-
Delft University of Technology
Advanced dynamic modeling of three-phase mutually-coupled
switched reluctancemachine
Dong, Jianning; Howey, Brock; Danen, Benjamin; Lin, Jianing;
Weisheng Jiang, James; Bilgin, Berker;Emadi,
AliDOI10.1109/TEC.2017.2724765Publication date2018Document
VersionAccepted author manuscriptPublished inIEEE Transactions on
Energy Conversion
Citation (APA)Dong, J., Howey, B., Danen, B., Lin, J., Weisheng
Jiang, J., Bilgin, B., & Emadi, A. (2018). Advanceddynamic
modeling of three-phase mutually-coupled switched reluctance
machine. IEEE Transactions onEnergy Conversion, 33(1), 146-154.
https://doi.org/10.1109/TEC.2017.2724765
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https://doi.org/10.1109/TEC.2017.2724765https://doi.org/10.1109/TEC.2017.2724765
-
1
Advanced Dynamic Modeling of Three-PhaseMutually-Coupled
Switched Reluctance Machine
Jianning Dong, Member, IEEE, Brock Howey, Benjamin Danen,
Jianing Lin, Student Member, IEEE, James Weisheng Jiang, Member,
IEEE, Berker Bilgin, Senior Member, IEEE, and Ali Emadi, Fellow,
IEEE
Abstract—This paper proposes an advanced dynamic mod-elling
approach of the mutually-coupled switched reluctance motor (MCSRM)
in the dq reference system which can con-sider saturation,
cross-coupling and spatial harmonics. Different topologies and
their operating principles are investigated and an idealized
dq-model considering the inductance harmonics is derived. A dynamic
model is built based on flux-current lookup tables (LUTs) obtained
from finite e lement a nalysis ( FEA). A simplified m ethod t o i
nverse t he 2 D L UTs i s p roposed. A fast computation approach is
used to reduce the number of FEA simulations and calculation time
to obtain the LUTs. Motor dynamic performances at different speeds
are simulated by using the proposed dynamic model and the results
are investigated and verified by FEA. The motor dynamic behavior
can be accurately obtained in a short simulation time by using the
proposed approach. Experiments are carried out on a 12/8 MCSRM,
showing good accuracy of the proposed model.
Index Terms—Dynamic modelling, finite-element m ethod,
in-ductance, mutual coupling, switched reluctance machine.
I. INTRODUCTION
SWITCHED reluctance motors are used in many industries including
automotive, textile, aerospace and hand-heldpower tools due to the
obvious merits in terms of robustness,low cost, and reliability.
The absence of excitation coils and permanent magnets on the rotor
also makes it a good alternative for harsh environments, such as
mining. However, the application of conventional switched
reluctance machines (CSRM) is also limited by its unconventional
power converter, as well as high acoustic noise and vibration
levels resultingfrom its pulsed excitation. These problems can be
partially eliminated by using mutually coupled SRMs (MCSRMs),which
can be powered with the widely available conventional three-phase
inverters [1], [2].
The CSRM uses a decoupled concentrated phase windingso that
torque is generated due to the rate of change of the
self-inductance of the excitation phase only, which limitsthe
utilization rate of the electrical circuits. The MCSRMoperates
based on the variation of mutual-inductance between
Manuscript submitted November 10, 2016, revised April 28, 2017,
accepted June 19, 2017.
Jianning Dong is with Delft University of Technology, Delft
2628CD, The Netherlands (e-mail: [email protected]). He was with
the McMaster Automotive Resource Centre, McMaster University,
Hamilton, ON L8P 0A6, Canada.
Brock Howey, Benjamin Danen, Jianing Lin, James Weisheng Jiang,
Berker Bilgin and Ali Emadi are with the McMaster Automotive
Re-source Centre, McMaster University, Hamilton, ON L8P 0A6,
Canada.(e-mail: [email protected];
[email protected];[email protected]; [email protected];
[email protected]; [email protected]).
different phases by rewinding the CSRM [3]. Previous re-search
has shown that MCSRM is less sensitive to saturation than the CSRM
[4] and has better thermal performance [5]. Multiphysics numerical
modeling and experiments also show that the MCSRM has lower
vibrations and sound power levels [6]. The torque ripple of the
MCSRM is found to be relatively higher, which can be reduced by
using a single-layer full-pitched winding [7]. However, its longer
end-windings increase the motor volume and deteriorate the rotor
stiffness, so concentrated windings are used, as in [1]. The
toroidal winding SRM (TWSRM) is proposed as it offers additional
winding space, while retaining the benefits of mutual coupling
excitation [8], [9].
Dynamic simulation of the CSRM based on offline lookup table
(LUT) of phase flux l inkage w ith r espect t o phase current and
rotor position has been extensively used. The LUT can be generated
either from finite e lement analysis (FEA) or experiments. This
nonlinear modeling approach is very fast but can provide accurate
dynamic waveforms of the CSRM considering electrical circuit
switching and magnetic circuit saturation, which is essential for
controller design, noise prediction, loss analysis, and
hardware-in-loop (HIL) experiments [10], [11]. However, for the
MCSRM, the phase flux l inkage L UT i s e xtremely c omplex t o p
roduce since the phases are not decoupled anymore. Most of the
current literature in this area analyze the dynamic performance of
the MCSRM using circuit-coupled time-stepping FEA [6], [12],[13].
This approach can give accurate results, but once the switching and
sampling of the digital voltage source inverter (VSI) are
considered, the simulation time-step is forced to become very
small, resulting in extremely long simulation time that is not
suitable for a batched analysis. Decoupling between phases are
achieved in [1] by using the dq transformation. However only the
static model in the dq reference system is proposed, the dynamic
performances are still evaluated directly by FEA; the
cross-coupling between the d-axis and q-axis and the inductance
spatial harmonics are not discussed.
This paper proposes a dynamic modeling approach based on LUTs,
which includes the effects of cross-coupling, saturation, and
spatial harmonics. A contour line based 2D LUT inversion method and
a simplified L UT c onstruction m ethod t o reduce the calculation
time is proposed. Comparing to the circuit coupling FEA model [6],
[12], the proposed model can predict the dynamic current and torque
waveforms of the MCSRM ac-curately with reduced calculation time.
The proposed 2D LUT inversion method and the simplified LUT
construction method can reduce the model preparation time further.
The topologies
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-
2
TABLE IMAIN PARAMETERS OF THE INVESTIGATED SRM
Stator poles 24 Air-gap length [mm] 0.5Rotor poles 16 Peak power
[kW] 60Stator outer diameter [mm] 264 Peak torque [Nm] 200Axial
length [mm] 92 Base speed [rpm] 2000
d-axis
q-axis
A+
B+
B-
C-
C+A+
A-
B-
B+
C+A- C-
(a)
d-axis
q-axis
A+
B+
B-
C-
C+A-
A+
B+
B-
C-A- C+
(b)
A+
B+
B-
C-
C+
A-
A+
B+
B-
C-
A-
C+
q-axis
d-axis
(c)
Fig. 1. Flux paths with phase A aligned with d-axis (a) CSRM,
(b) MCSRM,(c) TWSRM.
of the CSRM, MCSRM, and TWSRM and their operatingprinciples are
presented in Section II. Then in Section III,the dynamic modeling
process, an efficient 2D LUT inversionmethod and a fast approach to
obtain the LUTs is proposed.Section IV compares the dynamic
performance of SRMs withdifferent topologies by using the proposed
dynamic model.The results are verified by FEA and investigated to
showthe capabilities of different SRM topologies. In section
V,experiments are carried out on a 12/8 MCSRM prototype.
Themeasured current waveforms are compared with the simulatedones
from the dynamic model.
II. MOTOR TOPOLOGIES AND INDUCTANCES
A 24/16 SRM for a hybrid electric vehicle (HEV) ap-plication is
investigated in this paper. Table I shows thedesign parameters
[14]. The geometry was optimized forCSRM. However, in this paper,
the same motor geometryand conductors-per-slot are applied to the
CSRM, MCSRM,and TWSRM for validation of the proposed approach. Fig.
1compares the winding configurations and flux paths whenphase A is
aligned with the d-axis. The MCSRM is obtainedfrom the CSRM by
reversing the polarity of opposite poles[15] while the TWSRM uses
the single-layer full-pitch toroidalwinding [16]. The maximum phase
current for all cases is setat 140 Arms, but the DC bus voltages
have been adjusted toensure that all motors achieve the same 2000
rpm base speed.
Indu
ctan
ce (m
H)
20
15
10
5
0
-5
-10
-15
CSRM LaCSRM MabMCSRM LaMCSRM MabTWSRM LaTWSRM Mab
0 40 80 120 160 200 240 280 320 360Rotor Position (Electrical
Angle)
Fig. 2. Self and mutual inductance of CSRM, MCSRM, and
TWSRM.
Unlike the CSRM, it is apparent that the flux generatedby one
phase is coupled with the other two phases for bothMCSRM and TWSRM
configurations in Fig. 1. Obviously,the magnetic motive force (MMF)
of the TWSRM is moreconcentrated than that of the MCSRM, meaning
that theTWSRM is more sensitive to saturation.
Fig. 2 shows the self and mutual inductance variations
withrespect to rotor position, calculated when phase A is
suppliedwith DC current. The self-inductance of the TWSRM is
veryhigh but with extremely low ripples, which can be attributedto
the compensation effect of different poles. The mutual-inductance
is nearly sinusoidal which can lead to lower torqueripple with
sinusoidal current excitation. All the inductanceschange with two
cycles during one electrical cycle of theMCSRMs, which means that
they contain only even spatialharmonics.
III. DYNAMIC MODEL CONSIDERING SPATIALHARMONICS
A. DQ-Model of MCSRM and TWSRM
For a symmetric three-phase MCSRM or TWSRM withbalanced
excitation and without third harmonics, the three-phase state
variables (flux linkage ψ, current i, and voltage u)can be
converted into the static αβ reference system by usingthe Clarke
Transformation. If expressed in space vectors, thevoltage equation
in the αβ reference system is
uαβ = Rsiαβ +dψαβdt
(1)
where uαβ , iαβ and ψαβ are space vectors of the voltage,
cur-rent, and flux linkage in the αβ reference system
respectively.Rs is the phase resistance, and t is time. The
relationshipbetween the values in the static αβ reference system
and thosein the synchronous rotating dq reference system is
ψαβ = ψdqejθ (2)
where ψdq is the flux linkage space vector in the dq
referencesystem, θ is the rotor position in electrical angles. If
nonlin-earity of the magnetic circuit is neglected for clarity, the
fluxlinkage equation in the static abc reference system is
ψabc = Labciabc (3)
where ψabc and iabc are the column vector of the three-phaseflux
linkage and current respectively, Labc is the inductance
-
3
matrix. As it has been investigated in Section II, Labc of
theMCSRM and TWSRM contain only even spatial harmonics:
Labc =∑n
LnC(nθ) MnCn(θ + 2π3 ) MnCn(θ − 2π3 )MnCn(θ + 2π3 ) LnCn(θ − 2π3
) MnCn(θ)MnCn(θ − 2π3 ) MnCn(θ) LnCn(θ +
2π3 )
n = 0, 2, 4, 6, . . .
(4)
where C is the cosine function, Ln and Mn are the amplitudesof
the n-th harmonics of the self-inductance and mutual-inductance
respectively. By applying the amplitude invariantParks
Transformation K to (3) and substituting (4), the fluxlinkage
equation in the dq reference system becomes
ψdq =KLabcK−1Kiabc
= Ldqidq(5)
where Ldq is the inductance matrix in the dq reference systemand
is solved as
Ldq =∑
v=0,6,12,...
[Ldd,v cos(vθ) Ldq,v sin(vθ)Ldq,v sin(vθ) Lqq,v cos(vθ)
](6)
where Ldd,v , and Lqq,v are the amplitudes of the v-th
harmon-ics of the d-axis and q-axis inductances, Ldq,v are those
ofthe cross-coupling inductances. We can see that Ldq
containsspatial harmonics whose orders are multiples of 6 for
threephase MCSRM and TWSRM. The electromagnetic torqueequation in
the dq reference system is
T =3
2pψdq × idq + p
∂Wco∂θ
(7)
where p is the number of rotor pole pairs, Wco is the
magneticcoenergy [17]. The first term of (7) is labelled as Tdq and
isproduced by the dq axis alignment, while the second term
islabelled as Tco and is produced by the magnetic coenergy. Forthe
linear case, Wco can be computed as
Wco =1
2iTdqLdqidq (8)
B. Consideration of Saturation Based on LUTs
From (1)-(2) and (5)-(7) we can come up with a dynamicmodel of
the MCSRM. However, even though the spatial har-monics and the
cross-coupling are included, the nonlinearityof the magnetic
circuit is ignored. When saturation is included,the interpretation
of inductances becomes complex since nowthey depend on both idq and
θ. For calculation consideringcircuit switching, the incremental
inductance makes matterseven more complicated [18]. Therefore, the
LUT approachwhich gives the flux-current relationship directly is
morecommonly used.
The upper part of Fig. 3 illustrates the procedure to obtainthe
flux linkage LUT from FEA. The excitation current pairsare defined
as a 2D table in the dq plane. The calculated idqvalues are then
input into the static FEA model. Both 3Dor 2D FEA model can be
used. However, for machines withsmall rotor diameter/length ratio,
neglecting of ending effect
id
i q
…
id
i q
dq
0°
3°
6°
9°
…
d
q
idq
0°
3°
6°
9°
…
d
q
vidq,v
0
+6
-6
+12
FEA
Invert
FFT
…
id
i q
vdq,v
FFT
0
+6
-6
+12
Fig. 3. Procedure to generate the dq LUTs of the flux linkage
ψdq,v(idq , θ)and current idq(ψdq , θ).
-18 -12 -6 0 6 12 18Harmonic Order
0
0.2
0.4
0.6
0.8
ψdq,v
(Wb)Fig. 4. Spatial harmonic spectrum of ψdq(θ).
in 2D models may make significant difference. An estimationof
ending leakage will be essential in that case [19].
By evaluating the FEA model along one electrical cycle,the flux
linkage ψdq for each current excitation idq at eachrotor position
are solved, composing a 3D table. By using FastFourier
Transformation (FFT), the amplitude of each spatialharmonic of the
flux linkage ψdq,v is solved and saved asanother 3D LUT.
Considering only the first 2 harmonics isaccurate enough, as shown
in Fig. 4. The flux linkage ψdq forthe current excitation idq is
then calculated as
ψdq ≈∑
v=0,±6,±12ψdq,v(idq)e
jvθ (9)
where the parentheses stand for linear interpolation from
thesaid 3D LUT. The LUT for the coenergy torque Tco canalso be
derived in the same procedure, except that Tco is notsolved
directly from FEA, but obtained by subtracting Tdqfrom T using (7).
The dynamic model can be built directlybased on the flux linkage
LUT ψdq,v(idq). Then the currentsare used as the inputs while the
voltages and torque arethe outputs. The voltage space vector uαβ is
solved fromthe derivative of ψαβ as is expressed in (1), where ψαβ
isobtained from (2) and (9). This kind of dynamic model iscalled an
incremental inductance model, which is direct but thedifferential
coefficients in it can amplify errors. Moreover, it ismore
convenient to use the voltage as inputs when consideringthe voltage
source inverter. Therefore, a dynamic model basedon the inverted
current LUT idq,v(ψdq) is proposed here, asis presented in Fig.
5.
The flux linkage-current relationship is inverted at each
rotorposition and then a FFT is used to obtain the spatial
harmonics,
-
4
3/2
×
uabc u
3/2p
T
+Tco
ejv
idq
+
e-j idq,v( dq)dq
ej
2/3iuvw i
Tdq
v=0v=±6v=±12
Tco,v(idq)v=6v=12v=18
ejvRe( )
Rs
-
pr(idq, )pr( s)
Fig. 5. Dynamic model of the MCSRM based on idq,v(ψdq).
d, q
-0.8
-0.6
-0. 4
-0.2
0
0
0.2
0. 2
0.4
0. 4
0.6
0. 6
0.8
0.8
-0.4
-0.3
-0.3
-0.2
-0.2
-0.1
-0.1
0
0
0.1
0.1
0.2
0.2
0.30.30.4
-15 -10 -5 0 5 10 15id
(A)
-15
-10
-5
0
5
10
15
i q(A
)
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
B(8.0,-3.2)
(Wb)
Fig. 6. Isolines of the ψdq,v(idq) table at θ = π.
as is illustrated in Fig. 3. Now ψαβ is obtained from
theintegral of uαβ , and the current is reconstructed from the
linearinterpolation of the LUT idq,v(ψdq)
idq ≈∑
v=0,±6,±12idq,v(ψdq)e
jvθ (10)
The proposed dynamic model contains only one integrator butno
differentiator, which increases the accuracy when comparedto the
conventional incremental inductance model [10], [18].
C. LUT Inversion
A method based on contour plotting is proposed to invertthe LUT.
From the original ψdq(idq) table at a specific rotorposition, for
each specific flux linkage vector ψdq , we candraw a pair of
contour lines for ψd and ψq respectively. Each ofthe two contour
lines is joint by (id, iq) points which contributeto the same ψd or
ψq value, as shown in Fig. 6. The verticallines are isolines of ψd
while the horizontal lines are isolinesof ψq . The intersection
points of the two sets of lines representthe (id, iq) vector which
contributes the (ψd, ψq) vector. Forinstance, the point B in Fig. 6
indicates that the flux linkage
-1 -0.5 0 0.5 1
d(Vs)
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
q(Vs)
id(A)
-15
-10
-5
0
5
10
15
Fig. 7. Contour plot of the inverted id(ψdq) table at θ = π.
vector ψdq = 0.4− 0.1j [Wb] is related to the current vectoridq
= 8.0− 3.2j [A].
MATLAB function contourc is used to generate the contourlines of
specific ψd and ψq respectively, which are representedby a group of
arrays. Then the intersection point of the twocontour lines is
obtained by solving linear equations repsentingeach line
segment.
In this manner, we can map the corresponding points inthe dq
current plane onto a rectangular grid in the dq fluxlinkage plane,
resulting in a idq(ψdq) table by determiningthe intersection points
of the contour lines of the ψd(idq) tableand those of the ψq(idq)
table. For grid points in the dq fluxlinkage plane near the
boundary, the intersection points for thecontour lines of ψd and ψq
may not fall into the rectangulargrid range. In this case, their
values can be estimated byextrapolation from the neighboring grid
points. Fig. 7 presentsthe contour plot of the inverted id(ψdq)
table from Fig. 6 usingthe proposed approach.
D. Fast Computation Approach
As is presented in the last subsection, a vast number ofFEA
calculations are needed to obtain the LUTs required bythe dynamic
model. If we consider only the fundamental andfirst two harmonics
(0, ±6, ±12) for both the idq LUT and theTco LUT, 60 rotor
positions in one electrical cycle for eachgrid point in the dq
current plane is usually a good choice.For a 8 × 8 idq grid shown
in Fig. 4, 4 × 8 × 30 = 960FEA calculation steps are needed even
though the symmetryis used. This paper proposes a fast approach to
obtain theessential FEA data with reduced calculation steps. Since
weconsider only harmonic with order 0, ±6 and ±12, the
currentvectors at rotor positions of θ = 0, π/12, π/8, π/6 and
π/4can be expressed by harmonics
idq(θi) =∑
v=0,±6,±12idq,ve
jvθi , θi ∈{0,π
12,π
8,π
6,π
4
}(11)
-
5
Then the five harmonics idq,v can be solved from the
fiveequations in (11) as
[idq,v]v=0,±6,±12 = Ki2v [idq(θi)] (12)
where Ki2v is a 5× 5 matrix as expressed in (13). The
sameapproach can be applied to the torque results to get the LUTof
Tco harmonics. By using this fast computation approach,for the same
idq grid shown in Fig. 4, only 4 × 8 × 5 =160 FEA calculation steps
are needed to consider the first 2harmonic orders. Moreover, the
computation time of FFT istotally omitted. The overall calculation
time to generate theLUTs can be reduced by more than 83%.
IV. DYNAMIC SIMULATION AND RESULTS INVESTITATION
A. Torque-Speed Characteristics
The torque-speed curve in Fig. 8 for the CSRM are gener-ated
with a dynamic model and use multi-objective (maximumaverage torque
and minimum RMS torque ripple) geneticalgorithm optimization to
determine the firing angles [14],[20]. The MCSRM and TWSRM machines
use the maximumtorque per ampere (MTPA) excitation angle predicted
fromthe flux LUTs and the torque calculated from (7). All
threemotors use the same dimensions to validate the
proposedapproach. The motor geometry was optimized for CSRM.To run
MCSRM and TWSRM at the same voltage for thesame speed, geometrical
modification will be needed. HereDC-link voltages of each SRM type
have been adjusted toobtain the same base speed for each machine;
650V for theCSRM, 850V for the MCSRM, and 2000V for the
TWSRM.Considering that the TWSRM has roughly double the coilspace
compared to the other machines, twice the number ofturns have been
used. The contour labels in Fig. 8 representthe RMS phase
current.
Below the base speed, the TWSRM clearly provides thehighest
average torque, followed by the CSRM, and finallythe MCSRM. This is
due to the flux-focusing effect of theTWSRM machine, which is
reflected by the higher peak fluxlinkage (see Fig. 11) and the
higher torque for a given RMScurrent (see Fig. 8 ).
At higher speeds, the CSRM has superior torque output,followed
by the TWSRM, and finally the MCSRM. This canbe explained by the
higher peak current capability for the fixedDC bus voltages. It is
important to note that the TWSRMperformance at high speeds is
expected to be diminished dueto the high self-inductances.
In this comparison, the MCSRM performs poorly at bothlow speeds
and high speeds. The self-inductance in theMCSRM is a parasitic
effect, as the main torque producingmechanism is dependent on
mutual-inductance. Thus, it isexpected that the performance of the
MCSRM would be morecompetitive if the geometry were optimized
specially for theproduction of mutual-inductance. This particular
geometryappears to be unable to provide significant varying
mutual-inductance (see Fig. 2).
B. Transient Results and Validation
The proposed dynamic modelling technique is applied tothe MCSRM
and TWSRM to simulate the transient current.A 1D LUT based model
presented in [14] is used to simulatetransient performance of the
CSRM. FEA models fed byvoltage source waveforms with switching are
used to validatethe dynamic models. All the three motors are
controlled withMTPA control. The hysteresis current controller with
the samesampling frequency and relative hysteresis band is used.
Thereference currents are set at 140Arms and the rotating speedsare
set to 2000 rpm and 10 000 rpm respectively. Fig. 9 andFig. 10
compares the transient current waveforms obtainedfrom the proposed
dynamic model and those obtained by FEA,illustrating good agreement
at both low speed and high speed.
The phase ψ − i curves are plotted from the instantaneousdynamic
model phase current and phase flux-linkage at differ-ent speeds
(labeled as “Phase U” in Fig. 11). The boundarycurves (labeled as
“Machine Capability” in Fig. 11) are plottedfrom the flux-linkage
LUTs for the CSRM with the maximumRMS current excitation which
displays the machines torquecapability envelopes.
The flux focusing effect in the TWSRM results in a higherpeak
flux-linkage and a higher co-energy area. It is clearthat the TWSRM
is not utilizing the full co-energy area withsinusoidal excitation
below the base speed. A different currentwaveform shape is needed
to utilize the co-energy capabilitymore effectively.
The power factor comparison between the three topologiesshows
that the CSRM clearly has the highest power factor,with 650V
required to obtain the same base speed as the othertwo designs
(requiring 850V for the MCSRM and 2000Vfor the TWSRM,
respectively). However, it is not fair tocompare these machines by
the power factor, as the geometryis optimized for the CSRM.
From Fig. 11 we can see that the CSRM has a moderateco-energy
area available for use when compared with the co-energy area of the
MCSRM and TWSRM. It is also moresaturated than the MCSRM machine
when the same current isexcited. It is expected that the MCSRM
could perform betterif the motor is further saturated.
At high speed (5000 rpm), the CSRM has the highest
peakflux-linkage per turn, and it also exhibits signs of
significantsaturation, even at this higher speed. This allows the
powerfactor of the machine to stay at a reasonable value, whilethe
MCSRM cannot maintain this, and the high speed torqueand power
factor are impacted. However, if the geometry isoptimized to
properly saturate the MCSRM, it would improveboth the power factor
and torque output. The TWSRM, withthe benefit of flux focusing
design, can maintain adequatesaturation at higher speeds. However,
its higher self-inductancemakes it much more sensitive to
saturation in the back ironcompared to the MCSRM and CSRM (see Fig.
1(c)), and itsdesign must take this into account.
V. EXPERIMENTAL RESULTS AND COMPARISON
The winding of a downscaled 12/8 SRM prototype [21]is modified
to the MCSRM configuration to validate the
-
6
Ki2v =1
8
1 + (1−
√2)j −1 + (1 +
√2)j −4j 1 + (1 +
√2)j −1 + (1−
√2)j
2 2j 0 −2 −2j2 2 0 2 22 −2j 0 −2 2j
1 + (√2− 1)j −1− (1 +
√2)j 4j 1− (1 +
√2)j −1 + (
√2− 1)j
(13)
2000 6000 10000 1400020
60
100
140
180
220
30 30
40
40
50
50
60
70
80
90100
110120
Speed (rpm)
Ave
rage
Tor
que
(Nm
) 130140
60
(a)
Ave
rage
Tor
que
(Nm
)30
4050
60
70
80
90
100
110120
130
140
20
60
100
140
180
220
2000 6000 10000 14000Speed (rpm)
(b)
30
40
50
6070
80
100
120
140
90
Ave
rage
Tor
que
(Nm
)
20
60
100
140
180
220
2000 6000 10000 14000Speed (rpm)
110130
(c)
Fig. 8. Torque-speed curve of the motors under different current
constraints: (a) CSRM, (b) MCSRM and (c) TWSRM.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8Time (ms)
0
100
200
300
Curre
nt (A
)
Model UModel VModel WFEA UFEA VFEA W
(a)
0 0.5 1 1.5 2 2.5 3 3.5Time (ms)
-300
-200
-100
0
100
200
300
Curre
nt (A
)
Model UModel VModel WFEA UFEA VFEA W
(b)
0 0.5 1 1.5 2 2.5 3 3.5Time (ms)
-300
-200
-100
0
100
200
300
Curre
nt (A
)
Model UModel VModel WFEA UFEA VFEA W
(c)
Fig. 9. Current waveforms comparison between dynamic models and
FEAat 2000 rpm (a) CSRM, (b) MCSRM and (c) TWSRM.
proposed model by experiments. Table II shows the keyparameters
of the MCSRM prototype.
The shaft of the MCSRM prototype is coupled to a brushlessdc
(BLDC) machine which works as load. Fig. 12 shows thedrive system.
The MCSRM works as a motor while the BLDCmachine works as a
generator. They are driven by the back
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Time (ms)
-20
0
20
40
60
80
Curre
nt (A
) Model UModel VModel WFEA UFEA VFEA W
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time (ms)
-40
-20
0
20
40
Curre
nt (A
) Model UModel VModel WFEA UFEA VFEA W
(b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7Time (ms)
-30
-20
-10
0
10
20
30
Curre
nt (A
)
Model UModel VModel WFEA UFEA VFEA W
(c)
Fig. 10. Current waveforms comparison between dynamic models and
FEAat 10 000 rpm (a) CSRM, (b) MCSRM and (c) TWSRM.
TABLE IIKEY PARAMETERS OF THE 12/8 MCSRM PROTOTYPE.
Phase resistance [Ω] 0.44 Rated Torque [Nm] 3.2Air-gap length
[mm] 0.3 Stator outer diameter [mm] 136Active length [mm] 70 Stator
teeth height [mm] 14.9Rotor teeth height [mm] 11 Stator inner
diameter [mm] 83.6
-
7
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flux
Lin
kage
(Wb)
Phase Machine Capability
( )
( )
(a) (b) CSRM 2000 RPM
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flux
Lin
kage
(Wb)
Phase Machine Capability
( )
(b) (c) CSRM 5000 RPM
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flux
Lin
kage
(Wb)
Phase Machine Capability
(c)
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flux
Lin
kage
(Wb)
Phase Machine Capability
( )
( )
(d) (e) MCSRM 2000 RPM
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6Fl
ux L
inka
ge (W
b)
Phase Machine Capability
( )
(e) (f) MCSRM 5000 RPM
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flux
Lin
kage
(Wb)
Phase Machine Capability
(f)
(g) TWSRM 1000 rpm
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flux
Lin
kage
(Wb)
Phase Machine Capability
( )
( )
(g) (h) TWSRM 2000 rpm
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Flux
Lin
kage
(Wb)
Phase Machine Capability
( )
(h) (i) TWSRM 5000 rpm
-200 -100 0 100 200Current (A)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6Fl
ux L
inka
ge (W
b)
Phase Machine Capability
(i)
Fig. 11. Phase flux linkage-current curves of each motorat
different speeds: (a) CSRM at 1000 rpm (b) CSRM at 2000 rpm (c)
CSRM at 5000 rpm. (d)MCSRM at 1000 rpm (e) MCSRM at 2000 rpm (f)
MCSRM at 5000 rpm. (g) TWSRM at 1000 rpm (h) TWSRM at 2000 rpm (i)
TWSRM at 5000 rpm.
to back converter. The control scheme is implemented in theDSP
board. Rotor position of the MCSRM is detected by aresolver and fed
to the DSP.
The sinusoidal pulse width modulation (SPWM) methodis used in
the dSPACE controller to generate PWM signalsfor the inverter.
Experiments are carried out with differentreference speeds and
currents. The same excitation angle, sam-pling frequency, switching
frequency and current controller PIparameters are used in the
experimental setup and the dynamicmodels, as shown in Table
III.
Phase current waveforms are measured and saved usingan
oscilloscope during the experiment. Fig. 13 comparesthe current
waveforms of the MCSRM prototype measured
TABLE IIIPARAMETERS IN THE EXPERIMENTAL DRIVE AND DYNAMIC
MODELS
Parameter ValueDC link voltage 135 VReference excitation angle
72.6◦Current sampling frequency 10 kHzSwitching frequency 20
kHzCurrent controller parameters kp = 1, ki = 50
in the experiments and those obtaiend from the dynamicmodel
simulations at 1000 rpm, 10A and 1500 rpm, 20Arespectively.
Apparently, the simulated waveforms are in closeagreement with the
experimental ones at both cases. The small
-
8
Fig. 12. Experimental setup based on the 12/8 MCSRM
prototype.
deviations between them can be attributed to the neglect
ofend-winding leakge in the dynamic model and the manufac-turing
imperfection of the prototype.
-15 -10 -5 0 5 10 15Time (ms)
-15
-10
-5
0
5
10
15
Curre
nt (A
)
ExperimentSimulation
(a)
-10 -8 -6 -4 -2 0 2 4 6 8 10Time (ms)
-30
-20
-10
0
10
20
30
Curre
nt (A
)
ExperimentSimulation
(b)
Fig. 13. Comparison between the experimental and simulated phase
currentwaveforms. (a)1000 rpm, 10 A (b) 1500 rpm, 20 A
VI. CONCLUSIONA model for dynamic simulation of the MCSRM
based
on the current LUTs has been proposed in this paper, whichcan
consider the magnetic nonlinearity, spatial harmonics andswitching
effects. A method based on contour lines has beenraised for the
inversion of 2D flux linkage LUTs to currentLUTs. A fast
computation approach has been proposed toreduce the number of FEA
calculations and time to composethe LUTs.
The proposed model has been applied to both a conventional24/16
MCSRM and a TWSRM for validation based on ageometry which was
optimized for CSRM. Their dynamic per-formances have been
calculated using the proposed approach.FEM simulations coupled with
voltage source excitations have
been involved to validate the results, showing good
agreementsboth for current waveforms and torque waveforms.
Performance of the MCSRM and the TWSRM includingtorque-speed
curves and ψ − i curves under various speedshave been calculated
and compared with those of the CSRMusing the proposed model.
Results have shown that the CSRMgenerates highest torque at the
high speed region while theTWSRM have the highest torque capability
at the low speedregion, even though the co-energy capability of
which is notfully used when excited by sinusoidal current. To fully
take theadvantages of the MCSRM and the TWSRM, special
designcriteria which are different from the CSRM should be
used.
A drive system based on a 12/8 MCSRM prototype is usedfor
experimental validation of the proposed dynamic model.Experimental
results show that the dynamic model can predictthe current
waveforms accurately.
The works presented in this paper may be used for loss anal-ysis
and controller design for the MCSRM and TWSRM. Theconclusions give
some guidlines for the design of MCSRMand TWSRM.
ACKNOWLEDGMENTThis research was undertaken in part, thanks to
funding
from the Canada Excellence Research Chairs Program, andNatural
Sciences and Engineering Research Council of Canada(NSERC). The
authors gratefully acknowledge Powersys So-lutions for their
support with JMAG software in this research.
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Jianning Dong (IEEE S’10-M’17) received the B.S.and Ph.D.
degrees in electrical engineering fromSoutheast University,
Nanjing, China, in 2010 and2015, respectively. He is an Assistant
Professor atthe Delft University of Technology since 2016. Be-fore
joining TU Delft, he was a post-doc researcherat McMaster
Automotive Resource Centre (MARC),McMaster University, Hamilton,
Ontario, Canada.His main research interests are design,
modellingand control of electromechanical systems.
Brock Howey received his B.Tech degree in au-tomotive and
vehicle technology from McMasterUniversity (Hamilton, Canada) in
2014. During thisprogram he held various co-op positions relating
tosustainable energy production and vehicle engineer-ing. Brock is
currently a Ph.D candidate as part ofthe Canada Excellence Research
Chair (CERC) inhybrid powertrain program, at the McMaster
Auto-motive Resource Center (MARC). He is stronglypassionate about
vehicles and a firm believer inelectric vehicles. His research is
currently directed
to switched reluctance motor design for vehicle
applications.
Benjamin Danen received his Bachelor of Engi-neering and Masters
of Engineering in ElectricalEngineering from McMaster University in
2014 and2016. Ben joined MARC in September 2014 asa Research
Assistant for the Canada ExcellenceResearch Chair in Hybrid
Powertrain Program wherehe currently serves as a Research Engineer.
His areasof research include motor control, system modelingand
power electronics.
Jianing Lin (IEEE S’13) received her B.Eng. degreein electrical
engineering from Southeast Universityin 2011 and M.A.Sc. in
mechanical engineeringfrom McMaster University in 2014. She was
therecipient of national scholarship in China, as anoutstanding
graduate receiving several honors in thefields of electrical
engineering and business. Shehas experience in both electrical and
mechanicalengineering. Currently, she is a Ph.D. candidate inthe
Department of Electrical and Computer Engi-neering at McMaster
University. In 2009, she began
her research in electric machines and power electronics. Her
main researchinterests are power electronics, electric machine
design and controller designfor applications in electric vehicles
and home applications. She is currentlyfocused on high-speed
switched reluctance motor designs and control. Thesedesigns are
suitable for electric bicycles and home applications.
James Weisheng Jiang (IEEE S’12-M’16) receivedhis Bachelor’
Degree in vehicle engineering fromCollege of Automotive
Engineering, Jilin University,China, in 2009. He worked as a
research assistant atthe Clean Energy Automotive Engineering
ResearchCenter, Tongji University, China, from 2009 to 2011.He
received his Ph.D. in Mechanical Engineeringfrom McMaster
University in April 2016. Currently,he is working as a Principal
Research Engineer inthe Program of the Canada Excellence
ResearchChair in Hybrid Powertrain at McMaster Automotive
Resource Centre (MARC). His main research interests include
design ofelectric vehicle and hybrid electric vehicle powertrains,
design of interiorpermanent magnet motor, design and control of
switched reluctance motor,noise and vibration analysis of traction
motors.
Berker Bilgin (IEEE S’09-M’12-SM’16) is theResearch Program
Manager in Canada ExcellenceResearch Chair in Hybrid Powertrain
Program inMcMaster Institute for Automotive Research andTechnology
(MacAUTO) at McMaster University,Hamilton, Ontario, Canada. He
received his Ph.D.degree in Electrical Engineering from Illinois
In-stitute of Technology in Chicago, Illinois, USA.He is managing
many multidisciplinary projects onthe design of electric machines,
power electronics,electric motor drives, and electrified
powertrains. Dr.
Bilgin was the General Chair of the 2016 IEEE Transportation
ElectrificationConference and Expo (ITEC’16). He is now pursuing
his MBA degree inDeGroote School of Business at McMaster
University.
http://macsphere.mcmaster.ca/handle/11375/19087http://macsphere.mcmaster.ca/handle/11375/19087
-
10
Ali Emadi (IEEE S’98-M’00-SM’03-F’13) receivedthe B.S. and M.S.
degrees in electrical engineeringwith highest distinction from
Sharif University ofTechnology, Tehran, Iran, in 1995 and 1997,
respec-tively, and the Ph.D. degree in electrical engineer-ing from
Texas A&M University, College Station,TX, USA, in 2000. He is
the Canada ExcellenceResearch Chair in Hybrid Powertrain at
McMasterUniversity in Hamilton, Ontario, Canada. Beforejoining
McMaster University, Dr. Emadi was theHarris Perlstein Endowed
Chair Professor of Engi-
neering and Director of the Electric Power and Power Electronics
Center andGrainger Laboratories at Illinois Institute of Technology
in Chicago, Illinois,USA, where he established research and
teaching facilities as well as coursesin power electronics, motor
drives, and vehicular power systems. He was theFounder, Chairman,
and President of Hybrid Electric Vehicle Technologies,Inc. (HEVT) –
a university spin-off company of Illinois Tech. Dr. Emadi hasbeen
the recipient of numerous awards and recognitions. He was the
advisorfor the Formula Hybrid Teams at Illinois Tech and McMaster
University,which won the GM Best Engineered Hybrid System Award at
the 2010,2013, and 2015 competitions. He is the principal
author/coauthor of over 400journal and conference papers as well as
several books including VehicularElectric Power Systems (2003),
Energy Efficient Electric Motors (2004),Uninterruptible Power
Supplies and Active Filters (2004), Modern Electric,Hybrid
Electric, and Fuel Cell Vehicles (2nd ed, 2009), and Integrated
PowerElectronic Converters and Digital Control (2009). He is also
the editor ofthe Handbook of Automotive Power Electronics and Motor
Drives (2005)and Advanced Electric Drive Vehicles (2014). Dr. Emadi
was the InauguralGeneral Chair of the 2012 IEEE Transportation
Electrification Conference andExpo (ITEC) and has chaired several
IEEE and SAE conferences in the areasof vehicle power and
propulsion. He is the founding Editor-in-Chief of theIEEE
Transactions on Transportation Electrification.
IntroductionMotor Topologies and InductancesDynamic Model
Considering Spatial HarmonicsDQ-Model of MCSRM and
TWSRMConsideration of Saturation Based on LUTsLUT InversionFast
Computation Approach
Dynamic Simulation and Results InvestitationTorque-Speed
CharacteristicsTransient Results and Validation
Experimental Results and
ComparisonConclusionReferencesBiographiesJianning DongBrock
HoweyBenjamin DanenJianing LinJames Weisheng JiangBerker BilginAli
Emadi
