Peter Kascak, Ralph Jansen, and Timothy Dever

Glenn Research Center, Cleveland, Ohio

Aleksandr Nagorny

ResMed Motor Technologies Inc., Los Angeles, California

Kenneth Loparo

Case Western Reserve University, Cleveland, Ohio

Levitation Performance of Two Opposed

Permanent Magnet Pole-Pair Separated

Conical Bearingless Motors

NASA/TM—2013-217863

April 2013
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Peter Kascak, Ralph Jansen, and Timothy Dever

Glenn Research Center, Cleveland, Ohio

Aleksandr Nagorny

ResMed Motor Technologies Inc., Los Angeles, California

Kenneth Loparo

Case Western Reserve University, Cleveland, Ohio

Levitation Performance of Two Opposed

Permanent Magnet Pole-Pair Separated

Conical Bearingless Motors

NASA/TM—2013-217863

April 2013

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/TM—2013-217863 1 

Levitation Performance of Two Opposed Permanent Magnet Pole-Pair 

Separated Conical Bearingless Motors

Peter Kascak, Ralph Jansen, and Timothy Dever

National Aeronautics and Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Aleksandr Nagorny

ResMed Motor Technologies Inc.

Los Angeles, California 91311

Kenneth Loparo

Case Western Reserve University

Cleveland, Ohio 44106

Abstract

In standard motor applications, rotor suspension with 

traditional mechanical bearings represents the most economical solution. However, in certain high performance 

applications, rotor suspension without contacting bearings is 

either required or highly beneficial. Examples include 

applications requiring very high speed or extreme environment

operation, or with limited access for maintenance. This paper 

expands upon a novel bearingless motor concept, in which two 

motors with opposing conical air-gaps are used to achieve full 

five-axis levitation and rotation of the rotor. Force in this 

motor is created by deliberately leaving the motor’s pole-pairs 

unconnected, which allows the creation of different d-axis flux 

in each pole pair. This flux imbalance is used to create lateral 

force. This approach is different than previous bearingless 

motor designs, which require separate windings for levitation

and rotation. This paper examines the predicted and achieved 

suspension performance of a fully levitated prototype bearingless system. 

Introduction

Traditionally motor rotors are suspended from their stators 

with mechanical bearings. Mechanical bearings provide a very 

stable platform separating the rotating and stationary parts 

with relatively very little friction at a very small cost. In most 

applications this is the preferred solution; these bearings 

perform over an adequate lifetime, can be readily lubricated, 

and if necessary easily replaced.

However there are a growing number of high performance 

applications where the use of mechanical bearings is either 

unsatisfactory or impossible. These applications have one or 

more of the following attributes: extremely fast rotating speed, 

the need for almost no friction, limited or non-existent 

opportunity for maintenance, and operation in an extreme 

environment. Since these applications are very high 

performance, a higher cost rotor suspension system can be 

justified. 

Magnetic bearings have been used to provide higher 

performance rotor suspension. Separate magnetic actuators are 

designed which sense the position of the rotor and apply an 

appropriate magnetic force to stabilize the position of the 

rotor. This allows for contact free suspension of the rotor with

very little friction, no need for a lubrication system, and no 

need for maintenance.

One potential drawback of this method is that it adds iron 

and copper parts dedicated only to the function of rotor 

levitation, while also requiring the motor to contain iron and 

copper parts dedicated to only rotation. In addition to 

increasing parts count, this approach can also increase the 

axial length of the rotor, which lowers the bending mode 

frequencies and complicates suspension control at high 

speeds. Additionally, this separation of functions of the two 

actuators means that the motor must be sized for the largest 

torque load the machine will encounter, and the magnetic 

bearing system must be sized for the largest lateral force 

necessary to suspend the rotor.

For these reasons it was desired to develop a single actuator, 

i.e., a bearingless motor; which could produce lateral forces 

for rotor suspension in addition to the torque controlling the 

rotation of the rotor. Bearingless motors were demonstrated in 

synchronous reluctance (Refs. 1 and 2), induction (Ref. 3), 

permanent magnet (Ref. 4), and switched reluctance (Ref. 5)

motor types. These motors have a common iron structures 

carrying both levitation and rotation flux. However, they all 

use different windings for levitation and rotation, which means 

each function is still limited by windings and power 

electronics dedicated to a single function. A further limitation 

is that they are only capable of providing levitation forces in a 

radial plane in the case of radial motors, or in the axial 

direction for axial motors.
NASA/TM—2013-217863 2 

This paper examines the performance of a new bearingless 

motor concept, which was previously documented by the 

authors of this paper in References 6 to 11. Using this new 

approach the motor pole-pairs are separated, causing a flux 

imbalance which creates a levitation force. This allows all of 

the motor iron and copper to be used for either levitation or 

rotation. Additionally, these motors employ a conical shaped 

air-gap, enabling creation of axial as well as radial forces. A 

prototype controller and motor were built, using pair of these 

motors with opposing cones, and full 5-axis levitation was 

demonstrated. 

The contribution of this paper over (Refs. 6 to 11) is that it 

focuses on the suspension performance of the machine, where 

the previous literature was focused on presenting the basic 

concept. Furthermore, this paper introduces new parameters 

which are not commonly measured on electric machines, along

with the experimental methods used to measure them. These 

new parameters are figures of merit on magnetic bearing 

systems. 

Self Levitation by Two Pole-Pair 

Separated Conical Motors

Separate Control of Pole-Pairs

Standard electric machines are wound with pole-pairs 

connected either in series or parallel; series connected 

windings are shown in Figure 1. These windings do not allow 

independent control of the flux in each pole-pair separately; 

the flux of every pole-pair can only be increased or decreased 

together. For electric machines with rotors suspended using 

conventional mechanical bearings there is no reason to control 

the flux in each pole-pair individually; in fact, it is desired that 

each pole-pair carry the same flux, so as not to induce 

unwanted vibration.
NASA/TM—2013-217863 3 

The bearingless machine studied in this paper utilizes

unconnected pole-pairs, as seen in Figure 2. This configuration 

allows the flux in each individual pole pair to be varied, 

allowing a net force to be created on the rotor.

The electronics needed to drive a standard motor are shown 

in Figure 3. The pole-pairs are connected in series, and the 

motor can be controlled with three currents, which requires six 

switches.

More complex electronics are needed to drive the 

bearingless motor, as shown in Figure 4. Because the polepairs are now separate, nine currents have to be controlled.

This control requires eighteen switches.

The higher complexity of the bearingless motor’s drive 

electronics is not necessarily a disadvantage, and in some 

cases it may prove beneficial. For example, the bearingless 

drive will require the same silicon area as the standard drive, 

since the power rating of each switch is one third that of the 

standard motor. For high power applications, this could be 

used to eliminate the need to parallel switches, simplifying the 

system. Additionally, the bearingless drive offers fault tolerant 

capabilities—if a fault occurs in one of the pole-pairs, 

operation could continue using the other two pole-pairs.

The windings of the bearingless motor can be represented in 

the rotor reference frame; Figure 5 shows the fictional d and q 

axis windings rotating with the rotor. There are two types of 

forces which can be created with these windings: the first is 
NASA/TM—2013-217863 4 

the Lorentz force, which is the force associated with the 

production of torque, and the second is the Maxwell force, 

which acts in such a way as to minimize the reluctance of the 

magnetic circuit. The permanent magnet machine studied in 

this paper was intended for high speed operation; as such the 

Lorentz forces are much smaller than the Maxwell forces. 

The controller sets the q-axis currents equal in each polepair; this causes the net force induced by Lorenz type forces to 

be equal to zero. The d-axis currents in each pole-pair are set 

to be unequal, which creates a net Maxwell type force. The 

flux density plot in Figure 6, which was generated using finite 

element analysis, demonstrates this imbalance. In this test case 

a large d-axis current is present in one of the pole-pairs, while

the other pole-pairs have zero d-axis current. 

Axial Force Creation With Conical Air-Gap 

Motors

The Maxwell forces are directed straight across the air-gap; 

this means that a radial electric machine with a cylindrical airgap can create only radial forces. In order to fully suspend the 

rotor it is necessary to control 5-axes, which requires two 

planes of radial control in addition to axial control. Two 

cylindrical air-gap motors can’t actively control axial force.
NASA/TM—2013-217863 5 

In order to allow the active creation of axial forces, the 

machine studied in this paper employs a conical shaped airgap. Two motors of this type are arranged such that their 

cones are oriented in opposite directions; this is shown in 

Figure 7. In this configuration the forces created by d-axis flux 

are not purely radial, and have an axial component: the 

magnitude of the generated force is the same, but the axial 

component is determined by the cone angle. 

Figure 8 shows the forces created by the two motors (note 

that no attempt has been made in this diagram to show these 

forces in scale with the actual cone angle). The net axial force 

is proportional to the difference of the sum of the d-axis 

currents in the top motor and the sum of the d-axis currents in 

the bottom motor.

Prototype Machine

The motor was sized for a flywheel application with a full 

speed of 100 krpm. In order to keep material and construction

costs down, a low speed prototype of this motor was 

constructed to be used to prove the concept of combined 

levitation and rotation. The low speed requirement allowed the 

rotor and stator back iron of the machine to be constructed 

from solid M19 silicon steel. The windings were designed as if 

this machine would run at full speed in a flywheel application, 

and produce a back-emf of 110 V at 100 krpm. 

A summary of the motor design described in this paper is 

provided in Table 1. All of the parameters shown were 

experimentally confirmed.

TABLE 1.—PROTOTYPE CONICAL BEARINGLESS MOTOR

Parameter Design value

Soft magnetic material M19 Silicon Steel

Permanent magnets NdFe30

Poles 6

Turns/coil 10

Number of slots 18

Inner stator diameter 0.625 in.

Outer stator diameter (mid-point axially) 2.123 in.

Inner rotor diameter (mid-point axially) 2.148 in.

Outer rotor diameter 2.7 in.

Magnet thickness 0.025 in.

Mechanical air gap 0.025 in.

Slot depth 0.4 in.

Axial stack length 1.00 in.

Rotor yoke thickness (mid-point axially) 0.2420 in.

Stator back iron thickness (mid-point axially) 0.3455 in.

Slot pitch (tooth and slot) 0.3706 in.

Tooth width at air gap 0.2597 in.

Mutual Flux Linkage due to rotor flux (λaf) 3.4 mWb-t

q-axis inductance (Lq) 108 μH

d-axis inductance (Ld) 81 μH

Simulation Models

This paper employs three different models to obtain 

simulation results: a magnetic circuit model, a finite element 

model, and a control system model.

Magnetic Circuit Model

A magnetic circuit model was used to design and analyze 

the conical bearingless motor presented in this paper. In this 

model flux is assumed to travel straight through magnetic 

circuit elements; these can also be thought of as flux tubes. A 

magnetic circuit model can be created with as many elements 

as are necessary to accurately represent the device which they 

model; of course, the more elements that are used, the longer 

and more computationally intensive the simulation becomes.

The conical motor has dimensions which vary axially; 

however, for this analysis the average dimensions of the motor 

are used to simplify the analysis. The model used for analysis 

of this motor is shown in Figure 9. 
NASA/TM—2013-217863 6 

This model is solved by expressing 36 loop equations, 18 

stator loops and 18 rotor loops. Because the reluctance of the 

iron parts is a function of the flux which passes through them, 

this model is solved iteratively. The coils are represented by 

the “tooth” MMFs; the iron parts and the permanent magnets

are represented by an MMF source and a series reluctance; 

and the air gap and leakage are represented by a simple 

reluctance. 

First, an initial value of flux is first assumed for each loop 

flux. Next, using the model, the MMF source and series 

reluctance of each iron element is calculated at these flux 

levels. Then the loop equations are solved, and new flux 

values are obtained. The process is repeated until the 

difference between new and previous flux values converges to 

an appropriate tolerance.

Finite Element Model

A finite element model was also employed to analyze this 

motor. A 2-D magnetostatic model was used. Because the 

conical shaped motor has dimensions which vary axially, the 

plane in the middle of the motor was used for this analysis.

The finite element model does not use the simplifying 

assumptions of the magnetic circuit model; it solves for the 

actual field solution. It is more accurate than the magnetic 

circuit model; however, it takes much longer to run cases.

Additionally, relying only on this type of model can result in 

the loss of valuable design insight. For these reasons the finite 

element model was used to verify and fine tune the design 

obtained by magnetic circuit modeling. 

Control System Model 

In addition to the magnetics models, a control system model 

was created in MATLAB/Simulink. This model is of the entire 

controller and plant, including: a dynamic model of rotor

mechanics, inverter delay, both bearingless motors, and the 

controller (run at 0.3 ms sample time).

Results

Levitation and Rotation 

The prototype motor was successfully levitated and spun at 

a variety of speeds. Rotor position is measured and fed to the 

controller using non-contact eddy current sensors, which 

measured x and y radial positions at the top and bottom of the

machine, and axial position.

During 300 rpm operation, data was taken on the motor 

currents and the rotor position in 5 axes. The x-y plots of the 

top and bottom radial positions during operation are presented 

in Figures 10 and 11, and the axial position during operation is 

shown in Figure 12. 

Stability

Next, the loop gain of the bottom plane x controller was 

measured in hardware, and compared to the loop gain obtained 

through simulation. The comparison is shown in Figure 13. 

Note that the measured loop gain matches the simulated loop 

gain fairly well, except for around 32.5 Hz. 
NASA/TM—2013-217863 7 

To try to understand the origin of the extra mode seen in the 

experimental data, a standard rap test was performed on the 

motor. An accelerometer was attached in the center of the 

mounting plate, and the hammer was struck close by. The 

results, in Figure 14, show a clear dip in magnitude at 32.5 Hz 

which is consistent with the experimental loop gain data.

Based on the rap test results, it was concluded that the extra 

32.5 Hz mechanical mode was likely a rigid body mode of the 

motor baseplate interacting with the mounting isolation 

dampers. 

Finally, the experimental loop gain data was used to 

generate a transfer function, which represents the open loop 

system, and is useful in stability analysis.
NASA/TM—2013-217863 8 

First, the experimental loop gain data was fit to a ratio of 

complex polynomials (Refs. 12 and 13). Since the controller 

has a known sample time of 0.3 ms, this delay was divided out 

of the loop gain data, then the delay free data was curve fit.

This process yields the following:

( ) ( ) -×- 33 = × es no sdelay TsT e (1)

( ) ( )

( )sD

sN

sT

delayno

delayno

no delay = (2)

( )

392.031.1

1028.21088.1

1006.6 1012.5

35 23

511 48

s

s s

sN s s delayno

×+

××+××+

××+××-=

(3)

( )

ss

s s

sD s s delayno

-××-

××-××-

××+××=

25

48 36

614 510

1022.5

1060.71041.1

1037.2 1032.1

(4)

Using Equations (3) and (4), the zeros and poles of the open 

loop transfer function were calculated. The transfer function 

has the following zeros:

s j -±-= 3.0,2371.11,333,1160 (5)

And the following poles:

s -= ±- j2486.18,320,378,5650,0 (6)

Also using Equations (3) and (4), a Nyquist stability plot 

was generated, and is shown in Figure 15. Although the open 

loop transfer function has three poles in the right hand plane, 

the Nyquist stability plot shows that –1 is encircled three times 

in the counter clockwise direction, which guarantees that the 

closed loop system is stable.

Levitation Parameters

Now the suspension capabilities of the bearingless machine 

are examined. The two parameters of interest are the current 

stiffness and the negative position stiffness of the bearingless 

machine.

The current stiffness is the amount of force created for 

every amp of levitation current. To measure this, a known 

force is applied in the radial direction to one of the conical 

motors; this is done by hanging a known weight from a pulley 

and attaching it to the center of the rotor, seen in Figure 16. 

In order to determine the current stiffness from this test it is 

necessary to determine the current effort required to maintain 

levitation. This can be achieved by adding the d-axis current in 

each pole-pair vectorially. Note the forces created by each 

pole-pair are in directions that are separated by 120° from 

each other; this allows current effort of each pole-pair to be 

expressed as vectors: 

r

dd il = 1111

v

(7)

ail r

d rd

v v

1212 ×= (8)

2

1313 ail r

d rd

v v

×= (9)
NASA/TM—2013-217863 9 

Where v o ×120 = i

ea and r

di 11 , rdi 12 , rdi 13 are the d-axis current 

of each pole-pair in the first motor. Now the total levitation 

current effort can be expressed as:

r

d rd rd lllIeffort 131211

vvv

++= (10)

The current stiffness can now be expressed as:

weight-effort effort noweight

weight

current I = I

F

k (11)

Where Fweight is the force applied by the weight, Ieffort-weight is 

the current effort required for levitation with the applied force, 

and Ieffort-noweight is the current effort required to levitate the 

rotor with no applied force. 

Using the above approach, the current stiffness was 

measured with several different applied weights. The average 

value of current stiffness was found to be 0.59 lb/A. Current 

stiffness was also determined using both the magnetic circuit 

and finite element models; this was done by applying d-axis 

current to a pole-pair and noting the magnitude of the resulting 

force. Figure 17 shows force versus current for both 

simulation models; the slope of the line is the current stiffness.

Table 2 summarizes the current stiffness obtained by the two 

models, and experimentally. Note the excellent agreement 

between all three approaches.

TABLE 2.—CURRENT STIFFNESS

Current stiffness,

lb/A

Magnetic circuit 0.60

Finite element 0.59

Experimental 0.59

Next, the negative position stiffness is addressed. The 

negative position stiffness is the amount of force created on 

the rotor by moving it off of the magnetic center. Since the 

current stiffness is known, the force created by moving the 

rotor off center can be calculated as:

( ) position effort offcenter effort center current = IF I × k (12)

The position stiffness is then simply:

22

position

position

yx

F

k

D+D

= (13)

Where

=D offcenter act actcenter xx x (14)

=D offcenter act actcenter yy y (15)

Therefore the negative position stiffness can be measured 

by levitating the rotor, observing the current effort, then 

moving the rotor off center and observing the new current 

effort. There are two motors in this system; for this 

measurement we refer to them as the motor under test (MUT)

and the motor not under test (MNUT). During the 

measurement the MNUT is levitated, holding its position 

centered at its sensor plane. 

It is important to note that the above equations refer to 

movement at the center of the MUT, which is not coplanar 

with the sensors. Figure 18 shows the geometry of this 

measurement; the left most point of the triangle is the sensor 

plane of the MNUT. L2 is the distance from that sensor plane 

to the center of the MUT; L1 is the distance between the 

sensor planes of both motors. Movement of the rotor observed 

in the sensor plane can be expressed in its motor center plane as:

sensor 1

2

x

L

L

x D×=D (16)
NASA/TM—2013-217863 10

TABLE 3.—NEGATIVE POSITION STIFFNESS

Negative position stiffness,

lb/in.

Magnetic circuit 862

Finite element 842

Experimental 860

The measurement was performed on both the top and 

bottom motors, using various displacement values. The 

average measured negative position stiffness for both motors 

over the displacement range was 860 lb/in. The negative 

position stiffness was also determined using simulation by 

moving the rotor off center and observing the net force; 

simulations were performed using both models. Figure 19 

shows force versus rotor displacement for both the magnetic 

circuit model and the finite element model; the negative 

stiffness is the slope of these lines. A summary of the negative 

position stiffness using both simulation methods and 

experimental results is presented in Table 3. Note the close 

agreement between all three approaches.

Conclusion

This paper expands upon the conical bearingless concepts 

put forward previously by the authors. First, the basic concepts 

of the bearingless motor are briefly presented. Next the fully 

levitated, two-motor prototype rig is described. Three methods 

of simulation are described, including their specific uses.

Following this results are presented: position control while 

rotating, stability analysis, and levitation performance of the 

prototype machine. Experimental results closely match those 

obtained in simulation. The main contribution of this paper is 

in examining the bearingless motor’s performance as a rotor 

suspension system. 

References

1. A. Chiba, K. Chida and T. Fukao, “Principles and 

Characteristics of a Reluctance Motor with Windings 

of Magnetic Bearing,” in Proceedings of International 

Power Electronic Conference (IPEC-Tokyo), 1990, 

pp. 919-926.

2. A. Chiba, M.A. Rahman and T. Fukao, “Radial Force 

in a Bearingless Reluctance Motor,” IEEE 

Transactions on Magnetics, Volume 27, Issue 2, pp. 

786-790, March 1991.

3. A. Chiba, D.T. Power, M.A. Rahman, “No Load 

Characteristics of a Bearingless Induction Motor,”

IEEE Transactions on Magnetics, Volume 27, Issue 6, 

pp. 5199-5201, November 1991.

4. K. Dejima, T. Ohishi, and Y. Okada, “Analysis and 

control of a permanent magnet type levitated rotating 

motor,” in IEEJ Proceedings of Symposium Dynamics 

of Electro Magnetic Force, June 1992, pp. 251-256.

5. K. Shimada, M. Takemoto, A. Chiba, and T. Fukao, 

“A Stable Rotation in Switched Reluctance Type 

Bearingless Motors,” in Papers of IEEJ Technical 

Meeting on Linear Drive, 1997, LD-97-116.

6. P. Kascak, “Fully Levitated Rotor Magnetically 

Suspended by Two Pole-Pair Separated Conical 

Motors,” Ph.D. Thesis, Case Western Reserve 

University, Cleveland, OH, 2010.

7. P. Kascak, R. Jansen, and T. Dever, “Control System 

for Bearingless Motor-Generator,” U.S. Patent 

7,456,537 B1, Nov. 25, 2008.

8. P. Kascak, R. Jansen, and T. Dever, “Control System 

for Bearingless Motor-Generator,” U.S. Patent 

7,667,418, Feb. 23, 2010.

9. P. Kascak, R. Jansen, T. Dever, A. Nagorny, and K. 

Loparo, “Bearingless Five-Axis Rotor Levitation with 

Two Pole Pair Separated Conical Motors,” in the 

Conference Record of IEEE Industrial Applications 

Society Annual Meeting, October 2009.

10. P. Kascak, T. Dever, R. Jansen, “Conical bearingless 

motor-generators,” in McGraw-Hill Yearbook of 

Science & Technology, 2010 edition, pp. 81-84. 

11. P. Kascak, T. Dever, R. Jansen, A. Nagorny, and K. 

Loparo, “Motoring Performance of a Conical PolePair Separated Bearingless Electric Machine,” in the 

Conference Record of IEEE EnergyTech, May 2011.

12. E.C. Levy, “Complex-Curve Fitting,” in IRE 

transactions on automatic control, pp. 37-44, 1959.

13. C.K. Sanathanan and J. Koerner, “Transfer Function 

Synthesis as a Ratio of Two Complex Polynomials,” 

in IEEE Transactions on Automatic Control, vol. 8, 

issue 1, pp. 56-58, January, 1963.

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Levitation Performance of Two Opposed Permanent Magnet Pole-Pair Separated Conical 

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This publication is available from the NASA Center for AeroSpace Information, 443-757-5802

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14. ABSTRACT

In standard motor applications, rotor suspension with traditional mechanical bearings represents the most economical solution. However, in 

certain high performance applications, rotor suspension without contacting bearings is either required or highly beneficial. Examples include 

applications requiring very high speed or extreme environment operation, or with limited access for maintenance. This paper expands upon 

a novel bearingless motor concept, in which two motors with opposing conical air-gaps are used to achieve full five-axis levitation and 

rotation of the rotor. Force in this motor is created by deliberately leaving the motor’s pole-pairs unconnected, which allows the creation of 

different d-axis flux in each pole pair. This flux imbalance is used to create lateral force. This approach is different than previous bearingless 

motor designs, which require separate windings for levitation and rotation. This paper examines the predicted and achieved suspension 

performance of a fully levitated prototype bearingless system.

15. SUBJECT TERMS

Flywheels; Energy storage; Magnetic bearing; Motors; Controls 

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