Design and Simulation of Control System for Bearingless

Synchronous Reluctance Motor

Hannian Zhang, Huangqiu Zhu, Zhibao Zhang, Zhiyi Xie School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China

Abstract In this paper, the principle of bearingless synchronous reluctance motors is explained and a mathematic model of control system is deduced. The main problem in the control system is the coupling between the radial force and electromagnetic torque, and between the two radial forces in x- and y-direction. A decoupling method based on the feed-forward compensator is designed to decouple those variables. Proposed control system has been simulated in Matlab/Simulink environment. Simulation results have validated that stable suspension operation and the decoupling can be realized successfully with this method.

I. INTRODUCTION

Recently, Super high speed electrical drives are required for many special applications such as high speed machine tools, flywheels, turbomolecular pumps, compressors, etc. Advantages of no contact, no lubrication, and no maintenance are needed in the application areas of airplanes, bioengineering, pumps for pure substances, and so on. Magnetic bearings can suit for these applications, but there are some drawbacks, for example the sophisticated structure and high cost. The bearingless synchronous reluctance motor has combined characteristics of synchronous reluctance motor and magnetic bearings. Compared with the motor with magnetic bearings, the bearingless synchronous reluctance motors performance is obvious, the motors shaft length can be shorter, the critical speed is increased, and the motors structure is simplified. Because of the absence of windings and permanent magnets on the rotor, this type bearingless motor is advantageous in the high speed applications [1]-[6]. Research shows that the electromagnetic torque and the radial force are coupled by torque component fluxes [3], as may be seen from the mathematic model, the radial force generation is connected with the motor windings currents. Furthermore, the both two radial forces in x- and y-axis are coupled together, but it is difficult to decoupling those variables using flux orientation directly. So a decoupling control method based on the feed-forward compensator has been proposed and simulation results have been described in the following text. Computer simulation results have verified the validity of this decoupling control algorithm.

II. PRINCIPLE OF RADIAL FORCE GENERATION

In bearingless motor, a combination of two sets of windings with a difference in pole pair number of p 1 is needed to generate both the electromagnetic torque and radial force [5]. Fig. 1 has shown the principle of radial force production under no load condition. The motor

combined 4-pole motor windings and 2-pole suspension windings in the same stator slots, two-phase windings are exampled for simplicity. The currents in the 4-pole motor windings N generate 4-pole magnetic field a , and 2-pole magnetic fluxes y are produced by the currents in the 2-pole suspension windings yN . When the 2-pole winding currents are applied, the fluxes density in area 1 increases but the fluxes density in area 2 decreases, so the unbalanced revolving field results in producing the radial force yF in the positive y-direction. The motor has another 2-pole suspension windings which perpendicular to y windings, and then the radial force in any direction can be produced. And the motor has another 4-pole motor windings in order to produce the torque.

N

III. MATHEMATIC MODEL

A. Equation of Radial Force

Real-time control of radial force is required for the beaingless motors stable operation, so it is important to deduce the radial force mathematic model, and the mathematical model is the basis of the design for the control system. Usually, AC motor has uniform air gap, but for synchronous reluctance motor, the air gap variation caused by the rotor saliency has to be considered, so the equation of radial force is different from other typesbearignless motor. In this paper, the pole arc is assumed to be , the magnetic motive force distributions of both sets of windings are assumed to be sinusoidal, and magnetic saturation is neglected. The radial forces can be derived by the partial derivatives of the stored magnetic energy. In the d-q rotating reference frame, the stored energies in the windings may be calculated as in (1) [3].

30D

i

i

y

aN

yN

aN iN

aN

aN

yF

1

2

a ay yaa

xy

Fig. 1. Principle of radial force generation

Project supported by National Natural Science Foundation of China (50275067, 60174052), High technology research of Jiangsu Province (BG2005027), and by SRF for ROCS, SEM. 554

12

d dq dx dy d

qd q qx qy qm d q x y

xd xq x xy x

yd yq yx y y

L L L L iL L L L i

W i i i iL L L L iL L L L i

=

(1)

where d , q , xi and yi are the current components of the motor windings and suspension windings in the rotor coordinates, respectively. d

i i

L and qL are the d- and q-axis inductance of the motor windings, respectively. xL and

yL are the self-inductance of the suspension windings in the 2-phase coordinates, dxL , dyL , qxL and qyL are the mutual inductance between the motor windings and the suspension windings, respectively.

Because of the symmetrical winding, x yL L= , , . Equation (1) can be simplified as 0dq qdL L= = 0xy yxL L= =

1 1

2 2

1 2 2

1 2 2

12

00

00

m d q x y

dd m m

qq m m

xm m

ym m

W i i i iiL K x K yiL K y K xiK x K y LiK y K x L

=

(2)

where

( )( )

0 2 41 2

0

0 2 42 2

0

2 3 348

2 3 348

m

m

lrN NK

lrN NK

= +=

(3)

2L is the self-inductance of the suspension windings, 0 is

H/m, l is stack length, r is rotor radius, 2N and 4 are the per-phase effective turns in series of the motor

windings and suspension windings, respectively. is air gap length with centered rotor.

74 10N

0

The radial forces acting on the rotor can be derived from the differential of the magnetic energy and can be obtained as

1 2

2 1

mx m d x m q y

my m q x

W

m d y

F K i i K i ix

WF K i i K i iy

= = += = (4)

Equation (4) can be written in matrix form as shown

in (5).

1 2

2 1

x m d m q x

y m q m d y

F K i K i iF K i K i i =

(5)

Supposed that

0

cos sinsin cos

C

= (6)

where is mechanical rotor angle, the above equations are transformed into the stationary coordinate system and can be written as

1 2 2100

2 1 2

x m d m q

y m q m d

F K i K i iC C

F K i K i i

= (7)

where 2i and 2i are the currents of the xN and y suspension windings in the stationary coordinate system, equation (7) can be simplified as

N

1 2 22 1 2

cos 2 sin 2sin 2 cos 2

x m d m q

y m q m d

F K i K i iF K i K i i

= (8)

If the rotor is displaced, the magnetic tensile force

acting on the rotor produces and the expression is given as

2

0 0

sxs

sy

F x xrlBk KF y y

= = (9)

where k is the coefficient related to the motors structure, x and y are displacements, B is the flux density in the air gap at centered rotor, the other parameters meaning can be seen in (3).

zxF and zyF are assumed to the external interference forces applied on the rotor in x- and y- direction, and the gravity is included, so the motion equations of the rotor are described as

+ + = + + =

ii

iix sx zx

y sy zy

F F F m xF F F m y

(10)

where m is the mass of the rotor, based on (5), (9) and (10), the block diagram of the radial forces control system inside the motor is shown in Fig. 2

B. Equation of Rotation

The mathematic model of the motor is totally constructed with voltage, flux linkage and torque equations. In the rotor oriented coordinate system, the equations of the motors stator voltage, stator flux linkages and electromagnetic torque are given as in (11), (12) and (13).

+

+zxF

zyF

+

+

+

+

m1 dK i

m2 qK i

m2 qK i

m1 dK i

xi

yi

xii

yii

xi

yi

x

y

sK

sK

xF

yF

sxF

syF

+1/ m

1/ m

Fig. 2. The diagram of the radial forces control system

inside the motor

555

d s d d

q s q q

dU R idtdU R idt

q

d

= + = + +

(11)

d d

q q

L iL i

== dq (12)

( )32

= =

e p d q d

e Lp

T n L L iJ dT Tn dt

qi

(13)

d and qU are the d- and q-axis stator voltage components,

respectively. dU

and q are the d- and q- axis flux linkage components of the stator, respectively. sR is per-phase resistance of the stator, is synchronous angular speed,

p is pole pairs, J is the moment of inertia, eT is the motors electromagnetic torque, and n

LT is the load torque.

IV. DESIGN OF FEED-FORWARD COMPENSATOR

The bearingless synchronous reluctance motor is strongly coupled system. Under load condition, the motor could become unstable due to the coupling between the radial forces and electromagnetic torque, and between the two orthogonal radial forces. In order to keep the stable operation, the decoupling control of the motor is necessary. The bearingless synchronous reluctance motor is different from other types bearingless motor, it is difficult to realize the decoupling control based on the motor flux oriented directly. But research has shown that feed-forward compensation decoupling control is a very simple but effective method to decouple those variables. Based on the feed-forward compensator, the control system become simple and can be realized easily.

Matrix is defined as 1C

1

cos 2 sin 2sin 2 cos 2

C

= (14)

xF and yF are the references of radial forces in x- and

y- axis, so (15) can be obtained from (8) as

1 22 11 2 2 2 2

2 12 1 2

1

= + m d m q x

m q m dm d m q x

K i K ii

F

CK i K ii k i k i F

(15)

Substituting above equations into (8), the result can be expressed as

1 00 1

x x

y x

F FF F

=

(16)

The above16shows that the two orthogonal radial

forces are decoupled completely, furthermore, the radial

forces acting on the rotor are equal to the given reference values, and the radial forces are not affected by the electromagnetic torque.

0xF 0yF are the new references of radial forces

defined as

1 202 2 2 2

2 10 1 2

1 m d m qx xm q m dy ym d m q

K i K iF FK i K iF FK i K i

= +

(17) Substituting (17) into (15), the relationships between

the new references of radial forces and the currents in the suspension windings can be obtained as

2 0112 0

x

y

i FC

i F

=

(18)

Where

11cos 2 sin 2sin 2 cos 2

C

= (19)

Based on (17), the feed-forward compensator can be designed and the configuration is shown in Fig. 3

V. DESIGN OF CONTROL SYSTEM

The whole control system of the bearingless synchronous reluctance motor includes the motor control and the radial position control. The control system configuration is shown in Fig. 4, the motor control method is common, as can be seen that when the d-axis current dI is fixed, the torque is proportional to the torque

xF

yF

m1 dK I

m2 qK I

m1 dK Im2 qK I

+++

2 2 2 2m1 d m2 q

1K I K I +

2 2 2 2m1 d m2 q

1K I K I +

x0F

y0F

Fig.3. Block diagram of the compensator

++

+

x

yx

y

x0F

y0F

2i

2i

xF

yF

A2i

B2i

C2i

A2i

B2iC2i

A1i

B1i

C1i

B1i1i

1iqI

dI

A1i

C1i

2 dt 2

1Park

1Park

constant

PI

PID

PID

Decoupling CRPWMInverter

CRPWMInverter

Compensator

23

23

Fig. 4. Control system diagram of bearingless synchronous

reluctance motor

556

Fig. 5. Control system diagram in Matlab

component currents q

I , so the motor control system can

be simplified with this method. In the position control system, the displacements in x- and y-direction of the rotor are detected by the sensors and then compared with reference values, then the error signals are regulated by PID controllers to generated the radial forces references

xF and yF , the new reference values 0xF and 0yF can be

got after using the decoupling controller, then three phase reference currents 2Ai , 2 Bi and 2C are generated after coordination transformation of new reference values 0x

i

F and 0yF , then currents 2Ai , 2Bi and 2C are controlled by the inverter to follow their current references, and then the radial forces acting on the rotor

i

xF and yF are generated.

VI. SIMULATION OF CONTROL SYSTEM

In order to verify the validity of proposed control system in this paper, simulation is implemented based on the Matlab/Simulink. The configuration of the control system in Matlab environment is shown in Fig. 5. The motor parameters are given as follows;

The motor windings pole pairs 1p =2, =0.035 H, q

dLL =0.007 H, stator resistance 1sR =0.20 , =3 A. The suspension windings pole pairs 2

dip =1, x y 0.02 H,

stator resistance 2L L= =

sR =0.15 . The given reference values of speed Nn = 1500 r/min, rated power N

P =1 kW, the

rotor mass m=1 kg, the rotors moment of inertia J=0.002 , air-gap length 02kg m =0.3 mm, the touch down

bearing clearance =0.2 mm. The radial interference forces acting on the rotor in x-

and y-direction are given by zxF =20 N and zyF =20 N, respectively. The load torque ( LT =3 ) is exerted in 0.025 s. Fig. 6 - Fig. 11 show the simulation results of the control system.

N m

A. Simulation of Radial Position Control System

Fig. 6 and Fig. 7 show that the transient response during the process from starting-up to the stable suspension of the rotor, the initial values in x- and y-

direction are supposed that x=0.05 mm and mm, respectively, and the load torque is applied in 0.025 s. The figures show that the percent overshoot is small and the rotor can realize the stable suspension. Although the load torque is applied in 0.025 s, no displace is to be seen in x- and y-direction, thus the torque and the radial forces are decoupled.

0.20=y

Fig. 8 and Fig. 9 show the x- and y-axis step response of the rotor displacements. In Fig. 8, the initial value of x is x=0 mm, after 0.01 s the new set point value is

mm, after the quick dynamic regulation, the rotor suspended steadily in the given reference position, it

0.1=xshows the good regulation characteristic of the position

Fig. 6. x-axis displacement waveform

Fig. 7. y-axis displacement waveform

557

Fig. 8. x-axis step response

Fig. 10. Torque response waveform

Fig. 9. y-axis step response

Fig. 11. Step response of the speed

control system. In Fig. 9, the initial value of y is mm, the set point change is y=0.1 mm. after 0.015 s, the rotor can also suspended steadily in the given position, but in x-axis no deflection of the displacements can be seen, it shows that the coupling between x- and y-axis was canceled.

0.1=y

B. Simulation of Motor Control System

Fig. 10 shows the dynamic response of the torque. The motor starts with no load, after 0.025 s the load torque with 3 Nm is applied, the figure shows the good characteristic with the proposed torque control method. The step response characteristic of the speed for the bearingless synchronous reluctance motor is shown in Fig. 11, the figure shows that the percent overshoot is less than 5%, the response time is less than 0.015 s, and the steady-state error is zero, good performance can be also obtained.

VII. CONCLUSIONS

Based on the analyzing about the fundamental principle of bearingless synchronous reluctance motor, the motors mathematical model is given. Because of strongly coupled nonlinear system, a decoupling control method

for the motor is proposed, which is based on feed-forward compensator, to cancel the coupling between the torque and the radial forces, and the radial forces in both radial axis. Numerical simulation is implemented to shows that the decoupling control of those variables is realized using this control strategy, and good performance of torque control and speed adjusting is also shown.

REFERENCES [1] A. Chiba, K. Chida, and T. Fukao, Principles and characterstics of

a reluctance motor with windings of magnetic bearing, in Proc. of IPEC, Tokyo, 1990, pp. 919-926.

[2] A. Chiba, T. Deido, T. Fukao, M. A. Rahman, An analysis of bearingless AC motors, IEEE Trans. on Energy Conversion, vol. 9, no. 1, pp. 61-68, Mar. 1994.

[3] C. Michioka, T. Sakamoto, O. Ichikawa, A. Chiba, T. Kao, A decoupling control method of reluctance-type bearingless motors considering magnetic saturation, IEEE Trans. on Industry Applications, vol. 32, no. 5, pp. 1204-1210, Sep. 1996.

[4] L. Hertel, W. Hofmann, Hochtouriger lagerloser reluktanzantrieb, Kassel, 1999. Available: http://www.infotech.tu-chemnitz.de

[5] L. Hertel, W. Hofmann, Theory and test results of a high speed bearingless reluctance motor, in PCIM, Nuremberg, 1999, pp. 143-147.

[6] Huangqiu Zhu, Zhiquan Deng, Yangguang Yan, Principles of beaingless motors and research status, Micromotors Servo Technique, vol. 33, no. 6, 2000, pp. 29-31.

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