International Journal of Electrical Energy, Vol. 2, No. 3, September 2014
Effect of the Stator Mutual Leakage Reactance of
Dual Stator Induction Generator
M. B. Slimene and M. A. Khlifi
Department of Electrical Engineering, School of Sciences and Technology of Tunis, Tunisia University, Tunisia
Research Laboratory SIME. Team: ETEC (Conversion and Treatment of Electrical Energy)
Email: benslimene.marwa@gmail.com, mohamedarbi.khlifi@issatm.rnu.tn
M. Benfredj and H. Rhaoulia
Department of Electrical Engineering, School of Sciences and Technology of Tunis, Tunisia University, Tunisia
Abstract—this paper presents a mathematical model of dual
stator induction generator (DSIG) for analysis of its
transient behavior for self-excited induction generator. The
induction machine has two sets of three-phase stator
windings spatially shifted by 30 electrical degrees. In the
analytical model, impact of common mutual leakage
reactance between the two three-phase stator winding sets
has been considered with three cases. The proposed steady
state generalized model of DSIG self-excited dispenses with
tedious work of segregating real and imaginary components
of the complex impedance of dual stator induction generator
for deriving the specific models for each operating mode.
Predictions of various electrical characteristics have shown
the importance of taking into account the mutual leakage
fluxes in the DSIG modeling, especially in static operation.
Paper also discusses the applicability of DSIG for supplying
two individual loads by presenting the results of simulation
and experimental study of steady-state behavior under
various operating conditions.
Index Terms—dual stator induction generator, self excited
induction
generator,
mutual
leakage
reactance,
experimental study of steady-state
I.
INTRODUCTION
Self-excitation phenomenon in multi-phase induction
machines, although known for more than half a century
[1], [2], is still a subject of considerable attention. The
interest in this topic is sustained primarily due to
application of dual stator self-excited induction
generators in isolated power systems. A source of
reactive power, that is required for self-excitation and
subsequent generating operation, can be any of the
numerous types of static reactive power compensators [3].
Physical background of the self-excitation process has
been described in considerable depth in Ref. [4]. The
initiation of the excitation in the induction machine can
be viewed as a response of the resonant circuit, that is
composed of the machine and the variable capacitor bank
connected to its terminals.
The investigations spread over the last two decades
indicate the technical and economic viability of using the
number of phase higher than three in transmission [4],
Manuscript received April 10, 2014; revised June 28, 2014.
©2014 Engineering and Technology Publishing
doi: 10.12720/ijoee.2.3.189-193
189
multi-phase machines in general [5] and induction
machines [6]-[9] in particular. The research in this area is
still in its infancy, yet some extremely important findings
have been reported in the literature indicating general
feasibility of multi-phase systems.
Recently, Singh and Al, have presented the modeling
and analysis of six-phase SEIG where the effect of
common mutual leakage reactance between the two threephase stator winding sets has been included. In [10]-[14],
performance evaluation of simple shunt and series
compensated (short-shunt) six-phase SEIG is discussed
showing its practical feasibility, where as Singh deals
with the steady-state modeling and analysis of six-phase
SEIG. Similarly, Tessarolo in [15] proved the impact of
leakage inductances on multi-phase machine performance,
especially in PWM inverter-supplied synchronous motors
by using finite element techniques.
However, the study of the effect of the common
mutual leakage reactance in the self-excited dual stator
induction generator steady-state performance is nearly
nonexistent in the literature. For this purpose, the aim of
this paper is therefore to report on results of an extensive
experimental investigation of a SEIG operation. Selfexcitation under no-load conditions, with capacitor bank
connected in star loading of the generator is investigated.
The attention is focused at the impact of the common
mutual leakage reactance between the two three-phase
stator winding sets with this three cases: (a) mutual
leakage reactance ( X sm ) is correctly included, (b) X sm is
considered with self leakage reactance X s1 and (c) X sm
is ignored. Experimentations were also carried out to
judge the performance of the dual stator SEIG including
loading and unloading characteristic. Very good
correlation between theatricals and experimental results.
A common type of multiphase machine is the dual stator
induction machine (DSIM), where two sets of three-phase
windings, spatially phase shifted by an angle  , share a
common stator magnetic core.
II.
STEADY STATE ANALYSIS
Fig. 1 shows per phase equivalent circuit of a dual
stator SEIG under resistive load. Where Rs1 , Rs 2
International Journal of Electrical Energy, Vol. 2, No. 3, September 2014
Model 1 with (1 , 2 )  (0, 1) , corresponds to the case
where the stator mutual leakage inductance is correctly
included.
In model 2 with (1 , 2 )  (0, 1); X sm , is also
considered but as a self leakage impedance.
If both (1 , 2 ) are null, the mutual leakage
impedance is ignored in the modeling process.
At note “A” in Fig. 1, the relation between I s1 , I s 2 and
Rr' , X s1 , X s 2 , X sm , X m , X r' , X c , Rch1 , Rch 2 and g
represent the stator 1 and 2 resistance, rotor resistance
(referred to stator), stator 1 and 2 leakage reactance,
mutual leakage reactance between the two stator,
magnetizing reactance, rotor leakage reactance (referred
to stator), excitation capacitor reactance, load resistance
and the generator slip respectively.
A1
I ch1
R s1 j (X
I s1
jXc1
 1X
sm
)
A
I c1
Rs 2 j (X
A2 Is 2
Rch1
s1
V s1
I ch 2 I c 2
Rch 2
jXc2
s2
  1X
sm
Is
j  2 X sm
)
Ir
'
'
jX r
I s can be written as:
Im
R
jX m
'
r
I s  I s1  I s 2
g
Vs2
(2)
When the two sets of stator three-phase windings are
identical, then we can write:
Figure 1. Equivalent circuit of dual stator self-excited induction
generator.
I s1  I s 2 
Based on the equivalent circuit of a three winding
transformer, the equivalent circuit of the dual stator
induction generator having a double stator winding can
be realized as shown in Fig. 1 [11]. The common mutual
leakage reactance X sm , represents the fact that the two
sets of three-phase stator windings occupy the same slots
and are, therefore, mutually coupled by a component of
leakage flux. Note that although the circuit is termed ‘per
phase’; in reality the circuit is drawn with two stator
circuits, one per three-phase group. The equivalent circuit
based on the generalized mathematical model developed
by Singh [12], [13] is shown in Fig. 2. The coil pitch
affects the leakage reactance of the stator winding. Lipo
[2] has explained this in detail and has given the
technique for finding slot reactance.
When the dual stator SEIG is driven by dc machine in
which the shaft speed is maintained constant. All
parameters are fixed but both X m and g vary with the
load and hence they must be taken as variables for a
given load and excitation capacitance condition, [14]. The
parameters of the equivalent circuit are given below:
 Z c1   jX c1 ; Z c 2   jX c 2

 Z ch1  Rch1 ; Z ch 2  Rch 2

 Z s1  Rs1  j ( X s1  1 X sm )
 Z s 2  Rs 2  j ( X s 2  1 X sm )

 Z sm  j 2 X sm

R
 Z r  r  jX r ; Z m  jX m
g

TABLE I.
1
2
Model 1
Model 2
Model 3
0
1
0
1
0
0
and I s 1 can be written as:
I s 1  I ch1  I c 1
(4)
 1 


 Z c1 
I c1 
 1 
 

 I ch 1   V s 1 
 Z ch 1 
I 
 s1 
 1 

 Z s 1 
(5)
where:
Similarly, the same calculation procedures are adopted
to obtain the stator, excitation and load current for the
second stator ( I s 2 , I ch 2 , I c 2 ) . In order to simplify the
study, the following per phase circuit based on impedance
analysis are considered, Fig. 2.
I ch 1
A Is
Ic1
A2
Z ch 1
Z s1
I s1
A1
Z c1
(1)
V s1
I ch 2
Z sm
Zs 2
Is2
Ir
B
'
Im
Ic2
E
zch2
Z c 2V
s2
Zm
Z
r
Figure 2. Simple per phase equivalent circuit of dual stator SEIG
where, Z1 , Z 2 , Ys , Ym and Y r can be represented by
using the equivalent circuit as follows:

Z 1 


Y s 


Table I resumes the different values taken by the pair
(1 , 2 ) for each model version.
©2014 Engineering and Technology Publishing
(3)
At note “A1” in Fig. 2, the relation between I ch1 , I c1
VALUES OF (1 ,2 ) FOR EACH MODEL VERSION
i
Is
2
190
Z ch 1 Z c 1
Z Z
 Z s 1 ; Z 2  ch 2 c 2  Z s 2
Z ch 1  Z c 1
Z ch 2  Z c 2
1
Z 1Z 2
 Z sm
Z1  Z 2
;Ym 
1
; Yr 
Zm
1
Zr 
(6)
Rr
g
International Journal of Electrical Energy, Vol. 2, No. 3, September 2014
Terminal voltage set 'I' phase 'a' (V)
At note ‘B’ in Fig. 2, the relation between I m , I r and
I s can be written as:
Is  Ir  Im









(8)
(9)
50
0
50
100
150
200
250
speed (rad/s)
300
350
400
Xsm is correctly included
* Experimental
Xsm is considered with self leakage impedance
Xsm is ignored
1.5
1
0.5
100
150
200
250
speed (rad/s)
300
350
400
(10)
Stator voltage and current curves indicate that the case
1 ( X sm is correctly included) is the more realistic, as
expected. Another important feature is to be observed
from Fig. 4 and Fig. 5, more shaft speed increases more
the difference between curves relative to the different
cases is decreasing.
Fig. 4 Shows, the behavior obtained from the
experiment, of the output voltage as a function of speed.
These curves show two areas. The first, when the voltage
increases very rapidly with the speed, the corresponding
points are those obtained just after the self-excitation. In
the second area, voltage varies linearly with the speed to
a low coefficient. This area, where the characteristics are
substantially parallel, corresponds to the stable part of the
dual stator generator.
(11)
Terminal voltage set 'I' phase 'a' (V)
COMPUTER AND EXPERIMENTAL RESULTS
Alternatively, the values of star-capacitance at a given
speed to generate a particular terminal voltage can be
obtained experimentally by using a variable capacitor
bank.
A detailed study of steady-state performance of the
dual stator SEIG indicates that for three models. Selfexcitation under no-load condition and loading
performance under a typical resistive load are elaborated.
For simulation of no-load operation, Rch1 and Rch 2 in
the (1) are replaced by infinity. Fig. 3, Fig. 4, Fig. 5 and
Fig. 6 show experimental and simulation results
performed with the three cases at steady state self-excited
generator operation at no-load.
©2014 Engineering and Technology Publishing
100
Figure 4. Stator current per phase versus speed at no-load
Real (10) and Img (10) is solved by using “fzero”
MATLAB function. For a given excitation capacitor and
prime mover speed, the system of equation (11) has a one
unknown parameter, which is the frequency F, [15].
Total admittance is considered here as an objective
function, and the constrained function is applied to find
out simultaneously the value of F and Xm. Subsequently,
we can predict the necessary parameters to evaluate the
performance characteristics of the dual stator SEIG. The
value of F and Xm is simultaneously computed for
different operating condition by varying the load
resistance, and keeping the speed constant at desired
value or with fixed excitation capacitance.
III.
150
0
50
This implies that both the real and imaginary
components of (11) should be independently zero.

Re(Y s Y m Y r )  0


Im(Y s Y m Y r )  0
200
2
Under normal operating condition, the stator voltage
E  0 . Therefore, the total admittance must be equal to
zero.
Y s Y m Y r  0
250
Figure 3. Output stator voltage per phase versus speed at no-load
Hence, equation (7) can be written as:
E (Y s Y m Y r )  0
Xsm is correctly included
* Experimental
Xsm is considered with self leakage impedance
Xsm is ignored
300
Stator current set 'I' phase 'a' (A)
 1
 Zm
I m 
 1
 
I r   E 
 Zr
I 
 s 
 1

 Z s
(7)
350
300
250
200
150
Xsm is correctly included
* Experimental
Xsm is considered with self leakage impedance
Xsm is ignored
100
50
0
0.4
0.6
0.8
1
1.2
C(µF)
1.4
1.6
1.8
2
x 10
-5
Stator current set 'I' phase 'a' (V)
Figure 5. Variation of terminal voltage with capacitance at no-load
2
1.5
Xsm is correctly included
* Experimental
Xsm is considered with self leakage impedance
Xsm is ignored
1
0.5
0
0.4
0.6
0.8
1
1.2
C(µF)
1.4
1.6
1.8
2
x 10
-5
Figure 6. Variation of stator current with capacitance at no load
191
International Journal of Electrical Energy, Vol. 2, No. 3, September 2014
Terminal voltage set 'I' phase ''a'' (V)
Experimental and simulation were performed after
varying resistive load using the defined three types of
models. Characteristic curves shown in Fig. 7 and Fig. 8
depict the variation of terminal voltage and stator current
when two winding sets are equally loaded. It is worth
reminding that case 1, supposed to be the more accurate,
and was also taken as a reference for any comparison.
Im: magnetizing current per phase.
Is: common stator current.
Rs1: stator 1 resistance per phase.
Rs2: stator 2 resistance per phase.
Rr: rotor resistance per phase, referred to stator.
V: load voltage per phase.
Xs1: stator 1 reactance per phase.
Xs2: stator 2 reactance per phase.
Xsm: stator mutual leakage reactance per phase.
Xr’: rotor reactance per phase, referred to stator.
Xc1: capacitive reactance due to C1 at rated frequency.
Xc2: capacitive reactance due to C2 at rated frequency.
Xm: magnetizing reactance per phase at rated frequency.
300
Xsm is correctly included
* Experimental
Xsm is considered with self leakage impedance
Xsm is ignored
250
200
150
100
50
0
REFERENCES
0.05
0.1
0.15
Load current (A)
0.2
[1]
0.25
Figure 7. Terminal voltage for stator 1 with load current
[2]
Stator current set 'I' phase 'a' (A)
Terminal voltage and machine current both start
decreasing with the increase in load.
[3]
1
0.8
[4]
0.6
0.4
0.2
0
[5]
Xsm is correctly included
* Experimental
Xsm is considered with self leakage impedance
Xsm is ignored
[6]
0.05
0.1
0.15
Load current (A)
0.2
0.25
Figure 8. Stator current per phase versus load current
[7]
IV.
CONCLUSIONS
[8]
In this paper, a detailed structure of dual stator selfexcited induction generator (DS-SEIG) is presented. In
the analytical model, the effects of common mutual
leakage reactance X sm between the two three-phase
winding sets have been discussed with three possible
cases:
1) The mutual leakage reactance is correctly included.
2) The mutual leakage reactance is considered as a self
leakage reactance X s1 .
3) The mutual leakage reactance is neglected.
Paper also discusses the applicability of a dual stator
capacitor excited induction generator for supplying two
individual three-phase loads, by presenting results of an
experimental study of steady-state behavior for various
operating conditions.
[9]
[10]
[11]
[12]
APPENDIX NOMENCLATURE
[13]
C1: excitation capacitance per phase with stator1.
C2: excitation capacitance per phase with stator 2.
E: air gap voltage per phase at rated frequency F.
Is1: stator 1 current per phase.
Is2: stator 2 current per phase.
Ir’: rotor current per phase, referred to stator.
©2014 Engineering and Technology Publishing
[14]
192
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International Journal of Electrical Energy, Vol. 2, No. 3, September 2014
Mouldi Ben Fredj received the master’s
degree in electrical and electronic automatic
in 1985 and the doctoral degree in April 1989,
all from the University of Science and
Technology of Lille FRANCE. From 1987 to
1989 he taught at the School of Electronic
Nantes in France as assistant associate. In
1989, he was hired as an assistant at the
National Engineering School of Gabes
(ENIG)-TUNISIA. Since 2000 he is with the
ESSTT (Institute of sciences and technology
of Tunis). During his career, he was in training at Lille and Marseille in
France and Canada. His main research interests are power electronics
and control of electrical machines: modeling and simulation of
association’s electrical machines -static converters, renewable energy
sources. Dr Ben Fredj, is a member of SICISI (Research group on signal,
image and intelligent control of industrial process), ASET (association
for Tunisian electrical specialists).
[15] A. Tessarolo and D. Giulivo, “Analytical methods for the accurate
computation of stator leakage inductances in multi-phase
synchronous machines,” in Proc. International Symposium on
Power Electronics, Electrical Drives, Automation and Motion,
SPEEDAM, 2010.
Marwa Ben Slimene was born in Nabeul,
Tunisia, in 1988. She received the B.S. degree
in Electrical Engineering and Power
Electronics from the ESSTT (Institute of
sciences and technology) in Tunis, Tunisia, in
2009, the M.S. degree in electric systems
from the ENIT (National Institute of
Engineering of Tunis), in 2012. She joined the
Institute of sciences and technology, as an
Assistant in ISET in the Department of
Electrical Engineering. Her main research
interests are analysis and control of electrical machines, renewable
energy sources, and power electronics.
Habib Rehaoulia received the B.Sc. degree
in 1978, the M.Sc. degree in 1980 and the
doctoral degree in 1983, and the Ph.D. degree,
all in Electrical Engineering and from the
ENSET (Institute of Technical Sciences),
University of Tunis, Tunisia. He joined the
teaching staff of the ENSET in 1981. Since
1995, he is with the ESSTT (Institute of
sciences and technology) in Tunis. During his
career, he was on leave for several months at
WEMPEC (University of Madison Wisconsin
USA), ENSIEG (University of Grenoble France), “Lab.
d’électrotechnique” (University of Paris VI France), Member IEEE and
recently the CREA (University of Picardie France). His main research
interests are analysis and modeling of electrical machines.
Mohamed Arbi Khlifi was born in Tunisia,
in 1982. He received the M.S., and Ph.D. in
Electrical Engineering from the Tunisian
National University, Tunis, Tunisia, in 2005,
2007, and 2012, respectively. He joined the
National Institute of Engineering of Tunis, as
Professor in the Department of Electrical
Engineering. His current research interests
include power electronics and control, which
include ac machine drives, renewable energy,
and control of high efficiency integrated
electric energy conversion in various industrial fields. His main is
member IEEE.
©2014 Engineering and Technology Publishing
193