Equivalent circuit for single sided linear induction motors
-
Upload
jayasankarjnair -
Category
Documents
-
view
232 -
download
0
Transcript of Equivalent circuit for single sided linear induction motors
-
8/10/2019 Equivalent circuit for single sided linear induction motors
1/14
2410 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 6, NOVEMBER/DECEMBER 2010
Equivalent Circuits for Single-SidedLinear Induction Motors
Wei Xu,Member, IEEE, Jian Guo Zhu,Senior Member, IEEE, Yongchang Zhang,Member, IEEE,Zixin Li,Student Member, IEEE, Yaohua Li, Yi Wang, Youguang Guo, Senior Member, IEEE, and Yongjian Li
AbstractSingle-sided linear induction motors (SLIMs) havelately been applied in transportation system traction drives, par-ticularly in the intermediate speed range. This is because they havemerits, such as the ability to exert thrust on the secondary withoutmechanical contact, high acceleration or deceleration, less wheelwear, small turning circle radius, and flexible road line. The theoryof operation for these machines can be directly derived fromrotary induction motors (RIMs). However, while the cut-openprimary magnetic circuit has many inherent characteristics of theRIM equivalent circuits, several issues involving the transversaledge and longitudinal end effects and the half-filled slots at the
primary ends need to be investigated. In this paper, a T-modelequivalent circuit is proposed which is based on the 1-D magneticequations of the air gap, where half-filled slots are consideredby an equivalent pole number. Among the main five parameters,namely, the primary resistance, primary leakage inductance, mu-tual inductance, secondary resistance, and secondary inductance,the mutual inductance and the secondary resistance are influencedby the edge and end effects greatly, which can be revised byfour relative coefficients, i.e., Kr, Kx, Cr, and Cx. Moreover,two-axis equivalent circuits (dq or) according to the T-modelequivalent circuit are obtained using the power conversion rule,which are analogous with those of the RIM in a two-axis coor-dinate system. The linear induction motor dynamic performance,particularly the mutual inductance and the secondary resistance,can be analyzed by the four coefficients. Experimental verification
indicates that both the T-model and the new two-axis circuits arereasonable for describing the steady and dynamic performanceof the SLIM. These two models can provide good guidance forthe electromagnetic design and control scheme implementation forSLIM applications.
Index TermsCoefficient, control scheme, dynamic perfor-mance, equivalent circuit, longitudinal end effect, mutual induc-tance, secondary resistance, single-sided linear induction motor(SLIM), steady performance, transversal edge effect.
Manuscript received February 9, 2010; revised March 31, 2010; acceptedApril 9, 2010. Date of publication September 7, 2010; date of current version
November 19, 2010. Paper 2010-EMC-042.R1, presented at the 2009 IEEEEnergy Conversion Congress and Exposition, San Jose, CA, September 2024,and approved for publication in the IEEE TRANSACTIONS ON INDUSTRYAPPLICATIONS by the Electric Machines Committee of the IEEE IndustryApplications Society. This work was supported in part by the Chinese Academyof Sciences, China, under Grant 400012414-4.
W. Xu, J. G. Zhu, Y. C. Zhang, Y. Wang, Y. G. Guo, and Y. J. Liare with the School of Electrical, Mechanical and Mechatronic Systems,University of Technology Sydney, Sydney, N.S.W. 2007, Australia(e-mail: [email protected]; [email protected]; [email protected];[email protected]; [email protected]; [email protected]; [email protected]).
Z. X. Li and Y. H. Li are with the Institute of Electrical Engineering, ChineseAcademy of Sciences, Beijing 100864, China (e-mail: [email protected]).
Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIA.2010.2073434
NOMENCLATURE
1 Length of the entry-end-effect wave penetrationcoefficient.
2 Length of the exit-end-effect wave penetrationcoefficient.
Volume conductivity of the secondary sheet.e Surface conductivity of the secondary sheet. Primary winding short pitch.Cu Copper resistivity.
Fe Iron resistivity.0 Air permeability.Fe Iron permeability.e Angular frequency of the power supply.r Angular frequency of the rotor.s Slip angular frequency.g Flux root-mean-square value per pole pair. Pole pitch.e Half-wave length of the end-effect wave.a1 Half the width of the primary lamination.c Half the width of the secondary sheet overhanging the
primary lamination.
c2 Width of the secondary sheet.
d Secondary sheet thickness.dFe Depth of the flux density into the back iron.fs Primary frequency.ge Equivalent air-gap length.gm Mechanical air-gap length.hb Height of the secondary back iron.hy Primary height.l Primary lamination width.lav Half the average length of the primary winding coil.m1 Phase number.p Actual number of primary pole pairs.pe Equivalent number of primary pole pairs.
q Number of coil sides per phase per pole.s Per-unit slip.sf Slip frequency.t1 Pitch of the primary teeth.t2 Width of the primary teeth.B Magnetic flux density.Bg Air-gap flux density.Cr Transversal-edge-effect coefficient to the secondary
resistance.
Cx Transversal-edge-effect coefficient to the mutualinductance.
E Electrical field intensity.Fx
Thrust.
0093-9994/$26.00 2010 IEEE
-
8/10/2019 Equivalent circuit for single sided linear induction motors
2/14
XU et al.: EQUIVALENT CIRCUITS FOR SINGLE-SIDED LINEAR INDUCTION MOTORS 2411
Fig. 1. Structure of a SLIM.
G Goodness factor.H Magnetic field intensity.Ie Field current in the T-model circuit.Ids Field current in two-axis circuits.
Iqs Thrust current in two-axis circuits.Is Primary phase current.J Current density.Klam Silicon-steel stacking factor.Kr Longitudinal-end-effect coefficient to the secondary
resistance.
Kx Longitudinal-end-effect coefficient to the mutualinductance.
Kw1 Primary winding coefficient.Llr Secondary leakage inductance.Lls Primary leakage inductance.Lm Mutual inductance in the two-axis circuit.Lm1 Mutual inductance in the per-phase circuit.
Lmc Rectified mutual inductance in the two-axis circuit.Lr Secondary inductance in the two-axis circuit.Ls Primary inductance in the two-axis circuit.P2 Secondary active power.PFe Iron loss.Q3 Air-gap reactive power.RFe Iron resistance.Rr Secondary resistance.Rrc Rectified secondary resistance.Rs Primary resistance.SCu Effective cross-sectional area of the winding conductor.Vs Primary synchronous velocity.
W1 Number of primary winding turns in series.
I. INTRODUCTION
THE STRUCTURE diagram of a single-sided linear induc-
tion motor (SLIM) is shown in Fig. 1. The SLIM primary
can be simply regarded as a rotary cut-open stator and then
rolled flat [1]. The secondary, similar with the rotary induction
motor (RIM) rotor, often consists of a sheet conductor, such as
copper or aluminum, with a solid back iron acting as the return
path for the magnetic flux. The thrust corresponding to the RIM
torque can be produced by the reaction between the air-gap flux
density and the eddy current in the secondary sheet [2], [3].
A train driven by a SLIM, also called as a linear metro, hasbeen paid attention by academia and industry for more than
Fig. 2. Simple vehicle system diagrams propelled by the SLIM. (a) LIMinstalled under vehicle redirector. (b) Drive system.
20 years for its direct propulsive thrust which is dependent of
the friction between the wheel and the rail, for its smaller cross-
sectional area for the requirement of a tunnel, for its smaller
turning radius, for its larger acceleration, and for its stronger
climbing ability compared with that of the RIM [4], [5]. By
now, there are more than 20 commercial linear metro lines withmore than 400 km in the world, such as the Kennedy airline in
America, the linear metro in Japan, the Vancouver light train
in Canada, the Kuala Lumpur rapid train in Malaysia, and the
Guangzhou subway line 4 and Beijing airport rapid transport
line in China. The typical drive SLIM structure and system
diagrams are illustrated in Fig. 2. It can be seen that the SLIM
primary is hanged below the redirector, which is supplied by
a three-phase inverter on the vehicle. The secondary is flatted
on the railway track, which often consists of a 5-mm-thick
copper/aluminum conductance sheet and almost 20-mm-thick
back iron. When the primary three-phase windings are inputted
to an ac from a vehicular converter, they can build up air flux
linkage and induce eddy current in the secondary sheet. This
eddy current will react with the aforementioned air-gap flux
linkage so as to produce horizontal electromagnetic thrust that
can drive the vehicle forward directly without friction between
the wheel and the track [5], [6].
The SLIM special structure means that its performance is a
little different from that of an RIM. As we know, in the RIM,
an accurate equivalent circuit model can be derived easily by
simplifying the geometry per pole. Unfortunately, it is not as
straightforward to gain the equivalent circuit for a SLIM mainly
for the following three causes [7][22].
1) As the SLIM primary moves along the secondary sheet,
a new flux is continuously developed at the primaryentrance side while the air-gap flux disappears quickly
-
8/10/2019 Equivalent circuit for single sided linear induction motors
3/14
2412 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 6, NOVEMBER/DECEMBER 2010
Fig. 3. Profiles of secondary eddy current and air-gap flux linkage.
Fig. 4. Air-gap flux density amplitude alongz-axis.
at the exit side. By the influence from the sudden gen-
eration and disappearance of the air-gap penetrating flux
density, an amount of eddy current in antidirection to the
primary current will occur in the secondary sheet, which
correspondingly affects the air-gap flux profile along the
longitudinal direction (x-axis) as illustrated in Fig. 3.This phenomenon is called the longitudinal end effect,
which would cause extra copper loss and reduce the effec-
tive mutual inductance as the velocity goes up. In the end,
the effective electromagnetic thrust will decrease a little
because of the attenuating air-gap average flux linkage,
not similar to the constant flux linkage in the RIM whichis equal to the value of the exit position in Fig. 3 [3].
2) The different width between the primary lamination and
the secondary sheet can result in nonuniform flux density
distribution, i.e., the middle area flux density along the
z-axis is smaller than that of the terminal as indicatedin Fig. 4, where c is a half of the primary width lessthe secondary, l is the primary lamination thickness,Bx0 is the flux density amplitude with an equal widthbetween the primary and the secondary, and Bx is theflux density amplitude with a different width between the
primary and the secondary. The phenomenon is named
as the transversal edge effect, which may increase
the secondary equivalent resistivity and bring an inverseeffect to the neat thrust.
3) For the cut-open primary magnetic circuit, there exist
half-filled slots in the primary ends. Hence, the three-
phase magnetic circuits are not symmetric with each
other. The three-phase currents are not completely bal-
anced even when excited by three-phase balanced volt-
ages. The half-filled slots will affect the air-gap flux
density distribution so as to result in some alteration in themutual inductance, leakage inductance, and secondary
equivalent resistance.
During the past several decades, plenty of papers on the
SLIM performance analysis, involving steady and dynamic
states, have been available. Reference [1], based on the RIM
T-model equivalent circuit, provided one simple and useful
function expression f(q) according to the secondary eddycurrent average value by an energy conversion balance theorem.
Supposing that the air-gap flux linkage increased in one expo-
nential function form from the entrance end to the exit end, it
is affected by the SLIM operating speed, secondary resistance,
secondary inductance, and some other structure parameters,
such as the primary length, and so on. The per-phase simplified
model can be used to predict the SLIM output thrust, efficiency,
and power factor conveniently. Reference [3], according to the
result in [1], deduced the two-axis models (dqor ), whichcan be applied in vector control or direct torque control to
predict the SLIM dynamic performance. However, the deriva-
tion process of the revised function f(q) is very coarse onthe assumption that the eddy current in the secondary sheet
decreases from maximum to zero by exponential attenuation
only in the primary length range. It only considers the mutual
inductance influenced by the eddy current, disregarding its
effect on the secondary resistance. The analytic results, such
as mutual variance, suffer increasing error compared with themeasurement as the velocity goes up. Reference [10] derived
an equivalent circuit model from the pole-by-pole method based
on the winding functions of the SLIM primary windings. One
set of tenth-order differential equations is derived to describe a
basic model for a four-pole machine, and a higher order system
of equations can be provided as the pole number goes up.
Reference [11], based on the winding function method, divided
the SLIM air-gap flux density into three components, i.e.,
primary fundamental, secondary fundamental, and secondary
eddy-current parts. Then, it derived these three-group function
expressions and further achieved inductance, secondary resis-
tance, and other parameters so as to analyze the SLIM per-formance. The winding function method can study the steady,
transient, and dynamic performance. However, the expressions
of the secondary winding function distributions are achieved
by some approximate hypothesis. A field theory was utilized
in developing the lumped-parameter linear induction motor
(LIM) model [6] in which the end effect, the field diffusion in
the secondary sheet, the skin effect, and the back-iron satura-
tion were considered. However, the resultant lumped-parameter
models look very complicated for the practical use of modeling
and control. Gieras et al. developed an equivalent circuit by
supposing that the synchronous wave and the pulsating wave
were caused by the end effect [7]. However, more experience
should be made to verify the simulation investigation. Severaldifferent models were developed from the electromagnetic
-
8/10/2019 Equivalent circuit for single sided linear induction motors
4/14
XU et al.: EQUIVALENT CIRCUITS FOR SINGLE-SIDED LINEAR INDUCTION MOTORS 2413
Fig. 5. Structure and 1-D analytical model of SLIM. (a) Physical structure.(b) Longitudinal side view. (c) Transversal side view.
relation in the air gap through a Fourier-series approach [4],
[9], [16]. However, it requires more substantial computing time
to gain some useful data, whose accuracy depends closely on
many initially given parameters. If some key values are not
initialized rationally, the final solution cannot be succeeded for
nonconvergence.
This paper, based on the per-phase T-model equivalent circuitderived from 1-D flux density equations in [2], makes a steady
performance analysis of several SLIMs. Then, it deduces two-
axis equivalent circuits based on the per-phase equations so as
to study the SLIM dynamic performance. It is organized as
follows. The SLIM T-model equivalent circuit is described in
Section II, and its verification is described in Section III. Then,
the two-axis equivalent circuits are explained in Section IV.
Section V discusses four coefficients in detail. The simula-
tion and experimental verifications of the two-axis circuits are
given out in Section VI. Finally, this paper is summarized in
Section VII. The derivations on how to get the four coefficients
are indicated in the Appendix.
The notation used in this paper is fairly standard. The vector
is expressed in the form of A= ix + jy + kz, wherei, j,and k are the notations of the x-, y-, and z-axis directions,respectively. The complex number is expressed in the form of
A= x +jy , wherex is the real part, y is the imaginary part,andj is the notation of the imaginary part.
II. T-MODELE QUIVALENTC IRCUIT
Fig. 5 shows the SLIM longitudinal side view, the transversal
side view, and the 1-D analytical model. In Fig. 5(a),v2 is theprimary moving speed. In Fig. 5(b),j1is the primary equivalent
current, andj2is the secondary equivalent current. In Fig. 5(c),a1 is the half width of the primary lamination. In terms of 1-D
Fig. 6. T-model equivalent circuit of SLIM.
analysis, we can calculate the phase currents and the excita-
tion voltages. The air-gap flux linkage can be obtained using
Maxwells field equations and solved using the complex power
method with a conformal transformation which considers the
effects of the half-filled slots, magnetic saturation, and back-
iron resistance. Using the equal complex power relationship
between the magnetic field and the electrical circuit, we can
obtain several circuit parameters, such as mutual inductanceLm1, secondary resistanceRr, primary leakage inductanceLls,secondary leakage inductance Llr, longitudinal-end-effect coef-ficientsKr and Kx, and transversal-edge-effect coefficientsCrandCx. The comprehensive derivations of the four coefficientscan be referred to in the Appendix. The T-model equivalent
circuit is shown in Fig. 6, where the secondary equivalent resis-
tanceRr consists of the secondary conducting sheet resistanceR2Sheetand the secondary back iron R2Back. Some brief con-clusions are summarized in the following paragraphs [2], [9].
The longitudinal-end-effect coefficients Kr and Kx are de-noted by
Kr= sG
2pe
1 + (sG)2C21 + C
22
C1(1)
Kx= 1
2pe
1 + (sG)2C21 + C
22
C2(2)
where C1 and C2 are the functions of slip s and goodnessfactor G, is the primary pole pitch, and pe is the numberof equivalent pole pairs. For the existence of half-filled slots
in the primary ends, the expression of the primary equivalent
current sheet J1 can be divided into three regions, i.e., en-trance half-filled, full-filled, and exit half-filled slots. Then,
the expressions of the air-gap flux density can be gained.According to the electric machinery theory and complex power
conversion algorithm, the air-gap effective electromotive force
Em, air-gap reactive power Q3, secondary active power P2,mutual inductance, and secondary resistance can be deduced
by taking the half-filled slots into consideration. By the careful
comparison of those expressions without half-filled slots, the
number of equivalent pole pairspeis expressed by
pe = (2p 1)2
4p 3 + /(m1q) (3)
wherep is the actual number of the pole pairs, m1is the number
of primary phases,qis the number of coil sides per phase perpole, andis the length of the short pitch.
-
8/10/2019 Equivalent circuit for single sided linear induction motors
5/14
2414 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 6, NOVEMBER/DECEMBER 2010
The transversal-edge-effect coefficients Cr and Cx aregiven by
Cr =sG
R2e[T] + I
2m[T]
Re[T]
(4)
Cx =R2e[T] + I
2m[T]
Im[T] (5)
whereTis the function of the slip, goodness factor, and motorstructure parameters andReand Im are the real and imaginaryparts of the complexT, respectively. Here,Tis expressed by
T =j
2 + (1 2)
0.5lth(0.5l)
(6)
whereis the ratio ofcto andand can be obtained by
= 1
1 + 1
th(0.5l)th [0.5K(c2 l)] (7)
R2 = 11 +jsG (8)
whereKis the function of the slip and motor structure parame-ters and c2 is the width of the secondary sheet in the value of(l+ 2c).
The five parameters in the T-circuit, namely, the primary
resistanceRs, primary leakage inductance Lls, secondary re-sistance Rr, secondary leakage inductance Llr, and excitinginductanceLm1can be calculated as follows.
The primary resistanceRsis
Rs = Cu 2lavW1/SCu (9)
where Cu is the resistivity of copper, lav is half the averagelength of the primary winding coil, W1is the number of turns ofthe primary per phase winding in series, and SCuis the effectivecross-sectional area of the primary winding conductor.
The primary leakage inductanceLls is
Lls = 0.025W21
lq
sp
+t+ e+ d
pe
(10)
wheres is the primary slot leakage magnetic conductance,tis the primary tooth leakage magnetic conductance, e is theprimary winding end leakage magnetic conductance, andd is
the primary harmonic leakage magnetic conductance.The secondary resistance is composed of those of the con-
ducting sheet and back iron because the flux can penetrate
through the aluminum or copper sheet [6] and then enter the
back iron. The depth of the flux density into the back iron dFeis
dFe =
2FeseFe
(11)
whereFe is the back-iron resistivity, Fe is the permeabilityof the back iron, and e is the primary synchronous angularfrequency. The resistance of the secondary conducting sheet
R2Sheetis
R2Sheet= 4m1Sheet(W1Kw1)2
2pe
ld
(12)
where Sheet is the resistivity of the secondary conductancesheet andKw1is the primary winding coefficient.
The resistance of the secondary back iron R2Backis
R2Back = 4m1Fe(W1Kw1)
2
2pe
ldFe
. (13)
Therefore, the secondary equivalent resistanceRris
Rr= R2SheetR2BackR2Sheet+ R2Back
. (14)
The secondary leakage reactance is
Llr = Rr2fss
B1sh(2Kd) (15)
wherefsis the primary frequency andB1is the function of theslip, primary frequency, and machine structure parameters.
The exciting inductance is
Lm1= 4m10(W1Kw1)2 lVs
42fsgepe (16)
whereVsis the synchronous velocity of the primary side and geis the equivalent air-gap width.
The iron loss in the SLIM is composed of the primary yoke,
primary tooth, and secondary back-iron losses. These three
parts can be calculated as follows.
The primary yoke iron lossPFeyis
PFey = P10/50B2y
fs50
1.3Wy . (17)
The primary tooth iron lossPFetis
PFet = P10/50B2t
fs50
1.3Wt. (18)
The secondary back-iron lossPFebis
PFeb= P10/50B2b
sf
50
1.3Wb. (19)
Hence, the total iron loss PFeis
PFe= PFet+ PFey+ PFeb. (20)
In (16)(19), P10/50 is the iron loss value under 1.0 T and50 Hz; By, Bt, and Bb are the primary yoke, primary tooth,and secondary back-iron flux densities, respectively; Wy ,Wt,and Wb are the primary yoke, primary tooth, and secondaryback-iron weights, respectively; and sf is the slip frequencyin the secondary. According to the electromagnetic design
methods in [23] and [24], By, Bt, and Bb can be calculatedas follows:
By =
2g/(2lKlamhy) (21)
Bt= Bgt1/(Klamt2) (22)
Bb= g/(c2Klamhb) (23)
where gis the flux root-mean-square value per pole pair, Klamis the silicon-steel stacking factor, hy is the primary height,Bg
-
8/10/2019 Equivalent circuit for single sided linear induction motors
6/14
XU et al.: EQUIVALENT CIRCUITS FOR SINGLE-SIDED LINEAR INDUCTION MOTORS 2415
TABLE IDIMENSIONS OF THESLIM
is the air-gap flux density, t1 is the pitch of the primary teeth,t2 is the width of the primary teeth, and hbis the height of thesecondary back iron.
The iron loss resistance RFe in series with the excitationbranch can be calculated by
RFe = PFe/I2e (24)
whereIeis the field current.After the approximate regulation by the four coefficients
and the equivalent number of pole pairs, the SLIM T-model
equivalent circuit indicated in Fig. 6 is similar to that of the
RIM. The influence by the longitudinal and transversal end
effects and half-filled slots can be estimated by the corre-
sponding coefficients. The revised SLIM model based on the
equivalent pole pairs can be regarded as a three-phase magnetic
circuit symmetry structure. The characteristics of the SLIM are
reasonably considered in the circuit. When Kr = Kx= Cr =Cx = 1, i.e., when the longitudinal and transversal effects areneglected, the circuit can be simplified as the same as that
of the RIM. Therefore, it is very convenient to analyze the
performance of the SLIM in a similar way as that of the RIM
based on the T-model circuit [2], [9], [14], [15].
III. VERIFICATION OFT-M ODELE QUIVALENTC IRCUIT
In order to validate the T-model equivalent circuit in Fig. 6,
a lot of steady state performance analyses have been made
in [9]. Here, two kinds of SLIM performance evaluations are
given out, i.e., the 12 000 Japanese SLIM [4] and one arc SLIM
[9]. Their main dimensions are shown in Table I. In linear
metro drive machines, the magnetic saturation, particularly in
the low velocity or large slip region, might affect the equivalentlength of the electromagnetic air gap and further bring influence
on some parameters, such as the mutual inductance, etc. This
phenomenon can be described by one magnetic saturation
coefficientKsdefined as
Ks = MMFsum/MMFg (25)
where MMFsumis the total magnetomotive force (MMF) in one
pole pair, including the primary yoke MMFy , the primary teeth
MMFt, the air gap MMFg, and the secondary back iron MMFb.
The equivalent length of the electromagnetic air gapge canbe calculated by
ge = KsKc(gm+ d) (26)
Fig. 7. Magnetic saturation coefficient versus velocity.
where Kc is the Cater coefficient relative with different slotstructures [25]. As the level of the magnetic saturation in-
creases, the equivalent air-gap length ge will become larger.Take the 12 000 SLIM, for example. The Ks is about 1.26 at5 km/h and gradually decreased as the velocity goes up. It is
close to 1.01 above the base speed. Hence, the length of gedecreases approximately from 20 to 16.2 mm between 5 and
70 km/h, which is shown in Fig. 7.
Fig. 8 shows the 12 000 SLIM performance curves, involving
the thrust, slip frequency, power factor, efficiency, and current.
The base speed is 40 km/h. There are two working modes
in the operating regions, i.e., the constant current below the
base speed and the constant voltage beyond the base one.
Normally, the thrust below the base under a constant phase
current will decrease gradually for the longitudinal end effect.
In order to achieve a greater thrust by the drive requirement,it is necessary to increase the phase current to compensate
for the fading thrust [4]. The thrust and slip frequency in
Fig. 8(a) are mainly decided by the vehicle operating require-
ments, such as the acceleration, deceleration, road conditions,
windage resistance, etc. From Fig. 8(b)(d), the simulation
results about the power factor, efficiency, and phase current are
approximately in harmony with the measurements in [4]. By the
error analysis of both the simulation and measured values, the
average errors of the power factor, efficiency, and phase current
are 4.97%, 3.41%, and 2.46%, respectively, which are rational
to engineering applications.
Fig. 9 is the arc SLIM prototype experimental bench. It hasa rotor which is formed on the rim of the large-radius flywheel,
whose primary is fed by a converter. The load cell is a dc
machine which is connected to the shaft of the SLIM rig by
belts. The dc machine can operate at any desired speed and
load below their rating values to provide different working
points. The measurement sensors located between the SLIM
and the dc load can record the SLIM velocity, load power, and
thrust [9].
Fig. 10 presents different thrust curves with constant current
constant frequency or constant voltage constant frequency. It
can be seen obviously that the thrust will decrease gradually by
the influence of longitudinal end and transversal edge effects,
particularly in the high speed region. Moreover, the thrust ex-cited by constant voltage in Fig. 10(b) decreases more quickly
-
8/10/2019 Equivalent circuit for single sided linear induction motors
7/14
2416 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 6, NOVEMBER/DECEMBER 2010
Fig. 8. Steady performance of Japanese 12000 SLIM. (a) Thrust and slipfrequency. (b) Power factor. (c) Efficiency. (d) Primary phase current.
Fig. 9. Experimental setup of the arc SLIM.
Fig. 10. Steady thrusts of the arc SLIM prototype. (a) Constant currentconstant frequency (real lines are the thrusts without an end effect, dashedlines are the thrusts with end effects, and other shaped lines are the measuredones). (b) Constant voltage constant frequency (real lines are the thrusts withend effects, and the other shapes are the measured ones).
than that by constant current in Fig. 10(a) for its quicker air-
gap flux attenuation. The thrust simulation agrees with themeasurement reasonably in various velocities.
-
8/10/2019 Equivalent circuit for single sided linear induction motors
8/14
XU et al.: EQUIVALENT CIRCUITS FOR SINGLE-SIDED LINEAR INDUCTION MOTORS 2417
Fig. 11. Equivalent circuits of SLIM indq-axis coordinate. (a)d-axis circuit.(b)q-axis circuit.
Fig. 12. Four coefficients under different slip frequencies (solid lines denotethe simulation, and dashed lines denote the measurement).
IV. TWO-A XI SE QUIVALENTC IRCUITS
According to the Park coordinate transformation by the
power conversion rule, we can obtain the two-axis (dqor )equivalent circuits, which can be referred to in [9], [21], and
[22]. The flux linkage matrix is shown by (27) at the bottom of
the page, whereLm is the equivalent mutual inductance in thetwo-axis frame, which is equal to 1.5Lm1. In order to simplifythe flux matrix expression, we suppose that L
mc= K
xCx
Lm
,
Ls = KxCxLm+ Lls, and Lr = KxCxLm+ Llr. Then, thefour flux linkage expressions are further expressed by
ds = Lsids+ Lmcidrqs = Lsiqs+ Lmciqrdr =Lmcids+ Lridrqr =Lmciqs+ Lriqr .
(28)
Fig. 13. Modified mutual inductance Lmc and secondary resistance Rrcunder different slip frequencies.
Fig. 14. ThrustFx curve under different phase currents and slip frequencies.
From (28), it can be seen that the four SLIM flux linkage
equations are totally similar to those of the RIM. The special
performance traits resulting from the longitudinal end effect,
transversal edge effect, and half-filled slots are easily described
by the four coefficients and equivalent pole pairs. The SLIM
analysis procedure and algorithm in the two-axis coordinate
are also similar with those of the RIM. Fig. 10 indicates the
SLIM equivalent circuits in the dq-axis frame. Different fromthose of the RIM, four more rectification coefficients, i.e.,Kx,Cx, Kr, Cr, appear in the mutual inductance and secondaryresistance branch circuits. In Fig. 11, 11 and 12 are the
angular frequencies of the primary and secondary relative tothedq-axis.
V. ANALYSIS OFF OU RC OEFFICIENTS
The T-model circuit parameters for the three-phase arc SLIM
experimental prototype are Rs = 0.425 , Lls = 2.145 mH,
dsqsdrqr
=
KxCxLm+ Lls 0 KxCxLm 0
0 KxCxLm+ Lls 0 KxCxLmKxCxLm 0 KxCxLm+ Llr 0
0 KxCxLm 0 KxCxLm+ Llr
idsiqsidriqr
(27)
-
8/10/2019 Equivalent circuit for single sided linear induction motors
9/14
2418 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 6, NOVEMBER/DECEMBER 2010
Fig. 15. Indirect rotor field orientation control analysis diagram.
Lm1= 7.670 mH, Rr = 0.221 , and Llr= 0.550 mH. Inorder to investigate the end effects on the mutual inductance andsecondary resistance conveniently, we can analyze the different
variable curves in the arc SLIM start-up procedure.
The primary input frequency varies from 1 to 30 Hz. For each
input frequency, the phase current changes from 10 to 30 A. The
slip is one at start-up so that the slip frequency sf is equal tothe primary frequencyfs. The calculated and measured curvesof the four coefficients in different frequencies are shown in
Fig. 12. Fig. 13 shows the simulation curves for the rectified
mutual inductanceLmcin the amount ofKxCxLm1and for therectified secondary resistance Rrc that is equal to KrCrRr asthe slip frequency goes up. The thrust, with respect to the phase
current and excited frequency, is given in Fig. 14. Because ofthe end effects, the average air-gap flux density decreases with
growing velocity, which reduces the mutual inductance and
increases the secondary resistance [17][20].
VI. EXPERIMENTALV ERIFICATION OF
TWO-A XI SE QUIVALENTC IRCUITS
Fig. 15 is the indirect rotor field control scheme of the arc
linear induction motor (LIM) which is similar to a RIM control.
There are three closed loops, including speed and d- andq-axis current loops, which are modified by three plus integral
regulators.Fig. 16 shows the simulation and experiment curves of the
velocity v2, field current Ids, thrust current Iqs, and thrust.In the whole process, the velocity is given by 6.28 m/s, and
the field current is kept at a constant of 16 A. The SLIM is
accelerated from 0 to 6.28 m/s in the first 30 s, then operated
with constant velocity for approximately 20 s, and finally
decelerated to zero in about 20 s. Although excited by constant
currentIds, the thrust is attenuated a little as the speed goes upfor its longitudinal end and transversal edge effects. During the
regeneration, the braking torque correspondingly increases for
its continuously smaller end-effect influence. Considering dif-
ferent friction and windage resistances between the theoretical
simulation and practical experiment, the performance curves inFig. 16(a) validate those of Fig. 16(b) reasonably.
VII. CONCLUSIONCompared with some other models of the SLIM, the pro-
posed circuits have the following traits.
1) The T-model circuit is derived based on the 1-D air-gap
flux density equation, which gets four coefficients, i.e.,
Kx, Kr, Cx, and Cr, to describe the influence on themutual inductance and secondary resistance brought by
the longitudinal end and transversal edge effects. These
four coefficients have reasonable accuracy, combining
both the electromagnetic and numerical analyses. When
Kx= Kr = Cx= Cr = 1, i.e., when neglecting the endand edge effects, the T-model is the same as that of the
RIM. These coefficients have clear physical meaning soas to help researchers understand the end effects. Hence,
it is convenient to study the SLIM performance in a
similar way as that of the RIM.
2) By the Park coordination transformation, the SLIM spe-
cial traits brought by the end effects in the dq-axis can bealso expressed by the aforementioned four coefficients.
By brief simplification, the proposed two-axis circuits
have similar forms as those of the RIM except for the
four coefficients occurring in the mutual inductance and
secondary equivalent resistance. All the control schemes
of the RIM can be applied in the novel models directly,
which brings great convenience to the study of the SLIMdynamic performance.
3) For the linear metro application, the thickness of the sec-
ondary back iron placed on the track is always 2030 mm
to avoid the deformation of the conduct sheet which
resulted from the vertical force. The back-iron resistance,
particularly in the starting period, is considered in parallel
with the sheet resistance.
4) The half-filled slots in the primary ends could bring some
influence to the air-gap equivalent flux, particularly in the
cases where the number of the pole pairs is no more than
three. Based on the equations of the primary equivalent
current density, the number of equivalent pole pairs pesmaller than the actual value p is used to describe theinfluence by the half-filled slots.
-
8/10/2019 Equivalent circuit for single sided linear induction motors
10/14
XU et al.: EQUIVALENT CIRCUITS FOR SINGLE-SIDED LINEAR INDUCTION MOTORS 2419
Fig. 16. Dynamic performance curves. (a) Calculated. (b) Measured.
By the simulation and experimentation of the 12 000 and
arc SLIMs different steady working styles, including the vari-
able frequency variable voltage, constant current, and constant
voltage driving, the T-model circuit can describe the SLIM
steady performance reasonably, such as the thrust, power factor,
efficiency, phase current, etc. Moreover, the curves of the four
coefficients, the mutual inductance, the secondary resistance,
and the thrust in the arc SLIM variable excited frequency and
variable phase current at the starting state are analyzed in detail.
By the help of the SLIM dynamic mathematic equations in the
two-axis equivalent circuits, the simulations of the thrust and
velocity in the indirect rotor field orientation control schemeshow good correlation with the experiments.
APPENDIX
DERIVATION OFF OU RC OEFFICIENTS
A. One-Dimensional Physical Model
In order to simplify the derivation, some assumptions are
proposed in the following list [1], [2], [9], [20], [26].
1) The stator iron has infinite permeability.2) The skin effect is neglected in the secondary.
3) Winding space harmonics are negligible.
4) The primary and secondary currents flow in infinitesi-
mally thin sheets.
5) All magnetic variables are sinusoidal time functions.
The analytical model with one dimension is shown in Fig. 1,
where the primary is named as Area 1, the secondary sheet as
Area 2, the air gap as Area 3, the exit end as Area 4, and the
entrance end as Area 5.
For special structures, SLIMs have longitudinal end effects
which resulted from cut-open primary terminals and transversal
edge effects due to the different width between the primary andthe secondary. Their influences on the SLIM parameters can
be analyzed separately and then gathered up by a superposition
theorem.
B. Longitudinal-End-Effect CoefficientsKx andKr
Some fundamental electromagnetic equations applied in the
SLIM are summarized as
H
=J
(A-1)
E
= B
t (A-2)
B = 0 (A-3)B
=H
(A-4)
J
=(E
+ V B
) (A-5)
whereH
, J
, B
, E
, and V
are vectors for the magnetic field
intensity, current density, magnetic flux density, electrical field
intensity, and primary moving velocity, respectively; is thepermeability; and is the conductivity.
By the introduction of vector potential A
, we can get other
expressions forB
andE
B
=
A
(A-6)
E
= A
t . (A-7)
The primary equivalent current sheet is supposed to be
excited by the following current [20]
J1= J1exp [j(et kx)] , 0< x < p (A-8)
where J1 is the amplitude of the primary equivalent currentsheet andk is a constant that equals /.
From (A-1), similar to Amperes law with reference to the
rectangle in Fig. 5(b), we can get
ge0
B3yx
=J1+ J2 (A -9)
-
8/10/2019 Equivalent circuit for single sided linear induction motors
11/14
2420 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 6, NOVEMBER/DECEMBER 2010
whereJ2is in the complex form of the equivalent current sheetin the secondary and B3y is the y -axis component of the fluxdensity in Area 3.
Because only the z-axis component is in the primary cur-rent, the vector potential (ge/0)(B3y/x) =J2z J1 hasonly a z-axis part. Equations (A-6) and (A-7) can be further
expressed by
B3y = A3zx
(A -10)
E3z = A3zt
. (A-11)
Combining (A-10), (A-11), and (A-5), we can have
J2= eA3z
t + v2
A3zx (A-12)
wheree is the surface conductivity in the amount ofge andv2 is the primary moving velocity along thex-axis. The vectorpotentialA3z can be described as one function of time t andpositionx
A3z =A3z(x, t) =Az(x)exp(jet). (A-13)
Based on (A-8)(A-13), we can acquire
ge0
d2A3zdx2 ev2
dA3zdx jeeA3z = J1exp(jkx).
(A-14)
The solution of (A-14) is
A3z=csexp[j(etkx)]+ cc1expx
1+j
et
ex
+cc2exp
x
2+j
et+
ex
(A-15)
where cs, cc1, and cc2 are coefficients decided by boundary
conditions; 1and 2are attenuating coefficients of the air-gapentrance- and exit-flux density waves, respectively; and eis theend-effect half-wave length. The vector potential A3zis relativewith the air-gap flux density which includes three parts, b0,b1,andb2 [2], [20], where b0 is the normal traveling wave whichmoves forward in a similar manner to the fundamental flux
density wave in a rotating induction machine (the fundamental
wave b0 in the same amplitude transmits with synchronousspeed2f ) andb1 andb2 are the entrance and exit end-effectwaves. Generally speaking, b1 is a gradually attenuating wavetraveling along the x-axis. Its attenuation constant is 1/1.b2 travels along the x-axis in the negative direction with anattenuation constant of1/2. The transmitting speeds of the
forward and backward wavesb1 and b2 are 2f e. For the quickattenuation of1/2, the third part of (A-15) can be ignored.
More information can be found in [2] and [20]. The expressions
for the main parameters can be summarized as
cs = 0J1
k2ge(1 +jsG), 1=
gegeX 0ev2 ,
2= ge
geX+ 0ev2
,
e =2
Y , X=
0ev2ge
1 + (4ege/0ev22)2 + 12
,
Y =0ev2
ge
1 + (4ege/0ev22)2 12
,
cc1= jkcs 11
+j e
where s is the slip and G is the goodness factor described by [2]
G= 20efs2/(ge). (A-16)
By inserting (A-15) into (A-10) and (A-11), we can acquire
B3y = Bm
exp[j(et kx + s)]
expx
1+j
et
ex
(A-17)
E3z = eBmexp [j(et kx)]
1
k cos
s+ 1eexp(
x/1)2e + (1)2
cos
2
+ s
+
k
e
x
+jeBmexp [j(et kx)]
1k
sin s+
1eexp(x/1)
2e + (1)2
sin
2+ s
+
k
e
x
(A-18)
whereBm= GJ1/eVs
1 + (sG)2
,s= tan1
(1/sG), and= tan1(1/e).By electrical machinery knowledge, complex power S23
transmitted from the primary (Area 1) to the secondary (Area 2)
and the air gap (Area 3) is calculated by
S23= 2a1
p0
0.5 [j1E3z] dx
= J1Bma1Vs
pcos s NL
11 exp(p /1)sin(s + SLp)
+ SLexp(p /1)cos(s + SLp) 11 sin(s ) SLcos(s )
-
8/10/2019 Equivalent circuit for single sided linear induction motors
12/14
XU et al.: EQUIVALENT CIRCUITS FOR SINGLE-SIDED LINEAR INDUCTION MOTORS 2421
+jJ1Bma1Vs
psin s NL 11 exp(p /1)cos(s + SLp)
+ SLexp(p /1)sin(s + SLp) 11 cos(s ) SLsin(s )
= P2+ jQ3 (A-19)
where P2is the active power in the secondary, Q3is the reactivepower in the air gap, and coefficients SL, ML, and NL areexpressed by
SL= k e
, ML=
112
+ S2L,
NL= 1e
ML
2e + (1)2
,
s= tan1
1
SLG
, = tan1
1
e
.
The effective value of the primary per-phase current is
Is = p J1
2
2m1W1kw1. (A-20)
In terms of the air-gap effective electromotive force Em, wecan get the following equation according to the complex power
conversion theory
m1IsEm= P2+jQ3. (A-21)
By inserting (A-19) and (A-20) into (A-21), Em can besolved, and then, the secondary resistance Rr and mutual
reactance Xm per phase in reference to the primary can beexpressed by
Rr =m1|Em|2
P2
=8a1m1(W1kw1)
2
ep
sG
p
1 + (sG)2C21 + C
22
C1(A-22)
Xm=m1|Em|2
Q3
=16a1m10fs(W1kw1)
2
gep
1
p
1 + (sG)2C21 + C
22
C2(A-23)
where C1and C2are functions of the slip and machine structureparameters, described as
C1=pcos s NL
11 e
p/1 sin(s + SLp)+ SLe
p/1 cos(s + SLp) 11 sin(s ) SLcos(s )
C2=psin s NL
11 ep/1 cos(s + SLp)+ S
Lep/1 sin(
s + S
Lp)
+ 11 cos(s ) SLsin(s )
.
By the rotary induction machinery theory, the per-phase rotor
resistanceRrand mutual reactanceXmcan be expressed by
Rr=8a1m1(W1kw1)
2
ep (A-24)
Xm=
16a1m10fs(W1kw1)2
gep . (A-25)
By comparing (A-22) and (A-23) to (A-24) and (A-25), it
can be seen that Rr andXm in the SLIM are a little differentfrom those of the RIM, which are influenced by velocity, slip,
frequency, machine structures, and so on. Two coefficientsKrand Kxcan be used to describe the influence by the longitudinalend effects
Kr= sG
p
1 + (sG)2C21 + C
22
C1(A-26)
Kx= 1
p
1 + (sG)2C21 + C
22
C2 . (A-27)
C. Transversal-Edge-Effect CoefficientsCxandCr
Along the rectangle route in Fig. 5(c), the following relation-
ship based on (A-1) can be built by
ge0
B3yz
= J2x (A-28)
ge
0
B3y
x
= J2z
J1 (A-29)
where J2x and J2z are the x- and z-axis components of thesecondary equivalent current in Area 2, respectively, and B3yis they -axis component of the air-gap flux density in Area 3.By substituting (A-28) and (A-29) into (A-5), we can get the
following equation by curl calculation
2B3yx2
+2B3y
z2 0ev2
ge
B3yx
0ege
B3yt
=0ege
J1x
.
(A-30)
By supposing that B1y =B1y(x,z,t) =B(z)exp[j(et kx)], (A-30) can be simplified and further solved by
B1y(x,z,t) = j 0kge
J1R2
1 +
1 R2R2
cosh z
cosh a1
exp[j(et kx)] (A-31)
whereR2 = 1/1 +jsG and2 =k2 +j(e0e/ge)s.The per-pole flux linkage is described as
=
0
a1a1
B1y(x,z,t)dzdx
=40
ge J1R2
a1+
1
R2
R2
tanh a1
exp(jet).(A-32)
-
8/10/2019 Equivalent circuit for single sided linear induction motors
13/14
2422 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 46, NO. 6, NOVEMBER/DECEMBER 2010
The primary instantaneous electromotive force per phase is
em = W1kw1 ddt
[] =
2Emexp(jet) (A-33)
where
Em = 4
20fW1kw1a1
2
ge
J1
j
R2 + (1 R2)
a1tanh a1
. (A-34)
The complex power transmitted from the primary to the air
gap and secondary is
m1IsEm=P2+jQ3
=20a1fsp
3
ge
J1
Re
j
R2 + (1 R2)
a1tanh a1
+jIm
j
R2 + (1 R2)
a1tanh a1
.
(A-35)
According to the complex power theory, the secondary resis-
tance and mutual reactance per phase can be calculated by
Rr=m1|Em|2
P2=
8a1m1(W1kw1)2
ep
sG
R2e[T] + I2m[T]
Re[T]
(A-36)
Xm=m1|Em|2
Q3=
16a1m10fs(W1kw1)2
gep
R2e[T] + I2m[T]
Im[T]
(A-37)
where T =j [R2 + (1 R2)(/a1)tanh a1]. By compar-ing (A-24) and (A-25) to (A-36) and (A-37), two coefficients
Cr,Cx can be used to describe the influence by the transversaledge effects
Cr =sG
R2e[T] + I
2m[T]
Re[T]
(A-38)
Cx =
R2e[T] + I
2m[T]
Im[T]
. (A-39)
ACKNOWLEDGMENT
The authors would like to thank Prof. X. L. Long, Prof.
Y. M. Du, Dr. J. Q. Ren, Dr. K. Wang, and W. Wang at the
Institute of Electrical Engineering, Chinese Academy of Sci-
ences, China, and D. G. Dorrell at the University of TechnologySydney, Australia, for their kind help.
REFERENCES
[1] J. Duncan, Linear induction motorEquivalent-circuit model, Proc.Inst. Elect. Eng., vol. 130, no. 1, pt. B, pp. 5157, Jan. 1983.
[2] X. L. Long, Theory and Magnetic Design Method of Linear InductionMotor. Beijing, China: Science Publishing, 2006.
[3] G. Kang and K. Nam, Field-oriented control scheme for linear inductionmotor with the end effect, Proc. Inst. Elect. Eng.Elect. Power Appl.,
vol. 152, no. 6, pp. 15651572, Nov. 2005.[4] T. Higuchi and S. Nonaka, On the design of high efficiency linear induc-tion motors for linear metro, Elect. Eng. Jpn., vol. 137, no. 2, pp. 3643,Aug. 2001.
[5] P. Joon-Hyuk and B. Yoon Su, Design and analysis of a maglev planartransportation vehicle,IEEE Trans. Magn., vol. 44,no. 7, pp.18301836,Jul. 2008.
[6] K. Idir, G. E. Dawson, and A. R. Eastham, Modeling and performanceof linear induction motor with saturable primary,IEEE Trans. Ind. Appl.,vol. 29, no. 6, pp. 11231128, Nov./Dec. 1993.
[7] J. F. Gieras, G. E. Dawson, and A. R. Eastham, A new longitudinal endeffect factor for linear induction motors, IEEE Trans. Energy Convers.,vol. EC-2, no. 1, pp. 152159, Mar. 1987.
[8] Y. Mori, S. Torii, and D. Ebihara, End effect analysis of linear inductionmotor based on the wavelet transform technique, IEEE Trans. Magn.,vol. 35, no. 5, pp. 37393741, Sep. 1999.
[9] W. Xu, Research on the performance of single-sided linear inductionmotor, Ph.D. dissertation, Inst. Elect. Eng., Chinese Acad. Sci., Beijing,China, 2008.
[10] T. A. Lipo and T. A. Nondahl, Pole by pole dq model of a linearinduction machine, IEEE Trans. Power App. Syst., vol. PAS-98, no. 2,pp. 629642, Mar./Apr. 1979.
[11] C. A. Lu, A new coupled-circuit model of a linear induction motor and itsapplication to steady-state, transient, dynamic and control studies, Ph.D.dissertation, Dept. Elect. Eng., Queens Univ., Kingston, ON, Canada,1993.
[12] J. Faiz and H. Jafari, Accurate modeling of single-sided linear inductionmotor considers end effect and equivalent thickness,IEEE Trans. Magn.,vol. 36, no. 5, pp. 37853790, Sep. 2000.
[13] K. Yoshida, New transfer-matrix theory of linear induction machines,taking into account longitudinal and transverse ferromagnetic end ef-fects, Proc. Inst. Elect. Eng.Elect. Power Appl., vol. 128, no. 5,pp. 225236, Sep. 1981.
[14] W. Xu and J. Zhu, Optimal design of a linear induction motorapplied in transportation, in Proc. IEEE Int. Conf. Ind. Technol., 2009,pp. 790795.
[15] W. Xu, J. G. Zhu, Y. G. Guo, Y. C. Zhang, and Y. Wang, Equivalentcircuits for single-sided linear induction motors, in Proc. IEEE EnergyConvers. Congr. Expo., 2009, pp. 12881295.
[16] S. Nonaka, Investigation of equations for calculation of secondaryresistance and secondary leakage reactance of single sided linear in-duction motors, Elect. Eng. Jpn., vol. 117, no. 5, pp. 616621,Mar. 1997.
[17] R. Hellinger and P. Mnich, Linear motor-powered transportation: His-tory, present status, and future outlook, Proc. IEEE, vol. 97, no. 11,pp. 18921900, Nov. 2009.
[18] R. Thornton, M. T. Thompson, B. M. Perreault, andJ. Fang, Linear motorpowered transportation, Proc. IEEE, vol. 97, no. 11, pp. 17541757,Nov. 2009.
[19] J. Gustafsson, VectusIntelligent transport, Proc. IEEE, vol. 97,no. 11, pp. 18561863, Nov. 2009.
[20] S. Yamamura,Theory of Linear Induction Motors. Tokyo, Japan: TokyoPress, 1978.
[21] B. S. Chen,Electric Drive and Automatic Control System (the 3rd versionin Chinese). Beijing, China: Mech. Eng. Publishing Co., 2009.
[22] W. Xu, G. S. Sun, and Y. H. Li, A new two axis equivalent circuitsof single linear induction motor for linear metro, in Proc. IEEE Conf.
Maglev, Dec. 2008, pp. 16.[23] M. Poloujadoff, The Theory of Linear Induction Machinery. Oxford,
U.K.: Clarendon, 1980.[24] S. K. Chen,Electromagnetic Design of Electric Machinery (the 2nd ver-
sion in Chinese). Beijing, China: Mech. Eng. Publishing Co., 2004.[25] I. Boldea and S. A. Nasar, Linear Motion Electromagnetic Devices.
New York: Taylor & Francis, 2001.[26] W. Xu, J. G. Zhu, Y. C. Zhang, Y. H. Li, Y. Wang, and Y. G. Guo,
An improved equivalent circuit model of a single-sided linear induc-tion motor, IEEE Trans. Veh. Technol., vol. 59, no. 5, pp. 22772289,Jun. 2010.
-
8/10/2019 Equivalent circuit for single sided linear induction motors
14/14
XU et al.: EQUIVALENT CIRCUITS FOR SINGLE-SIDED LINEAR INDUCTION MOTORS 2423
Wei Xu (M09) was born in Chongqing, China, in1980. He received the B.E. and B.A. degrees, both in2002, and the M.E. degree in 2005 from Tianjin Uni-versity (TJU), China, and the Ph.D. degree in 2008from the Institute of Electrical Engineering, ChineseAcademy of Sciences, all in electrical engineering.
He is currently a Postdoctoral Fellow at theCenter for Electrical Machines and Power Elec-
tronics, University of Technology Sydney (UTS),Sydney, Australia, where his research is supportedin part by an Early Career Researcher Grant and in
part by the Chancellors Postdoctoral Research Fellowship, both from UTS. Hisresearch interests mainly include the electromagnetic design and performanceanalysis of linear/rotary machines, including induction, permanent-magnet,switched reluctance, and other emerging novel structure machines.
Jian Guo Zhu (S93M96SM03) received theB.E. degree from Jiangsu Institute of Technology,Nanjing, China, in 1982, the M.E. degree fromShanghai University of Technology, Shanghai,China, in 1987, and the Ph.D. degree from theUniversity of Technology Sydney (UTS), Sydney,Australia, in 1995.
He is currently with UTS, where he is a Profes-sor of electrical engineering and the Head of theSchool of Electrical, Mechanical and MechatronicSystems. His research interests include electromag-
netics, magnetic properties of materials, electrical machines and drives, powerelectronics, and renewable energy systems.
Yongchang Zhang (M10) received the B.S. de-gree from Chongqing University, Chongqing, China,in 2002, and the Ph.D. degree from Tsinghua Uni-versity, Beijing, China, in 2009, both in electricalengineering.
He is currently a Postdoctoral Fellow at the Uni-versity of Technology Sydney, Sydney, Australia.His research interests include sensorless and high-performance control of ac motor drives, control
of multilevel converters, pulsewidth modulation(PWM), PWM rectifiers, and advanced digital con-
trol with real-time implementation.
Zixin Li(S08) was born in Hebei Province, China,in 1981. He received the B.Eng. degree in industryautomation from North China University of Technol-ogy, Beijing, China, in 2001, and the Ph.D. degree inpower electronics and power drives from the Insti-tute of Electrical Engineering, Chinese Academy ofSciences, Beijing, in 2010.
Since 2010, he has been with the Institute of Elec-trical Engineering, Chinese Academy of Sciences,where he is currently an Assistant Research Profes-sor. His research interests include the design, control,
and analysis of power converters, particularly in high-power fields.Dr. Li has been the recipient of many honors and awards, including the schol-
arship awarded to the excellent Ph.D. candidates in the Graduate University ofChinese Academy of Sciences offered by the Australian company BHP Billitonin 2009 and the student scholarship at the IEEE International Symposium onIndustrial Electronics 2009.
Yaohua Li was born in Henan, China, in 1966. Hereceived the Ph.D. degree from Tsinghua University,Beijing, China, in 1994.
From 1995 to 1997, he was a Postdoctoral Re-search Fellow at the Institute of Electrical Ma-chines, Technical University of Berlin, Berlin,Germany. Since 1997, he has been with the Instituteof Electrical Engineering, Chinese Academy of Sci-
ences, Beijing, where he is currently a Professor andthe Director of the Laboratory of Power Electronicsand Electrical Drives. His research fields include
analysis and control of electrical machines, power electronics, etc.
Yi Wang received the B.Eng. and M.Sc. degreesin electrical engineering from Huazhong Universityof Science and Technology, Wuhan, China, in 2004and 2007, respectively. He is currently working to-ward the Ph.D. degree in the School of Electrical,Mechanical and Mechatronic Systems, University ofTechnology Sydney, Sydney, Australia.
His research interests include power electronics
andthe modeling andcontrolof electrical drives, par-ticularly permanent-magnet synchronous machines.
Youguang Guo (S02M05SM06) was born inHubei, China, in 1965. He received the B.E. degreefrom Huazhong University of Science and Tech-nology (HUST), Wuhan, China, in 1985, the M.E.degree from Zhejiang University, Hangzhou, China,in 1988, and the Ph.D degree from the Universityof Technology Sydney (UTS), Sydney, Australia, in2004, all in electrical engineering.
From 1988 to 1998, he waswith theDepartment ofElectric Power Engineering, HUST. He is currently
with UTS, where he was a Visiting Research Fellow,Ph.D. candidate, Postdoctoral Fellow, and Research Fellow in the Centerfor Electrical Machines and Power Electronics, Faculty of Engineering, fromMarch 1998 to July 2008 and where he is currently a Lecturer in the School ofElectrical, Mechanical and Mechatronic Systems. His research fields includemeasurement and modeling of magnetic properties of magnetic materials,numerical analysis of electromagnetic fields, electrical machine design andoptimization, and power electronic drives and control. In these fields, he haspublished over 230 refereed technical papers, including 110 journal articles.
Yongjian Li received the B.E. and M.E. degreesfrom Hebei University of Technology, Tianjin,China, in 2002 and 2007, respectively. He is cur-rently working toward the Ph.D. degree jointly atHebei University of Technology and the Universityof Technology Sydney, Sydney, Australia.
His research interests include measurement ofmagnetic properties, modeling of magnetic materi-als, and power electronics.