Internati
o
nal
Journal of P
o
wer Elect
roni
cs an
d
Drive
S
y
ste
m
(I
JPE
D
S)
Vol
.
6
,
No
. 2,
J
une
2
0
1
5
,
pp
. 22
5~
23
2
I
S
SN
: 208
8-8
6
9
4
2
25
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJPEDS
Math
ematical M
o
del of L
i
near Sw
itched Reluct
ance M
o
t
o
r with
Mutual Inductance Consideration
N.V. Grebennikov, A.V.
Kireev, N.M.
Koz
h
emyaka
S
c
ien
tifi
c
and T
echni
cal
Cen
t
er
“PRIVOD-N”, Rostov Region
,
Russia
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
Mar 11, 2015
Rev
i
sed
Ap
r
15
, 20
15
Accepted Apr 27, 2015
This paper pr
esents developing
an
m
a
them
ati
c
al m
odel for lin
ear switch
e
d
reluctance motor
(LSRM) with account of
the mutual indu
ctance
between
th
e
phases. Mutual
inter
act
ion betw
een th
e phases
of LSRM gives the positiv
e
effec
t
,
as
a ru
le
t
h
e power
of th
e
m
achine
is
in
cre
a
s
e
d b
y
5
-
15%.
Keyword:
Linear m
o
tor
Maglev system
Switch
e
d
relu
ctan
ce
m
o
to
r
Mutual inductance
Copyright ©
201
5 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
N.
V. Gre
b
e
n
niko
v,
Scien
tific and
t
ech
n
i
cal
ce
nter “PRIVOD-N”
,
K
r
i
v
o
s
h
l
yko
v
a
4A
, No
vo
ch
erk
a
ssk
,
R
o
stov
Reg
i
on
, 34
6428
,
R
u
ssia.
Em
a
il:
grebe
n
n
i
ko
vn
v
@
m
a
il
.ru
1.
INTRODUCTION
A linear s
w
itched relucta
n
ce drive with a large air gap
i
s
de
v
e
l
ope
d f
o
r
rai
l
w
ay
ve
hi
cl
es [1]
,
[
2
]
.
Th
e
pos
sibility to apply this type of el
ectric m
a
chines is de
fine
d by the great value of norm
a
l force c
o
m
pone
nt
bet
w
ee
n st
at
o
r
and
r
o
t
o
r,
w
h
i
c
h ca
n
be
use
d
f
o
r
ge
nerat
i
o
n
o
f
l
e
vi
t
a
t
i
o
n
an
d
assu
ra
nce
of
g
u
i
d
a
n
ce sy
st
em
.
Suc
h
kind of e
l
ectric
m
achine
obta
in
s t
h
e
passiv
e
ro
t
o
r con
s
isted
of ferromag
n
e
tic elemen
ts lo
cated
along trac
k st
ruct
ure. Rotors ele
m
ents ha
ve
great m
echanical strength, which elim
inates rest
rictions for
tran
sm
issio
n
of m
ech
an
ical tractio
n
fo
rce
an
d susp
en
sion
an
d
g
i
v
e
s t
h
e
p
o
s
sib
ility to
create th
e
p
a
ssiv
e
discrete track
structure with reduced m
a
te
rials cons
um
pt
i
on, a
nd at
t
h
e sam
e
tim
e,
t
h
e desi
g
n
o
f
st
at
or
wind
ing
with
co
n
c
en
trated
co
ils is ex
trem
ely
si
m
p
le.
Ho
we
ver
,
t
h
e
l
a
rge ai
r
gap
(
a
i
r
ga
p1
2 m
m
)
red
u
ces
the e
fficiency of the dr
i
v
e system (efficie
n
cy
76
%).
T
he
desi
gn
an
d c
o
nt
r
o
l
ob
ject
i
v
es
are
t
o
m
a
xim
i
ze t
h
e effi
ci
e
n
cy
at
t
h
e gi
ven
m
o
t
o
r
di
m
e
nsi
ons a
n
d
t
h
e
out
put
p
o
we
r.
The
pu
r
pose
o
f
t
h
i
s
pape
r i
s
t
h
e desi
gn
rat
i
onal
e
f
o
r t
h
e
choi
ce
of
t
h
e l
i
near m
o
t
o
r,
w
h
i
c
h
increases
efficiency.
Switch
e
d
reluctan
ce m
ach
in
es (SRM
) are d
e
si
g
n
e
d
as
a h
i
gh-qu
ality typ
e
o
f
elect
ro
m
ech
an
ical
ener
gy
co
nve
rt
er an
d can be a
ppl
i
e
d t
o
t
h
e i
n
dust
r
i
a
l
t
r
ans
p
ort
.
T
h
e m
a
i
n
di
st
i
n
g
u
i
s
hi
ng
feat
ure
of SR
M
i
s
t
h
e
l
ack of wi
ndi
ng at
t
h
e t
oot
hed r
o
t
o
r. I
n
com
p
ari
s
on
with electrical
machines of anothe
r types, SRM is
st
ruct
u
r
al
l
y
si
m
p
l
e
r and t
echn
o
l
o
gi
cal
l
y
effect
i
v
e, i
t
has
l
e
ss coppe
r a
nd i
n
sul
a
t
i
n
g m
a
t
e
ri
al
s consum
pt
i
on
whe
n
alm
o
st identical m
a
sse
s of elect
rical steel. As a res
u
lt it
makes possi
ble to ac
hieve highe
r
ene
r
gy and
weight-size
para
m
e
ters, to
reduce the
cost of
elect
rical
m
achines a
n
d expe
ns
es fo
r th
eir oper
a
tio
n.
Du
ri
n
g
SR
M
m
odel
l
i
ng a
nd
desi
gni
ng
i
t
i
s
usual
l
y
assum
e
d t
h
at
al
l
m
achi
n
e’s ph
ases ar
e
inde
pende
n
t both i
n
electrical and m
a
gnet
i
c relation.
It is accepted that
m
u
tual induct
ance is low in these
machines and it can be ignored. Howe
ve
r,
at the present
tim
e
researche
r
s are h
a
v
i
ng
ten
d
e
n
c
y to
take in
to
account the m
u
tual influe
nc
e of
phases for classical
SRM [3] –
[11]. This m
u
tu
al interaction
gives the
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6,
No
.
2,
Ju
ne 20
15
:
225
–
2
32
22
6
p
o
s
itiv
e effect, as a
ru
le t
h
e
po
wer
o
f
th
e m
ach
in
e is in
creased
b
y
5
-
1
5
%.
Math
em
a
tical
m
o
d
e
l o
f
lin
ear SRM
(LSR
M) is pres
en
ted
in th
e
p
a
p
e
r. Co
nfiguratio
n o
f
LSRM is
sim
i
l
a
r wi
t
h
ro
t
a
t
i
ng m
o
t
o
r w
h
i
c
h
has 1
8
st
a
t
or t
eet
h an
d
1
5
r
o
t
o
r t
eet
h. T
h
i
s
co
nfi
g
u
r
at
i
on
pr
o
v
i
d
es a s
t
ro
ng
m
a
gnet
i
c
co
up
l
i
ng bet
w
ee
n p
h
ases
of t
h
e m
achi
n
e t
h
at
m
u
st
be co
nsi
d
e
r
e
d
w
h
e
n
m
odel
l
i
ng.
The m
odel
al
l
o
w
to resea
r
c
h
ele
c
trom
echanical and
electrom
a
gnetic
processe
s in m
o
tor.
2.
MAT
H
EM
AT
ICAL
M
O
DE
L
Fo
r
m
a
g
n
e
tic ch
ar
acter
istics, calcu
latio
n
of
LSRM con
f
i
g
ur
ation
[6
, 7] Fin
ite Ele
m
en
t Method
Magnetics (FE
MM)
pac
k
a
g
e was use
d
[10].
LSRM
e
quati
on, taki
ng int
o
account
th
e m
u
tual
inducta
nce of
each pha
se,
is prese
n
ted
bel
o
w:
)
,
(
1
d
t
i
d
i
R
u
m
k
k
(1
)
whe
r
e
u
is th
e
p
h
a
se vo
ltag
e
,
R
is the phase
resistance,
i
is the pha
se current,
i
s
t
h
e pha
se fl
ux l
i
n
ka
ge
s,
(i
n o
u
r case i
t
i
s
a funct
i
o
n o
f
seve
n vari
a
b
l
e
s),
m
is the num
b
er of phas
es,
is the expressed in electrical
degrees line
a
r
shift
betwee
n t
r
anslator a
n
d st
ator,
k is the
phase
num
b
er of
electrical m
a
chine.
Th
e
d
i
stribu
tion
p
a
ttern of t
h
e m
a
g
n
e
tic field
lin
es to
th
e
LSRM co
nf
igu
r
at
io
n
is show
n in f
i
gu
r
e
1.
It is ob
v
i
o
u
s th
at all co
ils with
in
on
e
p
h
a
se are
co
nn
ected
in
op
po
site
d
i
rection
.
Magn
etic flow
gene
rated by
phase
A is c
o
m
p
letely run t
h
rough a
d
jacent
phases
(
B, C,
D, E, F
).
Fi
gu
re
1.
Lay
o
u
t
o
f
c
o
i
l
s
an
d
di
st
ri
b
u
t
i
o
n
o
f
m
a
gnet
i
c
fi
el
d
l
i
n
es i
n
L
S
R
M
un
de
r
ope
rat
i
o
n
of
p
h
ase
A
LSRM is p
r
ov
id
ed
with
three co
ils in
a
p
h
a
se g
e
n
e
rated
o
ppo
site mag
n
e
tic fl
o
w
s
clo
s
ing
v
i
a
adjace
nt phase
s
. The
r
efore
,
the m
odelling of this m
ach
in
e sh
ou
ld
tak
e
i
n
to
accoun
t the in
teractio
n
between
adjace
nt phase
s
.
The i
n
teraction betwee
n
phase
s
can be
classified
in
to two
categ
o
ries:
1
)
m
u
tu
al in
ductan
ce influ
e
n
c
e,
2)
m
u
t
u
al
sat
u
r
a
t
i
on i
n
fl
ue
nce
.
Mutual inductance is
due t
o
t
h
e
fi
eld
ov
erlap
thro
ugh
an
o
t
h
e
r ph
ase. Mu
t
u
al saturation
i
s
th
e im
p
act
of m
a
gnet
i
z
i
n
g
fo
rce o
f
o
n
e
p
h
ase o
n
t
h
e
sat
u
rat
i
o
n
of t
h
e
ot
he
r o
n
e. T
h
e
l
e
vel
of sat
u
ra
t
i
on i
m
pact
s on fl
u
x
l
i
nkage
s a
n
d
t
o
rq
ue at
t
h
e s
h
aft of electrical machine.
We can
wri
t
e
do
w
n
t
h
e sy
st
em
of equat
i
o
ns
for si
x-
phase machine. Let us a
ssum
e
that the adjace
nt
p
h
a
ses
h
a
v
e
si
g
n
i
f
i
can
t im
p
act o
n
th
e con
s
id
er
ed
p
h
a
ses
an
d th
e r
e
st
of
p
h
a
ses
d
o
no
t h
a
v
e
an
y st
r
ong
i
n
fl
ue
nce:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Mat
h
e
m
at
i
c
al
Mo
del
of
Li
ne
a
r
Sw
i
t
c
he
d Rel
u
ct
a
n
ce M
o
t
o
r
w
i
t
h
Mut
u
al
In
duct
ance
…
(
N
.V.
Gre
b
en
ni
ko
v)
22
7
dt
d
i
R
u
dt
d
i
R
u
dt
d
i
R
u
dt
d
i
R
u
dt
d
i
R
u
dt
d
i
R
u
F
F
F
F
E
E
E
E
D
D
D
D
C
C
C
C
B
B
B
B
A
A
A
A
(2
)
whe
r
e
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
ψ
AF
FF
FE
F
FE
EE
DE
E
ED
DD
CD
D
DC
CC
BC
C
CB
BB
AB
B
BA
A
А
FA
A
(3
)
Write (3) throug
h ind
u
c
tan
c
e
i
M
i
L
i
M
ψ
i
M
i
L
i
M
ψ
i
M
i
L
i
M
ψ
i
M
i
L
i
M
ψ
i
M
i
L
i
M
ψ
i
M
i
L
i
M
ψ
A
AF
F
F
E
FE
F
F
FE
E
E
D
DE
E
E
ED
D
D
C
CD
D
D
DC
C
C
B
BC
C
C
CB
B
B
A
AB
B
B
BA
A
A
F
FA
A
(4
)
whe
r
e M
j
k is t
h
e m
u
tual inductance betwee
n ph
ases, Lk
is t
h
e
p
h
a
se indu
ctiv
ity.
Since all phas
es of the
m
ach
in
e
con
s
ist o
f
id
en
ti
cal coils, it can
be ass
u
me
d that the
resistance of
each
pha
se is t
h
e sam
e
:
R
R
R
R
R
R
R
F
E
D
C
B
A
.
Give
n the
facts that ove
rlap i
n
the phase ope
r
ati
on i
s
1
2
0
de
g an
d t
h
e a
r
ea
of
gen
e
rat
o
r a
n
d t
r
act
i
o
n
m
o
d
e
s is 18
0 deg
,
t
h
en sim
u
lt
an
eou
s
ly
n
o
t
m
o
re th
an
three
p
h
a
ses
o
f
m
a
c
h
in
e
will op
erate in
no
m
i
n
a
l m
o
d
e
.
Thus,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-86
94
I
J
PED
S
Vo
l. 6,
No
.
2,
Ju
ne 20
15
:
225
–
2
32
22
8
3
3
2
2
1
1
dt
d
u
dt
d
u
dt
d
u
dt
d
i
R
u
dt
d
i
R
u
dt
d
i
R
u
Y
Y
Y
Y
Y
Y
Y
Y
Y
X
X
X
Z
Z
Z
(5
)
Where
flux linkage with anal
ogue
of (4)
will be the following:
Z
Y
Z
Y
Y
Y
X
Y
X
Y
Z
Y
Z
Y
Y
Y
X
Y
X
Y
Z
Y
Z
Y
Y
Y
X
Y
X
Y
Z
ZY
Y
Y
X
XY
Y
Y
YX
X
X
Z
ZX
X
X
XZ
Z
Z
Y
YZ
Z
i
M
i
M
i
M
ψ
i
M
i
M
i
M
ψ
i
M
i
M
i
M
ψ
i
M
i
L
i
M
ψ
i
M
i
L
i
M
ψ
i
M
i
L
i
M
ψ
)
3
(
)
3
(
)
3
(
)
3
(
)
2
(
)
2
(
)
2
(
)
2
(
)
1
(
)
1
(
)
1
(
)
1
(
(6
)
whe
r
e in
dexes
Z, X an
d Y c
o
rres
p
on
d with
com
b
ination o
f
ph
ase o
p
erati
on (
F
, A
,
B
)
, (
A
, B
,
C
)
, (B
,
C
,
D),
(
C
,
D
,
E)
,
(D
, E,
F)
и
(
E
,
F, A
)
.
As
was
m
e
ntione
d
ab
o
v
e, t
h
e c
o
nside
r
ed
m
achine has
a str
o
ng
m
u
tual
phase
in
fl
uence
,
it
is
therefore necessary
to re
view
the m
u
tual inductance
as a
function
of four varia
b
les:
)
(
)
(
)
(
)
(
)
(
)
(
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Z
Y
X
XZ
Z
Y
X
ZX
Z
Y
X
ZY
Z
Y
X
YZ
Z
Y
X
YX
Z
Y
X
XY
i
i
i
f
M
i
i
i
f
M
i
i
i
f
M
i
i
i
f
M
i
i
i
f
M
i
i
i
f
M
As far as the
m
a
gnetic syste
m
of the
m
achine is sy
m
m
e
tr
ic and the retu
rn pe
rio
d
is 60
deg.
, we ca
n
write the following:
))
60
(
(
))
300
(
(
)
(
,
,
,
,
,
,
,
,
,
Z
Y
X
X
Z
Z
Y
X
X
Y
Z
Y
X
X
i
i
i
L
L
i
i
i
L
L
i
i
i
f
L
))
360
(
(
))
120
(
(
))
120
(
(
))
60
(
(
))
300
(
(
)
(
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Z
Y
X
XY
XZ
Z
Y
X
XY
ZX
Z
Y
X
XY
ZY
Z
Y
X
XY
YZ
Z
Y
X
XY
YX
Z
Y
X
XY
i
i
i
M
M
i
i
i
M
M
i
i
i
M
M
i
i
i
M
M
i
i
i
M
M
i
i
i
f
M
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PEDS
I
S
SN:
208
8-8
6
9
4
M
a
the
m
atical
M
o
del of
Line
a
r
Sw
itche
d Rel
u
cta
n
ce M
o
tor
w
ith M
u
tu
al
In
duct
ance
…
(
N
.V.
Gre
b
en
niko
v)
22
9
There
f
ore, for
com
p
lete com
p
uter
sim
u
lation of this m
achine it is
necessa
ry to get a
d
ditionally by
m
eans of Fi
nite Elem
ent Met
h
od t
h
e
relation
of
)
,
,
,
(
Z
Y
X
X
i
i
i
f
or
)
(
,
,
,
Z
Y
X
X
i
i
i
f
L
,
)
,
,
,
(
Z
Y
X
XY
i
i
i
f
or
)
(
,
,
,
Z
Y
X
XY
i
i
i
f
M
. The
applic
ation
of FEM
M
pac
k
age
gi
ves the
possibility to
get im
m
e
diatel
y the following relations:
)
,
,
,
(
Z
Y
X
X
i
i
i
f
,
)
,
,
,
(
Z
Y
X
Z
i
i
i
f
,
)
,
,
,
(
Z
Y
X
Y
i
i
i
f
,
)
,
,
,
(
1
Z
Y
X
Y
i
i
i
f
,
)
,
,
,
(
2
Z
Y
X
Y
i
i
i
f
,
)
,
,
,
(
3
Z
Y
X
Y
i
i
i
f
.
Let differe
n
tiate the e
quati
on
(6):
i
M
dt
di
M
i
L
dt
di
L
i
M
dt
di
M
dt
d
ψ
i
M
dt
di
M
i
L
dt
di
L
i
M
dt
di
M
dt
d
ψ
i
M
dt
di
M
i
L
dt
di
L
i
M
dt
di
M
dt
d
ψ
Z
ZY
Z
ZY
Y
Y
Y
Y
YX
XY
X
XY
Y
Z
ZX
Z
ZX
X
X
X
X
Z
ZX
Z
ZX
X
X
XZ
X
XZ
Z
Z
Z
Z
Y
YZ
Y
YZ
Z
whe
r
e
dt
d
.
The propulsi
on
force depe
nding on
phase
curre
nt and
rotor
position
c
a
n
be e
x
presse
d in term
s of
coene
r
gy. In
our case the
coe
n
ergy
diffe
rent
ial is:
d
F
di
di
di
i
i
i
dW
e
Z
Z
Y
Y
X
X
Z
Y
X
c
)
,
,
,
(
.
(7
)
whe
r
e
F
е
is the
propulsion
force of m
achine.
The c
o
ene
r
gy
for t
h
e propose
d m
achine can
be foun
d by i
n
tegration (7) alon
g the
outline by analogy
with [4]. The
integra
tion pat
h
is selected i
n
t
h
e
following
way:
1)
Inte
grate
by rotation angle at
zero curre
nt in
all phases (
0
,
0
,
0
Z
Y
X
i
i
i
);
2)
Inte
grate by
Z
i
, keepi
ng ze
ro c
u
rrents i
n
the
othe
r two pha
s
es (
0
,
0
Y
X
i
i
), an
d r
o
t
a
tion
angle
θ
– as c
o
nstant;
3)
Inte
grate by
Y
i
, keeping ze
ro curre
nt in
phase
X (
0
X
i
), rotation a
ngle
θ
an
d
Z
i
as constant;
4)
Inte
grate by
X
i
, r
o
tation a
n
gle
θ
,
Y
i
and
Z
i
are const
a
nt .
At the
first sta
g
e
of i
n
tegration th
e t
o
rque
integral is ze
ro, si
nce t
h
e torque is ze
ro at
zero phase
currents (
0
,
0
,
0
Z
Y
X
i
i
i
); at the followi
ng
stages this inte
gr
al is ze
ro
be
cause the
rotation a
n
gle
θ
is
co
nstan
t
.
After inte
grati
n
g we
re
ceive
the e
x
pression
for
co
en
e
r
gy o
f
t
h
e c
o
n
s
id
e
r
e
d
m
achine
when three
pha
ses ope
rate sim
u
ltaneously
:
X
Y
Z
i
Z
Y
X
i
Z
Y
i
Z
e
Z
Y
X
c
d
i
i
d
i
d
d
F
i
i
i
W
0
0
0
0
)
,
,
,
(
)
,
,
,
0
(
)
,
,
0
,
0
(
)
,
0
,
0
,
0
(
)
,
,
,
(
X
Y
Z
i
Y
YX
X
i
Z
ZY
Y
i
Z
d
i
M
L
d
i
M
L
d
L
0
0
0
)
(
)
(
0
Y
X
YX
Z
Y
ZY
Z
Z
Y
Y
X
X
i
i
M
i
i
M
i
L
i
L
i
L
2
2
2
2
1
2
1
2
1
,
whe
r
e
– integration
varia
b
le takes t
h
e following
value
s
X
Y
Z
i
,
i
,
i
,
in o
r
de
r fo
r
inte
gra
l
s.
The
n
f
o
r
pr
op
u
l
sion
fo
rce calc
u
lation,
we
g
e
t the
final e
x
p
r
e
ssion:
Y
X
YX
Z
Y
ZY
Z
Z
Y
Y
X
X
i
i
i
Z
Y
X
c
e
i
i
M
i
i
M
i
L
i
L
i
L
i
i
i
W
F
Z
Y
X
2
2
2
,
,
2
1
2
1
2
1
)
,
,
,
(
(8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
J
PEDS Vo
l.
6,
No
.
2,
Ju
ne 20
15
:
225
–
2
32
23
0
There
f
ore,
the
pr
o
pulsio
n
f
o
r
ce o
f
si
x-
p
h
ase
is ex
p
r
esse
d i
n
term
s o
f
pha
se cu
rre
nts a
n
d
linear
shi
f
t
betwee
n tran
slator an
d stator
.
The effect
of
m
u
tual i
nductance is con
s
ider
ed in tw
o last m
e
m
b
ers of s
u
m
in
th
e expr
ession
(
8
)
.
3.
RESULTS
A
N
D
DI
SC
US
S
I
ON
As the
result of LSRM calcul
a
tion it was re
ceived th
e
rela
tion o
f
self a
n
d m
u
tual flux l
i
nka
ges wit
h
othe
r phase
s pr
esented
at
fi
gu
re 2. It
is ob
vio
u
s
t
h
at
in
tensit
y
of
m
u
tual flu
x
lin
k
age
s
rea
c
hes
5
0
%
of
its ow
n.
a) Fl
ux
linka
ge
s o
f
phas
e
A
b
)
Flux
link
ag
es of
p
h
a
se
A with
ph
ase B
c) Fl
ux
linka
ge
s o
f
phas
e
A w
ith p
h
ase C
d
)
Flux
link
ag
es of
p
h
a
se
A with
ph
ase
D
e) Fl
ux
linka
ge
s o
f
phas
e
A w
ith p
h
ase E
f
)
Flux
link
ag
es
of
p
h
a
se A
with
ph
ase F
Figu
re
2.
The
r
e
lation o
f
sel
f
and
m
u
tual flu
x
lin
k
age
s
of
L
S
R
M
B
a
sed o
n
diag
ram
s
analy
s
is
give
n ab
o
v
e, it follo
ws
that t
h
e consi
d
ere
d
LSRM has int
e
nse m
u
tual
influe
nce
between
pha
ses.
At
the sam
e
ti
m
e
, as ca
n
be
seen, t
h
ere
is a
signifi
cant im
pact on t
h
e a
d
j
acent
pha
ses of
the m
achine.
Prese
n
ted results allow concl
udi
ng that the
effect
s of m
u
t
u
al inductance
s are i
m
portant and shoul
d
be c
onsi
d
ere
d
in m
a
them
atical
m
odel o
f
L
R
SM
. The
m
u
tual in
ducta
nce
s
can
not
be
ne
glected in
a s
w
itch
e
d
reluctance m
achine
design. To esti
m
a
te of their i
n
fl
uence
o
n
the
m
achine perform
a
nce prel
i
m
inary
calculations a
n
d a
n
aly
s
es s
h
o
u
ld
be
d
one
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
PED
S
I
S
SN
:
208
8-8
6
9
4
Mat
h
e
m
at
i
c
al
Mo
del
of
Li
ne
a
r
Sw
i
t
c
he
d Rel
u
ct
a
n
ce M
o
t
o
r
w
i
t
h
Mut
u
al
In
duct
ance
…
(
N
.V.
Gre
b
en
ni
ko
v)
23
1
4.
CO
NCL
USI
O
N
Exp
r
essi
o
n
analysis (8
) shows th
e po
sitive effect
fro
m
t
h
e strong
m
u
t
u
al in
du
ctan
ce b
e
tween
the
pha
ses of L
S
R
M
. Ho
weve
r, t
h
e st
ro
n
g
m
u
t
u
al
i
nduct
a
nce
has t
h
e im
pact
on co
nt
r
o
l
par
a
m
e
t
e
rs and t
h
i
s
fact
requires t
h
e application of more c
o
m
p
licate
d
con
t
ro
l algo
rith
m
s
co
n
s
id
erin
g
th
e
p
r
o
cesses o
c
cu
rring
in
all
pha
ses.
ACKNOWLE
DGE
M
ENTS
The
prese
n
t
e
d
wo
r
k
ha
s be
en d
e
vel
ope
d
wi
t
h
s
u
p
p
o
rt
of R
u
ssi
an
M
i
ni
st
ry
o
f
E
duc
at
i
on,
gra
n
t
RFMEFI5
761
4X
004
0
REFERE
NC
ES
[1]
Kolom
e
its
ev L,
et al
.
Linear switched relu
ctan
ce motor as a high
e
ffi
cien
cy propu
lsion system for railway veh
icl
e
s
.
In the: SPEEDAM 2008 - Int
e
rnational S
y
mp
osium on
Po
wer Electronics, El
ectrical Drives, Automation and
Motion Ischia, 2
008: 155-160.
[2]
Kireev AV,
et
al.
Potential Devel
opment of V
e
hicle Tr
action
Levitation
S
y
stems with Ma
gne
tic
Suspe
n
sion.
International Jo
urnal of
Power
Elec
tronics and
Drive Systems (
I
JPEDS)
.
2015; 6
(
1): 26-31
[3]
F
l
eur
y
A, e
t
al
.
Experimen
t
al
Measurement a
nd Analysis
of
the Self and Mutual Inductan
ces
in
т
Two Differ
e
nt
Switched R
e
lu
ct
ance Machin
es
.
In the: Intern
ation
a
l Confere
n
ce on Renewa
ble Energ
i
es an
d Power Quality
(ICREPQ’10), Granada (Spain)
.
2010.
[4]
Fey
z
i R, et al. Direct Torqu
e
Control of
5-p
h
ase 10/
8 Switched R
e
luctance Motor b
y
Us
ing Fuzzy
Method.
International Jo
urnal of
Engineering and Techno
logy
. 2009; 3(1).
[5]
Alrifai M,
et al
.
Nonlinear Speed Control of Switched Relu
ctan
ce Motor Drives Taking into
Account Mutual
Inductan
c
e.
Hin
dawi Publishing
Corporation Jou
r
nal of Con
t
rol S
c
ien
c
e and Eng
i
neering.
2008; 1
1
.
[6]
Han-K
y
ung Bae. Control of Switched Relu
ctan
ce Motors C
onside
r
ing Mutual Inductan
c
e,
Dissertation subm
itted t
o
the f
acul
t
y
of
th
e Virgini
a
Pol
y
t
echni
c Insti
t
ute
and Stat
e Univ
er
sit
y
in p
a
rti
a
l fu
l
f
illm
ent of
the
re
quirem
e
nts for
t
h
e
degree of Docto
r
of Philosophy
In The Bradley
Department
of Electrical and C
o
mputer
Engineering, Blacksbur
g,
Virginia. 2000:
140.
[7]
Liu Y. Improved Torque Performance of
S
w
itch
e
d Reluct
ance M
achin
es
b
y
Redu
cing the M
u
tua
l
S
a
turation
Effe
c
t
.
IEEE
2004.
Transactions on
En
ergy Conversion
. 2004; 2(19): 25
1-257.
[8]
Grebennikov N.
Effect of
chang
e
s in
the number
of phases under
c
ar switch
e
d-i
ndu
ctor g
e
nerator o
n
his performance
at constan
t
stato
r
configuration.
Proceed
ings of t
h
e higher edu
c
a
tional inst
ituti
on
s. Ele
c
tromecha
n
ics
. 2011; 2: 1
7
-
21.
[9]
Grebennikov N
V
, et
al. Versio
ns of Sw
itched
Reluctance Gen
e
rator
Design at
a Constan
t
Stator Configuratio
n
.
International Jo
urnal of
Power
Elec
tronics and
Drive Systems (
I
JPEDS)
. 2015; 6
(
1): 65-69.
[10]
RU 2450410 C1. Reactiv
e switched
electrical machin
e
with rotation s
y
m
m
etr
y
, Rostov-
na-Donu, Russian
Federation. (Grebennikov, N.
V., A.D.
Petrushin)
Publ
. 10.05
.201
2. Bull. 13.
[11]
Finite Element Method
Magnetics. Date Views
2
5
.02.2015
www.femm.info.
BIOGRAP
HI
ES
OF AUTH
ORS
G
r
e
b
e
nnikov Nikolay
was born
in Russia, in
1
985. He r
e
ceived the Ph
.D. deg
r
ee
in 2012, in
railway
rolling stock from Rost
ov State Transp
ort
Unive
r
sity
.
His c
u
rre
nt re
sea
r
c
h
inte
re
sts:
trac
tion m
o
tors,
switched r
e
lu
cta
n
ce m
a
chines
an
d com
puter
sim
u
lation
.
E-mail: greb
ennikovnv@mail.ru.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN:
2
088
-86
94
I
J
PEDS Vo
l.
6,
No
.
2,
Ju
ne 20
15
:
225
–
2
32
23
2
Kir
eev Al
exan
d
er
was
born in
R
u
ssia in 1974
. H
e
received
the Ph.D. d
e
gree in
2
004 in
the area
of ele
c
tri
c
a
l
m
achines
from
S
outh-Rus
s
i
an S
t
a
t
e T
echni
ca
l Univers
i
t
y
.
His
c
u
rrent res
e
arch
inter
e
sts: electrical mach
ines, freque
ncy
conver
t
ers
and c
ontrol s
y
stems.
E-mail: akir
eev
@privod-n.ru.
Nikolay
M
i
khailovic
h
Koz
h
e
m
y
a
ka
was born in 1980. He finished South-Russian State
Techn
i
cal Univ
ersity
(NPI), Novocherkassk o
n
"Electrical tr
ansport" specialty
in 2002
. He
received PhD degree in
Techn
i
cal Scien
ces in 2
007. His main scien
tific
interests are related
to
power conv
ertor
s
for el
ec
tric
al
dr
ive,
e
l
ec
tric
al
tr
a
c
tion
s
y
s
t
em
s
,
a
nd el
ectr
i
c
a
l v
e
h
i
cl
es
.
Evaluation Warning : The document was created with Spire.PDF for Python.