Research Article Modeling and Analysis of Coupling ...downloads.hindawi.com/journals/mpe/2014/340140.pdfResearch Article Modeling and Analysis of Coupling Performance of Dynamic Stiffness

Research ArticleModeling and Analysis of Coupling Performance ofDynamic Stiffness Models for a Novel Combined Radial-AxialHybrid Magnetic Bearing

Bangcheng Han1,2,3 and Shiqiang Zheng1,2,3

1 Fundamental Sciency on Novel Inertial Instrument & Navitation System Technoloty Laboratory,Beijing 100191, China

2 Science and Technology on Inertial Laboratory, Beijing 100191, China3 School of Instrument Science and Opto-electronics Engineering, Beihang University, Beijing 100191, China

Correspondence should be addressed to Bangcheng Han; [email protected]

Received 27 June 2013; Accepted 18 November 2013; Published 12 January 2014

Academic Editor: Paulo Batista Goncalves

Copyright © 2014 B. Han and S. Zheng.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The combined radial-axialmagnetic bearing (CRAMB)with permanentmagnet creating bias flux can reduce the size, cost, andmassand save energy of the magnetic bearing. The CRAMB have three-degree-of-freedom control ability, so its structure and magneticcircuits are more complicated compared to those of the axial magnetic bearing (AMB) or radial magnetic bearing (RMB). And theeddy currents have a fundamental impact on the dynamic performance of the CRAMB.The dynamic stiffness model and its crosscoupling problems between different degrees of freedom affected for the CRAMB are proposed in this paper. The dynamic currentstiffness and the dynamic displacement stiffness models of the CRAMB are deduced by using the method of equivalent magneticcircuit including eddy current effect, but the dynamic current stiffness of the RMB unit is approximately equal to its static currentstiffness. The analytical results of an example show that the bandwidth of the dynamic current stiffness of the AMB unit and thedynamic displacement stiffness of the CRAMB is affected by the time-varying control currents or air gap, respectively. And thedynamic current stiffness and the dynamic displacement stiffness between the AMB unit and the RMB unit are decoupled due tofew coupling coefficients.

1. Introduction

Most of 5-axis active magnetic bearing systems (MBSs) areusually composed of two radial magnetic bearing (RMB)units and one axial magnetic bearing (AMB) unit [1–5].These magnetic bearing systems are the easiest way to beproduced, but they are also tending to bulky, high powerand high cost. In order to reduce the size, cost, and saveenergy and increase the high-power density of the MBS,which is important to reduce the number of the units bymeans of furthermore combination the AMB and RMB. Acombined radial-axial magnetic bearing (CRAMB) whichis named as 3-axis magnetic bearing (MB) is designed for

use in an ultra-high-speed machine [6]. The magnetic forcesand coupling problems of a combined AMB and RMB areanalyzed [7]. The integrated AMB and RMB with conicalrotor are designed and analyzed [8, 9], and one downside ofthe integrated bearings is a strong coupling problem betweenthe radial and axial degrees of freedom. An AC-DC 3-DOFhybrid magnetic bearing is proposed and designed [10].Structure and control method of an AC-DC 3-DOF hybridmagnetic bearing is introduced in the literature [11–14]. A3-DOF axial hybrid magnetic bearing [15] is proposed, butthe structure and processing technic are very complicated,and its rotational power loss will be large at high speed. Thestructure of a 3-DOF magnetic bearing without large thrust

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014, Article ID 340140, 17 pageshttp://dx.doi.org/10.1155/2014/340140

2 Mathematical Problems in Engineering

disk rotor is introduced and designed [16]. An integratedradial-axial magnetic bearing without axial disk is presentedin the literature, and its static performance is analyzed byusing 3-D FEM. The static mathematical model of a radial-axial magnetic bearing is introduced in the literatures [17, 18].

Most literaturesmentioned above described the structure,design, analysis, and control methods of the CRAMBs, whichare mainly consider the static mathematical model withoutincluding the eddy-current effects. But few literatures onthe dynamic stiffness modeling and coupling problems ofCRAMB with permanent magnet bias are described whenconsidering the dynamic characteristics affected by the eddycurrents caused by time-varying air gap and control currents.Some parts of the CRAMB are very difficult to be split intolaminated which can reduce the eddy-current loss, then theeddy currents will result in magnitude reduction and phaselag of radial and axial magnetic force and make the corehotter. That is to say, the eddy current has a serious impacton the dynamic performances of the CRAMB.

Due to the coupled magnetic, electrical, and mechanicaldomains, design and analysis of CRAMB are very difficult.In particular, its dynamic characteristics are affected by thecoupled domains mentioned above. As a result, to predictthe dynamic operating characteristics is necessary to avoidcostly trials. A few simple dynamic models including eddycurrents are described for magnetic actuators [19–25]. Thefinite element method (FEM) is also used to analyze theeddy-current effects on the active thrust MBs. The elec-tromechanical performance ofmagnetic actuators is analyzedusing a coupled structural and electromagnetic FEM [26, 27].The dynamic characteristics of MB based on field-circuitcoupled method are proposed [28]. The FEM is used toanalyze the operating characteristics of the electromagnetwith permanent magnets [29]. The dynamic stiffness modelsof active thrust magnetic bearing are also analyzed by FEMand analytical method [30, 31]. The eddy current loss insurface layer of a laminated core [32] and a solid core [33]for radial MB is analyzed. And the experimental methods areused to investigate the eddy-current effects.The force/currentrelationship of a solid thrust MB is tested by Allaire et al.[34]. DeWeese et al. investigated different rotor and statorconfigurations to find the one with minimal eddy currents[35].The limitations on the closed-loop performance of activethrust MB affected by the eddy currents are investigated [36].

The emphases of dynamic stiffness mentioned above arethe active thrust MB which has one degree of freedom(DOF), and the structure and magnetic flux path are easyto design and analyze. The structure and magnetic circuitsof the CRAMB are more complicated compared to those ofthe thrust MB or radial MB, and the cross-coupling problemsbetween different DOFsmay be produced. But few literatureson dynamic characteristics and the coupling problems of theCRAMB are published. So, it would have higher researchvalue to investigate the accurate dynamic factor model andthe coupling problems including eddy currents rather thanthe static mathematic model. The dynamic stiffness modelsand the dynamic coupling models including eddy-currenteffects for CRAMB with permanent magnet bias are pre-sented and derived analytically by using equivalent magnetic

Permanent magnet ring

Axial stator coreAxial control coil Radial control coils

Radial stator core

Magnetic ring

Thrust disk

Figure 1: Configuration of the CRAMB with permanent magnetring.

Permanent magnet ringNS

NSBias flux

Radial air gap

Radial control coils

Axial control coilRadial control flux Axial control flux

Axial air gapMagnetic ring

Axial disk

Figure 2: The magnetic circuits of the combined radial-axialmagnetic bearing with permanent magnet ring.

circuit method in this paper. The dynamic current stiffnessmodels and the dynamic displacement stiffness models ofthe RMB unit and AMB unit are given. And the dynamiccoupling models between axial and radial bearing units arealso given. An example is also given in this paper.

2. Analysis Model and Method

2.1. Analysis Model. Figure 1 shows the configuration of theCRAMB which combines the radial and axial magneticbearing unit. The axially magnetized permanent magnet ring(PMR) is used to produce bias flux for both the RMB and theAMB unit.The laminations are used only for radial stator androtor core, and the other parts are made of solid material.Smaller outside diameter of the rotor thrust disk is used.Therefor the lower rotational drag, the lower stresses, and thecompact structure are attained. The magnetic ring is used toeliminate the saturation of the magnetic circuit for the statorcore of the RMB unit.The bias flux and the control flux pathsof the RMB and AMB unit are depicted in Figure 2.The solidline indicates the bias flux paths generated by the PMR andthe dotted line denotes the control flux paths generated bythe control current of the AMB and the RMB unit. As canbe seen, their control flux produced by the control currentin coils will pass through air gap of the RAM and the AMBrather than the PMR, and therefore small control currents are

Mathematical Problems in Engineering 3

Radial stator core

13 (radial air gap) 6 (axial air gap)

(lamination stack)

Radial rotor core(lamination stack)

2 3

4

5

10

8

9 7

1 2

1

Axis

11

Figure 3: Division of the geometry of the CRAMB: 13 elementssimply divided.

enough since they do not have to compensate for the highmagnetic reluctance of the PMR. The control flux has to besuperimposed by the bias flux in air gap for producing theaxial and radial magnetic force.

The radial and axial magnetic bearing force and fluxcan be derived conveniently with the static magnetic circuitmodel without consideration of the eddy-current effect.However, the eddy currents caused by the time-varying airgap and the control currents will affect the dynamic char-acteristics of the CRAMB system. Based on an approximateanalysis method [22] considering the eddy-current effectinside a nonlaminated electromagnetic actuator, the CRAMBgeometry is divided into thirteen elements as shown inFigure 3. The four elements of the air gap between rotorand stator of the RMB are named as element 13. The eddy-current loss can be reduced by using laminated stator coreand rotor core of the RMB unit in CRAMB. According tothe equivalent magnetic circuit models of the CRAMB, theeffective reluctances are separated into two parts: one is thestatic magnetic reluctance of element, and the other one isthe dynamic magnetic reluctance which is a half-order termrelated to frequency.

The elements 1–5, 7, and 9–12 can be divided in the sameway as in Sun et al.’s models [30] and Zhu et al.’s models [23].The PMR is used for element 2 which is used to provide biasflux and can also be simplified in the same way.The elements1, 2, 11, and 12 are located in the bias flux path. Therefore,the eddy-current effect should be considered inmathematicalmodel. The simplified effective reluctance of each elementcan be used to design, analyze, and control purpose. Thesimplified calculation of the effectivemagnetic reluctances forthese elements is given as follow:

𝑅1= 𝑅0

1+ 𝑅𝑖

1=

ℎ1

𝜋𝜇0𝜇𝑟(𝑟2

6− 𝑟2

5)+

ℎ1

2𝜋𝑟5

√𝑗𝜔𝜎1

𝜇0𝜇𝑟

, (1)

where 𝑅01= ℎ1/𝜋𝜇0𝜇𝑟(𝑟2

6− 𝑟2

5), 𝑅𝑖

1= (ℎ1/2𝜋𝑟5)√𝑗𝜔𝜎

1/𝜇0𝜇𝑟

𝑅2= 𝑅0

2+ 𝑅𝑖

2=

ℎ2


6− 𝑟2

5)+

ℎ2

2𝜋𝑟5

√𝑗𝜔𝜎2

𝜇0𝜇𝑟

, (2)


6− 𝑟2

5), 𝑅𝑖

2= (ℎ2/2𝜋𝑟5)√𝑗𝜔𝜎

2/𝜇0𝜇𝑟

𝑅3= 𝑅0

3+ 𝑅𝑖

3=

ℎ4


6− 𝑟2

5)+

ℎ4

2𝜋𝑟5

√𝑗𝜔𝜎1

𝜇0𝜇𝑟

, (3)


6− 𝑟2

5), 𝑅𝑖3= (ℎ4/2𝜋𝑟5)√𝑗𝜔𝜎

1/𝜇0𝜇𝑟

𝑅4= 𝑅0

4+ 𝑅𝑖

4=

ln (𝑟6/𝑟4)

2𝜋𝜇0𝜇𝑟ℎ5

+ln (𝑟6/𝑟4)

2𝜋√𝑗𝜔𝜎1

𝜇0𝜇𝑟

, (4)

where 𝑅0

4= ln(𝑟

6/𝑟4)/2𝜋𝜇

0𝜇𝑟ℎ5, 𝑅𝑖

4= (ln(𝑟

6/𝑟4)/

2𝜋)√𝑗𝜔𝜎1/𝜇0𝜇𝑟

𝑅5= 𝑅0

5+ 𝑅𝑖

5=

ℎ6


4− 𝑟2

3)+

ℎ6

2𝜋𝑟3

√𝑗𝜔𝜎1

𝜇0𝜇𝑟

, (5)


4− 𝑟2

3), 𝑅𝑖5= (ℎ6/2𝜋𝑟3)√𝑗𝜔𝜎

1/𝜇0𝜇𝑟

𝑅𝑧1= 𝑅6=

𝑠0+ 𝑧

𝜋𝜇0(𝑟2

4− 𝑟2

3),

𝑅𝑧10

= 𝑅0

6=

𝑠0

𝜋𝜇0(𝑟2

4− 𝑟2

3), 𝑧 = 0,

𝑅7= 𝑅0

7+ 𝑅𝑖

7=

ln (𝑟4/𝑟3)


+ln (𝑟4/𝑟3)


𝜇0𝜇𝑟

,

(6)

where 𝑅0

7= ln(𝑟

4/𝑟3)/2𝜋𝜇

0𝜇𝑟ℎ7, 𝑅𝑖

7= (ln(𝑟

4/𝑟3)/


𝑅𝑧2= 𝑅8=

𝑠0− 𝑧

𝜋𝜇0(𝑟2

4− 𝑟2

3),

𝑅𝑧0= 𝑅𝑧20

= 𝑅0

8= 𝑅𝑧10

=𝑠0

𝜋𝜇0(𝑟2

4− 𝑟2

3), 𝑧 = 0,

𝑅9= 𝑅0

9+ 𝑅𝑖

9= 𝑅5= 𝑅0

5+ 𝑅𝑖

5,

(7)


4− 𝑟2

3), 𝑅𝑖9= (ℎ6/2𝜋𝑟3)√𝑗𝜔𝜎

1/𝜇0𝜇𝑟

𝑅10= 𝑅0

10+ 𝑅𝑖

10= 𝑅4= 𝑅0

4+ 𝑅𝑖

4, (8)

where 𝑅0

10= ln(𝑟

6/𝑟4)/2𝜋𝜇

0𝜇𝑟ℎ5, 𝑅𝑖10

= (ln(𝑟6/𝑟4)/


𝑅11= 𝑅0

11+ 𝑅𝑖

11=

ln (𝑟3/𝑟1)


+ln (𝑟3/𝑟1)


𝜇0𝜇𝑟

, (9)

where 𝑅0

11= ln(𝑟

3/𝑟1)/2𝜋𝜇

0𝜇𝑟ℎ7, 𝑅𝑖11

= (ln(𝑟3/𝑟1)/


𝑅12= 𝑅0

12+ 𝑅𝑖

12=

ℎ9


1− 𝑟2

8)+

ℎ9

2𝜋𝑟8

√𝑗𝜔𝜎1

𝜇0𝜇𝑟

, (10)


1− 𝑟2

8), 𝑅𝑖12= (ℎ9/2𝜋𝑟8)√𝑗𝜔𝜎

1/𝜇0𝜇𝑟

𝑅13= 𝑅0

13=

360 (𝑟3− 𝑟2)

𝜋𝜇0ℎ10(𝑟3+ 𝑟2), (11)

where𝑅0𝑗is the staticmagnetic reluctance (the first term in𝑅

𝑗

for element 𝑗) and𝑅𝑖𝑗is the eddy-currentmagnetic reluctance

(the second term in 𝑅𝑗for element 𝑗) with 𝑗 = 1, 2, . . . , 13,


Axis

h1 h2 h3 h4 h5

h6h8

h9

h10

s0s0

r1

r2 r3r4r5

r6

r7

r8

r9

h7

Figure 4: The dimension of the CRAMB.

and the parameters and their dimensions are given in Figure 4and Table 1. 𝑅

13is the magnetic reluctance of the radial air

gap. 𝑅6and 𝑅

8are the magnetic reluctances of the axial

air gap. The equivalent magnetic circuits including the biasmagnetic circuit and the control magnetic circuits of theCRAMB without the leakage flux are shown in Figure 5. Wefocus on the investigation of the eddy-current effects, andthe equivalent magnetic reluctances for the leakage flux pathswould be included in the future model.

2.2. Dynamic Current Stiffness Model for the AMB Unit ofCRAMB. It is assumed that the CRAMB system is linear, andthe AMB unit of the CRAMB excited a sinusoidal varyingac current 𝑖

𝑧= 𝐼𝑧sin(𝜔𝑡) (where the amplitude, 𝐼

𝑧, is very

small). When the thrust disk located its center position, thelength of air gap (the element 6, 8) is 𝑠

0.

The control flux 𝜙𝑧𝑖

produced by a sinusoidal varyingac current 𝑖

𝑧in the axial air gap of AMB unit and the bias

flux 𝜙𝑧𝑚0

in the axial air gap of AMB unit produced by thePMR, the control flux in the left air gap (element 8) will addto the bias flux, and will subtract from the bias flux in theright air gap (element 6) as shown in Figure 3. The total flux,𝜙𝑧sum𝑖 (𝑖 = 1, 2), in the left and the right air gap of AMB unit

is given as

𝜙𝑧sum1 = 𝜙

𝑧𝑚0+ 𝜙𝑧𝑖,

𝜙𝑧sum2 = 𝜙

𝑧𝑚0− 𝜙𝑧𝑖,

(12)

where the static bias flux in the axial air gap of AMB unit iscalculated as

𝜙𝑧𝑚

= 𝐹𝑚(𝑅0

1+ 𝑅0

2+ 𝑅0

7+ 𝑅0

11+ 𝑅0

12+𝑅13

4

+1

1/∑6

𝑘=3𝑅0

𝑘+ 1/∑

10

𝑗=8𝑅0

𝑗

)

−1

.

(13)

The permanent magnet material of the PMR is Nd-Fe-B, its magnetomotive force is 𝐹

𝑚= 𝐻𝑐ℎ2, and 𝐻

𝑐is the

Table 1: Parameters of The CRAMB.

Parameters Description Value𝑟1 Inner radius of the RMB rotor, (mm) 25𝑟2 Outer radius of the RMB rotor, (mm) 30𝑟7 Inner radius of the RMB stator, (mm) 30.5𝐴𝑟 Radial pole shoe area, (mm2) 1036

𝑁𝑟

The number of winding turns in coil ofRMB 220

𝑟9 Outer radius of the RMB stator, (mm) 60𝑟6 Outer radius of the yoke, (mm) 70

𝑟3

Inner radius of the axial pole shoe,(mm) 31

𝑟4

Outer radius of the axial pole shoe,(mm) 34

𝑟5 Inner radius of the yoke, (mm) 62𝑟8 Inner radius of the RMB rotor, (mm) 10

ℎ1

Length of the stator return ring ofRMB, (mm) 31

ℎ2

Thickness of the permanent ring,(mm) 10

ℎ3 Thickness of the element 10, (mm) 8ℎ4 Thickness of the element 3, (mm) 24ℎ5 Thickness of the element 4, (mm) 8ℎ6 Thickness of the element 5, (mm) 8ℎ7 Thickness of the thrust plane, (mm) 7ℎ8 Thickness of the element 9, (mm) 8ℎ9 Thickness of the element 12, (mm) 8𝑠0 radial and axial air gap, (mm) 0.5

𝑁𝑧

The number of winding turns in coil ofAMB 120

𝑚 Themass of the rotor, (kg) 7.2

𝜎1

The conductivity of yoke and thrustdisk, (S/m) 8 × 106

𝜇0 Permeability of free space, (H/m) 4𝜋 × 10

−7

𝜇𝑟

Relative permeability of the permanentmagnet 1.082

𝜎2 Resistivity of the magnet ring, (Ω⋅m) 0.9𝐸 − 6

coercive force of the PMR; ℎ2is thickness of the PMR. The

permeability of the magnetic ring and the thrust disk iscompared to that of the air and the PMB, so (13) can besimplified as

𝜙𝑧𝑚

≈𝐹𝑚

𝑅0

2+ 𝑅13/4 + 𝑅

𝑧0/2. (14)

And the static bias flux in the air gap of the AMB can becalculated as

𝜙𝑧𝑚0

= 𝜙0

𝑧1= 𝜙0

𝑧2=𝜙𝑧𝑚

2=

𝐹𝑚

2𝑅0

2+ 𝑅13/2 + 𝑅

𝑧0

. (15)


Rx2

Ry1

Ry2

𝜙rm

𝜙rm

𝜙rm

𝜙rm

𝜙zm1

𝜙zm2

𝜙mFm

R11R12R1 R2

R3 R4 R5 R6

R7

R10 R8R9

Rx1

(a)

Rx2

Rx1

Ry1

Ry2

Nrix

Nrix

Nriy

Nriy

𝜙ix1

𝜙ix2

𝜙iy1

𝜙iy2

(b)

𝜙zi

Nziz

R10

R3 R4 R5

R6

R7

R8R9

(c)

Figure 5: The equivalent magnetic circuits of the CRAMB. (a) The equivalent magnetic circuit of the bias flux paths. (b) The equivalentmagnetic circuit of control flux path of the RMB unit. (c) The equivalent magnetic circuit of control flux path of the AMB unit.

The dynamic flux in the air gap of AMB unit produced bythe ac current is calculated as

𝜙𝑧𝑖=𝑁𝑧𝐼𝑧sin (𝜔𝑡)

∑10

𝑘=1𝑅𝑘

=𝑁𝑧𝐼𝑧

𝑅0

𝑧𝑖

sin (𝜔𝑡)1 + 𝑅𝑒(𝜔) /𝑅

0

𝑧𝑖

=𝑁𝑧𝐼𝑧sin (𝜔𝑡)

𝑅0

𝑧𝑖(1 + 𝜆)

,

(16)

where 𝑅0

𝑧𝑖is the total static magnetic reluctance of the

equivalent magnetic circuit of control flux path for AMBunit, 𝑅

𝑒(𝜔) is the total eddy-current magnetic reluctance of

the equivalent magnetic circuit of control flux path for AMBunit, and 𝜆 is the ratio of the total eddy-current magneticreluctance to the total static magnetic reluctance 𝑅

𝑒(𝜔)/𝑅0

𝑧𝑖:

𝑅0

𝑧𝑖= 𝑅0

6+ 𝑅0

8,

𝑅𝑒(𝜔) = 𝑅

𝑖

3+ 𝑅𝑖

4+ 𝑅𝑖

5+ 𝑅𝑖

7+ 𝑅𝑖

9+ 𝑅𝑖

10.

(17)

𝑅𝑒(𝜔) is the magnetic reluctance including the eddy

current effect, and it will be increased with the additionof frequency of the time-varying air gap and the controlcurrents. The bandwidth will be increased with the additionof the air gap.

TheAMBunit of the CRAMBwill produce a net restoringforce 𝑓

𝑧𝑖on the thrust disk in 𝑧-axis direction. Since the net

force, 𝑓𝑧𝑖, is calculated by

𝑓𝑧𝑖=

𝜙2

𝑧1

2𝜇0𝐴𝑧

−𝜙2

𝑧2

2𝜇0𝐴𝑧

, (18)

where the constant 𝜇0is the permeability of free space, 𝐴

𝑧is

the area of pole face of the AMB unit; substituting (12) into(18) gives

𝑓𝑧𝑖=(𝜙𝑧𝑚0

+ 𝜙𝑧𝑖)2

2𝜇0𝐴𝑧

−(𝜙𝑧𝑚0

− 𝜙𝑧𝑖)2

2𝜇0𝐴𝑧

=2𝜙𝑧𝑚0

𝜙𝑧𝑖

𝜇0𝐴𝑧

. (19)

The dynamic current stiffness of the AMB unit can bederived as

𝑘𝑖

𝑧𝑖=𝑑𝑓𝑧𝑖

𝑑𝑖𝑧

=2𝜙𝑧𝑚0

𝑁𝑧

𝜇0𝐴𝑧𝑅0

𝑧𝑖(1 + 𝜆)

=𝑘0

𝑧𝑖

1 + 𝜆, (20)

where

𝑘0

𝑧𝑖=2𝜙𝑧𝑚0

𝑁𝑧

𝜇0𝐴𝑧𝑅0

𝑧𝑖

(21)

is the static current stiffness of the AMB unit.

2.3. Dynamic Displacement Stiffness Model for the AMB Unitof CRAMB. When the coil currents in the RMB and theAMBunit are zero, the air gap in the RMB remains stationary, butthe air gap in the AMB unit varies sinusoidally about thenominal value 𝑠

0; that is, 𝑧

1= 𝑠0− 𝛿 sin(𝜔𝑡) in left air gap,

and 𝑧2= 𝑠0+ 𝛿 sin(𝜔𝑡) in right air gap; then the magnetic

reluctance of the air gap will also vary sinusoidally about thenominal value. Thus, the reluctance of the two air gaps inAMB unit includes two parts, one is the static reluctance,𝑅0

𝑧𝑖(𝑖 = 1, 2), and the other one is the dynamic reluctance,

𝑅𝑑

𝑧𝑖(𝑖 = 1, 2). Consider the following:

𝑅𝑧1= 𝑅0

𝑧1+ 𝑅𝑑

𝑧1=

𝑠0

𝜇0𝐴𝑧

+𝛿 sin (𝜔𝑡)𝜇0𝐴𝑧

,

𝑅𝑧2= 𝑅0

𝑧2+ 𝑅𝑑

𝑧2=

𝑠0

𝜇0𝐴𝑧

−𝛿 sin (𝜔𝑡)𝜇0𝐴𝑧

,

(22)

where

𝑅0

𝑧1= 𝑅0

𝑧2=

𝑠0

𝜇0𝐴𝑧

,

𝑅𝑑

𝑧1= −𝑅𝑑

𝑧2=𝛿 sin (𝜔𝑡)𝜇0𝐴𝑧

.

(23)

If all the eddy-current reluctances and leakage reluctancesare ignored, the total static magnetic reluctance 𝑅met in thebias magnetic circuit is

𝑅met = 𝑅0

1+ 𝑅0

2+ 𝑅0

7+ 𝑅0

11+ 𝑅0

12+𝑅13

4

+1

1/∑6

𝑘=3𝑅0

𝑘+ 1/∑

10

𝑗=8𝑅0

𝑗

.

(24)

Since the relative permeability of the stator and rotor coreof the AMB and the RMB unit is larger, their static magnetic


reluctance can be ignored in the equivalent magnetic circuitof the bias flux path for calculating the dynamic displacementstiffness. The total static magnetic reluctance in (24) can besimplified as

𝑅met = 𝑅0

met + 𝑅𝑒

met ≈ 𝑅0

2+𝑅13

4+

1

1/𝑅𝑧1+ 1/𝑅

𝑧2

, (25)

where,

𝑅0

met = 𝑅0

2+𝑅13

4+

𝑅0

𝑧1𝑅0

𝑧2

𝑅0

𝑧1+ 𝑅0

𝑧2

,

𝑅𝑒

met =(𝑅0

𝑧2− 𝑅0

𝑧1) 𝑅𝑑

𝑧1− (𝑅𝑑

𝑧1)2

𝑅0

𝑧1+ 𝑅0

𝑧2

.

(26)

Since 𝛿 is smaller compared to 𝑠0, then 𝑅

𝑑

𝑧𝑖(𝑖 = 1, 2) is

smaller compared to 𝑅0

𝑧𝑖(𝑖 = 1, 2), and 𝑅

𝑒

met in (26) can beexpanded into the power series and approximated by the firsttwo terms as the higher order terms can be ignored. Considerthe following:

𝑅0

met = 𝑅0

2+𝑅13

4+

𝑅0

𝑧1𝑅0

𝑧2

𝑅0

𝑧1+ 𝑅0

𝑧2

,

𝑅𝑒

met ≈(𝑅0

𝑧2− 𝑅0

𝑧1) 𝑅𝑑

𝑧1

𝑅0

𝑧1+ 𝑅0

𝑧2

.

(27)

The flux in the air gap of AMB unit can be calculated as

𝜙𝑧1=

𝐹𝑚

𝑅0

met + 𝑅𝑒

met

1

1/ (𝑅0

𝑧1+ 𝑅𝑑

𝑧1) + 1/ (𝑅

0

𝑧2− 𝑅𝑑

𝑧1)

1

𝑅0

𝑧1+ 𝑅𝑑

𝑧1

,

𝜙𝑧2=

𝐹𝑚

𝑅0

met + 𝑅𝑒

met

1

1/ (𝑅0

𝑧1+ 𝑅𝑑

𝑧1) + 1/ (𝑅

0

𝑧2− 𝑅𝑑

𝑧1)

1

𝑅0

𝑧2− 𝑅𝑑

𝑧1

.

(28)

Substituting (27) for the right side of (28), since 𝑅𝑑𝑧𝑖is

smaller compared to𝑅0𝑧𝑖, (28) can be expanded into the power

series and approximated by the first two terms as the higherorder terms can be ignored. Consider the following:

𝜙𝑧1≈

𝐹𝑚

𝑅0

met

𝑅0

𝑧1

𝑅0

𝑧1+ 𝑅0

𝑧2

[1 −𝑅𝑑

𝑧1

𝑅0

met(𝑅0

𝑧2− 𝑅0

𝑧1

𝑅0

𝑧1+ 𝑅0

𝑧2

) −𝑅𝑑

𝑧1

𝑅0

𝑧2

] ,

𝜙𝑧2≈

𝐹𝑚

𝑅0

met

𝑅0

𝑧2

𝑅0

𝑧1+ 𝑅0

𝑧2

[1 −𝑅𝑑

𝑧1

𝑅0

met(𝑅0

𝑧2− 𝑅0

𝑧1

𝑅0

𝑧1+ 𝑅0

𝑧2

) +𝑅𝑑

𝑧1

𝑅0

𝑧1

] .

(29)

The Equation (29) can be modified due to 𝑅0𝑧1= 𝑅0

𝑧2.

𝜙𝑧1≈

𝐹𝑚

2𝑅0

met(1 −

𝐴𝑅𝑑

𝑧1

𝑅0

met−𝑅𝑑

𝑧1

𝑅0

𝑧2

) = 𝜙𝑧0(1 −

𝐴𝑅𝑑

𝑧1

𝑅0

met−𝑅𝑑

𝑧1

𝑅0

𝑧2

) ,

𝜙𝑧2≈

𝐹𝑚

2𝑅0

met(1 −

𝐴𝑅𝑑

𝑧1

𝑅0

met+𝑅𝑑

𝑧1

𝑅0

𝑧1

) = 𝜙𝑧0(1 −

𝐴𝑅𝑑

𝑧1

𝑅0

met+𝑅𝑑

𝑧1

𝑅0

𝑧1

) ,

(30)

where 𝐴 = (𝑅𝑒𝑡

𝑧2− 𝑅𝑒𝑡

𝑧1)/(𝑅0

𝑧1+ 𝑅0

𝑧2).

TheAMBunit of the CRAMBwill produce a net restoringforce𝑓

𝑧𝑖on the thrust disk in 𝑧-axis direction, since the force

is given by

𝑓𝑧𝑠=(𝜙𝑧1)2

2𝜇0𝐴𝑧

−(𝜙𝑧2)2

2𝜇0𝐴𝑧

=𝜙2

𝑧0

2𝜇0𝐴𝑧

× [

[

(1 −𝐴𝑅𝑑

𝑧1

𝑅0

met−𝑅𝑑

𝑧1

𝑅0

𝑧2

)

2

− (1 −𝐴𝑅𝑑

𝑧1

𝑅0

met+𝑅𝑑

𝑧1

𝑅0

𝑧1

)

2

]

]

,

(31)

where 𝜙𝑧0

= 𝜙0

𝑧1= 𝜙0

𝑧2= 𝐹𝑚/2𝑅0

met is the static bias flux inthe air gap of the AMB unit.

Since 𝑅𝑑𝑧𝑖(𝑖 = 1, 2) is smaller compared to 𝑅0

𝑧𝑖(𝑖 = 1, 2),

(31) can be expanded into the power series and approximatedby the first two terms as the higher order terms can beignored. Consider the following:

𝑓𝑧𝑠≈ −

2𝜙2

𝑧0

(𝜇0𝐴𝑧)2

𝛿 sin (𝜔𝑡)𝑅0

𝑧1

. (32)

Static displacement stiffness of the AMB unit is given as

𝑘0

𝑧𝑠=

𝑑𝑓𝑧𝑠

𝑑 (𝛿 sin (𝜔𝑡))= −

2𝜙2

𝑧0

𝜇0𝐴𝑧𝑠0

= −1

2𝜇0𝐴𝑧𝑠0

(𝐹𝑚

𝑅0

2+ 𝑅13/4 + 𝑅

0

𝑧1/2)

2

.

(33)

In the static analysis model mentioned above, the eddy-current magnetic reluctance is not considered. However, thedynamic field should also cause eddy current, and the staticmodel could be considered to include eddy-current magneticreluctance as in the dynamic model. Then the dynamic fluxincluding eddy-current effect is given as

𝜙𝑑

𝑧1= 𝜙0

𝑧1(1 −

𝐵𝑅𝑑

𝑧1

𝑅𝑒𝑡sum−𝑅𝑑

𝑧1

𝑅𝑒𝑡

𝑧2

) ,

𝜙𝑑

𝑧2= 𝜙0

𝑧2(1 −

𝐵𝑅𝑑

𝑧1

𝑅𝑒𝑡sum+𝑅𝑑

𝑧1

𝑅𝑒𝑡

𝑧1

) ,

(34)


where,

𝑅𝑒𝑡

sum = 𝑅0

sum + 𝑅𝜔

sum,

𝑅0

sum = 𝑅0

1+ 𝑅0

2+ 𝑅0

7+ 𝑅0

11+ 𝑅0

12+𝑅13

4+𝑅0

𝑧1

2,

𝑅𝜔

sum =

2

∑

𝑚=1

𝑅𝑖

𝑚+ 𝑅𝑖

7+

12

∑

𝑛=11

𝑅𝑖

𝑛

+1

1/∑6

𝑗=3𝑅𝑗+ 1/∑

10

𝑘=8𝑅𝑘

−𝑅0

𝑧1

2,

𝐵 =𝑅𝑒𝑡

𝑧2− 𝑅𝑒𝑡

𝑧1

𝑅𝑒𝑡

𝑧1+ 𝑅𝑒𝑡

𝑧2

,

𝑅𝑒𝑡

𝑧1=

6

∑

𝑗=3

𝑅𝑗, 𝑅

𝑒𝑡

𝑧2=

10

∑

𝑘=8

𝑅𝑘.

(35)

The permeability of the magnetic ring, stator core, rotorstator, and the thrust disk is compared to that of the air andthe PMR, so (35) can be modified as

𝑅0

sum ≈ 𝑅0

2+𝑅13

4+𝑅0

𝑧1

2,

𝑅𝜔

sum ≈

2

∑

𝑚=1

𝑅𝑖

𝑚+ 𝑅𝑖

7

+

12

∑

𝑛=11

𝑅𝑖

𝑛+

1

1/∑6

𝑗=3𝑅𝑖

𝑗+ 1/∑

10

𝑘=8𝑅𝑖

𝑘

−𝑅0

𝑧1

2,

𝑅𝑒𝑡

𝑧1= 𝑅𝑖

3+ 𝑅𝑖

4+ 𝑅𝑖

5+ 𝑅0

6,

𝑅𝑒𝑡

𝑧2= 𝑅0

8+ 𝑅𝑖

9+ 𝑅𝑖

10.

(36)

The net force of the AMB unit considering the eddy-current magnetic reluctance is given by

𝑓𝑧𝑠=

(𝜙𝑑

𝑧1)2

2𝜇0𝐴𝑧

−

(𝜙𝑑

𝑧2)2

2𝜇0𝐴𝑧

=𝜙2

𝑧0

2𝜇0𝐴𝑧

[

[

(1 −𝐵𝑅𝑑

𝑧1

𝑅𝑒𝑡sum−𝑅𝑑

𝑧1

𝑅𝑒𝑡

𝑧2

)

2

− (1 −𝐵𝑅𝑑

𝑧1

𝑅𝑒𝑡sum+𝑅𝑑

𝑧1

𝑅𝑒𝑡

𝑧1

)

2

]

]

.

(37)

Since𝑅𝑑𝑧𝑖is smaller compared to𝑅0

𝑧𝑖, (37) can be expanded

into the power series and approximated by the first twoterms as the higher order terms can be ignored. Consider thefollowing:

𝑓𝑑

𝑧𝑠≈ −

2𝜙2

𝑧0

(𝜇0𝐴𝑧)2(

1

2𝑅𝑒𝑡

𝑧2

+1

2𝑅𝑒𝑡

𝑧1

)𝛿 sin (𝜔𝑡) . (38)

The dynamic displacement stiffness considering theeddy-current magnetic reluctance is calculated as

𝑘𝑑

𝑧𝑠=

𝑑𝑓𝑑

𝑧𝑠


2𝜙2

𝑧0

𝜇0𝐴𝑧𝑠0

(𝑅0

𝑧2

2𝑅𝑒𝑡

𝑧2

+𝑅0

𝑧1

2𝑅𝑒𝑡

𝑧1

)

= 𝑘0

𝑧𝑠(𝑅0

𝑧2

2𝑅𝑒𝑡

𝑧2

+𝑅0

𝑧1

2𝑅𝑒𝑡

𝑧1

) .

(39)

2.4. Dynamic Current Stiffness Model for the RMB Unit ofCRAMB. The RMB unit controls the rotor movement along𝑥- or (/and) 𝑦-axis orthogonal to the spin axis (𝑧-axis).Whenthe rotor deviates radially from its “suspended” position at 𝑥-axis (the magnetomotive force at 𝑦-axis, 𝑁

𝑟𝑖𝑦= 0), the net

restoring force of the RMB unit at 𝑥-axis, 𝑓𝑥𝑖, can be given as

𝑓𝑥𝑖=

𝜙2

𝑥1

2𝜇0𝐴𝑟

−𝜙2

𝑥2

2𝜇0𝐴𝑟

, (40)

where,

𝜙𝑥1= 𝜙𝑟𝑚

+ 𝜙𝑖

𝑥1,

𝜙𝑥2= 𝜙𝑟𝑚

− 𝜙𝑖

𝑥2, 𝜙

𝑟𝑚=𝜙𝑚0

4.

(41)

The flux in the air gap of the RMB unit produced by the𝑥-axis coil current, 𝑖

𝑥, is calculated as

𝜙𝑖

𝑥1=

𝑁𝑟𝑖𝑥

𝑅𝑥1+ 1/ (1/𝑅

𝑥2+ 1/𝑅

𝑦1+ 1/𝑅

𝑦2)

+𝑁𝑟𝑖𝑥

𝑅𝑥2+ 1/ (1/𝑅

𝑥1+ 1/𝑅

𝑦1+ 1/𝑅

𝑦2)

×1

1/𝑅𝑥1+ 1/𝑅

𝑦1+ 1/𝑅

𝑦2

1

𝑅𝑥1

,

𝜙𝑖

𝑥2=

𝑁𝑟𝑖𝑥

𝑅𝑥2+ 1/ (1/𝑅

𝑥1+ 1/𝑅

𝑦1+ 1/𝑅

𝑦2)

+𝑁𝑟𝑖𝑥

𝑅𝑥1+ 1/ (1/𝑅

𝑥2+ 1/𝑅

𝑦1+ 1/𝑅

𝑦2)

×1

1/𝑅𝑥2+ 1/𝑅

𝑦1+ 1/𝑅

𝑦2

1

𝑅𝑥2

.

(42)

The nominal magnetic reluctances of the air gap in RMBunit are equal to each other when the rotor locates its centerposition, 𝑅

𝑥= 𝑅𝑥1

= 𝑅𝑥2

= 𝑅𝑦= 𝑅𝑦1

= 𝑅𝑦2. Then (42) can

be modified as

𝜙𝑖

𝑥= 𝜙𝑖

𝑥2= 𝜙𝑖

𝑥2=𝑁𝑟𝑖𝑥

𝑅𝑥

. (43)

Substituting (43) for the right side of (40), (40) can bemodified as

𝑓𝑥𝑖=2𝜙𝑟𝑚

𝜇0𝐴𝑟

𝑁𝑟𝑖𝑥

𝑅𝑥

. (44)


The eddy-current reluctance can be ignored due tolaminated rotor and stator core, and the controls current donot pass through the other nonlaminated materials. So thedynamic current stiffness of the RMB unit will be equal to itsstatic stiffness. Consider the following:

𝑘𝑖

𝑟𝑖= 𝑘0

𝑟𝑖=𝑑𝑓𝑥𝑖

𝑑𝑖𝑥

=2𝜙𝑟𝑚𝑁𝑟

𝜇0𝐴𝑟𝑅𝑥

=𝑁𝑟

2𝜇0𝐴𝑟𝑅𝑥

𝐹𝑚

𝑅0

2+ 𝑅13/4 + 𝑅

0

𝑧1/2.

(45)

2.5. Dynamic Displacement Stiffness Model for the RMB Unitof CRAMB. When the coil currents in the RMB and theAMBunit are zero, the air gap in the AMB unit remains stationary,but the air gap in the RMB unit varies sinusoidally about thenominal value 𝑠

0in 𝑥-axis; that is, 𝑥

1= 𝑠0+𝛿 sin(𝜔𝑡) in upper

air gap; and 𝑥2= 𝑠0− 𝛿 sin(𝜔𝑡) in lower air gap, then the

magnetic reluctance of the air gap will also vary sinusoidallyabout the nominal value. Thus, the reluctance of the two airgaps in the RMB unit includes two parts, one is the staticreluctance, 𝑅0

𝑥, and the other one is the dynamic reluctance,

𝑅𝑑

𝑥. Consider the following:

𝑅𝑥1= 𝑅0

𝑥+ 𝑅𝑑

𝑥=

𝑠0

𝜇0𝐴𝑟

+𝛿 sin (𝜔𝑡)𝜇0𝐴𝑟

,

𝑅𝑥2= 𝑅0

𝑥− 𝑅𝑑

𝑥=

𝑠0

𝜇0𝐴𝑟

−𝛿 sin (𝜔𝑡)𝜇0𝐴𝑟

,

(46)

where

𝑅0

𝑥=

𝑠0

𝜇0𝐴𝑟

, 𝑅𝑑

𝑥=𝛿 sin (𝜔𝑡)𝜇0𝐴𝑟

. (47)

If all the leakage reluctances and eddy-current reluctanceare ignored, the total reluctance 𝑅met in the bias magneticcircuit is

𝑅met =2

∑

𝑗=1

𝑅0

𝑗+ 𝑅0

7+

12

∑

𝑘=11

𝑅0

𝑘

+1

2/𝑅𝑦+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

+1

1/∑6

𝑘=3𝑅0

𝑘+ 1/∑

10

𝑗=8𝑅0

𝑗

.

(48)

The relative permeability of the stator and rotor core islager compared to the air gap, then their static reluctance canbe ignored in the equivalent magnetic circuit of the bias fluxpath for calculated the dynamic displacement stiffness. Thetotal reluctance in (48) can be simplified as

𝑅met = 𝑅0

met + 𝑅𝑒

met ≈ 𝑅0

2+

1

2/𝑅𝑦+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

+𝑅0

𝑧1

2,

(49)

where,

𝑅0

met = 𝑅0

2+𝑅0

𝑥

4+𝑅0

𝑧1

2,

𝑅𝑒

met =1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

−𝑅0

𝑥

4.

(50)

Since dynamic magnetic reluctance is smaller comparedto static magnetic reluctance of the air gap, 𝑅𝑒met can beexpanded into the power series and approximated by the firsttwo terms as the higher order terms can be ignored, and the𝑅𝑒

met will approximately equal to zero, 𝑅𝑒met ≈ 0.The flux in the upper and lower air gap of RMB in 𝑥-axis

is calculated, respectively. Consider the following:

𝜙𝑥1=

𝐹𝑚

𝑅met

1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

1

𝑅𝑥1

,

𝜙𝑥2=

𝐹𝑚

𝑅met

1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

1

𝑅𝑥2

.

(51)

Substituting (46) for the right side of (51), since 𝑅𝑑𝑥is

smaller compared to 𝑅0𝑥, (51) can be expanded into the power

series and approximated by the first two terms as the higherorder terms can be ignored. Consider the following:

𝜙𝑥1=

𝐹𝑚

4𝑅met(1 −

𝑅𝑑

𝑥

𝑅0𝑥

) = 𝜙𝑥0(1 −

𝑅𝑑

𝑥

𝑅0𝑥

)

𝜙𝑥2=

𝐹𝑚

4𝑅met(1 +

𝑅𝑑

𝑥

𝑅0𝑥

) = 𝜙𝑥0(1 +

𝑅𝑑

𝑥

𝑅0𝑥

) ,

(52)

where 𝜙𝑥0= 𝐹𝑚/4𝑅met.

The net force produced by RMB unit in 𝑥-axis is given by

𝑓𝑥𝑠=(𝜙𝑥1)2

2𝜇0𝐴𝑟

−(𝜙𝑥2)2

2𝜇0𝐴𝑟

=𝜙2

𝑥0

2𝜇0𝐴𝑟

[

[

(1 −𝑅𝑑

𝑥

𝑅0𝑥

)

2

− (1 +𝑅𝑑

𝑥

𝑅0𝑥

)

2

]

]

= −2𝜙2

𝑥0

(𝜇0𝐴𝑟)2

𝛿 sin (𝜔𝑡)𝑅0𝑥

.

(53)

The static displacement stiffness is calculated by

𝑘0

𝑟𝑠=

𝑑𝑓𝑥𝑠


2𝜙2

𝑥0

𝜇0𝐴𝑟𝑠0

. (54)

In the static analysis model mentioned above, the eddy-current magnetic reluctance is not considered. However, thedynamic field should also cause eddy current, and the staticmodel could be considered to include eddy-current magneticreluctance as in the dynamicmodel.Then the dynamic flux inair gap of RMB unit in 𝑥-axis including eddy-current effect isgiven as

𝜙𝑥1=

𝐹𝑚

𝑅𝑒𝑡sum

1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

1

𝑅𝑥1

,

𝜙𝑥2=

𝐹𝑚

𝑅𝑒𝑡sum

1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

1

𝑅𝑥2

,

(55)


where 𝑅𝑒𝑡sum = 𝑅0

sum + 𝑅𝜔

sum, 𝑅0

sum, and 𝑅𝜔

sum are given as

𝑅0

sum =

2

∑

𝑚=1

𝑅0

𝑚+ 𝑅0

7

+

12

∑

𝑛=11

𝑅0

𝑛+

1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

+𝑅0

𝑧1

2,

𝑅𝜔

sum =

2

∑

𝑚=1

𝑅𝑖

𝑚+ 𝑅𝑖

7

+

12

∑

𝑛=11

𝑅𝑖

𝑛+

1

1/∑6

𝑗=3𝑅𝑗+ 1/∑

10

𝑘=8𝑅𝑘

−𝑅0

𝑧1

2.

(56)

The permeability of the magnetic ring, stator core, rotorstator, and the thrust disk is compared to that of the air andthe PMB, so (56) can be modified as

𝑅0

sum = 𝑅0

2+

1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

+𝑅0

𝑧1

2,

𝑅𝜔

sum =

2

∑

𝑚=1

𝑅𝑖

𝑚+ 𝑅𝑖

7+

12

∑

𝑛=11

𝑅𝑖

𝑛

+1

1/∑6

𝑗=3𝑅𝑗+ 1/∑

10

𝑘=8𝑅𝑘

−𝑅0

𝑧1

2.

(57)

And (55) can be expanded into the power series andapproximated by the first two terms as the higher order termscan be ignored.

𝜙𝑥1=

𝐹𝑚

𝑅𝑒𝑡sum(1 −

𝑅𝑑

𝑥

𝑅0𝑥

) = 𝜙0

𝑥

𝑅met𝑅𝑒𝑡sum

(1 −𝑅𝑑

𝑥

𝑅0𝑥

) ,

𝜙𝑥2=

𝐹𝑚

𝑅𝑒𝑡sum(1 +

𝑅𝑑

𝑥

𝑅0𝑥

) = 𝜙0

𝑥

𝑅met𝑅𝑒𝑡sum

(1 +𝑅𝑑

𝑥

𝑅0𝑥

) .

(58)

The net force produced by the RMB unit considering theeddy-current magnetic reluctance is modified by

𝑓𝑑

𝑥𝑠=(𝜙𝑥1)2

2𝜇0𝐴𝑟

−(𝜙𝑥2)2

2𝜇0𝐴𝑟

. (59)

Substituting (58) for the right side of (59), (59) ismodifiedas

𝑓𝑑

𝑥𝑠=(𝜙𝑥1)2

2𝜇0𝐴𝑟

−(𝜙𝑥2)2

2𝜇0𝐴𝑟

= −2𝜙2

𝑥0

(𝜇0𝐴𝑟)2(𝑅met𝑅𝑒𝑡sum

)

2

𝛿 sin (𝜔𝑡)𝑅0𝑥

.

(60)

The dynamic displacement stiffness of RMB unit consid-ering the eddy-current effect is given by

𝑘𝑑

𝑟𝑠=

𝑑𝑓𝑥𝑠


2𝜙2

𝑥0

𝜇0𝐴𝑧𝑠0

(𝑅met𝑅𝑒𝑡sum

)

2

= 𝑘0

𝑟𝑠(𝑅met𝑅𝑒𝑡sum

)

2

.

(61)

3. Coupling Performance Analysis of theDynamic Stiffness Models of CRAMB

Since the equivalent magnetic circuits of control flux pathof the AMB and RMB unit do not influence each other,the current stiffness of AMB (or RMB) unit will not beaffected by the control current of the RMB (or AMB) unit.But the equivalent bias circuits of the AMB and the RMBpass through the PMB, and the dynamic models of the AMBand the RMB unit may influence each other due the air gapvarying of the AMB or RMB unit.

3.1. The Dynamic Current Stiffness of the AMB Unit Affectedby the Rotor Position of RMB Unit in 𝑥-Axis. Based on thesmall variation of the rotor position of the RMB unit in 𝑥-axis, 𝑥

1= 𝑠0+ 𝑥, 𝑥

2= 𝑠0− 𝑥, and the remains stationary in

𝑦-axis (𝑦1 = 𝑦2 = 𝑠0).The static bias flux in the air gap of the AMB unit is

modified as

𝜙𝑟

𝑧𝑚0= 𝜙𝑟0

𝑧1= 𝜙𝑟0

𝑧2=𝜙𝑟

𝑚0

2

=𝐹𝑚

𝑅0

2+ 1/ (2/𝑅0

𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2) + 𝑅0

𝑧1/2

=𝐹𝑚

𝑅0

2+ 1/ (2𝜇

0𝐴𝑟/𝑠0+ 2𝜇0𝐴𝑟𝑠0/ (𝑠2

0− 𝑥2)) + 𝑅

0

𝑧1/2.

(62)

And the dynamic current stiffness of the AMB unit ismodified due to rotor position varying in 𝑥-axis. Considerthe following:

𝑘𝑟𝑖

𝑧𝑖=

2𝑁𝑧

𝜇0𝐴𝑧𝑅0

𝑧𝑖(1 + 𝜆)

×𝐹𝑚

𝑅0

2+ 1/ (2𝜇


0− 𝑥2)) + 𝑅

0

𝑧1/2

= 𝑘𝑖

𝑧𝑖

𝑅0

2+ 𝑅13/4 + 𝑅

0

𝑧1/2

𝑅0

2+ 1/ (2𝜇


0− 𝑥2)) + 𝑅

0

𝑧1/2

= 𝑘𝑖

𝑧𝑖𝜆𝑟𝑧,

(63)

where 𝜆𝑟𝑧

= (𝑅0

2+ 𝑅13/4 + 𝑅

0

𝑧1/2)/(𝑅

0

2+ 1/(2𝜇

0𝐴𝑟/𝑠0+

2𝜇0𝐴𝑟𝑠0/(𝑠2

0− 𝑥2)) + 𝑅

0

𝑧1/2) is the coefficient which reflects

the dynamic current stiffness dependence of the AMB uniton the rotor position in 𝑥-axis (or 𝑦-axis). In other words, thecoefficient, 𝜆

𝑟𝑧, reflects the coupling extent of the dynamic

current stiffness of the AMB unit affected by the rotorposition in 𝑥-axis or 𝑦-axis.

From (63), the 𝑥 term is included in the dynamic currentstiffness model which will be affected by the rotor position ofthe RMB unit in 𝑥-axis or 𝑦-axis.

3.2. The Dynamic Displacement Stiffness of the AMB UnitAffected by the Rotor Position of RMB Unit in 𝑥-Axis. The


𝑅0

sum in (36) will be modified when considering the smallvariation of the rotor position of the RMB unit in 𝑥-axis.Consider the following:

𝑅0

sum = 𝑅0

2+

1

2𝜇0𝐴𝑟/𝑠0+ 2𝜇0𝐴𝑟𝑠0/ (𝑠2

0− 𝑥2)

+𝑅0

𝑧1𝑅0

𝑧2

𝑅0

𝑧1+ 𝑅0

𝑧2

.

(64)

And the dynamic displacement stiffness is modified dueto rotor position varying in 𝑥-axis. Consider the following:

𝑘𝑟𝑑

𝑧𝑠

= −1

2𝜇0𝐴𝑧𝑠0

× (𝐹𝑚

𝑅0

2+ 1/ (2𝜇


0− 𝑥2)) + 𝑅

0

𝑧1/2)

2

× (𝑅0

𝑧1

2𝑅𝑒𝑡

𝑧2

+𝑅0

𝑧2

2𝑅𝑒𝑡

𝑧1

)

= 𝑘𝑑

𝑧𝑠

(𝑅0

2+ 𝑅13/4 + 𝑅

0

𝑧1/2)2

(𝑅0

2+ 1/ (2𝜇


0− 𝑥2)) + 𝑅

0

𝑧1/2)2

= 𝑘𝑑

𝑧𝑠𝜆2

𝑟𝑧.

(65)

From (65), the 𝑥 term is included in the dynamicdisplacement stiffness model which will be affected by therotor position of the RMB unit in 𝑥-axis or 𝑦-axis. And 𝜆2

𝑟𝑧

is the coefficient which reflects the dynamic displacementstiffness of the AMB unit dependence on the rotor position in𝑥-axis (or 𝑦-axis). In other words, the coefficient, 𝜆2

𝑟𝑧, reflects

the coupling extent of the dynamic displacement stiffness ofthe AMB unit affected by the rotor position in 𝑥-axis or 𝑦-axis.

3.3. The Dynamic Current Stiffness of the RMB Unit Affectedby the Rotor Position of AMB Unit in 𝑧-Axis. Based on thesmall variation of the rotor position of the AMB unit in 𝑧-axis, 𝑧

1= 𝑠0+ 𝑧, 𝑧

2= 𝑠0− 𝑧, and the remains stationary in

𝑦-axis (𝑅𝑦1

= 𝑅𝑦2

≈ 𝑅0

𝑥). The magnetic reluctance of the air

gap for the AMB unit can be modified as

𝑅𝑧1=𝑠0+ 𝑧

𝜇0𝐴𝑧

, 𝑅𝑧2=𝑠0− 𝑧

𝜇0𝐴𝑧

. (66)

The static bias flux in the air gap of the RMB unit affectedby the rotor position in 𝑧-axis varying is modified as

𝜙𝑟𝑚

=𝜙𝑚0

4=

𝐹𝑚/4

𝑅0

2+ 𝑅13/4 + 1/ (1/𝑅

𝑧1+ 1/𝑅

𝑧2)

=𝐹𝑚/4

𝑅0

2+ 𝑅13/4 + (𝑠

2

0− 𝑧2) /2𝜇

0𝐴𝑧𝑠0

.

(67)

And the dynamic current stiffness of the RMB unitaffected by the rotor position varying in 𝑧-axis is modifiedas

𝑘𝑧𝑖

𝑟𝑖=𝑑𝑓𝑥𝑖

𝑑𝑖𝑥

=2𝜙𝑟𝑚𝑁𝑟

𝜇0𝐴𝑟𝑅𝑥

=𝑁𝑟

2𝜇0𝐴𝑟𝑅𝑥

𝐹𝑚

𝑅0

2+ 𝑅𝑥/4 + (𝑠

2

0− 𝑧2) /2𝜇

0𝐴𝑧𝑠0

= 𝑘𝑖

𝑟𝑖

𝑅0

2+ 𝑅𝑥/4 + 𝑅

0

𝑧1/2

𝑅0

2+ 𝑅𝑥/4 + (𝑠

2

0− 𝑧2) /2𝜇

0𝐴𝑧𝑠0

= 𝑘𝑖

𝑟𝑖𝜆𝑧𝑟,

(68)

where 𝜆𝑧𝑟

= (𝑅0

2+ 𝑅𝑥/4 + 𝑅

0

𝑧1/2)/(𝑅

0

2+ 𝑅𝑥/4 + (𝑠

2

0−

𝑧2)/2𝜇0𝐴𝑧𝑠0).

Base on (65), the 𝑧 term is included in the dynamiccurrent stiffness model which will be affected by the rotorposition varying in 𝑧-axis.

3.4. The Dynamic Displacement Stiffness of the RMB UnitAffected by the Rotor Position of AMB Unit in 𝑧-Axis.The total static magnetic reluctance will be modified whenconsidering the small variation of the rotor position of theAMB unit in 𝑧-axis. Consider the following:

𝑅met = 𝑅0

2+𝑅0

𝑥

4+

1

1/𝑅0

𝑧1+ 1/𝑅

0

𝑧2

= 𝑅0

2+𝑅0

𝑥

4+

𝑠2

0− 𝑧2

2𝜇0𝐴𝑧𝑠0

.

(69)

The static bias flux in the air gap of the RMB unit ismodified when considering the rotor position varying of theAMB unit in 𝑧-axis. Consider the following:

𝜙0

𝑥=

𝐹𝑚

4𝑅met=

𝐹𝑚

4 (𝑅0

2+ 𝑅0𝑥/4 + (𝑠

2

0− 𝑧2) /2𝜇

0𝐴𝑧𝑠0). (70)

The total dynamic magnetic reluctance affected by theeddy-current effect is modified as

𝑅𝑒𝑡

sum =

2

∑

𝑚=1

𝑅𝑚+ 𝑅7+

12

∑

𝑛=11

𝑅𝑛+

1

2/𝑅0𝑥+ 1/𝑅

𝑥1+ 1/𝑅

𝑥2

+1

1/∑6

𝑗=3𝑅𝑗+ 1/∑

10

𝑘=8𝑅𝑘

+𝑠2

0− 𝑧2

2𝜇0𝐴𝑧𝑠0

.

(71)

The relative permeability of the stator and rotor core islarger compared to that of the air gap and the permanentmagnet. Then (71) can be simplified.

Due to rotor position varying in 𝑧-axis, the dynamicdisplacement stiffness of the RMB unit is modified as

𝑘𝑧𝑑

𝑟𝑠=

𝑑𝑓𝑥𝑠

𝑑 (𝛿 sin (𝜔𝑡))

= −

𝐹2

𝑚/(𝑅0

2+ 𝑅0

𝑥/4 + (𝑠

2

0− 𝑧2) /2𝜇0𝐴𝑧𝑠0)2

8𝜇0𝐴𝑧𝑠0

(𝑅met𝑅𝑒𝑡sum

)

2

= 𝑘𝑑

𝑟𝑠𝜆2

𝑧𝑟.

(72)


200

300

400

500

600

700

800

Mag

nitu

de o

f cur

rent

stiff

ness

(N/A

)

Frequency (Hz)10−1 100 101 102 103

(a)

0

−10

−5

−15

−20

−25

−30

−35

20log(k

i zi/k0 zi)

Frequency (Hz)10−1 100 101 102 103

(b)

Phas

e (de

g)

0

−10

−5

−15

−20

−25

−30

−35

Frequency (Hz)10−1 100 101 102 103

(c)

Figure 6: Frequency response of the dynamic current stiffness of AMB unit. (a) The magnitude of the dynamic current stiffness of AMBunit. (b)The magnitude of the ratio of the dynamic current stiffness to the static current stiffness of the AMB unit. (c) The phase of the ratio,𝑘𝑖

𝑧𝑖/𝑘0

𝑧𝑖, of the dynamic current stiffness to the static current stiffness of the AMB unit.

0

1

2

3

4

5

6

7

8

9

10

Mag

nitu

de (d

B)

Frequency (Hz)10−1 100 101 102 103

Figure 7: The magnitude of the ratio of the total dynamic reluctance to the total static reluctance in equivalent control flux path of the AMBunit, 20log

10(𝑅𝑒(𝜔)/𝑅

0

𝑧𝑖).


3

4

5

6

7

8

Mag

nitu

de o

f dyn

amic

di

spla

cem

ent s

tiffne

ss (N

/m)

×105

Frequency (Hz)10−1 100 101 102 103

(a)

Frequency (Hz)10−1 100 101 102 103

0

−2

−4

−6

−8

−10

20log(k

d zs/k0 zs)

(b)

150

155

160

165

170

175

180

Phas

e (de

g)

Frequency (Hz)10−1 100 101 102 103

(c)

Figure 8: Frequency response of the dynamic displacement stiffness of the AMB unit. (a) The magnitude of the dynamic displacementstiffness of the AMB unit. (b)Themagnitude of the ratio of the dynamic displacement stiffness to the static displacement stiffness of the AMBunit. (c) The phase of the ratio, 𝑘𝑑

𝑧𝑠/𝑘0

𝑧𝑠, of the dynamic displacement stiffness to the static displacement stiffness of the AMB unit.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Mag

nitu

de (d

B)

Frequency (Hz)10−1 100 101 102 103

Figure 9:The magnitude of the ratio of the total dynamic reluctance to the total static reluctance in equivalent bias flux path of the CRAMB,20log

10(𝑅𝑒𝑡

sum/𝑅0

sum).


0.8

0.88

0.96

1.0

1.1

1.2

1.3

1.4

1.4

Mag

nitu

de o

f dyn

amic

di

spla

cem

ent s

tiffne

ss (N

/m)

Frequency (Hz)10−1 100 101 102 103

×106

(a)

0

−1

−2

−3

−4

−5

−6

20log(k

d rs/k

0 rs)

Frequency (Hz)10−1 100 101 102 103

(b)

150

155

160

165

170

175

180

Phas

e (de

g)

Frequency (Hz)10−1 100 101 102 103

(c)

Figure 10: Frequency response of the dynamic displacement stiffness of the RMB unit. (a) The dynamic displacement stiffness of the RMBunit. (b) The ratio of the dynamic displacement stiffness to the static displacement stiffness of the RMB unit (c) Phase of 𝑘𝑑

𝑟𝑠/𝑘0

𝑟𝑠.

Based on (72), the 𝑧 term is included in the dynamicdisplacement stiffness model which will be affected by therotor position varying in 𝑧-axis.

4. Example and Results

Based on the methods mentioned above, the CRAMB withpermanent magnet is designed. The configuration of theCRAMB studied is shown in Figure 1.The related parametersare shown in Table 1.

The static current stiffness and displacement stiffness ofthe CRAMB could be calculated by linearized model with asmall perturbation of the current or air gap.The static currentstiffness, 𝑘0

𝑧𝑖, and displacement stiffness, 𝑘0

𝑧𝑠, of the AMB unit

are 825.6N/A and −8.0067 × 105N/m, respectively.The staticcurrent stiffness, 𝑘0

𝑟𝑖, and displacement stiffness, 𝑘0

𝑟𝑠, of the

RMB unit are 756.8N/A and −1.5131 × 106N/m, respectively.

The magnitude and phase plots of the dynamic currentstiffness of the AMB unit are shown in Figure 6. It is obviousthat the dynamic current stiffness is affected by the varyingfrequency of the control current of the AMB unit. And thiscan be shown in Figure 7. Figure 7 gives the magnitude ofthe ratio of the total dynamic reluctance to the total staticreluctance in equivalent control flux path of the AMB unit,20log10(𝑅𝑒(𝜔)/𝑅

0

𝑧𝑖). The dynamic reluctance of the control

flux path of the AMB unit will increase with the varyingfrequency of the control current in the AMB coil current.

The magnitude and phase plots of the dynamic displace-ment stiffness of the AMB unit are shown in Figure 8. Itis obvious that the dynamic displacement stiffness is alsoaffected by the varying frequency of the rotor position ofthe AMB unit in 𝑧-axis. From Figures 6 and 8, we can seethat the response bandwidth of the dynamic displacementis higher compared to that of the dynamic current stiffness.And this can be explained by Figure 9, which shows themagnitude of the ratio of the total dynamic reluctance to


Figure 11: The magnitude of the dynamic current stiffness versusvarying frequency of the control current in AMB unit and the rotorposition of RMB unit in 𝑥-axis.

Figure 12: The magnitude of the dynamic displacement stiffnessversus varying frequency of the rotor position of the AMB unit in𝑧-axis and the rotor position of RMB unit in 𝑥-axis.

the total static reluctance in equivalent bias flux path of theCRAMB, 20log

10(𝑅𝑒𝑡

sum/𝑅0

sum). The dynamic reluctance of thebias flux path will increase with the varying frequency.

The magnitude and phase plots of the dynamic displace-ment stiffness of the RMB unit are shown in Figure 10. Itis obvious that the dynamic displacement stiffness is alsoaffected by the varying frequency of the rotor position of theRMBunit in𝑥-axis. FromFigures 8 and 10, we can see that theresponse bandwidth of the dynamic displacement of theAMBunit is higher compared to that of the dynamic displacementstiffness of the RMB unit.

Figure 13: The coefficients 𝜆𝑟𝑧and 𝜆2

𝑟𝑧.

Figure 14: The magnitude of the dynamic current stiffness of theRMB unit versus varying frequency of the control current of theRMB unit and the rotor position in 𝑧-axis.

The magnitude of the dynamic current stiffness of theAMB unit versus varying frequency of the control currentin AMB unit and the rotor position of RMB unit in 𝑥-axis(or 𝑦-axis) is shown in Figure 11. And the magnitude ofthe dynamic displacement stiffness of the AMB unit versusvarying frequency of the rotor position of the AMB unit in𝑧-axis and the rotor position of RMB unit in 𝑥-axis is shownin Figure 12. It is obvious that the dynamic current stiffnessand the dynamic displacement stiffness of the AMB unit aredetermined by the varying frequency of the control currentin the AMB unit or the rotor position in 𝑧-axis rather than


Figure 15: The magnitude of the dynamic displacement stiffness ofthe RMB versus varying frequency of the rotor position in 𝑥-axisand the rotor position of the AMB unit in 𝑧-axis.

Figure 16: The coefficients 𝜆𝑧𝑟and 𝜆2

𝑧𝑟.

the rotor position of the RMB unit in 𝑥-axis (or 𝑦-axis). Thereason is the coefficients (shown in Figure 13) which are littleaffected by the rotor position in 𝑥-axis (or 𝑦-axis) direction.

The magnitude of the dynamic current stiffness of theRMB unit versus varying frequency of the control currentin RMB unit and the rotor position in 𝑧-axis is shown inFigure 14. And the magnitude of the dynamic displacementstiffness of the RMB unit versus varying frequency of therotor position in 𝑥-axis (or 𝑦-axis) and the rotor position in𝑧-axis is shown in Figure 15. It is obvious that the dynamiccurrent stiffness and the dynamic displacement stiffness of

the RMB unit are determined by the varying frequency of thecontrol current in the RMB unit or the rotor position in 𝑥-axis (or𝑦-axis) rather than the rotor position of theAMBunitin 𝑧-axis. The reason is the coefficients (shown in Figure 16)which are little variation affected by the rotor position in 𝑧-axis.

5. Conclusion

The CRAMB with permanent magnet creating bias fluxcan reduce the size, cost, mass, and save energy of themagnetic bearing system [6, 7, 12, 16, 17]. The structure andits simplified dynamic stiffness models of a novel CRAMBwith permanent magnet bias are presented when consideringthe eddy-current effect in this paper. And the dynamic cross-coupling models between axial and radial bearing unitsare also given using the method of equivalent magneticcircuit including eddy-current effect. An example is givenand the analytical results show that the eddy-current effectshave a fundamental impact on the dynamic stiffness of theCRAMB, but the dynamic current stiffness of the RMB unitis approximately equal to its static current stiffness. Thebandwidths of the dynamic current stiffness of the AMBunit and the dynamic displacement stiffness of the CRAMBare affected by the time-varying control currents or air gaprespectively. The rotor position varying in 𝑥-axis (or 𝑦-axis)has little influence on the dynamic current stiffness and thedynamic displacement stiffness of the AMB unit due to thefew coupling coefficients. And the rotor position varyingin 𝑧-axis also has little influence on the dynamic currentstiffness and the dynamic displacement stiffness of the RMBunit due to the little coupling coefficients. The describedmethod can simulate the dynamic behavior of the CRAMBwith permanent magnet.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of the paper.

Acknowledgments

Thisworkwas supported in part by theAviation Science Fundof China under Grant 2012ZB51019 and by the Cultivationand Development Project of Science and Technology Inno-vation Base of Beijing under Grant Z131104002813105.

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