Ad ances in Elec ical and Elec onic Enginee ing
106
ON THE SEGMENTED ROTOR RELUCTANCE SYNCHRONOUS MOTOR
SALIENCY RATIO CALCULATION
I. A. Vio el, A. Banyai, C. S. Ma is, B. Ta a anu, I. Vin iloiu
Technical Uni e si y o Cluj-Napoca, Elec ical Machines Depa men ,
15 C. Daico iciu s . Cluj-Napoca , R-400020, ROMANIA
ioan.ad ian. io [email protected] cluj. o
Summa y
The eluc ance synch onous mo o pe o mance highly depends on i s saliency a io. The e o e an accu a e
analy ical calcula ion o he mo o ’s saliency is qui e a necessi y wi hin he designing p ocedu e. In his pape an analy ical
me hod o calcula e he main mo o induc ances and i s saliency a io is applied o a segmen ed o o eluc ance synch onous
mo o . The esul s ob ained o wo mo o a ian s, compa ed wi h he ones ob ained ia a 2-D FEM compu a ion and ia
es s, s and by o p o e he p oposed echnique accu acy.
1. INTRODUCTION
Con e e ed a iable eluc ance synch onous
mo o can be an impo an compe i o in he a iable
speed d i es ma ke due o i s educed p oduc ion
cos s, good con ollabili y a low speed and a qui e
high e iciency [1].
The a iable eluc ance synch onous mo o
(VRSM), also known as synch onous eluc ance
mo o , is a single salien machine. The VRSM
p oduces o que due o i s o o endency o mo e o
a posi ion whe e he lux o he exci ed winding and
i s induc ance, a e maximized. This ype o machine
does no equi e ield winding o pe manen magne s
on he o o . The s a o o a VRSM, iden ical wi h
ha o he induc ion machine, has uni o m
dis ibu ed slo s and usually a h ee-phase
dis ibu ed single o double laye winding.
The pe o mance ( o que, powe ac o and
e iciency) o a VRSM is dependen on he saliency
a io K=
qd
MM /
, which is he a io o he d-axis
(
d
M
) and q-axis (
q
M
) magne izing induc ances. A
la ge saliency a io means an impo an o o
magne ic asymme y, which can be ob ained by
adop ing a segmen al s uc u e, by employing he
lux ba ie s o by using he axially-lamina ed
(ALA) cons uc ion [1, 2, 3, 4]. The o o s wi h
mul iple lux ba ie s and axially lamina ed o o s
o e he possibili y o achie ing high saliency
a ios, bu ha e he disad an ages o hei qui e
complica ed echnology and high cos s. In some
applica ions, when he d i e cos has a majo
impo ance, he simples s uc u e wi h a smalle
saliency a io should be adop ed. Such a s uc u e, as
he segmen ed o o is, mus be designed e y
ca e ully in o de o ob ain he maximum saliency
a io, he e o e he pa ame e s calcula ion is
impo an .
The goal o his pape is o ob ain he mo o
pa ame e s and especially he saliency a io by a
qui e simple analy ical echnique. The ob ained
esul s, o wo mo o s, a e compa ed wi h he
nume ical (2D-FEM) and es calcula ed alues.
2. PARAMETERS AND SALIENCY RATIO
ANALYTICAL CALCULATION
In o de o calcula e he d- and q- axis
induc ances o a VRSM machine, a cylind ical o o
machine will be i s conside ed. I has he ai gap
leng h equal o he minimal ai -gap (g
1
) o VRSM
[5]. The magne izing induc ance o a cylind ical
o o is gi en by:
2
1
0
4.1
3







⋅
⋅⋅
⋅⋅⋅
⋅= p
Nk
g
QLD
M
swssii
mc
π
µ
(1)
wi h: D
i
– s a o in e io diame e , L
i
– axial leng h,
Q
s
– numbe o s a o slo s, N
s
– numbe o s a o
u ns, p – numbe o pole pai s, k
ws
– s a o winding
ac o .
Fo a mo o wi h a cylind ical o o , he ai -gap lux
densi y undamen al is:
'
0
1
g
F
B⋅
=
µ
(2)
whe e F is he esul an s a o mm and g’ is he
enla ged equi alen ai -gap, conside ing he s a o
slo s opening, ia Ca e ’s ac o k
C
and co e
sa u a ion :
gkkg Cs
⋅
⋅
=
'
(3)
The sa u a ion coe icien k
s
can be es ima ed in
he usual designing manne o calcula ed based on
2D-FEM calcula ions.
I he s a o MMF cen e ed on he d-axis has a co-
sinusoidal a ia ion in unc ion o he o o posi ion
angle , hen he undamen al componen o lux
densi y on he d-axis come as [5]:

⋅
⋅⋅=
2
0
2
0
)(
cos4
π
θ
θ
θ
µ
π
d
g
F
B
ld
(4)
Fo he MMF cen e ed on he q-axis:

⋅
⋅⋅=
2
0
2
0
)(
sin4
π
θ
θ
θ
µ
π
d
g
F
B
lq
(5)
On he segmen ed o o eluc ance synch onous mo o saliency a io calcula ion
107
a)
b)
Fig. 1. VRSM segmen ed o o
a) con igu a ion, b) simpli ied scheme
These ela ions, in he case o he o o s uc u e
gi en in Fig.1a, acco dingly o he simpli ied
scheme adop ed, Fig. 1b became:






⋅
⋅+⋅⋅⋅=

θ
θ
θ
θ
µ
π
π
α
α
d
gk
Fd
g
FB
s
ld
4
1
2
02
2
0
coscos4
(6)





⋅+⋅⋅⋅=

θ
θ
θ
θ
µ
π
π
α
α
d
g
Fd
g
FB
lq
4
1
2
02
2
0
sinsin4
(7)
whe e g
2
is he maximal ai gap equal o 30mm and
he angle  is 3.065 deg , Fig. 1b, and, as expec ed,
he sa u a ion is conside ed only on he d-axis, (6).
The lux densi y coe icien s a e:
1
B
B
k
ld
d
=
1
B
B
k
lq
q
=
(8)
which means o his case:








⋅
+
⋅
=
 
θ
θ
θ
θ
π
απ
α
d
gk
d
g
g
k
s
d0
4
1
2
2
2
1
coscos
4
(9)







+
⋅
=
 
θ
θ
θ
θ
π
απ
α
d
g
d
g
g
k
q0
4
1
2
2
2
1
sinsin
4
(10)
Simila equa ions we e ob ained o he o o
wi h mo e complica ed opology, gi en in Fig.2 [4].
The longi udinal and ans e sal magne izing
induc ances a e:
mcdd
MkM
⋅=
mcqq
MkM
⋅=
(11)
The saliency a io is gi en as:
q
d
q
d
k
k
M
M
K
==
(12)
a)
b)
Fig. 2. VRSM segmen ed o o wi h lux ba ie s
a) con igu a ion, b) simpli ied scheme
In ac he saliency a io is in luenced by he
s a o leakages and he e ec i e alue o he a io is
smalle :
q
q
Sq
Sd
q
d
K
KK
LM
LM
L
L
K
σ
σ
σ
σ
+
+
=
+
+
==
1
*
(13)
whe e:
q
S
q
M
L
K
σ
σ
=
(14)
Suppose ha
0
→
q
K
σ
, which is no he case
o his size o machines, han *
K
K
=, bu since M
q
and
σ
S
L alues a e qui e compa able i is clea
ha *
K
K
<.
Fo a designe h ee ules a e eme ging:
i) Make M
d
as la ge as possible and do no
sa u a e he i on-co e.
ii) Make M
q
as small as possible.
iii) Ob ain he smalle possible alue o
σ
S
L
The s a o phase leakage induc ance depends on
he s a o slo shape, on he s a o winding and on
he end winding opology. The calcula ions we e
done conside ing he usual o mulae [5, 6], bu wi h
a ce ain a en ion on he ai -gap leakages due o he
Ad ances in Elec ical and Elec onic Enginee ing
108
e y speci ic o o con igu a ion and o he highe
ha monic con en o he ai -gap ield densi y.
3. 2D-FEM COMPUTATION
A 2-D ini e elemen me hod (FEM) analysis
would o e di e en accu acy, bu i equi es longe
compu e ime. Two simpli ying assump ions we e
conside ed:
i) The o o ’s eddy cu en s a e neglec ed.
ii) The i on co e hys e ezis is neglec ed oo
Fo he de e mina ion o main d- and q-axis
induc ances, he cu en s we e imposed in he s a o
slo s in o de o gene a e a longi udinal, espec i e a
ans e sal eac ion magne ic ield. Fo he VRSM
wi h segmen ed o o , Fig. 1, he lux plo s ob ained
h ough a 2-D FEM analysis a e gi en in Fig. 3 (a,
b). The induc ances we e ob ained using qui e a
usual and simple me hod in nume ical calcula ion.
Fi s he ai -gap magne ic lux was calcula ed
on bo h axes,
d
Ψ
,
q
Ψ
,
hen he magne izing
induc ances,
d
d
d
i
M
Ψ
=
q
q
q
i
M
Ψ
=
(15)
a)
b)
Fig. 3. VRSM wi h segmen ed o o lux plo s
a) on d-axis, b) on q-axis
4. EXPERIMENTAL METHOD
The expe imen al me hod in ol ed ze o speed
ope a ion a a p ede e mined o o posi ion [7]. One
phase o he machine was ed wi h a low ol age a
he base equency and he o o was posi ioned in
an aligned and unaligned posi ion. Fo hese
posi ions o minimal and maximal eluc ances a se
o ol age, cu en and powe measu emen s we e
done.
Based on he measu ed alues he phase
impedances and esis ances, including he i on co e
losses equi alen esis ances, we e compu ed and he
eac ance o one phase is ob ained o he bo h axis.
22
ddd
RZX −=
(16)
22
qqq
RZX −=
(17)
The s a o winding leakages we e ob ained by
es ing he s a o winding wi hou he o o inside.
Respec ing he same p ocedu e like a he analy ical
compu a ion he saliency a io was ob ained ( ela ion
12).
5. RESULTS AND CONCLUSIONS
The calcula ions and es s we e ca ied on wo
h ee phase VRSMs wi h segmen ed o o . Thei
main da a a e gi en in Table 1 and he o o
opologies a e shown in Figu es 1a and 2a. A iew
o he segmen ed o o , he one shown in Figu e 1, is
gi en in Figu e 4, he s a o being iden ical o one o
an induc ion mo o , as a e he case and he o he
mechanical pa s. The main mo o s dimensions a e
gi en in Table 2.
The ob ained esul s ob ained o saliency a io
a e gi en, o bo h machines, in Table 3.
In he pape a echnique o analy ically
calcula ion o he d- and q-axis VRSM magne izing
induc ances and, consequen ly o he machine
saliency a io is p esen ed. The echnique, which is
qui e a hand and can be easily de eloped o o he
o o s uc u es, is applied on wo di e en o o
opologies induc ance and saliency a io calcula ion.
In bo h cases a 2-D FEM calcula ion was
ca ied on and he co esponding main induc ances
calcula ed a a ed cu en . The complex es s we e
conduc ed o he same pu pose.
Fig. 4. A iew o he segmen ed o o o
he conside ed VRSM
On he segmen ed o o eluc ance synch onous mo o saliency a io calcula ion
109
Tab. 1. VRSMs main da a
Ra ed induc ion base
machine ou pu powe , kW 2,2 5.5
Phase supply ol age, V 230 230
Ra ed induc ion base
machine phase cu en , A 5.1 12.4
Ra ed speed, pm 1425 1500
Phase connec ion s a s a
Numbe o s a o slo s 36 36
Numbe pole pai s 2 2
Ai -gap leng h, m 0.3510-3
3.510-3
The 2D-FEM compu ed saliency a io alues
a e la ge han he analy ically calcula ed ones in
bo h cases due o he simpli ica ions made in he
analy ical calcula ion. The es s ob ained alues
di e ences in compa ison wi h he 2D-FEM
calcula ed ones a e qui e la ge, bu he es
me hodology has some limi s.
As i can be no ed, he esul s a e qui e close, o
bo h o o con igu a ion conside ed, which p o e
ha he analy ical de eloped echnique leads o qui e
accu a e esul s, and ha i can be adequa ely
in oduced wi hin any design algo i hm o such
machines.
Tab. 2. Main dimensions o he es ed VRSMs
STATOR ROTOR
Ex e io
diame e 0.15m/0.2m Ex e io
diame e 92.3mm/
117mm
Inne
diame e 93 mm/
124mm Pole numbe 4/4
Tab. 3. Saliency a io coe icien s
K=M
d
/M
q
Model Analy ical 2D-FEM Tes s
Fig.1 3.183 3.584 3.062
Fig.2 3.98 4.14 4.36
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