04282107

2374 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007

Design of Multisectional Driver and Field-OrientedModeling of the Axial-Flux Linear Brushless

Motor for Railway TransportationsJian-Long Kuo, Associate Member, IEEE, Zen-Shan Chang, and Tzu-Shuang Fang

Abstract—A multisectional power driver design for thetwo-phase linear brushless motor will be proposed in this pa-per. The field-oriented modeling of the axial-flux linear brushlessmotor (AFLBM) will also be developed by using stationary andcosecant similarity transformation and singular value decomposi-tion. Stationary and cosecant coordinate models will be derivedbased on the proposed approach. Decoupled relation and con-stant torque property will be obtained from the proposed models.Switching logic table for the linear motor operation is providedto drive the AFLBM properly. The switching-mode analysis forthe power driver is also analyzed in detail. In order to increase theefficiency and the effectiveness of the electric motor applications, amultisectional driver circuit, which features the power driver withthe so-called N + 2 structure, will be investigated and compared.Experimental results show that the proposed circuit design canachieve better performance with the higher efficiency. It is believedthat the proposed driver circuit technique and system modelingcan be very helpful to the control of railway transportations.

Index Terms—Axial-flux linear brushless motor (AFLBM),blocking diode, cosecant model, multisectional power driver,power driver with N + 2 structure, similarity transformation(ST), singular value decomposition (SVD), stationary model.

I. INTRODUCTION

R ECENTLY, the axial-flux linear brushless motor(AFLBM) with two-phase parallel windings is widely

used in many information products in which the motive forceis required. The CPU cooling fan is often designed as theaxial-flux type. The disk rotation of the DVD-ROM will alsorequire such a motor to provide the required rotation. It belongsto a permanent-magnet synchronous motor. Axial-flux-typemotor is also widely used in many applications such as electricvehicles and transportations [1], [2]. This type motor has agood feature of controllable field current [3]. Field weakening

Manuscript received March 30, 2006; revised October 6, 2006. This workwas supported by the National Science Council, Taiwan, R.O.C., under ProjectNSC93-2213-E-327-019.

J.-L. Kuo and T.-S. Fang are with the Department of Mechanical andAutomation Engineering, National Kaohsiung First University of Science andTechnology, Kaohsiung 811, Taiwan, R.O.C. (e-mail: [email protected]).

Z.-S. Chang is with the Department of Electrical Engineering, Chang GungUniversity, Tao-Yuan 333, Taiwan, R.O.C.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIE.2007.900321

can be achieved to control this motor. Therefore, the axial-fluxmotor can operate under very wide speed range [3]. The fieldweakening can be easily carried out by eliminating the effectsof d-axis current injection [4].

Fig. 1 shows the entire system configuration for the AFLBM.DSP-based controller is used to control the linear motor. Asshown in Fig. 1, such a motor has winding configuration thatis different from the conventional one discussed in the generalthree-phase motor system. To simplify the power driver, the par-allel windings in Fig. 1(a) are designed to reduce the complexityof the driver, as shown in Fig. 2(b). It is a single-transistordriver type.

The flat-type brushless motors are thereby widely usedin many information product and industrial applications.Flat-type brushless motors have many different types of struc-tures. The common flat-type motor is possibly driven in manyways. Full-bridge driver is frequently illustrated to drive theproposed motor [5]. However, the full-bridge driver mightrequire an expensive cost. Only a simple driver is required todrive the mentioned motor in this paper. The parallel windingconfiguration is designed for the cost-down requirement ofthe motor driver. The driver structure can be selected as thesingle-transistor type instead of full-bridge type.

As shown in Fig. 1(a), the two-phase As- and Bs-parallelphase windings are not separated by π/2 between each otherinstead. The windings are in parallel by π between each otherin space. The polarity definition for the two windings is reversein the reference direction. The mutual inductance for thetwo-phase windings is fully coupled and cannot be neglectedto be zero.

Therefore, the dynamic system model for such a motoris quite different from the conventional model such as Parktransformation. Two orthogonal transformations can be usedfor a motor system [6]. One is similarity transformation (ST),and the other is singular value decomposition (SVD) as in [6].This paper intends to provide an alternative system modelingfor this motor. ST and SVD will be used to derive the localmodel for the linear motor.

After deriving the two-phase local model, the global system-integration modeling for the multisectional power driver will bediscussed, as shown in Fig. 1(b). The switching-mode analysisof the power driver for the proposed linear motor will beinvestigated in detail. Four kinds of power drivers are classifiedin Table I, which shows the four possible configurations for

0278-0046/$25.00 © 2007 IEEE

KUO et al.: DESIGN OF MULTISECTIONAL DRIVER AND FIELD-ORIENTED MODELING OF THE AFLBM 2375

Fig. 1. (a) Proposed AFLBM-04 system configuration and (b) investigation description.

the AFLBM. The railway is divided into N multisections inFig. 2(a) for the entire driver design. The illustrated possiblewinding configurations are N = 6, N = 3, N = 2, and N = 1,as classified in Table I.

The switching logic is defined in Table II, which indicatesthe moving bogie position by using 9-b signals coming fromthe absolute encoder circuit with Hall sensor signals. Thereare two kinds of power transistors: One is global sectional-power transistors for the multisections, and the other is localpower transistors for the two-phase windings. The switchinglogic coming from the Hall sensor signals can be classified asin Table II. The entire design concept can be summarized inFig. 1(b). From the integrated motor design concept, the powerdriver in Fig. 2(b) should be designed to be suitable for thespecific AFLBM system in Fig. 1(a).

Since there are two-phase windings in parallel configuration,the power driver is designed as a single-transistor driver withthe so-called sectional transistors SW1−SW6 in Fig. 2(b).

However, it does not belong to the conventional full-bridgedriver type. Since there are four kinds of winding configurationsillustrated in this paper for comparison, the power driver alsohas four types to provide the required motion control. Toidentify the modeling objectives, the local and global mod-els are defined in Fig. 1(b). The local model discusses onlyon the two-phase windings. The field-oriented modeling byusing ST–SVD is used to formulate the two-phase windingsin parallel. The global model discusses the system integra-tion, including the multisectional configuration. The configu-ration matrices will be defined to identify the multisectionalconfiguration.

II. MODELING OF THE TWO-PHASE FLAT-TYPE

BRUSHLESS MOTOR

The nonlinear dynamics for the electromechanical flightactuators were ever modeled based on the same approach such


Fig. 2. (a) Winding configuration and (b) simple driver for the investigated brushless motor.

TABLE ISUMMARY OF THE FOUR PROPOSED POWER DRIVERS

as ST [7]. The voltage and flux equations were derived for thementioned motor. The torque equation was also provided in thispaper [7]. In this paper, these formulations will be adopted forthe same derivation of the studied motor.

A. Local-Modeling Derivation of the Voltage andFlux Equations

The rotor reference frame [8] is frequently adopted for thesynchronous motor to transform the original system model.


TABLE IISWITCHING LOGIC OF THE MOTOR POSITION FROM ABSOLUTE

ENCODER CIRCUIT WITH HALL SENSOR SIGNALS

Similar to the rotor reference frame, the voltage and flux matrixequations can be expressed as the form in [8]. The voltageequation of the two-phase multiple P-pole flat-type motor canbe expressed as

[Vas

Vbs

]=

[rs 00 rs

] [iasibs

]+

[dλas/dtdλbs/dt

](1)

where the current vector is Iabs = [ias ibs]T, and the fluxlinkage vector is λabs = [λas λbs]T for the As- and Bs-phase windings. The resistance and inductance matrices aredefined as

Rabs =[

rs 00 rs

]

Labs =[

Lasas Lasbs

Lbsas Lbsbs

]

=[

Lss −Lm

−Lm Lss

]. (2)

Lij denotes the self- and mutual inductances between the ithand jth two windings. The minus notation indicates that thetwo phases are defined in the opposite direction. The inductancematrix includes the (Lss) for the diagonal elements and (−Lm)for the off-diagonal elements. The Lss = Lls + Lm is the self-inductance for the As- and Bs-phase windings. Lm is the mu-tual inductance between As- and Bs-phase windings. Leakageinductance Lls is roughly 10% of the magnetizing inductanceLss. By considering the principles of the magnetics in physics,the flux linkage equation for this motor system can be furtherwritten as

[λas

λbs

]=

[Lasasias+Lasbsibs+λ′

m cos(Pθrm/2)Lbsasias+Lbsbsibs−λ′

m cos(Pθrm/2)

]. (3-1)

For simplification, θrm = 2π(x/d) is defined as radian me-chanical angle in the linear motor, and x is the linear displace-ment in the moving direction. d is the pole pitch of the railway.The derivatives of the flux terms can be written as

d (λ′mcos(Pθrm/2))/dt= (−λ′

msin(Pθrm/2))((P/2)dθrmdt)

d (−λ′mcos(Pθrm/2))/dt= (λ′

msin(Pθrm/2))((P/2)dθrm/dt)

(3-2)

where there are magnetic flux terms λ′m resulting from the

permanent magnets. In order to identify the difference betweenthe mechanical and the electrical velocities in the multipleP-pole motors, two variables are defined. θrm is the mechanicalradian angle of the motor, and θr = (P/2)θrm is the radian elec-trical angle of the motor. The radian speed ωrm = dθrm/dt =2π(νx/d) is defined from the mechanical linear velocity ofthe motor νx. ωr = (P/2)(dθrm/dt) is the electrical radianvelocity of the motor. Two of the notations will be used inthis paper.

The voltage equation can be further expressed as follows:

[Vas

Vbs

]=Rabs

[iasibs

]+ Labs

[dias/dtdibs/dt

]+ ωrλ

′m

[− sin(θr)sin(θr)

]

(4)

where As- and Bs-phase windings are parallel except that thedefined reference polarities are reverse to each other due to theaxial-flux structure.

B. Local-Modeling Derivation of the Torque andMechanical Equation

The mechanical dynamics can be derived from the Newton’slaw of motion in [8]. The total magnetic field energy of thiselectromechanical system may be expressed as

Wf(ias, ibs, θrm) =12

∑j=as,bs

ijλj . (5)

The coenergy of the electromechanical system can bedefined as

Wc(ias, ibs, θrm) =∑

j=as,bs

ijλj − Wf(ias, ibs, θrm). (6)

The electromechanical force can be obtained from the deriva-tive of the coenergy

∂Wc(ias, ibs, θrm)∂θrm

=∑

j=as,bs

ij∂λj(ias, ibs, θrm)

∂θrm− ∂Wf(ias, ibs, θrm)

∂θrm. (7)


If the magnetic system is a linear problem, the relationWc = Wf can hold. By substituting all the related variables intothe coenergy function in (6)

Wc(ias, ibs, θrm) =12Lss

(i2as + i2bs

)+ Lss(iasibs)

+ λ′mias cos

(P

2θrm

)

− λ′mibs cos

(P

2θrm

)+ Wpm (8)

where Wpm relates to the offset level of the energy with respectto the permanent magnets, which is constant in this motor.

The differentiation of the coenergy can derive the torqueequation as follows:

Te(ias, ibs, θrm) =∂Wc(ias, ibs, θrm)

∂θrm

= −(

Pλ′m

2

)(ias−ibs)

(sin

P

2θrm

). (9)

The electromagnetic torque Te can be equal to the mechani-cal net force by the Newton’s second law of motion

Te(ias, ibs, θrm) =Jmαrm + Bmωrm + TL

=Jmθrm + Bmθrm + TL (10)

where the moment of inertia is Jm, and the damping coefficientis Bm. The load torque is defined as TL.

C. Global Modeling for System Integration

The generalized power-driver configurations associated withN -section cases are classified in Table I. The detailed circuitdiagram is shown in Fig. 2. To combine the motor model withthe proposed multisectional power driver, the following system-integration modeling is proposed.

The original local model can be expressed as in (4). Tosimplify the derivation of system integration, (4) can be writtenas the following compact form:

Vabs = RabsIabs + LabsIabs + Eabs (11)

where Eabs = ωrλ′m

[− sin(θr)sin(θr)

]. To expand the two-phase

local model into N -section generalized global model

Vglobal =ANVabs

Iglobal =ANIabs (12-1)

Lglobal =N∑

i=1

(BN)iLabs(BN)Ti (12-2)

where

Vglobal = [νAL1, νBL1, νAL2, νBL2, . . . , νAL6, νAL6]

and

Iglobal = [iAL1, iBL1, iAL2, iBL2, . . . , iAL6, iAL6]. (12-3)

AN and (BN)i are 2N by 2 matrices that are called globalconfiguration matrices. Vglobal is 2N by 1 matrix that is calledglobal system voltage. Lglobal is 2N by 2N that is called globalsystem inductance matrix. It should be noted that N is equal to6, 3, 2, and 1.

1) If N = 6, the matrices can be derived as

A6 =[

1 0 1 0 1 0 1 0 1 0 1 00 1 0 1 0 1 0 1 0 1 0 1

]T

(13-1)

(B6)1 =[

1 0 0 0 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 0 0 0 0

]T

(13-2)

(B6)2 =[

0 0 1 0 0 0 0 0 0 0 0 00 0 0 1 0 0 0 0 0 0 0 0

]T

(13-3)

· · ·

(B6)6 =[

0 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 1

]T

. (13-4)

The other cases for N = 3, N = 2, and N = 1 can bederived in the same way.

2) In case if N = 3, the matrices can be derived as

A3 =[

1 0 1 0 1 00 1 0 1 0 1

]T

(14-1)

(B3)1 =[

1 0 0 0 0 00 1 0 0 0 0

]T

(14-2)

(B3)2 =[

0 0 1 0 0 00 0 0 1 0 0

]T

(14-3)

(B3)3 =[

0 0 0 0 1 00 0 0 0 0 1

]T

. (14-4)


A2 =[

1 0 1 00 1 0 1

]T

(15-1)

(B2)1 =[

1 0 0 00 1 0 0

]T

(15-2)

(B2)2 =[

0 0 1 00 0 0 1

]T

. (15-3)


A1 =[

1 00 1

]T

(16-1)

(B1)1 =[

1 00 1

]T

. (16-2)

In particular, this N = 1 case can be reduced from the globalmodel for the system integration to the original two-phasemodel with 2 × 2 matrices. With the aforementioned definition


of the configuration matrices Vglobal, Lglobal can be derivedfrom the local two-phase model into global system-integrationmodel.

III. REALIZATION OF THE PROPOSED FORMULATION

To derive the modal matrix for the ST, the eigenvalues for themotor system have to be derived first. By the definition of theeigenvalue λ, the equation can be expressed as

|Labs − λI| = 0 (17)

where I is the identity matrix defined in linear algebra. Bysolving the aforementioned equation, the eigenvalues can bederived as the form of λ1 = Lss + Lm, λ2 = Lss − Lm. Then,the eigenvectors can be selected with respect to the two eigen-values. The eigenvectors X1 and X2 can be used to define themodal matrix for the associated transformation

X = c1X1 + c2X2 = c1[1 1]T + c2[−1 1]T (18)

where c1 and c2 are the dummy coefficients for the X vectors.The flux linkage terms for the As- and Bs-phase windings arein the opposite directions. Therefore, too simple transformationcannot be derived to obtain the linear torque relation by theconventional form. The orthogonal eigenvectors can span thespecific linear space to describe the motor model in a differentpoint of view.

The famous Park transformation can be expressed as the formin [8] that has good compact transformation matrix. In thispaper, two kinds of new coordinate systems will be developed tofulfill the different requirements for vector-control applications.One is the SVD method, and the other is the ST method.The two transformations are useful in the linear algebra. Inthis paper, they will be used to derive the motor model in adifferent way.

1) The stationary coordinate system is defined as

Mt =1√2

[1 −11 1

], NT

t =1√2

[1 1−1 1

]. (19)

This can be also named as the stationary coordinate trans-formation. The scaling normalization factor is selectedas 1/

√2.

2) The cosecant coordinate system is defined as

Kt =12

[sin θr sin θr

− sin θr sin θr

](20)

where θr = (P/2)θrm and ωr = (P/2)ωrm. The eigen-vectors can be reasonably selected as the other sets of thefollowing combination:

X = c′2X′2 + c′1X

′1 = c′2[1 − 1]T + c′1[1 1]T (21)

where c′1 = c1, c′2 = −c2. c′1 and c′2 are the dummy co-efficients for the X vectors. With this set of eigenvectors,

the scaling coefficient is selected as sin θr which is thefunction of rotating angle. The inverse of the transforma-tion matrix is

K−1t =

[1/ sin θr −1/ sin θr

1/ sin θr 1/ sin θr

]. (22)

The matrix can be used anywhere except for the casethat leads the denominator to zero. The eigenvectorsX′

2 and X′1 can be used to define the modal matrix

Kt. It should be noted that these two proposed sets ofeigenvectors are not only linearly independent but alsoorthogonal in linear space. In order to derive the compactform to analyze such a motor with parallel windings, theproposed orthogonal property is required. Nevertheless,the linear independent eigenvectors can satisfy the suffi-cient condition in the eigenspace.

IV. COSECANT MODELING BY USING THE ST METHOD

In the cosecant modeling, this paper will discuss how toderive the constant torque relation. The mentioned “constant”means that the torque can be derived to be independent ofthe rotating angle. The torque will be derived into the linearfunction of β-axis current iβs. The eigenvectors that are used toform the modal matrix are the cosecant coordinate form in (20).The ST can keep the diagonalized property inherently

Lαβs = K−1t LabsKt =

[Lss + Lm 0

0 Lss − Lm

]. (23)

The transformation will derive the linear relation between thetorque and current variables. Under such a modeling, the trans-formation can be derived to keep the constant torque relationthat is proportional to the iβs current variable. However, theflux linkage terms will have time-varying relation

λ′αβm = Ktλ

′abm =

[0

(−λ′m sin 2θr) /2

]. (24)

Therefore, the torque control can be easily implemented to belike the conventional dc motor control. The linear function canbe guaranteed under the proposed model. A detailed derivationwill be formulated in the following.

A. Derivation for the Flux and Voltage Equation

In this derivation, the modal matrix turns out to be (20)with the cosecant form. The flux equation for the α−β modelcan be derived by substituting the a−b variables into α−βvariables. The flux equation will be related to the rotating anglein (24). The voltage equation can be derived into (25) from theoriginal form in (4). The derivatives of the product term can bederived further by applying the product rule for the derivativein calculus. The voltage equation can be derived as follows:

Vαβs = RαβsIαβs + Ktp(LabsK−1

t

)Iαβs + Ktpλ′

abm

(25)


TABLE IIISUMMARY OF THE TWO PROPOSED ST AND SVD MODELS

where p = d/dt is defined as the differential operator. Vαβs

and Iαβs are the voltage and current vectors defined in the α−βvariables

Rαβs =K−1t RabsKt

=[

rs 00 rs

](26)

Ktp(LabsK−1

t

)=−ωr cos2 θr

2

[Lss+Lm 0

0 Lss−Lm

]

+12

[p(Lss+Lm) 0

0 p(Lss−Lm)

](27)

Ktpλ′abm =

[0

ωrλ′m sin2 θr

]. (28)

With the aforementioned detailed derivation, (25) can be finallyrepresented as (41) from (26)–(28).

B. Derivation for the Torque and Mechanical Equation

The torque equation can also be derived by substituting thea−b variables into α−β variables. The α−β form for the torqueequation can be obtained as

Te = − (Pλ′m/2) (ias − ibs)(sin 2θr) = − (Pλ′

m/2) (−2iβs)(29)

where (ias − ibs) = (1/ sin 2θr)(−2iβs). It should be notedthat the torque equation becomes one compact form whichis proportional to the β-axis current iβs only. The model isfinally derived to keep the torque relation independent of therotating angle.

V. STATIONARY MODELING BY USING THE SVD METHOD

The modal matrix Kt for the ST has the singular case whichhappened at a specific angle at 0 or π. The transformationswill fail at these specific angles. Stationary coordinate trans-formation will be proposed to solve the singular problem. Thistransformation can be used when the ST fails.

The modal matrix Kt can be any possible a combinationof the eigenvectors. First, the modal matrix Kt for the STshould be derived independent of the rotating angle to avoid

the singularity. It is necessary to adopt the so-called stationarymodel due to the singular problem for the STs.

A. Derivation for the Flux and Voltage Equation

The flux linkage vector in (3-1) can be represented as

λabs = Lsiabs + λ′abm (30)

where Labs =[

Lss −Lm

−Lm Lss

]is the system inductance ma-

trix. The flux linkage equation has to be derived first. Since thecurrent and flux vectors in α−β form can be expressed as

Iαβs = MtIabs, λαβs = Mtλabs. (31)

To express the a−b variables as function of the α−β variables,two matrices are defined as follows:

Iabs = NHt Iαβs, λabs = NH

t λαβs. (32)

Mt = (1/√

2)[

1 −11 1

]is the left modal matrix for the SVD

method, and NHt = (1/

√2)

[1 1−1 1

]is the right modal ma-

trix for the SVD method. The flux relation can be expressed interms of the α−β variables

λαβs = Lαβsiαβs + λ′αβm (33)

where

Lαβs =NHt LabsMt

=[

Lss + Lm 00 Lss − Lm

](34)

λ′αβm =Mtλ

′abm

=1√2

[2λ′

m cos θr

0

]. (35)

Then, the voltage is the derivative of the flux linkage

Vαβs = RαβsIαβs + Mtp(LabsNH

t

)Iαβs + Mtpλ′

abm

(36)


Fig. 3. Switching modes when the linear motor is passing through one section. (a) Mode 1 with blocking diodes. (b) Mode 2 with blocking diodes.(c) Mode 1 without blocking diodes. (d) Mode 2 without blocking diodes.

where

Rαβs =NHt RabsMt

=[

rs 00 rs

](37)

Mtp(LabsNH

t

)=

[p(Lss + Lm) 0

0 p(Lss − Lm)

](38)

Mtpλ′abm =

1√2

[−2ωrλ

′m sin θr

0

]. (39)

With the same formulation in the previous discussion ofST method, the aforementioned equation can be furtherderived. Eventually, (36) can be rewritten as (42) from(37)–(39).

B. Derivation for the Torque and Mechanical Equation

The torque equation can be transformed from the originala−b form into the new α−β form

Te = ∂Wc/∂θr = −(√

2/2)

(Pλ′m) (iαs sin θr). (40)


For the sake of clarity, the resultant voltage equations areexpressed in (41) and (42), which are shown at the bottomof the page. Obviously, the decoupled relation will make thesystem model easy to be applied to the associated decoupledvector control. Table III also summarizes the comparison be-tween the stationary and cosecant models by using the ST andSVD methods.

VI. SWITCHING-MODE ANALYSIS FOR POWER

DRIVER OF THE AFLBM-04

For the design of the multisectional driver circuit for thelinear brushless motor, it is very important to achieve anadjustable-speed control with higher efficiency and lowerswitching loss. The fixed power source is provided on theground side. The moving bogie with permanent magnets movesback and forth along the railway with proper switching logicproposed in Table II. The switching analysis of driving circuitwill be discussed in detail.

When the blocking diodes are not used in series with phasewindings, multisectional switching current will result in someinteraction problems [9]–[11]. Therefore, the topological circuitanalysis of linear motor passing through a specific section andtwo sections will be discussed in this paper.

Finally, switching current path under different modes willbe verified further. The multisectional driver circuit is used toprovide the required propulsion force for the moving bogie. Thepower diodes RD1−RD6 and D19−D30 in Fig. 2(b) are used toprovide the power device a current path to release energy whenswitching off the power device. The power diodes D7−D18 areused to block the possible unexpected interaction for the As-and Bs-phase windings.

The driver circuit could provide an appropriate excitation forthe specific phase winding to interact with the moving bogie.In the following cases, the Kirchhoff’s current law is used toexplain the current path for different switching modes. For theconvenience of discussion, only two sections SW1 and SW2

are illustrated for the associated switching-mode analysis.

A. Analysis of Passing Through One Section Only

The first type circuit operates under the condition of themoving bogie passing through a specific section, as shown inFig. 3(a). According to the aforementioned local modeling,the propulsion force is proportional to the current of the As-and Bs-phase windings. The switching current analysis forthe proposed power driver with blocking diodes is shown inFig. 3(a) and (b), respectively.

Fig. 4. Current waveform of mode 2 with blocking diodes for the electricmotor passing through a specific section. Ch1: iSW1 1 A/100 mv, ch2: iBL1

1 A/100 mv, and ch3: iAL1 1 A/100 mv.

With the blocking diodes placed between the SW1, AL1,and BL1 windings, the current path for the As- and Bs-phasewindings are independent of each other. Fig. 3(c) and (d)shows the other two cases without blocking diodes. As shownin Fig. 3(c) and (d), the current path is dependent with eachother. This will cause abnormal interaction for the two-phasewindings.

When the motor is passing through a specific section, theSW1 is on, and the Hall sensor signals begin toggling betweenthe cases of Fig. 3(a) and (b). The switching-mode analysis isshown in Fig. 3(a) to represent the possible current path in themultisectional driver

iSW1 = iAL1 + 0. (43)

It shows the current path when transistor Q1 is on, where apositive flux comes from the As-phase winding and interactswith the north pole of the permanent magnet on the movingbogie. Similarly, the switching mode in Fig. 3(c) reveals thefollowing relation:

iSW1 + (−iBL1) = iAL1. (44)

It shows that if the blocking diodes of the As-phase and theBs-phase windings are removed, the As-phase and Bs-phasewindings will have dependent current relation in mode 1, asshown in Fig. 3(c). The mode 2 in Fig. 3(b) also has thefollowing relation:

iSW1 = iBL1 + 0. (45)

[Vαs

Vβs

]=

[rs− ωr

2 cos2 θr(Lss+Lm)+ 12p(Lss+Lm) 0

0 rs − ωr2 cos2 θr(Lss−Lm)+ 1

2p(Lss−Lm)

][iαs

iβs

]+

[0

ωrλ′m sin2 θr

]

(41)[Vαs

Vβs

]=

[rs+p(Lss+Lm) 0

0 rs+p(Lss+Lm)

] [iαs

iβs

]+

[−(2/√

2)ωrλ′m sin θr

0

](42)


Fig. 5. Switching modes with blocking diodes for the electric motor passing through the two adjacent sections. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.

It shows that the negative flux comes from the Bs-phase wind-ing and interacts with the south pole of the permanent magnet.Finally, the mode 2 in Fig. 3(d) has the following relation:

iSW1 + (−iAL1) = iBL1. (46)

The current relation in the mode 2 in Fig. 3(d) without blockingdiodes is provided in (46). The current-dependent path affectsthe performance of linear thrust for motor operation.

For experimental verification of the power driver, the currentwaveform is shown in Fig. 4 when the electric motor is passingthrough one section. The current waveform in Fig. 4 only haspositive current in the mode 2. The current condition matchesthe relation in (45). It can be clearly observed that the current

of the As-phase winding is zero and the current of the Bs-phasewinding is equal to the current of the sectional switch SW1.

B. Analysis of Passing Through the Two Adjacent Sections

Following the aforementioned discussion, this paper willanalyze the case of the moving bogie passing through the twoadjacent sections. This paper will focus on the current analysisof power switches and the As- and the Bs-phase windings.In this paper, two types of driving circuit are illustrated forcomparison as follows:

1) Type I: four switching modes of driver circuit with block-ing diodes, as shown in Fig. 5;

2) Type II: four switching modes of driver circuit withoutblocking diodes, as shown in Fig. 6.


Fig. 6. Switching modes without blocking diode for the electric motor passing through the two sections. (a) Mode 1. (b) Mode 2. (c) Mode 3. (d) Mode 4.

By comparing Type I with Type II, the current-dependent prob-lem in Type II is the drawback for the linear motor operation.As shown in Fig. 5, the power driver with blocking diodes canprevent instant reverse current when linear motor passes fromone section to another adjacent section.

These phenomena can be very helpful in designing a powerdriver for the AFLBM. By using the blocking diodes, there isno current-dependent problem when the linear motor movesbetween the two adjacent sections. In the following, the powerdriver without blocking diodes will be discussed further. Dif-ferent from Type I in Fig. 5, Fig. 6 shows that these switchingcurrents for Type II will create unstable thrust for multisectionalpower driver without blocking diodes.

Therefore, the proposed circuit analysis can provide a gooddesign guidance to the linear motion application. For the

mode l, as shown in Fig. 6(a), the current path of the powerdriver is illustrated when the linear motor passes throughsection 1. The current relation can be expressed as

iSW1 = iAL1. (47)

When the linear motor is passing through the toggle phase, theBs-phase winding current will join with the As-phase windingcurrent for the mode 2, as shown in Fig. 6(b). If there is nopower diode blocked in series with the As-phase winding, theAs-phase reverse current can be observed in the mode 2. Therelationship of current can be further expressed as

iSW1 + (−iAL1) = iBL1. (48)


Fig. 7. Current waveform of mode 2 without blocking diodes for theelectric motor passing through the two sections. Ch1: iBL1 1 A/100 mv,ch2: iSW1 1 A/100 mv, and ch3:iAL1 1 A/100 mv.

When the motor is passing through section 2, the current iAL2

begins flowing for the mode 3, as shown in Fig. 6(c) and theiBL1 disappeared. Therefore, the current will flow upward tothe SW1 and then go back to the power source. The currentrelation can be written as follows:

−iSW1 = −iBL1, iSW2 = iAL2. (49)

For the mode 4 [Fig. 6(d)], the same inference still holds whenthe motor is passing through the toggle Bs-phase winding onsection 2 instead of the original section 1. Thus, the currentequation can be derived as

iSW2 + (−iAL2) = −iBL2. (50)

For experimental verification, the current waveform of themultisectional power driver in the mode 2 of type II is shownin Fig. 7 when the linear motor passes through the two adjacentsections. The current condition in Fig. 7 matches the relationin (48). It can be clearly concluded that, if there is no blockingdiode, the current of As-phase winding will have an unexpectedinteraction with the current of the Bs-phase winding and thepower switch SW1.

VII. VERIFICATIONS

A. System Construction and Hardware Implementation

To validate both the proposed multisectional driver and mod-eling of the AFLBM, simulation and experimental results areprovided for further verification. The adjustable-speed charac-teristics of AFLBM, switching current for the multisectionaldriver, efficiency assessment, acceleration, and decelerationperformance will be demonstrated. The railway is constructedwith the length of 6.24 m, as shown in Fig. 8(a). Power driver isimplemented, as shown in Fig. 8(b). The adjustable-speed char-acteristics by using the pulsewidth-modulation (PWM) dutycycle control of the AFLBM under different loads are shownin Fig. 9. The PWM duty ratio is illustrated as 100%, 80%70%, and 60% for the speed adjustment of the AFLBM system.

Fig. 8. (a) Constructed AFLBM railway system with the length of 6.24 m.(b) Hardware implementation for the power driver.

The simulation results come from the proposed mathematicalmodeling. The experimental results are measured by using theexperimental setup with parameters in Table IV.

By comparing two of the characteristics, the results arematched with each other, as shown in Fig. 9. There are threewinding configurations N = 6, N = 3, and N = 2 for com-parison in Fig. 9. It can be observed that the N = 6 case hasmaximum speed curve and the N = 2 has minimum speedcurve.

B. Speed Performance of Acceleration and Deceleration forthe AFLBM System

The reciprocating motion is demonstrated to show the dy-namic performance for the AFLBM. The railway is 6.24 min length. The moving bogie moves back and forth to showthe acceleration and deceleration. Simulation result and exper-imental result are compared, as shown in Fig. 10. The testingspeed ranges from 147.22 to 187.89 cm/s. The simulation andmeasured speed characteristics are also quite matched witheach other.

C. Blocking-Diode Effect on the Switching Waveform

The switching waveforms without blocking diodes are shownin Fig. 11. νQ1 is the voltage drop on the drain and source of theQ1 transistor in Fig. 3. νQ2 is the voltage drop on the drain and


Fig. 9. Adjustable-speed diagram by using the PWM duty cycle control of the AFLBM under different loads. The cases N = 6, N = 3, and N = 2 are testedfor comparison. Sim: (dashed) simulation result, exp: (solid) experimental result. (a) Duty = 100%. (b) Duty = 80%. (c) Duty = 70%. (d) Duty = 60%.

TABLE IVPARAMETERS OF THE MULTISECTIONAL AFLBM-04 SYSTEM

source of the Q2 transistor in Fig. 3. The switching waveformsindicate that the voltage drop for the power transistors exhibithigher surge up to the peak value of 200 V and then down to80 V when the power switch is off, as shown in Fig. 11(a).Furthermore, it is an abnormal switching condition for thepower driver. By inserting the blocking diodes in series with thephase windings, the surge disappears, as shown in Fig. 11(b).The maximum turn-off voltage for the power transistor is200 V without the surge shape. Normal switching condition canbe measured as expected.

D. Efficiency Assessment for the AFLBM System

The efficiency for the entire system is also measured, asshown in Fig. 12. The AFLBM can work under higher ef-

ficiency by using the proposed power driver. This efficiencyassessment validates the effectiveness of the multisectionalpower driver. The maximum efficiency is 0.41 for the casewithout blocking diodes in Fig. 12(a). The maximum efficiencyis 0.91 for the case with blocking diodes in Fig. 12(b). As shownin Fig. 12, the efficiency can be guaranteed with the help of theblocking diodes. The results also indicate the importance of theblocking diodes, which are emphasized in this paper.

VIII. CONCLUSION

The proposed transformation technique is quite differentfrom the conventional technique due to the parallel windingconfiguration for such a brushless motor. To provide a bettercontrol theoretical basis for this motor, a field-oriented model-ing technique for analyzing such a motor has been proposed.The alternative ST–SVD method has been used as the majortechnique for the associated system modeling. To describe ahighly nonlinear time-varying system for the motor, this paperhas proposed the generalized ST–SVD method to obtain thelinearized relation such as constant torque form. Decoupledrelation has been derived as well. Switching modes for themultisectional power driver have been investigated in detail.The global modeling for N sections and local modeling byusing ST–SVD for the AFLBM has also been formulated.This paper has successfully developed the detailed mathe-matical modeling for system simulation of the linear mo-tor. It will be helpful to the associated further field-orientedcontrol of such a flat-type brushless motor and design ofmultisectional power driver that is applied onto the railwaytransportations.


Fig. 10. Simulation and experimental results for acceleration and deceleration of the AFLBM. (a) Simulation results. (b) Experimental results.

Fig. 11. Blocking-diode effect on the illustrated experiment waveform of the multisectional driver. Ch1: νQ1 10 V/div × 10, ch2: νQ2 10 V/div × 10, ch3:νAL1 2 V/div × 50, and ch4: νAL2 2 V/div × 50. (a) Without blocking diodes in series with windings. (b) With blocking diodes in series with windings.

Fig. 12. Relationship between the output power and the efficiency for the linear motor combined with power driver with respect to the different power losses.(a) N = 6 case with blocking diodes. (b) N = 6 case without blocking diodes.


ACKNOWLEDGMENT

The authors would like to thank Yuan-Giey Tech., Inc., forproviding the testing equipment. The authors would also like tothank J. D. Lee and T. Tseng for the valuable help.

REFERENCES

[1] C. T. Liu, T. S. Chiang, J. F. Díaz Zamora, and S. C. Lin, “Field-orientedcontrol evaluations of a single-sided permanent magnet axial-flux motorfor an electric vehicle,” IEEE Trans. Magn., vol. 39, no. 5, pp. 3280–3282,Sep. 2003.

[2] C. T. Liu, S. C. Lin, J. F. Díaz Zamora, and T. S. Chiang, “Optimaloperational strategy design of a single-sided permanent magnet axial-fluxmotor for electrical vehicle application,” in Proc. IEEE Ind. Appl. Soc.Annu. Meeting, Oct. 2003, vol. 3, pp. 1677–1683.

[3] J. A. Tapia, M. Aydin, S. Huang, and T. A. Lipo, “Sizing equation analysisfor field controlled PM machines: A unified approach,” in Proc. IEEE Int.Elect. Mach. and Drives Conf., Jun. 2003, vol. 2, pp. 1111–1116.

[4] M. Aydin, S. Huang, and T. A. Lipo, “A new axial flux surface mountedpermanent magnet machine capable of field control,” in Proc. IEEE Ind.Appl. Soc. Annu. Meeting, Oct. 2002, vol. 2, pp. 1250–1257.

[5] E. A. Mendrela and M. Jagiela, “Analysis of torque developed in axialflux, single-phase brushless DC motor with salient-pole stator,” IEEETrans. Energy Convers., vol. 19, no. 2, pp. 271–277, Jun. 2004.

[6] C. P. Therapos, “Minimal realization of transfer function matrices via oneorthogonal transformation,” IEEE Trans. Autom. Control, vol. 34, no. 8,pp. 893–895, Aug. 1989.

[7] S. E. Lyshevski, V. A. Skormin, and R. D. Colgren, “High-torque den-sity integrated electro-mechanical flight actuators,” IEEE Trans. Aerosp.Electron. Syst., vol. 38, no. 1, pp. 174–182, Jan. 2002.

[8] S. E. Lyshevski and A. Nazarov, “Control and analysis of synchronousreluctance motors,” in Proc. Amer. Control Conf., Jun. 1999, vol. 3,pp. 1682–1686.

[9] S. W. Leung, T. W. S. Chow, and Y. S. Zhu, “Circuit analysis of a novelloudspeaker system based on linear motor principle,” in Proc. Int. Conf.Circuits and Syst., China, Jun. 1991, vol. 1, pp. 382–384.

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Jian-Long Kuo (S’92–S’93–A’95) received the B.S.degree in electrical engineering and the Ph.D. de-gree from the National Sun Yat-Sen University,Kaohsiung, Taiwan, R.O.C., in 1991 and 1995,respectively.

Currently, he is with the Faculty of Institute ofSystems and Control Engineering, Mechanical andAutomation Engineering, National Kaohsiung FirstUniversity of Science and Technology, Kaohsiung.His research interests include analog and digital cir-cuit design, system control and integration, the motor

driver system for the information products, and electric vehicles.Dr. Kuo was the recipient of the Prize for the Excellent Engineering

Student from the Chinese Institute of Engineers in June 1991. He was alsoawarded with three national patents and two excellent scholastic prizes. Heis currently a member of the Institution of Electrical Engineers, U.K., theInstitute of Electronics, Information, and Communication Engineers, and thePHI-TOU-PHI.

Zen-Shan Chang received the M.S. degree in elec-trical engineering from Chang Gung University,Tao-Yuan, Taiwan, R.O.C., in 2001. Since September2001, he has been working toward the Ph.D. degreeat the Department of Electrical Engineering, ChangGung University.

His research interests include system control,fuzzy logic control, power electronics, integratedcircuit system, and motor control.

Tzu-Shuang Fang is currently working toward theM.S. degree at the Department of Mechanical andAutomation Engineering, National Kaohsiung FirstUniversity of Science and Technology, Kaohsiung,Taiwan, R.O.C.

His current research interests focus on the designof the power electronics and driver circuit.

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