IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 1, MARCH 2011
1
Fast Estimation of Line-Start Reluctance Machine
Parameters by Finite Element Analysis
Samad Taghipour Boroujeni, Nicola Bianchi, Senior Member, IEEE, and Luigi Alberti, Member, IEEE
Abstract—This paper compares two methods to compute the parameters of a line-start synchronous reluctance (LSSR) machine,
by means of finite element analysis. In both methods, both the
impact of the skin effect on the rotor parameters and the impact
of saturation on magnetizing inductances are taken into account.
The first method is based on an equivalent single-phase model of
the machine. A pulsating flux is imposed along the d- and q-axes
separately, so as to compute the machine parameters along both
d- and q-axes. Magnetizing inductances are achieved as a function
of the flux, and the rotor resistances and leakage inductances are
obtained as a function of rotor frequency. The second method is
based on the d-q axis model of the LSSR machine. A rotating magnetic field is imposed, and the d-q axis parameters are estimated
simultaneously from the field solution. In addition, such a method
allows the cross-saturation to be considered as well.
Index Terms—Equivalent circuit, finite element (FE) simulation,
line-start, parameter estimation, synchronous reluctance machine.
I. INTRODUCTION
HE line-start synchronous reluctance (LSSR) machines
are more and more used in several applications replacing
induction machines (IM). The main reason is that their efficiency
results to be higher than the IM, since there are no rotor current
at steady state. Since there is no permanent magnets (PMs)
included in the rotor, the LSSR cost is lower than PM machines.
In addition, the rotor cage allows the machine to start up directly
with line voltage, without inverter.
The main purpose of this paper is to estimate all parameters of an LSSR machine, using a finite element (FE) analysis.
Although the LSSR machine has been detailed in literature,
few paper cover the estimation of its parameters. Many efforts
are found about the IM: analytical studies, FE analysis, experimental methods, and combined FE and circuit-models have
been suggested for the estimation of IM parameters [1], [2].
Similarly, FE analysis approaches have been proposed for the
estimation of the parameters of PM machines [3]. Since LSSR
machine is a subgroup of buried PM machines with rotor cage,
all parameter estimation methods for buried PM machine can be
adopted to achieve LSSR machine parameters. In [4], a method
T
Manuscript received February 22, 2010; revised June 10, 2010; accepted
July 13, 2010. Date of publication September 2, 2010; date of current version
February 18, 2011. Paper no. TEC-00079-2010.
S. T. Boroujeni is with the Department of Engineering, Shahrekord University, Sharekord 88186, Iran (e-mail: samadtb@yahoo.com).
N. Bianchi and L. Alberti are with the Department of Electrical Engineering, University of Padova, I-35131 Padova, Italy (e-mail: bianchi@die.unipd.it;
luigi.alberti@unipd.it).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TEC.2010.2061851
is presented to predict the parameters of buried PM machines,
even though the effect of rotor cage is not considered. Therefore,
it can be applied to determine the d- and q-axis magnetizing inductances for the steady-state analysis of the machine. Also,
in [5] an approach is presented for the analytical estimation of
the magnetizing inductances in the LSSR machines.
Steady-state performance of an LSSR machine depends on
d–q magnetizing inductances [6]. Although rotor parameters do
not affect steady-state performance of the LSSR machine, they
determine its transient behavior. In some applications, where the
motor has to start up under load, such as pumps or compressors,
the successful starting and synchronization strongly depend on
the LSSR rotor parameters [7]. The effect of rotor cage is also
considered in [8]: the rotor d–q parameters are used to analyze
the dynamic stability of the machine. They show that the rotor
parameters are a function of rotor frequency because of the different distribution of the current within the rotor bars at different
frequency.
A typical approach to compute the IM parameters is based
on the computation of integral quantities, e.g., magnetic energy
and losses, from the results of FE simulations. For example, it is
straightforward to compute the rotor resistance from the Joule
losses in the rotor bars and the inductance from the magnetic
energy. This allows to compute the frequency effects on the
parameters since the skin effects in the rotor bars is considered
for various slip frequency. This is possible since the machine is
isotropic and the contribution to the total losses of each phase
is equal.
In an anisotropic machine, as the LSSR motor, it is not always
possible to compute the parameters from integral quantities,
e.g., from Joule losses and magnetic energy, and alternative
approaches have to be adopted.
In this paper, two methods are proposed to estimate the parameters of the LSSR machine. Both the iron saturation and
the non-uniformly distributed current density in the rotor cage
are taken into account. Both methods combine FE simulation
results with analytical approach to the aim of reaching precise
predictions and computational speed at the same time.
The first method is based on an equivalent single-phase model
of the machine. A pulsating field is imposed separately along
the d-axis and the q-axis. From the FE field solutions, the parameters of the LSSR machine are estimated. The d- and q-axis
magnetizing inductances are computed as a function of the iron
saturation. Similarly, the rotor resistances and leakage inductances are obtained as a function of rotor frequency.
Conversely, in the second method, a rotating field is imposed.
Then, adopting the d–q model of the LSSR machine, the d- and
q-axis parameters are computed from the field solutions. The
advantages of this second method are that it allows to compute
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 1, MARCH 2011
Fig. 1.
Geometry of the considered LSSR machine.
the d- and q-axis parameters from the same field solution, reducing the computational time, and that it allows to considering
the cross-saturation effect between the two axes.
In addition, both methods allow to identify the rotor parameters along the d- and the q-axis. Therefore, the equivalent circuits
exhibit different rotor parameters according to the considered
axis. This is generally not considered in literature, where the
rotor parameters are considered equal and computed from the
corresponding IM structure [10].
The results obtained by means of these two methods are
compared together. Then, for the sake of verifying the effectiveness of the proposed methods, they are used to estimate the
parameters of an IM. The obtained results are compared with
those obtained in a previous research [2]. All computations refer
to a 2.2-kW two-pole LSSR motor, with nominal voltage Vn =
230/400 V and frequency f = 50 Hz. The structure of the LSSR
machine is sketched in Fig. 1.
II. PARAMETER ESTIMATION BASED ON SINGLE-PHASE
MODEL WITH PULSATING FLUX
This method is based on a single-phase model of the machine. The three-phase winding is rearranged so as to obtain
a corresponding single-phase winding. So that the three-phase
machine is transformed into a single-phase machine.
This is carried out attempting to reproduce with the singlephase winding the air gap magneto motive force (MMF) distribution of the three-phase winding. Therefore, a pulsating magnetic field is obtained when an alternate current flows through
the equivalent single-phase winding. The rotor is at standstill
and positioned so that such a pulsating field is only along the
d-axis, or only along the q-axis. Hence, the LSSR machine parameters are investigated separately.
Changing the frequency of the pulsating field, the frequency
effect on the motor parameters is achieved. At zero frequency,
the steady-state inductances are got, then, at higher frequency
the rotor parameters can be predicted.
A. Corresponding Single-Phase Winding
Fig. 2(a) shows a sketch of the three-phase winding in the
time instant when
Ia = I
Ib = Ic = −
I
2
(1)
Fig. 2. Equivalence between three-phase and single-phase windings.
(a) Three-phase winding with currents Ia = I and Ib = Ic = −I/2. (b) Singlephase winding: I = I.
so that the main field is along the a-axis. The number of turns
of each phase is Nt .
Fig. 2(b) shows the equivalent single-phase winding. The
current of such a winding is set to be equal to I = I. Therefore,
in order to make the MMF equivalence between single-phase
and three-phase, the turns are rearranged, as shown in Fig. 2(b):
the single-phase winding is composed by all the turns of the
phase a, and half turns (with opposite sign) of the phases b and
c. The number of turns of the single-phase winding results in
N1 = 2 N3 , where N3 is the number of turns of each phase of
the original three-phase winding.
The ratio between the effective number of turns of the threephase and the single-phase winding results in
c=
kw 3 N3
kw 1 N1
(2)
where kw 1 and kw 3 are the winding factors of the single-phase
winding and the three-phase winding, respectively.
Let αse be the slot angle in electrical degrees, i.e., αse =
π/(3qp), where p is the number of pole pairs and q is the
number of slots per pole per phase of the original three-phase
motor. As well known, the three-phase winding (distribution)
factor is computed as
kw 3 =
sin(qαse /2)
q sin(αse /2)
(3)
BOROUJENI et al.: FAST ESTIMATION OF LINE-START RELUCTANCE MACHINE PARAMETERS BY FINITE ELEMENT ANALYSIS
Fig. 5.
Fig. 3. Analysis of LSSR machine using single-phase winding, using pulsating
field along the d- or the q-axes.
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Single-phase equivalent circuit at standstill.
still, i.e., the slip s is unity. Therefore, the equivalent circuits of
Fig. 4 have the upper part to be equal to the lower part. They
can be simplified as reported in Fig. 5(a) and (b).
The parameters of the equivalent circuits are computed as
described in the following.
C. d–q Magnetizing Inductances
Fig. 4.
Selecting a frequency equal to zero, magnetostatic FE simulations are carried out. So that the steady-state d- and q-axis
magnetizing inductances are determined. The magnetizing inductances are computed imposing different magnetizing currents, so that the iron saturation is carefully taken into account
during the analysis.
The magnetizing inductance of the single-phase model of the
LSSR machine (a star will be used to distinguish the parameter
of such a machine) is expressed as
Single-phase equivalent circuits.
while the single-phase winding factor is computed as
q −1
2q
−1
3q
−1
1 jiα es
e
e jiα s
jiα s e
+2
e
+
e
kw 1 =
.
4q i=0
i=q
i=2q
WA J
Λ
(5)
2 = I
I
indicates the integral of A · J over all stator slots.
WA J = Lstk
A · JdS
(6)
Lm =
(4)
It is worth noticing that, with the assumption of the winding
correspondence defined earlier (see Fig. 2), the ratio (2) results
in c = 2/3.
Once the winding is defined, the rotor is positioned so that the
pulsating field due to the single-phase winding current results
to be along the d-axis only, or along the q-axis only. This is
sketched in Fig. 3. In Fig. 3(a), the d-axis is placed along the axis
of the equivalent single-phase winding. Similarly, in Fig. 3(b),
rotor is rotated of 90 electrical degrees, so that the q-axis is
placed along the axis of the single-phase winding.
B. Equivalent Circuits
The equivalent circuits corresponding to the single-phase
LSSR machine are shown in Fig. 4, according to the analysis along the d-axis, Fig. 4(a), and along the q-axis, Fig. 4(b),
respectively. These circuits are achieved directly from the singlephase IM circuit, considering the field to be split in its forward
and backward rotating components. The upper part refers to the
forward rotating field and the lower part refers to the backward
rotating field. The rotor slip s is highlighted in the circuits of
Fig. 4.
The FE simulations are carried out imposing different frequencies, so as to consider the frequency effect on the circuit
parameters. However, in each simulation, the rotor is at stand-
where WA J
Ss
I and Λ are, respectively, the current imposed and the flux
linkage computed in the single-phase winding during the simulation. Lstk is the stack length of the machine, and Ss in the
integral indicates the slot-cross-sectional area.
The computation of (5) is repeated for various currents along
the d-axis and the q-axis, achieving the magnetizing inductances
Lm d and Lm q .
Once the d–q magnetizing inductances are computed with the
equivalent single-phase winding, the effective d–q magnetizing
inductances of the three-phase LSSR machine are computed by
means of the transformation ratio (2), yielding
Lm d = c Lm d
Lm q = c Lm q .
(7)
The saliency of the rotor is defined by the ratio between the
two inductances. With this analysis strategy, the cross saturation
in the magnetizing inductance cannot be considered.
D. Rotor Parameters
In order to estimate the rotor parameters, the simulations are
carried out at frequency higher than zero, so that currents are
induced in the rotor bars. It is verified that the rotor parameters
depend mainly on the rotor geometry. In particular, they vary
according to the pulsating field axis. The single-phase model is
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 1, MARCH 2011
well suited for this computation, allowing pulsating field to be
imposed separately along the d- and the q-axes.
In addition, the rotor parameters might depend on the rotor
frequency, due to the nonuniform current distribution within
the rotor bars, especially for higher frequencies and larger machines. Therefore, the parameters are computed from a set of
time harmonic FE simulations carried out at different excitation
frequencies.
In these FE simulations, a linearized iron is considered in the
lamination, freezing the magnetic permeability in the various
machine parts. A particular care has to be taken for the iron
bridges, also called iron ribs, which are saturated even at low
value of the flux of the machine. They are considered to be linear
as well; however, their relative permeability is chosen to be quite
small, e.g., in the range between 5 and 20. The assumption of
carrying out simulations with linear iron is particularly useful in
determining the machine parameters, since the superposition of
the effects can be applied, as will be clarified in the following.
As said earlier, since these simulations are carried out with the
machine at standstill, the machine equivalent circuits in Fig. 5
can be used. From the field solutions, the equivalent parameters
of the circuits are determined as follows:
Leq =
2Wm
2
Irm
s
(8)
=
Req
Pj r
2
Irm
s
(9)
and
where Wm is the average magnetic energy in a period, computed
in the entire machine (let us remember that a linear model is
considered), Pj r is the average Joule loss in the rotor in a period,
and Irm
s is the rms current imposed in the simulation.
Then, the leakage inductance and rotor resistance are computed as
Ll = Lm
Leq (Lm − Leq ) − (Req
/ω)2
/ω)2
(Lm − Leq )2 + (Req
(10)
Rr = Req
Lm + Ll
.
Lm − Leq
(11)
Rr q = c Rr q
(12)
and
Lld = c Lld
Dynamic d–q equivalent circuits of the LSSR machine.
III. PARAMETER ESTIMATION BASED ON dq MODEL
The analysis method described in this section is based on the
d- and q-axis model of the LSSR machine. A rotating magnetic
field is imposed, since d- and q- axes are excited together. Both dand q-axis parameters of the model are obtained simultaneously
from each field solution.
Let us consider a d–q reference frame fixed to the rotor with
d- and q-axes as specified in Fig. 3. Then, it rotates at the
e
e
= ωm
in electrical radians per
same speed of the rotor, i.e., ωdq
second. In such a reference frame, the stator voltages are
dλsd
e
− ωm
λsq
dt
dλsq
e
+ ωm
vsq = Rs isq +
λsd .
(14)
dt
The corresponding d–q model of LSSR machine is shown
in Fig. 6, where all parameters are referred to the stator. The
Γ-type equivalent circuits are used, with all leakage inductances
considered in the rotor circuit. The stator resistances are omitted
in the circuits of Fig. 6. Such stator resistance as well as all 3-D
parameters are not included in the FE model of the motor, but
they are computed analytically and added to the circuits in a
second time [1].
At steady state, the voltages and currents are considered to
be sinusoidal with time. In the d–q reference frame, they vary
e
), where ω = 2πf , with f is the
at the electrical speed (ω − ωdq
line frequency. It follows that the complex notation can be used.
Overline symbols will be used to highlight complex quantities.
Thus, (14) are rewritten as
vsd = Rs isd +
e
e
V sd = Rs I sd + j(ω − ωdq
)Λsd − ωdq
Λsq
This analysis is based on the superposition of the effects, allowed
by the choice of adopting linear iron in these simulations. As
mentioned earlier, the FE simulations are carried out separately
for the d- and the q-axes, yielding the parameters Lld and Llq ,
as well as Rr d and Rr q .
Finally, the parameters of equivalent single-phase machine
have to be converted for the three-phase machine, using the
effective turn ratio (2), yielding
Rr d = c Rr d
Fig. 6.
Llq = c Llq .
(13)
e
e
V sq = Rs I sq + j(ω − ωdq
)Λsq + ωdq
Λsd .
(15)
A. d–q Magnetizing Inductances
For the computation of the magnetizing inductances, it is
convenient to choose the speed of the d–q reference frame (that
e
= ω.
is the rotor speed) equal to the line electrical speed, i.e., ωdq
It results that the electrical quantities (currents, voltages, flux
linkages) exhibit zero frequency, i.e., they have constant values.
The voltage equations (15) are rewritten as
Vsd = Rs Isd − ωΛsq
Vsq = Rs Isq + ωΛsd
(16)
without overline, they being constant.
Since the frequency is zero, there are no currents induced in
the rotor. The d–q flux linkages correspond to the magnetizing
BOROUJENI et al.: FAST ESTIMATION OF LINE-START RELUCTANCE MACHINE PARAMETERS BY FINITE ELEMENT ANALYSIS
d–q flux linkages of the motor. They depend on the d–q currents
imposed in the stator, since the iron is nonlinear.
It is worth noticing that d- and q-axis currents can be imposed
simultaneously. Then, both d- and q-axis flux linkages, and dand q-axis inductances, can be computed from the same field
solution. Therefore, not only the saturation effect is considered,
but also the mutual effect between the d- and the q-axes, that is
the d–q cross-saturation effect. In this condition, the d–q circuits
of LSSR model (see Fig. 6) are not independent.
Starting from the magnetizing currents Id and Iq , the phase
currents are obtained from the Park transformation as
e
e
) + Iq sin(θm
)
Ia = Id cos(θm
2π
2π
e
e
Ib = Id cos θm
−
−
+ Iq sin θm
3
3
4π
4π
e
e
Ic = Id cos θm −
+ Iq sin θm −
3
3
(17)
e
is the electrical angle of the d-q reference frame with
where θm
respect to the stator reference axis (axis of phase a).
The d–q flux linkages are obtained from field solutions. The
magnetizing inductances are defined as
Lm d (Id , Iq ) =
Λm d (Id , Iq )
Id
Lm q (Id , Iq ) =
Λm q (Id , Iq )
Iq
(18)
and they are functions of both magnetizing currents. Of course,
without considering cross-saturation effect, the magnetizing inductances are simplified as
Λm d Λm q Lm q =
(19)
Lm d =
Id I q =0
Iq I d =0
B. Rotor Parameters
For the computation of the rotor parameters, it is convenient
to consider the rotor frequency. In the adopted reference frame,
e
e
= ωm
, such a rotor frequency results in ωr =
in which ωdq
e
e
).
(ω − ωdq ) = (ω − ωm
Neglecting the stator resistance Rs , the voltages (15) are
rewritten as
e
V sd = +jωr Λsd − ωm
Λsq
e
V sq = +jωr Λsq + ωm
Λsd
Fig. 7.
5
Steady-state d–q equivalent circuits. (a) d-axis. (b) q-axis.
in the previous section. Hence, superposition of the effects can
be applied.
In all FE simulations, the rotor is at standstill and the d–q
circuits of Fig. 6 reduce to those shown in Fig 7.
In order to estimate the rotor parameters, the d–q stator currents in the simulations are fixed to
I sd = I
√
I sq = −jI
(21)
where |I| = 2 Irm s is the current amplitude.
Thus, the magnetic field is rotating, but modulated by the
rotor anisotropy. Currents are induced in the rotor bars, and the
rotor parameters of the d–q model of Fig. 7 can be computed.
The flux linkages Λsd and Λsq are determined from the field
solution. The voltages can be achieved from (20).
Therefore, the equivalent parameters of the circuits of Fig. 7
are computed from flux linkages as
Λsd
Λsd
Req,d = −ωr m ag
(22)
Leq,d = eal
I sd
I sd
and
Leq,q = eal
Λsq
I sq
Req,q = −ωr m ag
Λsq
I sq
. (23)
Finally, the rotor parameters Rr d , Lld , Rr q , and Llq are then
computed as suggested in the previous section.
It is worth noticing that |Isd | = |Isq | are imposed in the FE
simulation. As a consequence |Vsd | = |Vsq |, since the flux linkages and the corresponding voltage are modulated by the rotor
anisotropy. In the actual operating conditions, it is |Vsd | = |Vsq |,
while |Isd | = |Isq |.
IV. CASE OF STUDY
(20)
and they vary at the electrical frequency ωr .
Therefore, a time-harmonic FE simulation is carried out, at
frequency fr = ωr /2π, so as to compute the rotor parameters.
The FE simulations are carried out with fixed rotor and stator
(as at standstill), supplying the d–q stator windings at the rotor frequency fr . Since rotor parameters depend on the rotor
frequency, they are obtained from various time-harmonic FE
simulations imposing different rotor frequencies.
As mentioned earlier, the B–H curve of the iron laminations
are considered to be linear in the FE time harmonic simulations,
freezing the magnetic permeability in the different parts of the
machine. Care has to be taken to the iron bridges, as described
In this section, the results obtained by means of the proposed
methods are compared. In particular, the main focus is on parameters of the LSSR machine, computed applying both analysis
methods and then compared.
In addition, since the three-phase IM is a special case of
the LSSR machine, characterized by the same d and q circuit,
the proposed methods are used to estimate the IM parameters.
The effectiveness of the proposed procedures is then tested in
predicting the motor parameters of an IM, comparing the results
with those achieved by means of the indirect test method [2].
In the next figures, circles refer to the single-phase model
with pulsating flux, while solid lines refer to the d–q model with
rotating field.
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Fig. 8.
IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 1, MARCH 2011
d–q Magnetizing inductances versus d–q magnetizing currents.
Fig. 10. Rotor parameters versus rotor frequency. (a) d–q rotor resistances.
(b) d–q leakage inductances.
Fig. 9. d- and q-axis flux linkages versus both d- and q-axis magnetizing
currents.
A. Parameter Estimation of an LSSR Machine
Fig. 8 shows the d–q magnetizing inductances computed by
means of the two methods proposed earlier: Ld is a function of
Id , and Lq is a function of Iq .
These d–q inductances achieved from the flux linkages (second method, with rotating field) are computed without considering the cross-saturation, since cross-saturation effect is not
considered by the first method (with pulsating field). In order
to highlight the dependence of the d–q fluxes on both d–q magnetizing currents, Fig. 9(a) and (b) shows d- and q-axis flux
linkages versus d- and q-axis magnetizing currents.
Fig. 10(a) shows the obtained d–q rotor resistances versus
rotor frequency of the considered LSSR machine with the two
proposed methods. Because of skin effect, d–q rotor resistances
increased with rotor frequency. The q-axis rotor resistance is
more affected by skin effect and more dependent on rotor
frequency.
Fig. 10(b) shows the leakage inductances versus rotor frequency computed with both methods. Because of skin effect,
d–q leakage inductances are expected to decrease with rotor
frequency.
From Fig. 10(a) and (b), a satisfactory agreement between
the two computations is evident. The slight difference is due
to the different MMF distribution, resulting in the simulations
with the two methods. In the first method (pulsating field), the
currents are imposed as defined in (1) for both the computation
along the d- and q-axis and the rotor is rotated to consider the
right axis. So that the current in the stator winding is the same
for the simulations along both d- and q-axes, yielding the same
stator MMF distribution.
In the second method (the d–q model with rotating field), the
d–q parameters are obtained from the same simulation. Therefore, the MMF distribution along the d-axis and q-axis is not
the same, depending on the rotor position. This consideration
suggests that, even adopting the second method, the computation of the average value of the d- and q-axis parameters, should
be carried out according to different positions of the rotor. As
an example, Fig. 11 shows the LSSR rotor parameters versus
the rotor position computed at a frequency of 5 Hz. The rotor
parameters are computed from the flux linkage, see (22) and
(23), and the computation is affected by the MMF harmonics.
In Fig. 11, it is clearly visible the sixth harmonic effect with a
period of 60 mechanical degrees and the superimposed effect
of the slotting. The curves reported in Fig. 10 correspond to the
average values of the parameters computed according to various
rotor position.
BOROUJENI et al.: FAST ESTIMATION OF LINE-START RELUCTANCE MACHINE PARAMETERS BY FINITE ELEMENT ANALYSIS
Fig. 12.
IM magnetizing inductance, obtained from different methods.
Fig. 13.
IM rotor resistance, obtained from different methods.
Fig. 14.
IM leakage inductance, obtained from different methods.
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Fig. 11. Rotor parameters versus rotor position, simulation at 5 Hz. (a) d–q
Rotor resistances. (b) d–q Leakage inductances.
TABLE I
COMPARISON OF LSSR PERFORMANCE COMPUTED BY THE PROPOSED
STRATEGY (EC) AND NONLINEAR FE
C. Parameter Estimation of IM
B. Comparison Between Equivalent Circuit and FE Results
For the sake of testing equivalent circuit model achieved by
the proposed strategy, it is solved for various frequencies, and the
results compared with the predictions achieved by a nonlinear
time-harmonic FE simulations in the same working point. In
this case, the actual iron B–H characteristic is considered in
the non-linear FE simulation. The currents are fixed as I sd = 2,
I sq = −j2 (rms values). Then the nonlinear FE simulations are
carried out and the machine performance computed for various
frequencies. The same currents are then imposed in the d–q
equivalent circuit computed, as described in Section IV-A.
Some results of both computations are reported in Table I. FE
refers to the nonlinear FE simulations, EC refers to the equivalent circuit whose parameters are computed with the proposed
strategy. The Joule losses and both the d- and q-axis flux linkages are reported. There is a satisfactory agreement between EC
and nonlinear FE results.
As stated earlier, the IM is a subgroup of LSSR machine
with the same d- and q-axis circuit parameters. Therefore, the
proposed methods can be applied also to the study of the IM
parameters.
The purpose of this section is to predict the parameters of
the IM and to compare them to the same parameters computed
using the indirect test method. Such a method is based on the
IM no-load test and the locked-rotor test [2]. Figs. 12 to 14
show the computed magnetizing inductance, rotor resistance,
and rotor leakage inductance, respectively, achieved by means
of three different methods. The two methods presented earlier
are reported using marked-points curves: circle marks are used
for the method with pulsating flux described in Section II, star
and square marks are used for the method with rotating flux
described in Section III. The parameters computed by means of
the indirect test method, as described in [2], are reported in solid
line.
From the comparison, it is possible to observe that there is a
good agreement among the results of the three methods. Also
considering the IM, there is a slight difference in the q-axis
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IEEE TRANSACTIONS ON ENERGY CONVERSION, VOL. 26, NO. 1, MARCH 2011
parameters, computed by the two methods. The same consideration reported in the previous Section is still valid. The MMF
distribution affects the computation of the rotor parameters of
the IM as well.
At last, it is worth noticing that the parameters obtained by
means of the indirect test method (solid lines) result to be the
mean values between d- and q-axis parameters. This is because
the parameters in the three-phase indirect test method are computed from integral quantities (Joule losses and magnetic energy), which are almost constant with a symmetric three-phase
current supply.
[7] J. Soulard and H. P. Nee, “Study of the synchronization of line-start
permanent magnet synchronous motors,” in Proc. IEEE Ind. Appl. Conf.
Rec. 2000, vol. 1, Oct. 2000, pp. 424–431.
[8] D. W. Shimmin, J. Wang, N. Bennett, and K. J. Binns, “Modelling and
stability analysis of a permanent-magnet synchronous machine taking into
account the effect of cage bars,” in Proc. IEE Electr. Power Appl., vol. 142,
Mar. 1995, pp. 137–144.
[9] S. Williamson and D. R. Gersh, “Finite element calculation of double-cage
rotor equivalent circuit parameters,” IEEE Trans. Energy Conversion,
vol. 11, no. 1, pp. 41–48, Mar. 1996.
[10] P. Krause, O. Wasynczuk, and S. Sudhoff, Analysis of Electric Machinery
and Drive System. New York: Wiley, 2002.
V. CONCLUSION
In this paper, two methods are proposed for LSSR parameter
estimation. The first method is based on steady-state singlephase model of the machine and a pulsating field. The second
method is based on the d–q model of the machine and a rotating
field. Both methods compute the rotor parameters as a function
of rotor frequency and estimate the magnetizing inductances as
a function of magnetizing currents. So skin effect and magnetic
saturation are considered in both methods proposed. However,
in contrast with the first method, the cross-saturation effect can
only be considered by means of the model based on the d–q
model. In addition, the impact of the MMF distribution is more
evident. A computation of the d- and q-axis parameters with at
least two positions of the rotor is advised.
The effectiveness of the proposed methods is verified by
means of estimating the parameters of an LSSR and comparing
the results together. Then, the parameters of an IM are estimated
using the proposed methods and compared to the results of a
well-known procedure, achieving a satisfactory agreement.
REFERENCES
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and performance of rare-earth permanent-magnet motors avoiding measurement of load angle,” in Electr Power Appl., IEE Proc. B, vol. 138,
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Samad Taghipour Boroujeni was born in 1981. He
received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from the University of Amirkabir
(Tehran Polytechnic), Iran, in 2003, 2005, and 2010
respectively.
He is currently Assistant Professor at Electrical Engineering Faculty of Shahrekord University
(SKU), Shahrekord, Iran. He is working on design,
analysis and optimization of Electric Machines.
Nicola Bianchi (M’98–SM’10) was born in Verona,
Italy, in 1967. He received the M.S. and Ph.D. degrees in electrical engineering from the University of
Padova, Padova, Italy, in 1991 and 1995, respectively.
Since 1998, he has been with the Electric Drives
Laboratory, Department of Electrical Engineering,
University of Padova, as an Assistant Professor. Since
2005, he has been an Associate Professor at the same
Department. He is the author or coauthor of several papers on the subject of electrical machines and
drives, and the author of the international book Electrical Machine Analysis using Finite Elements (CRC Press, Taylor & Francis
Group, Boca Raton, FL) and two Italian textbooks. His current research interests
include design of electrical motors for electric drive applications.
Luigi Alberti (S’07–M’09) received the Laurea degree and the Ph.D. degree in electrical engineering
from the University of Padova, Padova, Italy, in 2005
and 2009, respectively.
He is currently an Assistant Researcher at the
Electric Drive Laboratory, Department of Electrical
Engineering, University of Padova, where he is engaged in research on design, analysis, and control of
Electric Machines. He is also a Consultant for various
industries.