Description

Dynamic Analysis of Synchronous Reluctance Motor Drives Based on
Simulink
®
and Finite Element Model
Francesco Parasiliti, Marco Villani Alessandro Tassi
Department of Electrical and Information Engineering Spin Applicazioni Magnetiche S.r.l.
University of L’Aquila
67100 Poggio di Roio, L’Aquila 29010 Pianello Val Tidone, Piacenza
ITALY ITALY
villani@ing.univaq.it info@spinmag.it
Abstract – A fine motor analysis that takes the driving control
into account a

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Dynamic Analysis of Synchronous Reluctance Motor Drives Based on Simulink
and Finite Element Model
Francesco Parasiliti, Marco Villani
Alessandro Tassi
Department of Electrical and Information Engineering
Spin Applicazioni Magnetiche S.r.l.
University of L’Aquila
67100 Poggio di Roio, L’Aquila
29010 Pianello Val Tidone, Piacenza
ITALY
ITALY
villani@ing.univaq.it info@spinmag.it
Abstract
– A fine motor analysis that takes the driving control into account allows to evaluate with a good accuracy the dynamic performance. The use of Simulink
together with FEA has allowed to investigate deeply the dynamic behaviour of the Synchronous Reluctance Motor in different operating conditions, giving results closer to the actual motor performance.
I. INTRODUCTION Synchronous Reluctance Motors (SRM) could be considered as alternative to its counterparts, namely Permanent Magnet, Switched Reluctance and Induction Motors. Moreover, it has been demonstrated that SRM has the same or higher power density than the Induction Motors in the low and medium power level. The fine analyses of these motors are very difficult because of their highly saturated operation conditions and their salient structure. The use of linear models in evaluating the performances of the considered motors can lead to serious errors since the SRM present notable nonlinear characteristics due to the effects of the saturation and cross-coupling phenomena occurring in the magnetic circuits. These phenomena can be taken into account only by an accurate nonlinear analysis which can be performed by Finite Element programs that allow an accurate prediction of machine parameters and performances [1]. Moreover, the growing demand of high dynamic performance motors (e.g. with fast torque response) requires a fine motor analysis taking into account the driving control. Usually, to simplify the controller, constant values of axis inductances (
L
d
and
L
q
) are considered, but it is well known that the axis currents and the stator-rotor relative position have a considerable influence on the inductances, with significant effects on torque behaviour. For these reasons, in order to predict with a good accuracy the dynamic performance, it is suitable to develop a model that allows to link the drive scheme with a fine motor model. With this aim, the link between Simulink
®
code (by Mathworks) and a Finite Element software Flux2D
®
by Cedrat has been employed to simulate the SRM drive. This approach allows to simulate, with good accuracy, the motor behaviour in different operating conditions related to imposed control strategy [2], giving to the designer useful indications (e.g. the torque ripple) in view of the motor and control design refinements. The proposed study concerns with a Synchronous Reluctance Motor with two flux barriers 4 pole, 200 V, 20 Nm: a view of the rotor structure is shown in Fig.1 The paper presents a preliminary analysis with constant values of axis inductances; then, more accurate models have been carried out tacking into account the parameters variation.
Fig. 1 - View of the two flux barriers rotor
II. THE CONTROL STRATEGIES OF THE SRM The considered torque expressions of the SRM, with reference to the rotating d-q frame synchronized with the rotor, are:
( )
ε
223
2
sin I L L pT
sqd
−=
, (1)
( )
qd qd
I I L L pT
−=
23
, (2) where
p
are the pole pairs,
I
s
the space vector of the stator current,
I
d
and
I
q
the direct and quadrature axis currents and
ε
the angle between d-axis current and the vector
I
s
. A suitable control strategy is to maximize the “Torque-Current” ratio and this can be achieved if
ε
is equal 45°: it corresponds to impose the same values for the axis currents. Since for this type of motors
L
d
is usually 7÷8 times higher than
L
q
, the above mentioned strategy cannot be adopted when
I
d
component increases; then, the d-axis current should be kept constant and only the q-axis component increases.
15161-4244-0136-4/06/$20.00 '2006 IEEE
Alternatively, the d-axis current can be always fixed constant. Therefore, the control strategies could be (Fig.2): - d-q current angle control (A, B, C trajectory); - constant
I
d
control (A’, B, C trajectory). In this paper the “constant
I
d
control” only has been investigated, and the control scheme is shown in Fig.3.
ε
=45°ADCBi
d
i
q
i
dmax
Fig. 2 - Locus of the stator current vector
III. THE FE ANALYSIS OF THE SRM The proposed study concerns the SRM whose main data are presented in Tab.I. The investigated motor has a flux barriers rotor that presents, respect to the axially laminated one [2], a simplicity in mechanical construction, lower manufacturing cost, and the rotor skewing possibility; on the other hand it has a quite large level of torque ripple due to the inductances variation with rotor position that can cause, particularly at low-speed, inaccuracy in motion control, noise and vibrations.
TABLE I MAIN DATA OF THE MOTOR Number of poles 4 Stack length (mm) 130 Outside stator diameter (mm) 152 Inner stator diameter (mm) 90 Stator slots 36 Number of flux barriers per pole 2 Number of turns per phase 198 Phase voltage (V) 200 Phase current (rms) (A) 7.7 Torque (Nm) 20 Frequency (Hz) 50
An accurate bi-dimensional FE model has been carried out by the software Flux2D
ver.9
by Cedrat, [3], introducing a parametric model of motor in order to modify the geometric dimensions of stator and rotor shape, the rotor position (
θ
) and the currents. The magnetostatic analyses allow to evaluate the axis inductances, whose values could depend on the axis currents only (if one rotor position only is analyzed): - L
d
=
f
(I
d
, I
q
); (3) - L
q
=
f
(I
d
, I
q
); or on currents and rotor position (see Fig.4): - L
d
=
f
(
θ
, I
d
, I
q
); (4) - L
q
=
f
(
θ
, I
d
, I
q
). If a transient analysis is carried out, the calculated inductances depend also on the variation of currents with time: - L
d
=
f
(
θ
, I
d
(t), I
q
(t)); (5) - L
q
=
f
(
θ
, I
d
(t), I
q
(t)). A further FE model has been developed, taking into account the rotor skewing. This design solution allows to drastically reduce the high torque pulsation in flux barriers type SRM [1], [4]. Referring to Finite Element analysis, skewing presents difficulties in computing the magnetic field distribution, as well as machine parameters and performances evaluation. These difficulties are overcome considering a simplified several-blocks equivalent rotor and by using several cross sectional 2D FEA [5]. As alternative the “Skew” module of Flux2D can also be used. Fig.4 shows the axis inductances profiles for a skewed rotor (one stator slot pitch): in this case a skewing of one pole pitch has been considered and the rotor has been divided into 10 blocks. The inductances profiles have been compared with the no-skewed rotor ones (4). A’
Fig. 3 - Scheme of the constant I
d
control
1517
Comparison Ld
0,000,050,100,150,200,250 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90
Position [degrees]
I n d u c t a n c e [ H ]
LdLd Skewed
Comparison Lq
0,0000,0050,0100,0150,0200,0250,0300,035051015202530354045505560657075808590
Position [degrees]
I n d u c t a n c e [ H ]
LqLq Skewed
Fig. 4 - Axis inductances profiles (I
d
=3.7 A; I
q
=10.2 A)
IV. DINAMIC ANALYSIS OF SRM BY SIMULINK
In this step the SRM drive (Fig.3) is simulated by Simulink code, solving the dynamic equations of the SRM in
d-q
reference frame, where inductances values are those evaluated by preliminary (out of line) FEA; the SMR is then simulated by lumped parameters like inductances and resistances. The analyses concern the no-skewed rotor. Two different tests have been carried out on the grounds of two inductance evaluation hypotheses: a)
variable inductances with currents but independent of stator-rotor relative position (3); b)
variable inductances with currents and stator-rotor relative position (4). Simulink solves the SRM model using the preliminary evaluated inductances values corresponding to the reference currents imposed by the controller. Speed responses between 0 and 1500 rpm with constant 22 Nm load torque have been simulated. Steady-state axis currents were: I
d
=3.7 A and I
q
=10.2 A. The time torque response for both cases are shown in Fig.5. In the case (a), since the axis inductances are constant, the torque pulsation is absent. In the case (b) a significant ripple appears, that is about 11% (defined like as the ratio between the difference of the maximum and minimum values of torque and the average one). V. DINAMIC ANALYSIS BY SIMULINK AND FEM With the aim to link the drive scheme with a fine motor model, the Simulink control scheme has been directly interfaced with the FE model of the SRM.
Fig. 5 - Torque waveforms (I
d
=3.7 A; I
q
=10.2 A).
This analysis has been carried out thanks to the link between the FE software Flux2D and Simulink code [3]. By such co-simulation method, the transitory effects and the cross-coupling effects are accurately reproduced. Respect to the analysis in the Section IV, the SMR is not simulated by lumped parameters but it is accurately modelled by on-line transient FE analysis where the axis inductances depend on the rotor position and the currents both variable with time (5). The drive scheme of the SRM with “constant
I
d
control” is shown in Fig.6, where it is evident the block for the FEA of the Synchronous Reluctance Motor. This co-simulation between Simulink (for the drive) and Flux2D (for the motor) needs an external circuit that allows to link the two codes. Particularly, in order to fed the FE model of the SRM, three voltage generators have been introduced, whose values (variable with time) depend on the controller. Then, starting from the motor phase resistance and the inductances values (evaluated on-line by FEA), the phase currents are calculated by Flux2D. From these values, the currents in each stator slot are imposed according to the winding distribution. The time currents, torque and speed responses are presented in Fig.7 and Fig.8, where a load torque of 22 Nm has been imposed. After the transient operation, the speed reaches 1500 rpm and the axis currents settle to the imposed values.
case (b) case (a)
d-axis inductances q-axis inductances skewed no-skewed
skewed no-skewed
1518
In this case, the torque ripple is higher than the previous analysis one (Fig.5b), giving result closer to the actual motor torque. It is important to underline that knowledge of the torque ripple it is very important for the SRM since it can cause,
Current Id=costant
-25-20-15-10-5051015200,00 0,05 0,10 0,15 0,20 0,25
Time [s]
C u r r e n t [ A ]
Ia Ib Ic
Current dq Id=costant
05101520250,000,050,100,150,200,25
Time [s]
C u r r e n t [ A ]
IdIqIdrif
Fig. 7 – Phase currents and axis currents vs. time
particularly at low-speed, inaccuracy in motion control, noise and vibrations. This example demonstrates how an accurate analysis of the motor by a suitable tool, allows to predict the real motor performance and verify the dynamic behaviour for different operating conditions.
Torque Id=costant
0510152025303540450,00 0,05 0,10 0,15 0,20 0,25
Time [s]
T o r q u e [ N m ]
TorqueDrag Torque
Angular Velocity Id=costant
-2000200400600800100012001400160018000,00 0,05 0,10 0,15 0,20 0,25
Time [s]
A n g u l a r V e l o c i t y [ r p m ]
Angular VelocityReference
Fig. 8– Torque and speed vs. time
Fig. 6 - SRM drive with “constant I
d
control”
FEA
Phase currents Speed
speed
Axis currents
1519

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