equivalent circuit for single sided linear induction motors

8/10/2019 Equivalent circuit for single sided linear induction motors

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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


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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


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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


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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.


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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


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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


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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.


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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)


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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.


10/14


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)


11/14


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 )


12/14


+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)


13/14


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.

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[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.

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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.

equivalent circuit for single sided linear induction motors

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