06547672.pdf

338 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 50, NO. 1, JANUARY/FEBRUARY 2014

Stability Consideration of Magnetic Suspension inTwo-Axis Actively Positioned Bearingless

Motor With Collocation ProblemHiroya Sugimoto, Member, IEEE, and Akira Chiba, Fellow, IEEE

AbstractThis paper presents a comprehensive stabilizationtechnique for a two-axis actively regulated bearingless motor.Two-axis actively regulated bearingless motors have the advan-tages of low cost and compactness; however, only two degrees offreedom are actively regulated in the radial xy directions, thusa tilting motion sometimes causes serious interference resultingto instability. The interference occurs in non-collocated structure.In this paper, novel mathematical analysis is carried out and ablock diagram including the interference is presented. It is alsoshown that stable suspension can be successfully realized by meansof adjustments of an integral gain. The analytical results areevaluated by the experiments.

Index TermsBearingless motor, collocation problem, magneticbearing, stability, two-axis active regulation.

I. INTRODUCTION

B EARINGLESS motors combine the functions of a mag-netic bearing and a motor within the same stator/rotorframe. These motors can generate torque as well as a magneticsuspension force on the rotor so that there is no contact be-tween the stator and the rotor. It is anticipated that their mainapplications will lie in centrifugal pumps, contamination-freeventricular assist devices, rotating stages, highly purity phar-maceutical mixing device, flywheels, and other fields requiringno contaminations, no wear, no lubrication, and maintenance-free [1][5].

Literatures report on bearingless motors with one to fiveaxes of active magnetic suspension. An active regulation offive degrees of freedom (5DOF) has been employed in manybearingless or self-bearing motors [6][8] to acquire high out-put power or high reliability. However, if the disturbance is notsignificant, then, magnetic suspension can be realized by onlyactive 2DOF regulation in radial directions [3], [4], [9][18].The 2DOF bearingless motors are compact and can be manu-factured at low cost.

Manuscript received December 5, 2012; revised February 24, 2013; acceptedApril 3, 2013. Date of publication June 26, 2013; date of current versionJanuary 16, 2014. Paper 2012-EMC-692.R1 presented at the 2012 IEEE EnergyConversion Congress and Exposition, Raleigh, NC, USA, September 1520,and approved for publication in the IEEE TRANSACTIONS ON INDUSTRYAPPLICATIONS by the Electric Machines Committee of the IEEE IndustryApplications Society. This work was supported by MEXT KAKENHI Grant-in-Aid for Scientific Research (A) 24246046.

The authors are with Tokyo Institute of Technology, Tokyo 226-0026, Japan(e-mail: [email protected]; [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.2013.2271251

Fig. 1. Principle of the magnetic suspension force generation in the 2DOFbearingless motor. (a) Radial displacement. (b) Axial displacement. (c) Tiltingdisplacement.

In the 2DOF bearingless motors, the axial and tilting direc-tions are passively positioned. A flat disc rotor shape enhancesthe stiffness of the passive positioning. However, the authorshave seen some cases when instability occurs. For example, thetilting rotor displacement influences the rotor positions detectedby the displacement sensors when the position sensors and thesensor target ring are separately installed in tandem to the rotor[17]. The interference occurs when the following three axialpositions are not exactly collocated: 1) the axial position ofthe center of gravity of the rotor (position G), 2) the axialposition of active radial suspension force (position F), and3) the axial position of radial displacement sensors (position S).The definition and the detail explanation will be described inSection II with respect to Fig. 1.

The interference between the magnetic bearing unit and themotor in the tilting direction is considered in [11]. Tilting torqueis generated by the radial magnetic attraction force becauseof a discrepancy between position F and position G in theaxial direction. It is shown that the tilting restoring torquethat overcomes the tilting torque is necessary in the magneticbearing unit. In several previous reports [3], [9], [10] and

0093-9994 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

SUGIMOTO AND CHIBA: MAGNETIC SUSPENSION IN TWO-AXIS ACTIVELY POSITIONED BEARINGLESS MOTOR 339

Fig. 2. Positions of the center of gravity of the rotor (position G), the application position of the radial suspension force (position F) and the position ofthe displacement sensors (position S) in typical applications. (a) Top weight at axially aligned position zS = 0, ZF0 < 0, ZS0 < 0. (b) Top weight at axiallydisplaced position zS = z,ZF < 0, ZS > 0. (c) Top hole at axially aligned position zS = 0, ZF0 > 0, ZS0 > 0. (d) Top hole at axially displaced positionzS = z,ZF > 0, ZS > 0. (e) Outer rotor at axially aligned position zS = 0, ZF0 = ZS0 = 0. (f) Outer rotor at axially displaced position zS = z,ZF >0, ZS > 0.

[15], [16], various centrifugal pumps (e.g., liquid and bloodpumps) with the 2DOF bearingless motors were presented.Each of these pumps has an impeller on the rotor, and thus,the center of gravity of the rotor is moved to the impeller side.Therefore, position G is not collocated to position F. In [11][14], rotating stages in a process chamber were presented. Therotor has asymmetry structure in the axial direction because awork object is installed in the upper surface of the rotor. Thus,position G is moved from position F in the axial direction,thus, these positions are not collocated. Therefore, in mostof the applications, the 2DOF bearingless motors have non-collocated structure. Practically efforts have been done to easethe non-collocation problem. However, dynamics analysis ofthe 2DOF bearingless motor with the non-collocated structureand the instability have never presented in the literature to theauthors best knowledge. In [5], a block diagram of the basicmagnetic bearing is shown. But the block diagram of the 2DOFbearingless motor including the interferences is not shown.

In this paper, novel dynamics of the radial magnetic suspen-sion system including the tilting rotor movement is shown. Itis shown that the adjustment of the integral gain in the PIDcontroller has a significant effect. A comprehensive conditionof the integral gain for stable magnetic suspension is derived.

II. COLLOCATION PROBLEM

A. Definitions of Three PositionsFig. 1(a) and (b) show xz cross-sectional views when the

axial positions G, F, and S are collocated and non-collocated,respectively. Let us define the three positions. Position G isthe axial position of the center of gravity of the rotor. In caseof Fig. 1, position G is center of the rotor height because therotor consists of the cylindrical rotor core and the permanentmagnets. When some attachments are installed on the rotor asshown in Fig. 2, position G is moved from the center of therotor height. Position F is the axial position of active radialsuspension force, hence, the center of confronting area betweenthe rotor and stator. In case of Fig. 1(a), the rotor surface is fullyaligned with the stator surface in the air-gap so that position F

Fig. 3. Analysis models of the rotor tilting stability. (a) A test machine withouta counter weight ZF > 0, ZS < 0; (b) a test machine with a counter weightZF < 0, ZS < 0.

is the center of the rotor height. When the rotor is displacedin the axial direction as shown in Fig. 1(b), the confrontingare is reduced so that position F is moved by z from thecenter of the rotor height. Position S is the axial position of theradial displacement sensors. In case of Fig. 1, the displacementsensor in the x-direction is installed in the center of the statorheight between the stator teeth arranged in the circumferentialdirection. Therefore, position S is the center of the stator height.When the displacement sensors are installed off from the centerof the stator height as shown in Fig. 3, position S is moved tothe axial sensor position.

B. Magnetic Suspension Forces

Fig. 1(a) shows the case when the rotor is displaced inthe positive x-direction. The displacements in the x- and


y-direction are detected by two displacement sensors. InFig. 1(a), only the sensor in the x-direction is illustrated. Letus suppose that the sensor is ideally installed on the horizontalplane at the center of the lamination stack. The lamination stackconsists of magnetic steel sheets laminated in the axial directionas shown in Fig. 1. The magnetic attraction force is generatedin the positive x-direction because of the rotor permanentmagnets (PMs) flux. The radial magnetic suspension force isautomatically generated to overcome the magnetic attractionforce in the negative x-direction by a negative feedback of thedisplacement signal. The axial movement z, tilting rotationalmovements x and y around x- and y-axes, respectively, arepassively stable.

Fig. 1(b) show the case when the rotor is displaced in thenegative z-direction. This displacement is generally caused bythe rotor weight. Positions G, F, and S are not collocated, thus,the non-collocated structure is general in the 2DOF bearinglessmotors. Let us explain axial positions. Position G is moveddown in the negative z-direction along to the rotor becauseposition G is the center of gravity of the rotor. Position F ismoved up at the red broken line because position F is thecenter of confronting area between the rotor and stator. Notethat the distances A and B, these are from the rotor and statoredges, are equal. Position S is also moved up at the bluebroken line because position S is the axial position of the radialdisplacement sensors fixed to the stator. The axial restoringforce is generated in the positive z-direction by fringing fluxesat the edges of the PMs. Thus, magnetic suspension is stable.

Fig. 1(c) shows the case when the rotor is displaced in thepositive y-direction. The tilting restoring torque is generatedin the negative y-direction by the fringing fluxes. Hence, themagnetic suspension in the y-direction is stable in the steadystate.

The positions G,S, and F are located at exactly the sameaxial positions in Fig. 1(a) and (c). However, in Fig. 1(b), thepositions G, F and S are not collocated. Note that position G isthe origin of x-, y- and z-axes. The z-axis positions of S andF with respect to that of position G are defined as ZS and ZF ,respectively. Note that the ZS and ZF in Fig. 1(a) and (c) areboth zero. In Fig. 1(b), ZS and ZF are both positive.

C. Bearingless Motor Applications

Fig. 2 shows the xz cross-sectional views of the threepossible application forms of the bearingless motors. Fig. 2(a)and (b) illustrate a case when a weight is added on the top ofthe suspended rotor. These cases may be centrifugal pumps,which has an impeller at the top. In Fig. 2(b), the rotor is axiallydisplaced in the negative zs-direction, typically caused by therotor weight. Note that zs is the axis fixed to the stator. Anadditional weight is attached on the rotor, and thus, positionG is moved to the upper side. Both ZF and ZS are negativein Fig. 2(a). This case is general in horizontal rotational axismotors. The gravity force direction is perpendicular to therotational z-axis. On the other hand, in Fig. 2(b), ZF and ZSare negative and positive, respectively. This is a case of verticalaxis motors: the rotational z-axis corresponds to the gravityaxis. Let us define ZF0 and ZS0 as the ZF and ZS values inaligned position as shown in Fig. 2(a). Note that ZF0 = ZS0

in this case. Let us also define the rotor axial displacement asz in Fig. 2(b). Note that z is negative in Fig. 2(b). The axialdisplacements are given by

ZS =ZS0 z (1)ZF =ZF0 z/2. (2)

From (1) and (2), the condition of ZS > 0 and ZF < 0 occurswhen z is in a range below

2ZF0 < z < ZF0. (3)Fig. 2(c) and (d) show a case when the mass is eliminated

from the upper surface of the suspended rotor. Thus, position Gis moved to the low side from the center of the lamination stack.This may be a case of rotating stage applications, and a workobject can be installed on the upper surface of the rotor. Notethat the nominal distances of positions F and S, ZF0 and ZS0,respectively from position G are positive in Fig. 2(c). This caseis possible when the rotational z-axis is horizontal. In Fig. 2(d),the z-axis is vertical, thus, the rotor is moved because of itsweight. The positions ZF and ZS are also positive.

Fig. 2(e) and (f) show an outer rotor type bearingless motor.The outer rotor type is useful in the applications of flywheels[18] and stirring devices [4]. In Fig. 2(f), the outer rotor ismoved because of its weight. The positions ZF and ZS are bothpositive. In the three cases, the signs of ZF and ZS are differentin Fig. 2(b); however, the signs are the same in Fig. 2(a) and(c)(f). These signs are shown to be depending on the machinestructure. In the next section, it is shown that stable magneticsuspension is not very easy if the signs of ZF and ZS aredifferent.

D. Tilting Stability Problem

In this section, a mathematical representation of dynamicsequations based on a structure close to that of a test machineis derived. Fig. 3(a) shows the analysis model of a test machinewith the rotor tilting displacement without an additional counterweight. In the test machine, a sensor target ring is constructedunder the rotor, and the displacement sensors are installed underthe stator. Position S is located rather low compared to thestructures shown in Fig. 2. The center of gravity of the rotoris moved under the center of the lamination stack. The axialpositions ZF and ZS are positive and negative, respectively.The signs are different. On the other hand, in Fig. 3(b), a counterweight is attached on the rotor to move the center of gravity inthe z-axis direction. Thus, both ZF and ZS are in the samesigns, i.e., negative.

Fig. 3(a) is similar to Fig. 2(b) because the signs of ZF andZS are different. Fig. 3(b) is similar to Fig. 2(d) and (f) becausethe signs of ZF and ZS are the same.

The center of gravity is the origin of x-, y- and z-coordinates.Let us also suppose that the rotor only tilts around they-axis for simplicity. Due to the tilting displacement, the radialdisplacement occurs at position S as one can see in Fig. 3. By2DOF active feedback loops, the active force fx is generated atposition F. This additional force generates a tilting torque in they-direction in Fig. 3(a), although it is opposite to y-directionin Fig. 3(b). In the case of Fig. 3(a), the stability criterion


Fig. 4. Waveforms in the y , xs and feedback currents of a PD or PID controller with a collocation problem. (a) In case of Fig. 3(b), i.e., ZF and ZS areboth negative, y and xs are converged by the current of the proportional and derivative gains. (b) In case of Fig. 3(a), i.e., ZF and ZS is positive and negative,respectively, y and xs are oscillated by the current of the proportional and derivative gains. (c) In case of Fig. 3(a), y and xs are converged by the current of thecarefully selected integral gain.

is serious, because the generated tilting torque enlarges thetilting displacement, i.e., a positive feedback. Thus, interactionbetween the radial and the tilting movements occur, whichresults in unstable in the magnetic suspension in the worst case.

Fig. 4(a) shows the waveforms of y, xs, and currents IKPand IKD when the initial tilting displacement y = 1 mrad isprovided. The structure is close to that shown in Fig. 3(b), i.e.,ZF = 2.05 mm and ZS = 19.55 mm. In the calculation,position G is fixed in the radial and axial directions. Only rotortilting movement around position G is assumed. The positions Fand S are displaced in the radial and tilting directions becauseof the non-collocated structure, thus, force is generated by theradial negative stiffness. The structure of the block diagramin these calculation will be shown later in the description ofFig. 6. The xs is the radial displacement at position S from thenominal position. IKP and IKD are the current componentsof the proportional and derivative controller, respectively. Thecurrents IKP and IKD are given by

IKP = KPxs (4)IKD = KD dxs

dt(5)

where KP and KD are the proportional and derivative gains,respectively. The suspension force is proportional to thesecurrents. At the time indicated by I in Fig. 4(a), IKP is atmaximum because IKP provides spring force with respect tothe displacement xs. As for the rotational moment y , theoscillation is opposite to xs. Fortunately, the force fx generatesnegative torque in y as one can see in Fig. 3(b), so, the currentIKP also effectively generates a stable spring force in tiltingmovement. The force generated by IKD also provides effectivedamping force for the tilting displacement. In short, stable forceis generated by the displacement in y through radial activepositioning. Therefore, the tilting displacement is converged bythe PD feedback loop.

Fig. 4(b) shows similar waveforms for the condition of ZF =2.05 mm and ZS = 15.45 mm. This is the case of Fig. 3(a)when the sign of a product of ZF and ZS is negative, i.e.,

ZFZS < 0. The feedback currents IKP and IKD are generatedbased on xs. Thus radial movement is actively regulated. How-ever, the feedback radial force generates unstable spring anddamping torque in tilting movement. Therefore, the oscillationof y is gradually enlarged.

Fig. 4(c) shows the similar waveforms when an integralcontroller is added. IKI is the current output of the integralcontroller, given as

IKI = KI

xsdt (6)

where KI is the integral gain. The integral function is generallyused to eliminate steady-state error in active radial positioning.From the waveform of the current IKI , it is seen that IKIoscillation is opposite to IKD. Thus, damping force is gen-erated in the tilting oscillation. In this case, the value of KImust be carefully selected. The value must be high enoughto be effective, but low enough to realize stable radial activepositioning. The aim of this paper is to derive stable conditionsincluding the effect of the integral controller.

III. MODELING AND ANALYSIS OF MAGNETICSUSPENSION SYSTEM

In this chapter, the condition of stable magnetic suspensionat non-collocated structure is mathematically derived.

A. Dynamics

The radial displacement xF of position F due to the rotortilting displacement y is given, assuming siny and cosy areapproximately equal to y and 1, respectively, as

xF = ZF y. (7)

The radial force fx applied to position F is represented as

fx = kiix + kx(x+ xF ) (8)


Fig. 5. Block diagram of the radial suspension system with collocation, i.e.,ZF = ZS = 0.

where ki is the currentforce factor, kx is the displacement-force factor, ix is the x-axis suspension current and x is theradial displacement of position G. Here, fx is applied to therotor surface confronting the air gaps. From fx, the radial forceis generated at position G as a functions of the rotor radius rand the distance ZF . This force is equal to the product of therotor mass m and the second-order derivative of x with respectto the time, as follows:

mx =r2

r2 + Z2Ffx. (9)

A tilting moment is also generated by force fx due to thediscrepancy between positions F and G. The moment is given asa product of ZF and fx. The equation of the rotational motionis given as follows:

Jy = kty + ZF fx (10)

where J is the rotor inertia around the tilting axis and kt is theratio between the tilting angle and the restoring torque.

In addition, the radial displacement xS of position S isexpressed as

xS = ZSy. (11)

Therefore, a feedback signal is the sum of x and xS .Fig. 5 shows a block diagram of the radial suspension control

system when ZF and ZS are zero, exactly collocated condition.A PID controller is connected to the plant, which has a typicalone-axis magnetic suspension system with a positive feedbackloop of the displacement force factor kx [5]. Negative feedbackis provided comparing the position reference x and the de-tected position x and processing the position error in the PIDcontroller.

Fig. 6 shows the block diagram of the case when ZF andZS are not zero, that is, when positions G, F, and S are notcollocated. The corresponding equation numbers from (7)(11)are written in the corresponding block diagram. One can seethat there are two additional feedback signals A and B indicatedin the figure. These signals are caused by the non-collocationof positions F and S with respect to G. Feedback signal A isunstable positive feedback that does not depend on the sign ofZF . Feedback signal B can be negative or positive feedbackdepending on the sign of the product of ZF and ZS . Thefeedback signal would be unstable positive feedback when thesign of ZFZS is negative. In this case, the system becomesunstable if the controller feedback is properly designed. If thesign is positive, this loop provides negative feedback.

The blocks in the area C is removed in the calculation inFig. 4 because the rotor radial displacement is fixed to zero.

Fig. 6. Block diagram of the radial suspension system with a non-collocationproblem, i.e., at ZF = 0, ZS = 0.

TABLE ICOEFFICIENTS OF DENOMINATOR AND NUMERATOR POLYNOMIALS

B. Derivation of Stability ConditionsThe transfer function of the closed-loop system in Fig. 6,

including the PID controller and the whole plant including thenon-collocated tilting movement, is represented by the fifth-order system as follows:

x

x=

N(s)

D(s)=

b0s4 + b1s

3 + b2s2 + b3s+ b4

a0s5 + a1s4 + a2s3 + a3s2 + a4s+ a5(12)

where D(s) and N(s) are the denominator and the numeratorpolynomials, respectively. For simplicity, it is assumed that therotor radius is large enough with respect to ZF so that r2/(r2 +Z2F ) can approximate unity. The coefficients a0 a5 and b0 b4 are listed in Table I.

To derive the stability condition, the Hurwitz stability cri-terion is applied to the characteristics polynomial. The fol-lowing conditions of the coefficients and the determinants arenecessary:

1) All of the signs of the coefficients a0, a1, a2, a3, a4 anda5 of the denominator polynomial are positive.

2) All of the determinants calculated from the Hurwitzmatrices are positive.

The Hurwitz matrix is given by

Hn =

a1 a3 0a0 a2 a4 00

.

.

.

.

.

. ...0 an3 an1 00 an4 an2 an

> 0, n = 1, 2, 3, 4, 5.

(13)


The necessary and sufficient conditions for the system to bestable are calculated. As space is limited, we list only the mostimportant three conditions, as follows:

ZFZS < Jm

(14)

KP >kxki

(15)H5 =mk

2t k

3iKIQ(ZFZS)R(KI) > 0 (16)

where

Q(ZFZS)

= mkx(ZFZS)2 +

(Jkx +mkt mZ2F kx

)ZFZS

JkxZ2F (17)R(KI)

= JmK2I +KD

(Jkx +mZ

2F kx mkt JkiKP

mZFZSkiKP)KI + ktK

2D(kiKP kx). (18)

Equation (14) becomes a critical problem only when a productof ZF and ZS is negative. However, in the test machine,the right term is approximately one hundredth times as largeas the left term, so (14) can be realized in most machines.Equation (15) provides a minimum proportional gain condition.Equation (15) can be rewritten as KP ki > kx, thus, the nega-tive feedback force must cancel the magnetic attraction forcegenerated in a radial displacement. This is typical requirementof stable magnetic suspension even though the machine hascollocation. Thus, (16) is the most important condition of thenon-collocated structure. For (16), Q(ZFZS) and R(KI) arewritten as the functions of the ZFZS and KI , respectively.These are second-order polynomials. Thus, the condition of theKI is classified by the signs of Q(ZFZS) so that there are onlytwo cases.

The equation Q(ZFZS) is equated to zero, and two roots forthe product of ZF and ZS , hereafter referred as (ZFZS)1 and(ZFZS)2 of ZFZS are derived as follows:

(ZFZS)1, (ZFZS)2 =1

2mkx

[ (Jkx +mkt mZ2F kx)

(Jkx +mkt)2 + 2mZ2F kx(Jkx mkt) + (mkxZ2F )2

].

(19)

Note that the (ZFZS)1 is defined smaller than (ZFZS)2.As Q(ZFZS) is a quadric function, Q(ZFZS) is positivewhen (ZFZS) < (ZFZS)1 or (ZFZS)2 < (ZFZS). On thecontrary, it is negative when (ZFZS)1 < (ZFZS) < (ZFZS)2.The equation R(KI) is also equal to zero. The roots KI1 andKI2 are derived as follows:

KI1,KI2 = KD

(A+

kiKP kx2m

B2Jm

)(20)

TABLE IICONDITIONS OF INTEGRAL GAIN

TABLE IIIPARAMETERS OF THE TEST MACHINE

where

A =ZFZSKiKP Z2F kx + kt

2J(21)

B =

{(kiKP )

2(J + ZFZS)2

2kiKP{Jkx

(J +mZ2F

)+mZFZS

[kx(J +mZ2F

)mkt]}+ (Jkx +mkt)

2

+mZ2F[2J +m(kxZF )

2 2mkt]}1/2

. (22)

Further calculation reveals that (16) is satisfied for certainvalues of KI . The conditions are summarized in Table II. TheHurwitz criterion is satisfied in two cases, the case 1 and case 2.In case 1, KI can be zero, thus this case corresponds tostructures in Fig. 2(a), (c), (d), (f) and Fig. 3(a). The case 2corresponds to the structures in Fig. 2(b), Fig. 3(b) and test ma-chine. Let us examine the conditions substituting parameters.

In Table III, parameters of the prototype machine are sum-marized. The case 2 is adapted in the test machine becauseZFZS = 31.67 mm2. Substituting parameters into (19), weobtain (ZFZS)1 = 1934 mm2 and (ZFZS)2 = 3.737 mm2.Substituting parameters in (20), we obtain KI1 = 2.526106 A/(m s) and KI2 = 2.908 105 A/(m s). Hence, the sta-ble condition of KI is given as 2.908 105 KI 2.526106. Note that the system is at its stability limit when KI is thelowest or the highest values. Note that (ZFZS)2 is not exactlyzero, thus, the correspondence of the structures in Fig. 2(a), (c),(d), and (f) should be carefully checked numerically.


Fig. 7. Root-locus of the system.

Fig. 8. Detailed view of Pole A.

C. Root Locus Analysis

Fig. 7 shows the whole root locus of the system for 0 KI 6 106 at the test machine parameters. Five poles existin the complex plane because the characteristic polynomial ofthe system is a fifth-order polynomial. The pole A and thepole B are conjugate complex poles close to the imaginary axis.These two poles exist only in the case of the non-collocatedstructure. Another two poles are located at the imaginary axiswhen KI = 2.526 106. If KI is increased, then, the polesmove to the right-half plane and magnetic suspension is un-stable. This unstable state occurs even in the exactly collocatedcondition. Thus, the poles A and B should be examined for thenon-collocation problem.

Fig. 8 shows the detailed view of the pole A. The pole Ais in the right half plane when KI = 0. The pole moves fromthe right-half plane to the left-half plane as KI is increased.From Figs. 7 and 8, it can be seen that all poles are locatedin the left-half plane when 2.908 105 < KI < 2.526 106.In the next section, the experimental results are shown in casesof KI = 2 104 and KI = 7 105, i.e., unstable and stablecases.

IV. EXPERIMENTAL RESULTS

Fig. 9(a) and (b) show a prototype 2DOF test machine.The rotor has consequent-pole PM structure. The suspensionwindings are wound in toroidal windings around the statoryoke. The motor windings are wound around the stator teeth asthe concentrated windings. Fig. 9(b) shows the rotor structure.A sensor target ring is constructed below the rotor as previouslyshown in Fig. 3.

Fig. 10 shows the start-up waveforms of x, y and y with thelow integral gain of KI = 2 104 in the x-direction. The inte-gral gain in the y-direction is 7 105. Theses x and y integralgains are located outside and inside of the stable conditions,respectively as previously shown in Fig. 8. The x and y aremeasured by the displacement sensors. The y is calculated bytwo monitoring eddy-current sensors attached above the rotor

Fig. 9. (a) 2DOF bearingless motor and controller and (b) consequent-polerotor with ZF = 2.05 mm and ZS = 15.45 mm.

Fig. 10. Start-up waveforms in the x, y, and y when KI in the x- andy-directions are 2 104 and 7 105, respectively.

Fig. 11. Start-up waveforms in the x-, y-, and y-directions when KI in thex- and y-directions is both 7 105.

as described in [18]. The position references in the x- andy-direction are 0 mm. The rotor is at touch-down before thesuspension control is activated. The radial displacements x andy follow the references as soon as the suspension control isstarted. However, x and y begin to oscillate and then resultin touch-down.

Fig. 11 shows the waveforms with the increased integral gainof KI = 7 105 in the x-direction. Both integral gains are setin the theoretically stable condition. The x and y immediatelyfollow the references at start-up. The y converges in 0 mrad.Hence, it is seen that the adjustment of the integral gainsrealizes stable magnetic suspension. The theoretical calculationis confirmed by the start-up test.

Fig. 12 shows the combination of the proportional andintegral gains when the derivative gain is fixed to KD =18.5 A/(m/s). The black curve is calculated by (20). Thearea enclosed by the curve is stable. A series of experimentsare carried out, these are close to the start-up test presentedin Figs. 10 and 11. The minimum and maximum integralgains are found for each proportional gain, then, these are


Fig. 12. Combination of the proportional and integral gains in the testmachine.

plotted in Fig. 12. The previous two cases in Figs. 10 and11 are also plotted. The stable gain area in the experimentand theoretical calculation is mostly corresponding. Therefore,theoretical analysis is confirmed by the experiments in thecase 2 in Table II.

V. CONCLUSION

This paper has presented a comprehensive regulation tech-nique for the stabilization of radial magnetic suspension ina system with the two-axis active positioning. With respectto the center of gravity G of the rotor, the axial position ofactive suspension force F and the axial position of displacementsensors S are not always collocated in the axial direction. Themagnetic suspension has a serious problem when the sign ofa product of ZF and ZS is negative. Even in this case, themagnetic suspension can be stabilized when the integral gainis properly selected.

The dynamic analysis of the suspension control system in-cluding the interference was presented. By the Hurwitz stabilitycriterion, conditions of the integral gain were derived for stablemagnetic suspension. It is found that adjustment of the integralgain has a significant effect on stable magnetic suspension. Thederived suspension stabilization is verified experimentally witha 2DOF prototype machine. In the further studies, other caseswill be verified.

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Hiroya Sugimoto (M12) was born in Tokyo, Japan,in 1985. He received the B.S. and M.S. degreesin electrical engineering from Tokyo University ofScience, Tokyo, in 2007 and 2009, respectively.

In 2009, he joined the Honda R&D Company,Ltd., Automobile R&D Center, Tochigi, Japan. Since2011, he has been Research Associate in the De-partment of Electrical and Electronic Engineering,Graduate School of Science and Engineering, TokyoInstitute of Technology, Tokyo. He has been studyingmagnetically suspended bearingless motors.

Mr. Sugimoto is a member of the Institute of Electrical Engineering of Japan.

Akira Chiba (S82M88SM97F07) was bornin Tokyo, Japan, in 1960. He received the B.S., M.S.,and Ph.D. degrees in electrical engineering fromTokyo Institute of Technology, Tokyo, in 1983, 1985,and 1988, respectively.

In 1988, he joined Tokyo University of Science,Tokyo, as a Research Associate in the Departmentof Electrical Engineering, Faculty of Science andTechnology. From 1992 to 1993, from 1993 to 1997,from 1997 to 2004, and from 2004 to 2010 he wasResearch Lecturer, Senior Lecturer, Associate Pro-

fessor, and Professor, respectively. Since 2010, he has been a Professor in theDepartment of Electrical and Electronic Engineering in the Graduate School ofScience and Engineering, Tokyo Institute of Technology. In 19901991, he wasa Natural Science and Engineering Research Council of Canada InternationalPost-Doctoral Fellow at the Memorial University of Newfoundland, Canada.He has been studying magnetically suspended bearingless ac motors, superhigh-speed motor drives, and rare-earth-free-motors for hybrid and pure elec-trical vehicles. He has published more than 884 papers and the first book onmagnetic bearings and bearingless drives in 2005, and submitted 60 patents.He received IEEJ Paper Awards in 1998 and 2005. He received the First PrizePaper Award from the Electric Machines Committee of the IEEE IndustryApplications Society (IAS) in 2011. He was a member and Chair of theIEEE Nikola Tesla Award Committee in 20102012 and 2013, respectively. Hehas served as Secretary, Vice-Chair, Vice-Chair-elect, and Chair of the MotorSub-Committee of the Electric Machinery Committee of the IEEE PES in20072008, 20092010, 20112012, and 2013, respectively. He served as Vice-Chair and Chair of the IEEE IAS Japan Chapter in 20082009 and 20102011,respectively. He has been a member of the Electric Machines Committee andthe Industrial Drives Committee of the IEEE IAS.

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