A Saturated Synchronous Machine Study for the
Converter-Machine-Command Set Simulation
S. Lasquellee, M. Benkhoris, M. Féliachi
To cite this version:
S. Lasquellee, M. Benkhoris, M. Féliachi. A Saturated Synchronous Machine Study for the
Converter-Machine-Command Set Simulation. Journal de Physique III, EDP Sciences, 1997, 7
(11), pp.2239-2249. <10.1051/jp3:1997255>. <jpa-00249714>
HAL Id: jpa-00249714
https://hal.archives-ouvertes.fr/jpa-00249714
Submitted on 1 Jan 1997
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J.
Phys.
III
IYance
(1997)
7
2239-2249
NOVEMBER1g97,
Saturated
Synchronous
Converter-Machine-Colnlnand
A
S. Lasquellec
/
GE44
(*"),
LARGE,
LRTI
(Received
PACS.02.60.Cb
PACS.84
50 +d
Electric
on
modified
a
the
GDJ
Finite
model
Element
results
field
be
with
a
model
the
based
synchronous
the
Finally
saturation
a
synchronous
method
proposed
Model:
machine
saturated
The
GDJ
[1, 2j)
model
calculating
when
been
model
measured
saturated
On
machine
the
whereas
state
the
other
conventional
the
The
model
improved
validity
model
is
to
obtained
elaborated
is
on
the
the
when
in
order
algorithm to simulate the CMC set In comparison
proposed one avoids complex tests on high power
hand, this study reveals Potier
variations
parameter
synchronous
saturated
The
reactances
experiments,
on
estimated
based
is
has
and
the
Jaeger
De
and
models
simulation.
set
improvement
machines.
develop
we
presented
study
based
using a field computation
parameters
model (GDJ model) is modified in
accordance
the
into
equations
of
solution
first
GDJ
the
(FEM).
computed
the
France
The
and
4.8%
introduced
range
with
laws
Cedex,
Nazaire
(CMC)
(Garndo
model
Method
and
comparing
to
Park
(*)
motors
electromagnetic
The
the
July 1997)
18
Converter-Machine-Command
the
Saint
simulation,
Numerical
Abstract.
for
accepted
for
Silnulation
FAliachi
M.
406, 44602
BP
1997,
March
20
Benkhoris,
M-F-
Study
Machine
Set
2239
PAGE
methods
machine
model
consider
associated
it
as
with
constant
a
the
value.
converter.
Introduction
1.
phenomenon in electrical machines has already been investigated m
literature, the saturation
networks [4], the
modified
approaches: the Finite
Element
Method [3] and
reluctance
dq (Park) model [1, 2,5]... For a machine supplied by a static converter, the machine model
the
One of the adapted
which
take into
requires quickness
constraints
methods
account
saturation
and De Jaeger (GDJ model:
in synchronous machines, has been proposed by Garrido
firstly it uses the Park reference, secondly
[1, 2]). The model has the following advantages:
cylindrical and salient poles machines and finally it needs not too many
it applies for both
besides the
classical
like the
saturated
inductances, the model requires
parameters,
ones
non
(ap
the
Potier
flux
law K~(Imd).
The
and
and
the
magnetizing
saturation
parameters
tp)
load
and
authors
obtained
them with
experimental
short-circuit,
inductive
no-load
tests:
some
In
different
tests.
This
model
methodology
take
(*)
(**
into
This
@ Les
for
iditions
was
propose
saturation
presented
correspondence
de
experiments
such
on
we
the
account
paper
Author
based
which
Physique
at
is
based
on
phenomenon
nevertheless
finite
with
element
accuracy
alternate
precise enough. An
computations and allows us to
recalls
The first part of this paper
not
field
NUMELEC'97
(e-mail.
1997
is
sophie©large.crttsn.univ-nantes.fr)
JOURNAL
2240
PHYSIQUE
DE
III
N°11
flux
Ks=AB/AC
Fig.
Definition
I.
mathematical
the
proposed
basis
method
and
finite
with
GDJ
and
measured
results
the
in
in the
last
model, and the
GDJ
second
modej
limits
are
underlined.
The
he~ coupling of the GDJ model
improvement when comparing
validated~ when comparing computed
associated jwith the
is finally
converter
part: it is based on
third part, the precision
In the
given.
The
reactances.
Mathematical
2.
is
K~
factor
machine
calculations.
element
developed
of the
presented
is
model
saturation
of the
coupling
This
saturated
method
is
model
machine
part.
Basis
of
Machine
the
Model
machine, the total flux (qi) is assumed to be divided into two components:
the
saturated
leakage flux~(qii). The magnetizing flux
magnetizing flux (aim) and the non
factor Ks defined as the ratio
saturated
is represented by the
saturation
between the
saturation
saturated
value [1,)2] (Fig. I). The factor Ks
magnetizing flux d-component qimd and its non
saturated
evaluated
expressed by the magnetizing)current Imd.
is
state
at every
In the steady state, the magnetizing
is given by:
current
In
saturated
a
saturated
~
Imd
=
Ill
I
currents
in
the
ratio
(ki
With
(i)
~~~il~/°~
where:
id
and
If
is the
ki
is the
op
is
iq
are
the
field
stator
saliency
as
the
transform
d-component of the
For
the
constant
The
the
Ev
model, Ks(Imd)
Potier
GDJ
versus
is
stator
ratio
Xqo/Xdo)
=
which
relates
field
the
current
to
the
one.
obtained
from
the
Ev(If)
no-load
and
ap is the
parameter
model
If
Potier
asumption:
accepts the classical
no-load
from the
relationship
obtained
(Imd)
the
test.
As
magnetizing flux qimd produced by the field windings is
This
asumption
qimd(iq) relationships of the other windings.
of the
and
GDJ
reference
current
machine
defined
Park
result.
variation
the
represent
not
repeats
relationship
the qimd(id)
reflect
fully the
SATURATED
A
N°11
SYNCHRONOUS
MACHINE
STUDY
2241
reality. Effects are obviously different when the
of
saturation
in rotor
because
occurs
areas
field windings high
when
in
teeth
due
supply
it
in
stator
stator
current
to
occurs
or
a
case
of a salient poles machine.
The
rotor
geometry influence will be analysed on the machine
qimd(id), qimd(If) and op will be determined.
method
The proposed
improvement
saturation:
estimated by comparing it to the GDJ
method.
will be
Coupling
3.
The
previous
the
method
Model
model
parameters
variations
parameter
section
includes
FLUX-EXPERT
The
software
by
obtained
are
with
the
package
is
used
rot(ur
rot
A)
field
a
saturation
solve
to
based
calculation
the
on
FEM;
state.
the
magnetostatic
2D
linear
non
equation:
Maxwell
(2)
~Lo3ex
=
with
A
the
magnetic
ur
the
relative
the
vacuum
~Lo
3ex
Material
hnearities
non
reluctivity
permeability
field
the
potential
vector
density
current
imbedded
are
ur(B~),
in
obtained
from
ferromagnetic
the
characteris-
field density B (Bx; By)
airgap
cartesian
extracted
they
the
flux
and
used
calculate
airgap
magnetizing
to
qimd and then the GDJ
are
are
(saturation factor Ks and the Potier parameter ap).
model
modified
parameters
B(H)
tics
The
The
magnetic
machine
studied
is
Ks
factor
ration
is
the
as
fundamental
duces
the
torque;
the
have
been
filtered
by
qi~m are
fundamental
d-axis
Only
the
id
current
interpolated
to
a
supplied
are
PARAMETER
POTIER
3.2.
fixes
value
The
in
the
field
calculation
has
with
respect
the
ap,
calculated
and is in
In
a
current
load
first
id.
step,
and
for
magnetizing
the
that
so
AiImd
are
machine
in
useful
Imd is
current
Ks(Imd)
law
and
pro-
shown
is
reduced
Figure
in
to
2
the
and
is
fitting technique:
curve
A2I$~
AgI$~.
+
+
ap plays
several
the
~ersus
ap
classical
the
is
steps
magnetic
machine
with
ap
parameter
+
parameter
completed
saturated
law
a
+
The
been
calculations
Field
condition)
the
is
this
under
a
great role
(3)
the
in
model
because
Imd (Eq. (i)).
state
agreement
of the
determination
the
to
Ao
=
op.
saturation
component
asumption. Space harmonics
fundamental
flux qiam, qibm
The
three-phase
statoric
transform
the
is applied to obtain
magnetizing flux
possible
(Eq. (i)). The
factor
saturation
function by a
9 degree polynomial
K~(Imd)
its
flux is
fundamental
qimd.
component
windings
stator
the
in
saturated
flux
the
is also
FFT
function.
a
calculated
and a Park
first
and
Only
second section, the satumagnetizing flux and its non
calculated
in the airgap by the radial
the
magnetizing
transform
Park
explained
As
between
ratio
value
the
d-position
K~(Imd).
LAW
defined
and
A
reference
Park
11, 2j (Fig. i). The
field density integration.
saturated
stator
the
in
FACTOR
SATURATION
3.I.
potential
vector
Potier
machine
studied
First, the study
state.
parameter.
saturated
versus
The
the
mean
A
showed
a
variation
of ap has been
step consisted in
value
second
state.
field
If
current
performed for a machine supplied by
supplied by both
If and id
currents
a
and
field
the
current
d-stator
Ifo (no
JOURNAL
2242
PHYSIQUE
DE
III
N°11
0
------I--
0
----I----
0
0
20
10
30
50
40
lmd (A)
Fig
Saturation
2
factor
law
K~(Imd)
'
u
~
I.
1,
02
U15
~
~~
IO
I(=
A
005
~
~U
5
lo
~
20
15
25
35
id (A)
Fig
op,variations
3.
The
for
each
EXPERT
files
(no
The
the
the
saturation
equation (2) is solved
Maxwell
calculated
flux in
versus
several
problem. A post
solved
load
for
state
condition
and
the
current
treatment
two
currents
parameter
ap is
then
obtained
by
the
currents.
the
magnetizing flux
the
is
FLUX-
magnetizing
following expression:
~~~~ ~
shows
Its
ap
mean
variations
value
on
~ersus
operating
(4)
=
j~~,
J
3
ajid
operated it order to find
is
suppljr) which jive the same
airgap.
ap
Figure
intensities
the
currents
points
<
op
id
range
>=
and If
has
0.318.
been
We
notile
that
calcula(ed:
ap
depends
on
the
N°11
A
SYNCHRONOUS
SATURATED
MACHINE
STUDY
2243
1
+
+
+
I
j
~
4w(Ir'°p)
+
§
+
io
2s
2o
is
Comparison
4.
corresponds
Which
The
the
to
of ap
variation
pmd(id)
between
Potier
the
versus
~md (If lap
and
experimental
magnetizing
3o
35
(A)
current
Fig.
(id)
~
in
bend.
the
parameter.
Imd has
current
been
determined
in
second
a
step.
the magnetizing
Imd is equal to If lap if the machine
current
If
machine
steady
In this formula, the
in the
is
parameter
current
state.
a
If
value:
the
Potier
the
machine,
in d-position, is supplied by
constant
parameter.
ap is a
id, the magnetizing
The
Imd is equal to id.
magnetizing flux qimd
stator
current
current
a
We used field
for
because
the
level
Imd
the
law
qimd(Imd)
is
unique.
remains
given
same
a
(id)
(If
lap)
lap
for
If
calculations
for
Imd
id
and
Imd
where
obtain
to
qimd
qimd
ap is the
According
[1,2],
authors
the
to
supplied by
field
=
=
authors
bend,
The
is
a
calculations
function
2)
fitting
curve
The
for
3) op
for
qimd
method
expression
the
two
range
is
each qimd~
is
used
to
noticeable
more
obtain
the
calculated
is
from
field
the
happens
function
to
interpolate
to
the
as
in
op
non
that
so
first
the
We
saturated
in the
the qimd (id) law
calculations
laws.
two
m
case
operating
the
the
notice
two
two
that
the
in
the
but
points
range
match.
curves
previous
id(qimd) expression and then
Two 9-degree polynomials
results.
cases,
the
are
laws.
divided
into
by using the
at
used
it
correct
which brings the qimd (If lop law
by following the next methodology:
If lop (qimd)
chosen
all the
validated
is
the
represents
4
op
determined
1) A
are
value)
constant
a
Figure
them.
compare
we
assumption (op as
discrepancy occurs,
field
The
and
parameter,
Potier
each
two
k
intervals
and
interpolated
magnetizing
~XP~
flux
<
the
currents
If
lap
and
id
are
calculated
polynomials.
level qimd~
al~i
by the
~~~~
i~~
expression:
15)
JOURNAL
2244
PHYSIQUE
DE
III
N°11
o
u
o
x=o
o
o
u
~
~
2J
lo
40
3l
53
Imd (Al
Fig
Comparison
5
op(Imd, If)
the
between
magnetizing
is reduced
current
by field calculations
determined
The
be
as
fitting
curve
method
again
used
is
op(Imd)
A
g-degree polynomial
The
latter
rotor
supply,
but
the
effect
current
Imd and the field
result, the function
a
greater
on
4.
example, Figure
5
interp~[Imd~
=
the op
because
0.
=
function.
The
(6)
law.
obtaine)
was
supplies)
rotor
law:
idki op~
=
it
and
ap(Ijd)
the
ap(Imd)
the
to
(stator
case
supply or a
winding
magnetizing
stator
a
that
dipends
function
final
for
shows
field
the
the
on
If.
current
ap(Imd, If
fg(Imd)
Where
an
for If
results
=
interpolate
to
suits
ap(Imd, If)
As
field
the
idk. Note thit the Potier parameter can
explained before; it corresponds to the field results
Imd~
modified
was
intermediate
has
a
fitted
is
ap(Imd)
function
current
As
and
value.
mean
4) A
to:
law
compares
~
the
has
the
following
=<
op
>
80
fg(Imd)
x
BIImd
+
op(Imd, If
form:
B2I$d
+
law
~
the
and
(7)
If
+ A
~
field
BgI$d.
results
for
the
case.
If
0.
=
Applications
The study has been applied to a tetrapolar synchronous
machine
windings (rated power: 7.2 kW) The studied
Figure
6.
is drawn
on
The
and
reactances
model
The
mental
calculations
field
the
on
the
GDJ
numerical
one
at
50
have
computed
been
ones
used
to
first
validate
secondly
and
the
with
of the
the
estimate
to
machine
salient
in
comparing
model
of
the
24
Park
the
the
damper
reference
measured
numerical
model.
synchronous
reactance
frequency-
Xd
has
been
compared
to
the
manufacturer
experi-
Hertz
Xd
=
KsXmdo
+
Xp
(8)
Fig
Isovalues
6
Xmdo is the
Xp
machine
non
linear
potential
vector
for
magnetizing
reactance
(the
the
state,
Xp
are:
synchronous
the
SYNCHRONOUS
saturated
Xmdo
and
Then
of the
leakage
is the
the
In
Xp
SATURATED
A
N°11
3.3 Q
=
factor
saturation
Xmdo
experimental
value
saturated
a
than
one.
the
numerical
model
GDJ
The
the
GDJ
dilatation
could
by
involves
it
two
validated
the
to
the
from
law
an
model
in
order
Adjustments are
qimd(If) relationship by
the
inaccuracy
of
is less
on
the Imd
magnetizing
to
estimate
necessary
an
level
the
because
abscissa
axis
calculation.
ways:
ered
ap is the
value
by modifying op (Eq. (7) ). op
saturation
factor
is the
experimental
=
the
is
used
to
field
similar
are
one)
the
Imd is estimated at
parameter. In this case,
computed function (3).
current
Potier
Ks(Imd)
results
is
values
factor
saturation
model.
magnetizing
first
the
numerical
GDJ
The
While
A),
computed
the
Q.
29.6
=
one:
by
current
ap
and
13.7
=
magnetizing level Imd is calculated to its correct
considered
field computed
function (7) and the
Ks(Imd)
Ks (If lap). with op constant.
The
is the
(the
replaced by
brings
the qimd(id)
new
corrected
be
and
is:
reactance
model
obtains
model
constant
It
this
i
Q.
32.9
=
operating point (id
rated
d-axis
been
has
which
x.
31.8 Q.
=
Xd
precision
to
x
is:
for
state,
The
Ks is equal
100
x
model).
28 5 Q.
"
Xd
In
(§imdo/Imdo)
=
GDJ
is:
reactance
Xd
The
Xmdo
for the
reactance
2245
machine.
reactance:
Potier
and
studied
the
STUDY
MACHINE
only
calculate
in
machine
the
accurate
the
method
incremental
an
approximative level (i) where the considreal magnetizing flux qimd is corrected
the
steady state,
in
the
inductances
the
transient
magnetizing
state
because
current
this
Md, Mq and &Idq [1,2].
modification
magnetizing
JOURNAL
2246
PHYSIQUE
DE
III
N°11
(=
~j=I
j=2
j=3
I=2
Fig
diagram
The
7
Compared
5.i%
between
the
method.
of
will
and
at
the
model
the
rated
is due
to
coupling
=
interpolating
for
model
studied
is
b,IACHINE
functionl.
CMC
the
Inverter-Synchronous
the
set
is improved by 4.8% w.ith the op
modification
and
difference
operating point range (imd
20 A). The
the
introduced
be
the
SYNCHRONOUS
THE
5.I.
model
Converter-Machine-Control
of the
modifications
(inverter)
Modeling
5.
block
modification
two
coupling
This
converter
Control
GDJ
the
to
the Ks
with
Motor
set
the
in
Machine
MODEL.
simulajion.
last fiart
Set
The
link
Saturated
in
machine
The
is
issumed
steady state.
When the synchronous
is supplied by a DC-voltage sour(e ma
motor
input voltages are not
sinusoidal.
Generally, in this case,
auth6rs
use
it linear
the dq reference [6] But they
that
the
motor
operates
assume
between
the
State
work
to
on
a
me-
chanical
the
in
inductance
For
tion,
to
the
matrix
saturated
a
the
field
GDJ
is
Vd
machine,
is
the
electrical
assumed
(Fig. 7),
machine
state,
therefore
model
the
(fp
Md)
+
and
equations
~~
complicated. As a first simplificaare
more
damper windings are neglected. According
the following form:'
equations
constant
electrical
the
=
the
constaiit.
current
model.
inverter
an
the
have
~~
+
Mdq
+
Mq) ~~
+
Raid
+
pa
pa
x
(fp
+
Ks
x
Lmdo)iq
n~do)id
Vq
=
fifdq~~
+
(ip
+Raiq
+
pa
x
~s
x
x
(ip
Lmdo
+
K~
~~~
x
~
°p
with
Id~
~~
~~~
~~
~'
Id
~
dK~
Imd dImd
x
Imd
Iq
~~~°'
dKs
dImd~~~°
~~
~
~
Iq~
dKs
Imd
dImd
~~~°'
SATURATED
A
N°11
SYNCHRONOUS
MACHINE
STUDY
2247
where:
Ra
is the
stator
p
is the
number
Q
is the
machine
Lmdo
is the
magnetizing
of poles pairs,
speed,
constant
iiiductance.
require the previous field results: the Ks(Imd) which is obtained by the field
of the K~(Imd) eletromagnetic
the dK~ ldImd law which is the
derivative
equations
The
(3)
calculations
resistance,
and
law:
dK~(Imd)
Note
magnetizing
the
that
Ai
=
Imd is
current
2A2Imd
+
calculated
9AgI$~.
+
+
from
the op
(10)
(Imd, If) electromagnetic
law
(7).
general
The
the
equation (9)
linear
K~
case-
remains
and
i
=
for
valid
dK~
=
following
the
cases-
0.
corresponds to the classical Park model. For a sinusoidal voltages
The stator
supplied machine, the voltages Vd and Vq are
constant.
constant
currents
are
(9).
well
and
their
derivatives
vanish
When
the
machine
is
associated
time
in
equation
as
sinusoidal
and the Vd and Vq voltages are not
inverter, the input voltages are not
to an
The
model
machine
Then
constant.
the
saturated
changes
system
state
voltage
vector
The
is
connects
valid
and
solved
be
to
to
the
lead
with
the
and
currents
stator
their
time
by
a
inverter
highly
inverter
contains
linear.
non
equation
size
I-input
to
model
connexion
the
~'on/off" switching
non-linear
configuration changes
to
(2,3)
0.
becomes
particular
a
modelled
dKsldImd #
The
devices
corresponds
[Vd Vqj~.
inverter
fiIC(I, j)
and
MODEL.
these
of
i
<
remains
INVERTER
THE
State
fis
case:
Equation (9)
5.2.
equation (9) has
the
[6, 9].
derivatives
system
equations.
[10] of the
inverter
and
matrix
MC
j-output
inverter
elements.
of the
[8~
9].
Each
(Fig. 7).
It
set
input
element
matrix
is
Each
of
calculated
as
follows:
MC(I, ii
MC(I, j)
5. 3.
THE
lems
of
The
=
=
switch
the
which
connects
electronic
to
imposes
digital
sinusoidal
non
input
machine
voltages
simulation
j-output
the
~~~~
The
summarized
is
on
are
and T(~
are
~~u6
general principles
in
(P(-6)lT321~
-1
2
-1
lmcl~
j
given by:
~~~
~~~ ~~~~~
i~
and
prob-
the
Park
[10].
voltage supplies to the machine inputs.
calculated
by the equation (ii)
are
i ~
~
P(6)
MODEL
MACHINE-INVERTER"
system
=
matrix
i-input
~
~j~
The
the
otherwise
~~SYNCHRONOUS
power
inverter
reference,
i if the
0
~~~ -~~2
In
(lo
JOURNAL
2248
PHYSIQUE
DE
III
N°11
Beginning
control
Inverter
Connexion
calculation
matrix
yes
t
?
Tfinal
<
no
Fig
Synchronous
8.
where:
9 is
The
the
electrical
the
angle
E is the
and
have
required to calculate
Figure 8 illustrates the
the
electrical
is
currents
in
flowchart
model
equations
machines
presented
machine-inverter
domain
time
in
currents
to
DC
voltage supply.
be
solved
saturated
computation
according
tj
equation
(9).
Iteration
of
magnetic state' [7]. The global flowchart
lachine current in the time
principle of the
domain.
Conclusion
6.
coupling model
experimental tests.
A
comparison
has
been
On
the
other
the
This
This
measured
the
model
to
model
a
displays op
reactances
saturated
synchronous
variations
with
has
validated
the
requiring
without
the
saturation
and
a
4.8%
The
state
improvement
estimated.
Potier
in the
with
proposed
is
hand, the
parameter
model
op
adapted to
taking into
is well
simulation
set
presented coupling model can also be us~d separately
performing experiments.
to
determine
without
the
converter-machine
account
the
set
simulati)n
and
it
will be
included
phenomenon.
saturation
References
[ii
De
ences
Jaeger E.,
ModAlisation
Appliqu6es
de
des
l'Universit6
machines
Cathohque
synchrones
de
Louvain
satur6es,
Thbse
de
(PhD),~octobre
Doctorat
1991.
en
Sci-
N°11
[2]
[3]
F.
Piriou
and
A.,
Razek
Numerical
equations,
circuits
Electric
simulation
2249
analysis of
matrix
Machines
and
saturated
Power
electrical
Systems (IMACS,
electromagnetic
considering
systems
(LiAge 28-30 September 1992)
of
Magnetic
and
STUDY
MACHINE
Fields
5.i-5.6.
pp.
Delforge C.,
rAseaux
Lille,
[5]
SYNCHRONOUS
Garrido MS-,
Pierrat L. and De Jaeger E., The
machines, Modelling and Simulation of Electrical
North Holland, 1988) pp.
129-136.
electric
[4]
SATURATED
A
Janvier
d'un
asynchrone
actionneur
Thbse
de
de
Doctorat
de
et
l'Universit6
commande
sa
des
Sciences
et
vectorielle
Technologies
par
de
1995.
J., A simplified
Robert
and
ModAlisation
perm6ances,
de
Simulation
of
method
Electrical
for
the
Machines
study of
and
Power
saturation
Systems
in
Modelling
Proceeding (North
machines,
AC
Conference
Holland, 1988) pp. 129-136.
Doeuff R.,
Contribution
simulation
d'ensembles
h la
mod61isation
convertiset h la
National
statiques-machines
thAse
de
Doctorat
Sciences
de
l'lnstitut
As
toumantes,
seurs
Polytechnique de Lorraine, Juin 1981.
M. F., F61iachi M. and Le Doeuff R., Modelling of voltage
[7] Lasquellec S., Benkhoris
source
machine
saturated
Electric
and Magnetic Fields (Liege
inverter-synchronous
set in
state,
6-9 May 1996) pp.
363-368.
Doeuff
A
mathematical
model for static
allowing digital simulation of
Le
R.,
converters
[8]
[6] Le
[9]
IECI 24 (1977) 35-38.
IEEE
7Fansactions
transients,
on
F., Le Doeuff R. and Saadate S., Modelling and simulation of a synchronous
Conference
supplied by a GTO voltage inverter, IEE Fourth
International
Power
on
associate
machine
Benkhoris
M.
motor
Electronics
[10] Le
Doeuff
tion
of
pp.
263-270.
and
R.
static
Variable
and
Speed
Benkhoris
converters
and
Drives
M-F-,
drives,
(London, 1990)
General
principles
Mathematics
and
190-195.
pp.
and
new
Computers
trends
in
in
the
Simulation
simula-
(1995)