollowing three methods and confirmed their effectiveness: (A) a virtual rotor magnetic flux is generated by controlling the zerophase current with a PI controller, which realizes the V/f control for the SRM, (B) the damping control gain is designed to be stable
at all operating points since the rotor magnetic flux and inductance value change dynamically according to the operating conditions,and (C) a high torque/ampere is achieved by applying both i0 = iq control and id = 0 control. In the experiment, the proposed method
achieved the stable operation in all N-T regions for the tested SRM. In addition, the current RMS value is reduced by 39% at thetorque of 0.05p.u. compared with no MTPA control, which is the almost performance as the drive with the position sensor.
Keywords : Switched reluctance motor (SRM), MTPA control, V/f control, Stabilization method, Magnetic saturation1.

miniaturization of a motor. Furthermore, the adjustment of thecontrol parameters, such as the current amplitude, turn-on angle,
and turn-off angle, is required for the design of the current andspeed controllers with a high torque/ampere ratio, which requires
pre-experiment or FEA [9][10].
This paper proposes a V/f control method for the SRM drive
without the position sensor in order to solve the above problems.
The V/f control essentially does not require the rotor position
information due to its control strategy based on the rotating
coordinate system relative to the converter output voltage.
Therefore, the V/f control does not have the problem of theinductance detection accuracy and the conversion accuracy to therotor position. Besides, the maximum torque/ampere (MTPA)
control based on the maximum torque/ampere ratio in the fieldoriented control for the SRM [11] is applied into the V/f control that
achieves both simple speed control and high-efficiency drive
without the rotor position sensor.
Although the principle of the proposed V/f control method for
the SRM drive is based on the V/f control with the damping control
and MTPA control proposed for the PMSM [12], the following
features are mainly different. (A) It is necessary to generate a virtual
rotor magnetic flux corresponding to the rotor magnet magnetic
flux in the PMSM. (B) It is revealed for the damping control gain
design method, which stabilizes the SRM in the whole operatingarea because the rotor magnetic flux and inductance change
according to the operating conditions. Note that the effect of themagnetic saturation is simply considered by the magnetic
characteristic at the aligned position which is obtained by simple
pre-measurement without a locked rotor test, i.e., unnecessary ofcomplicated pre-measurement and FEA. (C) It is necessary to apply
not only id=0 but also i0=iq for the MTPA control.
In this paper, first, the proposed method for the above features of(A), (B), and (C) is explained. Next, the usefulness of the proposed

Introduction

A switched reluctance motor (SRM) is composed of only an ironcore and windings without a permanent magnet using a rare earth,which is inexpensive to manufacture. The SRM has attracted
attention as a variable-speed motor for industrial applications dueto the robust salient pole structure of its rotor [1][2]. The variablespeed drive without a position sensor is expected for the SRM totake advantage of the high-temperature and high-speed durability
and cost-effectiveness.
Many studies for the drive method without the rotor position
sensor have been examined, which uses the position dependence on
the inductance or flux linkage [3-8]. In this method, first, theinstantaneous value of the inductance is detected from the voltage
and the current. Next, the detected inductance is converted to therotor position information based on the look-up table or theapproximation formula of the relationship between the inductanceand the rotor position. However, complicated control, such as the
harmonic voltage superposition [3][4] or the additional analog
circuit [5][6], is required to accurately detect the instantaneous
value of the inductance. In addition, complicated pre-measurement
or finite element analysis (FEA) is necessary to obtain the nonlinear
magnetic model of the SRM [7][8] considering magnetic saturation.
Recently, the driving method without a rotor position sensor
including magnetic saturation region is practically essential for thea) Correspondence to: Hirotaka Kato. E-mail:s203139+kato@nagaokaut.ac.jp＊
Nagaoka University of Technology.
1603-1, Kamitomioka-machi, Nagaoka-shi, Niigata 940-2188,
Japan＊＊
Nagaoka Motor Development Co., Ltd.
16-2085, Nagaoka Business Incubation Center, Kukasawamachi, Nagaoka-shi, Niigata 940-2135, Japan© 200● The Institute of Electrical Engineers of Japan.

1V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)V/f control method for the SRM drive is confirmed by theexperimental results. In this paper, the parameters are designed
under the assumption of a steady-state, because the SRM is
expected to apply into fans, pumps and so on, which does not
require a high torque response but only a speed control.

2.

(A) Zero-phase current control
i0*
PI

m*

(B) V/f control*Pf

f/V conv.

v*V/f control for Switched Reluctance Motor
HPF
K1
Damping control

Fig.1 shows the control diagram of the proposed V/f control for
the SRM. The proposed method consists of (A) zero-phase current
control, (B) V/f control, and (C) MTPA control. The details of eachcontrol are explained and designed in the following subsections.
2.1 Zero-phase Current Control
This paper develops a theory based on the mathematical modelin the Field-Oriented Control (FOC, i.e. vector control) for the SRM,
which is interpreted about, a virtual rotor magnetic flux is generated
by DC component of an excitation current, and rotating magnetic
field of the stator is generated by AC component of the excitation
current.
In the zero-phase current control, the virtual rotor flux is
generated by controlling the zero-phase current with a PI controller,
which realizes the V/f control for the SRM described in Section 2.2.
2.2 V/f Control
Fig.2 shows the relationship between dq-axis and -axis. The
FOC of the SRM uses the dq-axis Cartesian coordinate system withthe d-axis defined as the direction of the virtual rotor flux vector
and the q-axis defined as the direction 90 degrees behind the d-axis.
However, the V/f control uses the -axis Cartesian coordinate
system with the -axis defined as the direction of the converter
output voltage and the -axis defined as the direction 90 degrees
behind the -axis. Note that the defined -axis in this paper isdifferent from one in a sensor-less FOC. In the V/f control for the
PMSM, the torque oscillation occurs due to the resonance between
the moment of inertia and the armature inductance. Then, the torqueoscillation is suppressed by the damping control, which feeds backthe active current i to the electric angular frequency command *.
In the SRM, the same damping control as that for the PMSM isachieved by generating the virtual rotor flux by the zero-phasecurrent control.
This subsection first presents the simplified state equation of theSRM. Then, the inductance considering nonlinearity is obtained
from simple approximation method. Finally, the cut-off frequencyof the HPF and the damping gain are designed from the simplifiedstate equation and the inductance.
The voltage equation and the torque equation in the -axis
Cartesian coordinate system are expressed as in (1) and (2), which
are transformed from the equations in the dq-axis Cartesian
coordinate system.
v = Ri + L1 pi + 2 L2 i + 2 p ( ) L3 i ..................................... (1)wherev = v

v

v0 

i = i

i

i0 

 L

L1 =  L
 L0


L
L
L0

1* 1
s

vu
vv
vw

1*

i
i
3f/0
i0

iu
iv
iw

Fig. 1. V/f control method based on  axis.
q-axis-axis

Iac

iq

i
v
vd

-axisvq

i
d-axisv =0 id

Relationship between dq-axis and -axis.

Fig. 2.

 − L

L3 =  L − Ldc L 0L − Ldc
L− L 0L 0 
− L 0 
0 

Ldc
cos ( 61 − 2 )2LL = Ldc − dc cos ( 61 − 2 )2
L0 = Ldc
L = Ldc +L = L = −Ldc
sin ( 61 − 2 )2

Ldc
cos ( 2 )2
LL 0 = L0 = − dc sin ( 2 )2
L 0 = L0 =T = 2 PLaci0iq = 2 PLaci0 (i sin(2 ) + i cos(2 ))...............................................................................................(2)R is the winding resistance, Ldc and Lac are the dc and ac componentsof the self-inductance, respectively, p is the differential operator, φis the deviation angle between the -axis Cartesian coordinate andthe dq-axis Cartesian coordinate, P is the number of pole pairs, andJ is the moment of inertia. Note that the absolute transformation isapplied.
The state equation is derived by first ignoring the asynchronous
term appearing as sixth harmonic for simplification and thenperforming a linear approximation near the steady-state in the same
way as Ref. [12], and expressed as in (3),TL 0 
L 0 
L0 
L − L − Ldc−2 L
0

3f/0(C) MTPA
controlT

2 L
L2 =  L − L + Ldc

0

v0*
v*
0 *
v

 i   A11 i   A
    21
 i   A
p  0  =  31
    A 41
    A
   51    A 61

− L 0 L 0 
0 

A12
A 22

A13
A 23

A14
A 24

A15
A 25

A 32
A 42
A 52

A 33
A 43
A 53

A 34
A 44
A 54

A 35
A 45
A 55

A 62

A 63

A 64

A 65

A16   i   B11

A 26   i   B21
A 36   i0   B31
  + 
A 46      B41
A 56      B51
  
A 66      B61

B12
B22

B13
B23

B32
B42
B52

B33
B43
B53

B62

B63

B14 

B24   v 
B34   v B44   v0 
B54    * 
B64 

.............................................................................................(3)2

IEEJ Trans. ●●, Vol.●●, No.●, ●●●V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)Note that the specific formulas for each element are shown in the
appendix.
The deviation angle φ0 at each operating point is expressed as in(4), assuming the MTPA control (id=0, i0=iq) is achieved which will
be detailed below (The derivation is shown in the appendix).

1
2

Table 1.


0 Ldc ........................................ (4)
0.5
R
+

L
0 ac 


0 = arctan 

where subscript "0" of the deviation angle φ and the electric angularfrequency , represent that they are the steady-state value.
Hereinafter subscript "0" shows the steady-state value. The -axis
current i, i at each operating point is expressed as in (5), using thezero-phase current i00 at its operating point and the deviation angle
expressed at (4) (The derivation is shown in the appendix).

4.5Hz

9.9Hz

20

Gain [dB]

-20
-40

2.2kW
4800rpm, 7200rpm
4.38Nm300V
Stator:18, Rotor:12
62.3×10-4kgm2
0.66W
7.82mH
5.19mH
7.7A(i0M ax, iqMax)

15.9Hz960Hz
i0=1p.u.

0

 i00

sin(20 ) i 0   2
 .................................................. (5)i  =  i  0   00 cos(2 ) 
0 
 2



i0=0.33p.u.
i0=0.66p.u.

-60-80 Line: Calculation from Eq.(3)-100 Dot: Simulation results
10
100
1000
Angular frequency [rad/s]

The state equation of the SRM is determined by the electric
angular frequency , the zero-phase current i0, the motor
parameters P, R, Ldc, Lac, and cut-off frequency of the HPF c ateach operating point.
Table 1 shows the motor parameters. Fig.3 shows the frequency
characteristics of the command speed to the motor speed under thebase speed, which are obtained from (3) and the simulation. The test
motor is an 18S/12P type SRM of 2.2 kW, 4800 rpm (=1p.u.), and4.38 Nm (=1p.u.). In the simulation, the mathematical model
presented in Reds. [11] is applied. Fig.3 assumes id=0 and i0=iq
thanks to the MTPA control and the figure shows the three conditionof the zero-phase current of 0.33p.u., 0.66p.u., and 1p.u. Besides, inthe simulation results in Fig. 3, the characteristics are obtained bygiving the motor the rated mechanical angular frequency m0 as the
input to the mechanical angular frequency command m* andobserving the response of the output speed m. On the other hand,the analysis using (3) is conducted with the steady-state value of thestate variable , denoted as the rated electrical angular frequency
0 (Pm0*=0). Note that m0 and m* are not state variables. Note
that, in Fig. 3, the characteristics were obtained with the damping
control disabled in order to verify the frequency characteristics
including the resonance. As shown in Fig.3, the calculation resultsby (3) and the simulation results are almost the same, i.e., thederived sixth-order equation of the state-model is valid.
In order to simplify the equation, the following conditions are
assumed to discuss the stability and response.
(ⅰ) No-load condition: φ = 0, i = 0, i = 0(ⅱ) High-speed rotation condition: 0Ldc, 0Lac >> R, 0Ldc >> K1i(ⅲ) The mechanical time constant is sufficiently larger than theelectrical time constant: p(∆i) = 0, p(∆i) = 0, p(∆i0) = 0(ⅳ) The effect of the HPF on the stability is small: p(∆x) = 0
The natural angular frequency n and the damping factor  arecalculated from the characteristic equation of the second-ordersystem, which is derived by approximating the derived sixth-orderstate equation expressed as in (1) as a second-order system from theassumption of from (ⅰ) to (ⅳ), and expressed as in (6), 2 Pr 2[n ,  ] = , JLdc P

Motor parameters of test motor.

Rated power
Base speed(1p.u.), Max. speed
Max. torque(1p.u.)
DC voltage
Poles
Inertia
Resistance
DC inductanceAC inductance
Rated current(1p.u.)

10000

Fig. 3. Frequency characteristics of speed command to
motor speed at base speed (1p.u.) with i0=iq, id=0.
where

r =

Lac
i0 .....................................................................(7)2where K1 is the damping gain, r is the virtual rotor flux, and i0
is the zero-phase current. As shown in (6), the damping factor  isthe function of the feedback gain K1. In other words, the proposed
V/f control is stabilized by setting K1 appropriately. Comparing thenatural angular frequency and the damping factor in the IPMSM[12], the rotor magnet magnetic flux corresponds to the virtual rotor
magnetic flux in case of considering the difference of the coefficientby the transformation methods which are the absolutetransformation and the reactive transformation. In addition, the qaxis inductance corresponds to the dc components of the selfinductance Ldc.
In this paper, the cut-off frequency of the HPF for the damping
control is set to 1/20 of the natural frequency n [13][14]. The
sufficiently small HPF cutoff frequency prevents steady speed
deviation while suppressing oscillation of the effective current(torque) due to the resonance. The HPF rejects the DC component
and extracts the resonance component. Note that in case of the SRM,
n is changed by the zero-phase current as shown in (6). Therefore,
the cut-off frequency of the HPF is designed from the lower-limit
value of i0 (i0Low) in order to be always lower enough compared withthe n, even when i0 is changed. In this paper, i0Low is defined to
1/10 of i0Max in order that phase-current is always bigger than zeroeven under the light loading condition. The cut-off frequency of the
HPF c is expressed as in (8).

c =

n
20

=

1 2 P Lac
1 2 P Lac i0 Max
i0 Low =.......(8)20 JLdc 2
20 JLdc 2 10

The cut-off frequency of the HPF is determined by motor
parameters P, Ldc, Lac, and rated zero-phase current i0.
The self-inductance Ldc and Lac are necessary to obtain thecontroller parameters. This paper adopts simple approximation
method for the inductance to consider the magnetic saturation. The
magnetic characteristic at the aligned position, at which themagnetic characteristic has the strongest nonlinearity, is acquiredJK1  ...................................... (6)Ldc 

3

IEEJ Trans. ●●, Vol.●●, No.●, ●●●V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)Imaginary partfrom the measured current when one pulse voltage is injected. The
magnetic characteristic at the aligned position is expressed as in (9),using a polynomial approximation.
Na =  kni n ................................................................... (9)n =1where kn is each coefficient of the polynomial approximation and N
is a maximum order of the polynomial approximation. N have to berelatively high order in order that the magnetic characteristic at thealigned position is expressed accurately. N is determined by theapproximation accuracy of the magnetization curve and is 10 in thispater. Since the maximum current value Imax is expressed as twice
the zero-phase current i00 (Imax=2*i00), assuming the MTPA control(id=0, i0=iq) is achieved as detailed in detail below, the average
inductance at the aligned position is expressed as in (10),

I Max

N

=  kn I Max
n =1

n −1

N

=  kn ( 2i00 )

n −1

60
40

K1=0
No.1

20
0
-20

No.5

K1=0.013p.u.
No.2
K1=0

-40

0

20

-60
-20 -10 0 10

Fig. 4. Loot locus of sixth-order state equation when damping
gain K1 is increased at base speed (1.0p.u.) with i0=iq, id=0.
0.06

Damping gain K1 [p.u.]

La (i00 ) =a ( I Max )

8000
K1=0
6000
4000
No.3
2000 No.6
K1=0.013p.u.
0
K1=0
-2000
-4000
No.4
-6000
K =0
-8000 1-120 -100 -80 -60 -40 -20
Real part......... (10)

n =1

K1Up
0.04

0.02

K1Stb.

Adapted K1

4×10-52×10-5

DC and AC components of self-inductance considering themagnetic saturation Ldc, Lac are expressed from the inductance at thealigned and unaligned position La, Lu as in (11),K1L ow

00.2

Fig. 5.

 L (i ) − Lu La (i00 ) + Lu 
[ Lac (i00 ), Ldc (i00 )] =  a 00,
 .......... (11)
2
2



0.4
0.6
Zero-phase current i0 [p.u.]

0.8

1

Relationship between zero-phase current
and damping gain K1.

The analysis is performed by substituting (11) into (1) to (8), whichallows us to take magnetic saturation into account.
Fig.4 shows the loot locus of the sixth-order state equation when
the damping gain K1 is increased at the rated-load and the basespeed. Note that from No. 1 to No.6 are the characteristic roots ofthe sixth-order state equation. As shown in Fig.4, poles named No.1
and 2 move toward the right-half plane as K1 decreases. On the other
hand, No.3 and 4 move toward the right-half plane as K1 increases.

StartJ, P, i0Max
i0Low=0.1i0Max

Measurement of R and Lu
Measurement of i-a(i)with one-pulse voltage test
Determination of coefficients kn byfitting measured i-a(i) to Eq.(9)Cal. c with Eq.(8) and i0Low

From Fig.4, following three variables are define:(ⅰ) Lower-limit value of K1, K1Low: The SRM is at the stability limit caused by No.1 and 2, when
K1=K1Low.
(ⅰⅰ) Most stable value of K1, K1Stb.
: The dominant poles of the SRM have the farthest from theimaginary axis when K1= K1Stb.. In this condition, the real part ofthe poles No. 1, 2, 3, and 4 matches.
(ⅰⅰⅰ) Upper-limit value of K1, K1Up
: The SRM is at the stability limit caused by No.3 and 4 when
K1=K1Up.

i00 =i0Low

i00 =i00 +d

i00 >i0Max
No
Cal. Lac (i00 ) and Ldc (i00 )with Eq.(11) and determined kn

YesCal. K1L ow(i00 ), K1Opt(i00 ), K1Up (i00 )with Eq.(3), Lac (i00 ), and Ldc (i00 )Selection of K1 based on calculated
K1L ow(i00 ), K1Stb.(i00 ), K1Up (i00 )EndThis is beca use phase-current has to be alwa ys bigger
than z ero even under the light loading condition.

Note that suppression of the overshoot of the motor rotation speed
is achieved by the same way as Refs. [13][14].
Fig.5 shows the relationship with the zero-phase current and theK1Low, K1Stb., and K1Up. In case of the SRM, K1Stb. is changed by thezero-phase current. Although the transient response is improved incase that K1 is adjusted to KStb. according the zero-phase current,
control system would be complex. Therefore, K1 is set to be within
the stable value in all conditions and the closest value with KStb..
Fig.6 shows the flowchart of the design of the damping gain K1.
The moment of inertia J, the number of pole pairs P, and rated zerophase current i0 is input constants. First, the inductance at theunaligned position and the resistance, which is independent of thecurrent is measured. Second, the i-a() characteristic at the aligned
position is derived with the approximation, as shown in (9). This
characteristic is acquired by the injection of the one pulse voltage.

Fig. 6.

Design flow of damping gain K1.

This measurement relatively easy compared to the complicated premeasurement including the locked rotor test or the FEA. Third, thecut-off frequency of the HPF is calculated from the lower-limit
value of the zero-phase current i0LOW and (8). Then, lower-limit
value K1Low, most stable value K1Stb., and upper-limit value K1Up ofthe damping gain are calculated in the different zero-phase current
from the lower-limit value i0LOW to the rated value i0Max. Finally, K1
to stabilize the SRM is adopted.

4

IEEJ Trans. ●●, Vol.●●, No.●, ●●●V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)

2.3 Maximum Torque/Ampere (MTPA) Control
Fig.7 shows the control diagram of the MTPA control for the
SRM. The MTPA control improves the torque/ampere ratio by
adjusting the converter output voltage. The following conditions
achieve the maximum torque/ampere ratio in the SRM.

i0=iq control

i

2

P

LPF

Qdq *
2Ldc

i

Q

v*

id = 0 ........................................................................... (13)id=0 control

Note that the zero-phase current is larger than AC current amplitude
because this paper uses the asymmetric half-bridge converter for the
unipolar driving and negative current does not flow (√2 in (12) isthe coefficient of the park transformation.). Therefore, it isnecessary to apply not only id=0 but also i0=iq for the MTPA control.
First, the i0=iq control is explained. iq is equal to the AC current
amplitude Iac when i0=0. Thus, i0 should be controlled to Iac not toflow the negative current. The command value of zero-phasecurrent control is expressed as (14).

Fig. 7.

Control diagram of MTPA control.

Test SR motor(V/f control)

Load motor(torque control)
Torque meteri0* = 2iq = 2 I ac = 2 i 2 + i 2 ................................ (14)Coupling

where i and i are the -axis and -axis current, respectively.
Next, the id=0 control is explained. The id=0 control is achieved
indirectly in the -axis coordinate by focusing on the reactive powerof the SRM. In the dq-axis coordinate, the reactive power Qdq input
into the SRM is expressed as in (15).

(a) Experimental systemQdq = vqid − vd iq = 2 Ldc (id 2 + iq 2 ) + Laci0id

v*
Limit
[0, Vdc ]1*i0 = 2iq ...................................................................... (12)

i0*

sqrt

Converter



6

300V

2 ...... (15)where vd and vq are the d-axis and q-axis voltage respectively. The
reactive power Qdq under the condition of id=0 is expressed as in (16),3

Gate
DriverQdq = 2Ldciq 2 = 2Ldc I ac 2 = 2Ldc (i 2 + i 2 ) .............. (16)i (u, v, w)

Controller

On the other hand, in the -axis coordinate, the reactive powerQ is expressed as in (17),SRMInterface Board
PWM
ModuleQ = v i .............................................................................. (17)Therefore, the v should be adjusted by the P controller in orderto match the reactive power Q calculated by (17) to the reactive
power Qdq calculated by (16). As shown in Fig.7, the value
calculated by (14) is applied as the command value for zero-phasecurrent control for the i0=iq control. On the other hand, the deviation
between (16) and (17) is regulated to be zero by adjusting the v
with the P controller for the id=0 control. Note that the LPF in thesecond stage of the P controller shown in Fig.7 is to achieve the id=0
control at a steady-state by reducing the gain for high frequencies[12]-[14]. In addition, the limit is provided in order to prevent
overcompensation.

v*

Torque [p.u.]
U-phasecurrent [A]

i
12bit
A/D

(b) Control system
Configuration of test system.

Fig. 8.

3. Experimental Results

DSP
TMS320F28379D

2

500ms/div

Vibration

1
0
-1
30

Stabilization control
ON
OFF

20
Over current

100Fig. 9.

Fig. 8 shows the experimental system. As shown in Fig.8(a), theload motor outputs the arbitrary constant torque. In addition, theinstantaneous torque is measured by a high-response torque meter(UTMII-10 Nm, 1 kHz bandwidth, UNIPULSE). As shown in
Fig.8(b), the controller consists of an interface board, 12bit Analog
Digital converter (A/D), Digital Signal Processor (DSP), and Pulse
Width Modulation (PWM) module. The greatest part of the proposed
V/f control is implemented on the DSP. Exceptionally, only A/Dconversion of the detected current and the generation of PWM signal
based on voltage command are performed by dedicated parts. On theother words, the proposed method composed only of a general
purpose micro-computer and does not require a high-speed

Effect of damping control.

computing device such as Field Programmable Gate Array (FPGA).
In “V/f conv” in Fig.1, the V/f ratio is set so that the provisional
modulation index is 1 at the base speed (1p.u.). Note that its
modulation index is adjusted in the MTPA control.
Fig. 9 shows the effect of the damping control. Speed and torqueare 1p.u. In the section with the damping control, there is no
oscillation in the U-phase current and torque. On the other hand, theoscillation occurs after disabling the damping control. These results
indicate the effectiveness of the damping control.

5

IEEJ Trans. ●●, Vol.●●, No.●, ●●●V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)

2
1
0

-1
1.5

Torque [p.u.]

1

250ms/div

0.5
0.5p.u.

0U-phasecurrent [A]

-0.5
15
10
5
0

Fig. 11. Step response for 0.5p.u. torque.
0.1p.u. speed(lowest)Fig.11(a)Torque [p.u.]1
Fig.11(b)

0.8
0.6
0.4
0.2
0

0.25

0.5

0.75
1
1.25
Speed [p.u.]
Speed and torque curves.

1.5

1.5
1
0.5

U-phasecurrent [A]Torque [p.u.]

Fig. 12.

1.0p.u.(4.38Nm)

0
20
15
10
5
0

500us/div

1.5
1
0.5

U-phasecurrent [A]Torque [p.u.](a) 1p.u. torque, 1p.u. speed(b)0.67p.u.(2.92Nm)

0
20
15
10
5
0

500us/div(b) 0.67p.u. torque, 1.5p.u. speed
Fig. 13. Torque and U-phase current waveform.
8
Current RMS value [A]Torque [p.u.]

Fig.10 shows the experiment results that startup is achieved with
1p.u. torque from zero-speed to base speed. The motor is accelerated
by the boost voltage after the zero-phase current of the rated value
i0Max was generated by the PI control at the stop condition. Note thatthe fine vibration, which is different from the mechanical resonancefrequency n, is superimposed on the torque waveform. This is dueto the torsional resonance in the experimental system excited by thetorque step in the startup. (The detail is explained in the appendix).
As a result, the acceleration with the rated torque (i.e. rated
acceleration time) is achieved. Although the step-out does not occur,
the torque is oscillating because the torque changes rapidly at thestart and finish of acceleration. This is because each control system
is designed under the steady-state. The design of the control system
considering including the transient response is a future issue.
Fig.11 shows the step response for 0.5p.u. torque at rated speed.
The motor follows the load without stepping out. Note that the motor
will step out with this control at step torques of 0.5p.u. or morebecause the control system is designed with a low bandwidthassuming the steady-state.
Fig.12 shows the speed and torque curves and Fig.13 shows thetorque and U-phase current at the rated speed and Max. speed
condition. The proposed method achieved to drive at 1p.u. torqueand at from 0.1p.u. to 1p.u. speed. In addition, the proposed method
drives with the rated power at from 1p.u. to 1.5p.u. speed. Note thatthe rated torque is not output at the 0.1p.u. speed or lower becauseof the output voltage error due to the dead time.
Fig.14 shows the effect of the MTPA control at the speed of
0.1p.u. The current of the proposed method (V/f control with i0=iq
and id=0) is compared with that of the FOC with i0=iq and id=0 andthat of V/f control with only i0=iq. The current RMS value isreduced by 39.0% compared with the only i0=iq control at the torqueof 0.05p.u. The proposed drive without the position sensor bringsout the same performance as the drive with the position sensor. On
the other hand, the current RMS value is increased by 14.6%
compared with the FOC with the position sensor. There are threereasons why the current of the proposed method increases morethan that of the FOC in the low-torque region. One reason is thatthe control accuracy deteriorates as the input power decreases andthe reactive power decreases in the low torque conditions. The otherreasons are the use of the proportional controller in Fig. 7 andsetting the lower limit on the zero-phase current. In this paper, thecomparison of the current RMS value with the torque is presented
in order to validate the effect of the MTPA control. Since the ironloss accounts for a larger proportion of the total loss at the highspeeds, minimizing the loss including the iron loss is necessary to
maximize the efficiency. The exploration of the loss minimization
control is a subject for future work.
Fine Vibration occurs due to experimental system.
(See appendix for details)
200ms/div

Startup

6 Reduced
by 39.0%4
2

01

Increasedby 14.6%
0.05 0.1

V/f control(only i0=iq)
FOC with position sensor(i0=iq, id=0)
V/f control with high efficiency control(i0=iq, id=0)0.2
0.3
Torque [p.u.]

0.4

0.5

Fig. 14. Effect of MTPA control (0.1p.u. speed).

0.5
0Fig. 10.

Startup at i0, iq=1.0p.u.

6

IEEJ Trans. ●●, Vol.●●, No.●, ●●●V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)

4. Conclusions

Appendix

This paper proposed a novel V/f control method for the SRM.
This paper proposed the following three methods and confirmed
their effectiveness: (ⅰ) the virtual rotor magnetic flux is generated
by controlling the zero-phase current with the PI controller, which
realizes the V/f control for the SRM, (ⅱ) the damping control gainis designed to be stable at all operating points even when the rotor
magnetic flux and the inductance value changes dynamicallyaccording to the operating conditions, and (iii) the hightorque/ampere is achieved by applying both the i0 = iq control and
id = 0 control. In the experiment, the proposed method achieved thestable operation in all N-T regions with the tested SRM. In addition,
the current RMS value is reduced by 39% at the 0.05p.u. torquecompared with the case of no MTPA control, which is the almost
same performance as the driving with the position sensor.

1.
State equation of the SRM
Each element of the state equation (3) are expressed as follows.
Note that the subscript "0" indicates the value of the operating pointin each variable.
 L L 2L 2
− R  Ldc 2 − ac sin 2 (20 )  − 20  − dc ac sin(20 ) cos(20 ) 
2
2




A11 =
.......................... (A-1)
2
L L
Ldc3 − dc ac
2
 L 2



L L 2
L L 2
− R  − ac sin(20 ) cos(20 )  − 20  − Ldc3 + dc ac sin 2 (20 )  − 2 K1 ( Ldc3 − dc ac )i 0
2
2
2




A12 =
L L 2
Ldc3 − dc ac2

Lac 3 
1  2
 L L

− R  − dc ac cos(20 )  − 20 Ldc Lac − sin(20 )
2 
2
2 


............................ (A-3)
A13 =
2
L
L
Ldc3 − dc ac
2

References
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)
(13)

(14)

(A-2)

2



 L L 2

Lac 3 1  2
 L L−2  − dc ac sin(20 ) cos(20 )  i 0 +  − dc ac cos 2 (20 )  i 0 + Ldc Lac − sin(20 )i00 
2
2
2 
2 








A14 =
2
L L
Ldc 3 − dc ac
2

T. Kosaka, A. Kume, H. Wakayama, and N. Matsui: “Development of hightorque density and efficiency vswitched reluctance motor with 0.1mm short
airgap”, 2007 European Conference on Power Electronics and Applications,pp. 1-9 (2007)
H. Yamai, Y. Sawada, and K. Ohyama: “Applying Switched ReluctanceMotor to Oil Hydraulic Pump Use”, IEEJ Trans. on Ind. Applicat., Vol.123,
No.2, pp.96-104 (2003)
K.-W. Hu, Y.-Y. Chen, and C.-M. Liaw: “A reversible position sensorless
controlled switched-reluctance motor drive with adaptive and intuitive
commutation tunings”, IEEE Trans. Power Electron., Vol.30, No.7, pp.37813793 (2015)
K. Ha, R.Y. Kim, and R. Krishnan: “Position estimation in switched
reluctancemotor drive using the first switching harmonics through fourier
series”, IEEE Trans. Ind. Electron., Vol.58, No.12, pp.5352–5360 (2011)
G.G. Lopez, P.C. Kjaer, and T.J.E. Miller: “A new sensorless method forswitched reluctancemotor drives”, IEEE Trans. Ind. Applicat., Vol.34, No.4,
pp.832–840 (1998)
S. Sumita, K. Deguchi, and Y. Iwaji: “Position Sensorless Control Method
for SRM Using Analog Circuit of Type-1 Control System”, IEEJ Trans. Ind.
Applicat., Vol.137, No. 8, pp.612-621 (2017)
T. Kosaka, K. Ochiai, and N. Matsui: “Sensorless Control of SRM using
Magnetizing Curves”, IEEJ Trans. Ind. Applicat., Vol.120, No.2, pp.216-222(2000)
T. Kosaka, Y. Nabeya, K. Ohyama, and N. Matsui: “Position Sensorless Driveof SRM Mounted on Hydraulic Pump Unit”, IEEJ Trans. Ind. Applicat.,Vol.123, No.2, pp.103-111 (2003)
K.M. Rahman, B. Fahimi, G. Suresh, A.V. Rajarathnam, and M. Ehsani:“Advantages of Switched Reluctance Motor Applications to EV and HEV:
Design and Control Issues”, IEEE Trans. on Ind. Applicat., Vol.36, No.1,
pp.111–121 (2000)
I. Husain and S.A. Hossain: “Modeling, Simulation, and Control of Switched
Reluctance Motor Drives”, IEEE Trans. on Ind. Electron., Vol.52, No.6,
pp.1625–1634 (2005)
N. Nakao and K. Akatsu: “Vector Control Specialized for Switched
Reluctance Motor Drives”, IEEE Trans. on Ind. Applicat., Vol.134, No.12,
pp.1006–1015 (2014)
J. Itoh, J. Toyosaki, and H. Ohsawa: “High performance V/f control method
for PM Motor”, Vol.122, No.3, pp.235-259 (2002)
T. Toi, M. Kato, and J. Itoh: “Parameter Design Considering Loot Locus for
Damping Control Based on V/f control of IPMSM”, 2016 Kansai-section
Joint Convention of Institutes of Electrical Engineering, No. G4-22 (2016)
T. Toi, K. Nishizawa, and J. Itoh: "Stabilization Method for IPMSM withLong Electrical Time Constant Using Equivalent Resistance Gain Based on
V/f Control", IEEJ Transactions on Industry Applications, Vol. 8, No. 4, pp.
592-599 (2019)................................................................................................ (A-4)
 L 2 L−40  dc ac cos(20 )i00  2............................................................. (A-5)
A15 =
L L 2
Ldc3 − dc ac
2

A16 = 2K1i 0 ................................................................................. (A-6)
 L 2



L L 2
− R  − ac sin(20 ) cos(20 )  − 20  Ldc3 − dc ac cos 2 (20 ) 
2
2



 ......................... (A-7)
A21 =
2
L L
Ldc3 − dc ac2L L 2
L 2
L L 2− R  Ldc 2 − ac cos2 (20 )  − 20  dc ac sin(20 ) cos(20 )  + 2K1 ( Ldc3 − dc ac )i 0
2
2
2




.. (A-8)
A22 =
L L 2
Ldc3 − dc ac2
Lac3 
1  2
L L

− R  dc ac sin(20 )  − 20 Ldc Lac − cos(20 )
2 
2 
 2

............................... (A-9)
A23 =
2
L
L
Ldc3 − dc ac
2
2



L L 2

Lac 3 1  2
 L L−2  dc ac sin 2 (20 )  i 0 +  dc ac sin(20 ) cos(20 )  i 0 + Ldc Lac − cos(20 )i00 
2 
2 


 2

 2



A 24 =2
L L
Ldc3 − dc ac
2.............................................................................................. (A-10)
A 25 =

 L 2 L−40 − dc ac sin(20 )i00 2........................................................ (A-11)
L L 2
Ldc3 − dc ac
2

A 26 = −2K1i 0 .............................................................................. (A-12)

7

 L 2L L L

− R  − dc ac cos(20 )  − 20  dc ac sin(20 ) 
2
2




A31 =
2
L L
Ldc3 − dc ac
2..................................... (A-13)

 L 2L
L L

− R  dc ac sin(20 )  − 20  dc ac cos(20 ) 
2
 2



A32 =
L L 2
Ldc3 − dc ac
2...................................... (A-14)

IEEJ Trans. ●●, Vol.●●, No.●, ●●●V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)A33 =− RLdc 2
L L 2
Ldc − dc ac
2

B24 = −2i 0 ................................................................................. (A-38)....................................................................... (A-15)

3

B54 = 1 ..................................................................................... (A-39)2 L 2L  L L−2  dc ac sin(20 )  i 0 +  dc ac cos(20 )  i 0 
2
2




 


A34 =
L L 2
Ldc 3 − dc ac
2

Botherwise = 0 ................................................................................ (A-40).................................. (A-16)

2.
Derivation of (4) and (5)Relationship between the dq-axis current and the dq-axis current
is expressed as in (A41),

 L L 2 
−40 − dc ac i00 2................................................................. (A-17)
A35 =
L L 2
Ldc3 − dc ac
2
A 41 =

2 P 2 Lac
i00 sin(20 )J................................................................. (A-18)

A 42 =

2 P 2 Lac
i00 cos(20 )J................................................................ (A-19)

A 43 =

2 P 2 Lac
( i 0 sin(20 ) + i 0 cos(20 ) )J

A 45 =

2 P 2 Lac
( 2i 0 i00 cos(20 ) − 2i 0 i00 sin(20 ) )Ji   cos(20 ) sin(20 ) 0  id 
i  =  − sin(2 ) cos(2 ) 0   i 
0
0
 
  q  ........................... (A-41)
 i0  
00
1   i0 

Equation (5) is derived, substituting (12) and (13) into (A-41) under
the assumption that the MTPA control is achieved. The simplified
-axis voltage is derived by ignoring the asynchronous term
appearing as the sixth harmonic for simplification in the steadystate, which is expressed as in (A-42),................................................ (A-20).......................................... (A-21)

Lv = Ri + 2  Ldci + ac cos(20 )i0 
2


i0
i0
=R
cos(20 ) + 2( Ldc sin(20 ) + Lac cos(20 ) )2
2

A52 = − K1 ................................................................................. (A-22)

A54 = −1 ................................................................................... (A-23)

Assuming that the reactive power Qdq in the dq-axis expressed as
expressed in (16) is equal to that in the -axis as in (17), (A43) isobtained from (14) and (5).

A56 = K ................................................................................... (A-24)
A 62 =

1.................................................................................... (A-25)
 Ldci0 2 = v1
A 66 = − .................................................................................. (A-26)


B11 =

Lac 2
sin 2 (20 )2
Ldc Lac 2
3
Ldc −
2

 Ldc
sin(20 )== tan(20 ) .............................(A-44)
0.5R +  Lac cos(20 )................................................................ (A-28)

Thus, (4) is derived by solving (A-44) for φ0.

L 2− ac sin(20 ) cos(20 )2.............................................................. (A-29)
B12 =
L L 2
Ldc 3 − dc ac
2

Ldc Lac2
Ldc3 −

B21 =

B22 =

−

.................................................................... (A-30)

Lac 2
sin(20 ) cos(20 )2.............................................................. (A-31)
L L 2
Ldc3 − dc ac2

Lac 2
cos 2 (20 )2
L L 2
Ldc3 − dc ac
2

Ldc Lac
2
Ldc −
3

−

sin(20 )Ldc Lac 2
2

Ldc Lac2

B31 =Ldc 3 −
Ldc Lac

B33 =Ldc Lac 22

Ldc 2 −

B23 =

B32 =

cos(20 )3.
Fine vibration in Fig. 10
App. Fig.1 shows an enlarged waveform of the fine vibrations in
the torque waveform immediately after startup in Fig. 10. As shownin App. Fig.1, the frequency of the fine vibrations is 200 Hz.
App. Fig.2 shows the impulse response of the current and torquein the experimental system. The torsional resonance is excited by
impulse current, giving a torque shock, as can be seen from the App.
Fig.2, the torsional resonance frequency of the experimental system
is 200 Hz. This frequency matches the frequency of fine vibrationsin the torque waveform immediately after startup. Therefore, thefine vibrations, which differ from n, result from the excitation of
the torsional resonance in the experimental system due to the torquestep in the startup.

2....................................................................... (A-33)

cos(20 )Ldc Lac 22

sin(20 )L L 2
Ldc3 − dc ac
2................................................................ (A-32).................................................................... (A-34)

Torque [p.u.]

−
B13 =

i0
sin(20 ) ...............................................(A-43)
2

Substituting (A-42) into (A-43), the deviation angle between the dqaxis and the -axis is expressed as

Aotherwise = 0 ................................................................................ (A-27)
Ldc 2 −

(A-42)....................................................................... (A-35)

2200ms/div

1
0-1

Startup
10ms/div

Ldc 2........................................................................ (A-36)
L L 2
Ldc 3 − dc ac2

5ms(200Hz)
B14 = 2i 0 .................................................................................. (A-37)

App. Fig.1.
8

Enlarged torque waveform at startup.
IEEJ Trans. ●●, Vol.●●, No.●, ●●●V/f Control for Switched Reluctance Motor (Hirotaka Kato et al.)

6
4
2
0
-2
0.04
0.02
0
-0.02
-0.04

U-phasecurrent [A]Torque [p.u.]

an associate professor. Since 2017, he has been a professor. His researchinterests are matrix converters, dc/dc converters, power factor correction
techniques, energy storage system and adjustable speed drive systems. He
received IEEJ Academic Promotion Award (IEEJ Technical Development
Award) in 2007. In addition, he also received Isao Takahashi PowerElectronics 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 Institute of Electrical and Electronics Engineers.

5ms/div

5ms(200Hz)

App. Fig.2. Measurement results of resonant frequency.

Hirotaka Kato (Student member) received the B.S.
degree in electrical, electronics and informationengineering from Nagaoka University of Technology,
Niigata, Japan, in 2022. He has been with Nagaoka
University of Technology as a M.S. course student inelectrical, electronics and information engineering.
He is a student member of IEEJ.

Masakazu Kato (Member) received the B.S. and
M.S. degrees in electrical, electronics andinformation from Nagaoka University ofTechnology, Niigata, Japan in 2012, 2014,
respectively. In August 2016, he established
Nagaoka Motor Development Co., Ltd. In 2017, hereceived the Ph.D. degree in energy and environment
science from Nagaoka University of Technology. His current researchinterests include control and simulation of electrical motor.

Takahiro Kumagai (Member) received the B.S.
degree in electrical, electronics and informationengineering from Nagaoka University of Technology,
Niigata, Japan, in 2017. From June to December in
2021, he was with Yokohama National University
M&E Energy Conversion Laboratory as a special
research student. In 2022, he received the Ph.D.
degree in 5-year integrated doctoral program of
department of science of technology innovation from Nagaoka Universityof Technology. He is a member of IEEJ and IEEE. Since 2022, he has beenwith DENSO Corporation, Aichi, Japan. His current research focuses on thedriving method and the motor design method for switched reluctance motor.
He received IEEJ Industry Applications Society Distinguished Transaction
Paper Award in 2022.
Keisuke Kusaka (Member) received the B.S. and
M.S. degrees in electrical, electronics and informationfrom Nagaoka University of Technology, Niigata,Japan in 2011, 2013, respectively. From 2015 to 2016,he was with Swiss Federal Institute of Technology in
Lausanne (EPFL), Switzerland as a trainee. In 2016,
he received the Ph.D. degree in energy andenvironment science from Nagaoka University of Technology. He was with
Nagaoka University of Technology, Niigata, Japan as a researcher from2016 to 2018. Since 2018, he has been with Nagaoka University ofTechnology, Niigata, Japan as an assistant professor. His current researchinterests include the areas of IPT systems and high-frequency converters.
He is a member of Institute of Electrical Engineers of Japan, Society ofAutomotive Engineers of Japan and the IEEE.
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 was with Fuji Electric Corporate Researchand Development Ltd., Tokyo, Japan. He was with
Nagaoka University of Technology, Niigata, Japan as
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