CONTROL ANALYSIS OF BEARINGLESS ELECTRIC MOTOR

CONTROL ANALYSIS OF BEARINGLESS ELECTRIC MOTOR

Paulina Mazurek

, Maciej Henzel

, Krzysztof Falkowski

1 Department of Mechatronics and Aerospace, Military University of Technology

apaulina.mazurek@wat.edu.pl, ^bmaciej.henzel@wat.edu.pl, ^ckrzysztof.falkowski@wat.edu.pl

Summary

The paper deals with issue of bearingless electric motor control. Bearingless electric motors (BEM) compound in one construction active magnetic bearing (AMB) and electric motor. New motor structure eliminates disadvan- tages of conventional motors, e.g. friction forces, heat abstraction, noise. Control system parameters decides about static and dynamic motor characteristics. It will assure proper bandwidth, damping factor, diagnostic flexibility and high reliability.

In the paper, mathematical and simulation model of bearingless electric motor with control system will be pre- sented. The control systems is designed with use of Matlab – Simulink program. Time and frequency characteris- tics of bearingless motor are presented as well.

Keywords: bearingless electric motor, control system

ANALIZA STEROWANIA BEZŁOŻYSKOWYM SILNIKIEM ELEKTRYCZNYM

Streszczenie

Artykuł podejmuje tematykę sterowania bezłożyskowym silnikiem elektrycznym (BEM – ang. Bearingless Elec- tric Motor). W konstrukcji takiego silnika połączono aktywne łożysko magnetyczne (AMB – ang. Active Magnetic Bearing) oraz silnik elektryczny. Konstrukcja taka eliminuje wady klasycznych silników, tj. siły tarcia, nagrzewa- nie, hałas. Parametry systemu sterowania mają wpływ na statyczne i dynamiczne charakterystyki silnika. System sterowania musi zapewniać odpowiednie pasmo przenoszenia, współczynnik tłumienia, oraz musi cechować się po- datnością diagnostyczną i wysoką niezawodnością działania.

W artykule zostanie przedstawiony model matematyczny i symulacyjny BEM wraz z układem sterowania. Pa- rametry systemu sterowania zostaną zaprojektowane z wykorzystaniem środowiska Matlab-Simulink. W artykule przedstawiono charakterystyki czasowe oraz częstotliwościowe silnika bezłożyskowego.

Słowa kluczowe: bezłożyskowy silnik elektryczny, system sterowania

1. INTRODUCTION

In the paper, the control system of the bearingless electric motor (BEM) are presented. This motor com- pound in one construction active magnetic bearing (AMB) and electric motor. AMB technology use attrac- tion force to ensure rotor stable levitation at the work point. The phenomenon of magnetic levitation depends on forces equilibrium between the electromagnetic force Fm and the gravity force Fg (Fig.1.)[1].

The AMB control systems are consisted of electro- mechanical actuator with stator and rotor, displacement sensor, controller and power amplifier. Change of rotor

position is measured by sensor. The position signal ZS

activates a controller to make a output signal. If rotor is above the work point, control system reduces current control signal im on electromagnet. So, electromagnetic force Fm is reduced. However, if the rotor is occurred below the work point, control signal im is increased to generate bigger force Fm. The electromagnetic force Fm is equal to the gravity force Fg, if the rotor is in the work point (central position). So, attraction or pushing away forces compensate changes of rotor position in an air gap.[1]

Fig. 1. The control system of the AMB

In the paper BEM magnetic suspension control sys- tem was analyzed. There are presented time and fre- quency characteristics. There are also implemented the advanced control algorithm to improve static and dy- namic parameters of the system.

2. MATHEMATICAL MODEL OF BEARINGLESS

ELECTRIC MOTOR

Rotor of bearingless electric motor can perform a li- near movement (in air gap along Ox and Oy axis) and the rotational movement (around axis Oz). Fig. 2 presents directions of electromagnetic forces interaction in BEM.

Fig. 2. Directions of electromagnetic forces interaction in BEM Movement equations of electromechanical systems have been derived using Hamilton`s minimal action rule, which is based on energy functional extreme searching by means of variation method.

Based on Lagrange`s equations were derived BEM electric equation (1) and BEM mechanical equation (2).

Moments balance equation is given by (3).

(1)

(2)

!"

! # $ % & $' (3)

where:

ms – coils number in suspension windings;

mm – coils number in motor windings;

R – resistance diagonal square matrix (dimension ms + mm);

M – inductance square matrix (dimension ms + mm);

u(t) – phase voltages column matrix with (ms + mm)-rows;

i(t) – phase current column matrix with (ms + mm)-rows;

ψ – magnetic flux;

Fx,Fy – electromagnetic force generated in Ox and Oy axis;

Mz – electromagnetic moment;

m – rotor mass;

J – inertia moment;

b – friction coefficient;

k – stiffness coefficient;

Fzew x,y, – external disturbance force in Ox and Oy axis;

Mzzew–external disturbance moment;

x – movement in x-axis;

x – acceleration in x-axis;

x – linear velocity in x-axis;

y – movement in y-axis;

y – acceleration in y-axis;

y– linear velocity in y-axis;

ω,– rotor rotary speed.

For two phase motor equation (1), can be derived by

- ./0

./1

.20

.21

3 4

5/0 0 0

0 5/1 0

0 0 520

0 0 0

00 5027

8 4 9/0

9/1

9₂₀ 921

8 4

:/0

:/1

:₂₀ :21

8(4)

where:

uma, umb – voltage of four-pole motor windings;

ima, imb– current of four-pole motor windings;

usa, usb – voltage of two-pole suspension windings;

isa, isb – current of two-pole suspension windings;

Rma, Rmb– resistance of four-pole motor winding;

Rsa, Rsb– resistance of two-pole suspension windings;

:/0, :/1– magnetic flux from motor windingss;

:20, :21– magnetic flux from suspension windingss.

In equation (2) <= >element describes inertial force,

?= >element describes friction forces and @= >element determinates coupling from a displacement occurring in a closed-loop system. Electromagnetic forces occurring on the right side of this equation can be calculated by designating energy derivative (6) with respect to the direction of force (5).

AB

AC -

DE DBDE DC

3 (5)

F

1 2

-

9_<K 9<?

9_LK 9_L?

3

I

°

-

N_<K J_<K<? J_<KLK J<?<K N<? J<?LK

J_LK<K J_LK<? N_LK J_L?<K J_L?<? J_L?LK

J_<KL?

J<?L?

J_LKL?

N_L?

3 -

9_<K 9<?

9_LK 9_L?

3

where:

Lma Lmb– self-inductance of four-pole motor windings;

Lsa Lsb– self-inductance of two-pole suspension windings;

Msasb– mutual inductance between two-pole suspension wind- ings;

Mmamb– mutual inductance between four-pole motor windings;

Mmasa Mmbsa Mmasb Mmbsb– mutual inductance between two-pole suspension windings and four-pole motor windings.

Thus, we obtain in coordinates associated with stator:

AB

AC -

DE DEDB DC

3 J 9^/0 9/1

9/1 >9/0

920

921 (7)

where:

J ^πµ0RlN_8g^<Ns

02 ; g0– nominal air gap;

WX–vacuum permeability;

5–rotor radius;

Y–axial rotor length;

Z/–turns number of motor windings;

Z2–turns number of suspension windings.

3. MATHEMATICAL MODEL OF CONTROL SYSTEM

In Fig. 3 are presented the control loop of a bearingless electric motor. The control system is divided into two parts: one is active magnetic bearing control loop and second is a motor control loop. The active magnetic bearing controller counteract the rotor radial displace- ment. In x- and y-directions the rotor position is de- tected by displacement sensors. The differences between detected displacements and the reference signals x’ and y’ are calculated by AMB controller. Therefore radial suspension forces F’x and F’y in proper direction are correctly determined. The control signal is modulate according to rotor angular position. Then they are inverted from two-phase system to three-phase system (from coordinate system isa, isb to isu, isv, isw).

Fig. 3. The control system of the bearingless motor.[2,5]

Equation (7) can be expressed in coordinate system associated with rotor (equation 8).

AB

AC J[\/]^_`L2 a ^b_c L9d2 a ^b_c

L9d2 a ^b_c >_`L2 a ^b_c e 99^c0c1 (8) As a result of the unbalance force, radial vibrations occur. These are synchronized to the shaft rotation. The frequency of the vibrations increases with angular speed of motor. Current amplifiers have to provide high fre- quency currents to magnetic bearings. The impedance of the magnetic bearing winding is approximated to ωL, where L is winding self-inductance. The electromotive

force (EMF) is proportional to the rotational speed so that the voltage requirement increases. [3]

The DC current is used to excite the motor winding in the primitive bearingless motor. Since this current generate only a static magnetic field, there is no rota- tional torque production. Hence windings acts as a radial magnetic bearing. A basic controller configuration is introduced. It is shown that the system has good characteristics and can operate as 3-phase inverter- driven magnetic bearing. [3]

Let us consider that one of 4-pole winding currents i^ma is set to DC current Im, while imb is zero. The Eq. (7) and (8) will be reduced to form Eq. (9), because ωt = 0.

AB

AC J[\/ 1 0 0 >1 920

921 (9)

In a radial magnetic suspension system the radial force commands of the two perpendicular axes are given by the position controllers (e.g. PID controller). Current commands are generated from the controller so that the actual radial forces precisely follow the force commands.

Suppose that the radial force commands are generated on the x- and y- axes as F’x and F’y forces. The current commands i’sa and i’sb describes equation (10). The derivative of mutual M’ is a constant determined by the machine dimensions and number of windings turns. The excitation DC current Im can be kept constant.

9[20

9[21 f ghij

A[B

>A[C (10)

4. ROBUST CONTROL SYSTEM

Robust control method is one of the adaptive control method. It takes into account the object dynamic prop- erties during operation, the disturbances dynamics, uncertainty of object parameters determining and the acceptable range of signal changes occurring in the system. This method allows to overcome some limita- tions of available technologies, materials, operating conditions, or errors arising from mathematical models of objects.

Fig. 4. Diagram of control system (K(s) - transfer function of controller; G(s) - transfer function of control plant; r - order signal; e - error control signal; u/up- control signal; di - distur- bance of object input signal; do - disturbance of object output

signal; y- output signal; n – noise signal)

The advantage of this method is augmentation of the control quality in the relation to classic methods. The structure of the control in the Linear Fractional Trans- formation (LFT) is determined during the designing of the robust controller. The control structure are consisted

from the control plant and the weight functions limita- tive the value of the respective signals. Before plotting robust control functions it is necessary to build state space of control plant and classic controller based on knowledge of theirs transfer functions. Current con- trolled active magnetic bearing is described in state space as:

= k= l.

m n= o. (11)

where:

x – state vector = =

= ;

x&– derivative of state vector = =

= ; u – input vector;

A – state matrix k 0 1 1 0; B – input matrix l J p \_//<

1/< ; y – output vector y x x C – output matrix n &0 1'; x ; D – direct transmission matrix o &0'.

Weight functions are calculated on of closed system basis proprieties with classic controller (e.g. PD) en- gaged in the main track of control loop (Fig.4). Based on this there are described the open-loop function L(s), sensitivity function S(s), complementary sensitivity function T(s) and control function R(s) (Fig.5). [2]

Fig. 5. Bode characteristic of robust control functions Sensitivity function is a very good indicator of closed-loop system performance. Mentioned functions are given by (12), where indexes mean: i – input signal, o – output signal. There are designed frequency charac- teristics of these robust functions and there are calcu- lated weight functions based on (13) (Fig.6.). So, there is estimated bandwidth of control system, magnitude, sensitivity on external disturbances, etc. [5]

rs L \ Ns tf;rv L \ Nv tf

Is L \ > rs Ns \ Ns tf; Iv L \ > rv Nv \ Nv tf

5s L w \ Ns tf;5v L w \ Nv tf

(12)

The weight functions are chosen in such a way to create the system parameters which are able to fulfill

the assumptions. The weight We(s) is reflected the requirements put on the shape of the closed-loop trans- fer functions, e.g. on the shape of the output sensitivity function. Similarly, the weight Wu(s) is described some restrictions put on the control or actuator signals. The last weight Wy(s) is reflected to the output signal.

Magnitude [dB]

Fig. 5. The magnitude frequency characteristics of the function S(s), R(s), T(s) and the weight functions. [4]

x] L ^z{_2|}^y2|}~{

~{•_{; x€ L _•^{2|•~‚z‚}

‚2|}_~‚; x€ L ^2|

•~ƒ zƒ

•_ƒ2|}_~ƒ;

(13)

where:

Me – the peak of the sensitivity function S(jω) for the frequency ω> ωbe;

ω_be – the bandwidth of sensitivity function S(jω) for which S(jω) = 1;

„] – the peak of the sensitivity function S(jω) for the frequency ω< ωbe;

Mu – the peak of the control function R(jω) for the frequency ω< ωbu;

ω_bu – the bandwidth of control function R(jω) for which R(jω) = 1;

„^u– the peak of the control function R(jω) for the frequency ω > ωbu;

M^y – the peak of the complementary sensitivity function T(jω) for the frequency ω> ωby;

ω_by – the bandwidth of complementary sensi- tivity function T(jω) for which T(jω) = 1;

„^y – the peak of the complementary sensitivity function T(jω) for the frequency ω < ωby. The next step of the designing control system with robust controller is determine the value of the aug- mented control plant P(s), which is described by (14).

The parameters of the plant are based on the analysis of the frequency characteristics of the function L(s), S(s), T(s), R(s) and their weights. This is named the mixed sensitivity analysis. [2]

… L ^k lf lc

nf off ofc

nc ocf occ

e (14)

Fig. 7. Control system model for the robust control realization Control system model adopted for robust control method implementation are presented in fig. 7.

The robust controllers are determined by three dif- ferent optimization methods: optimal H2 control – min||T(s)||2, standard H∞ control – min||T(s)||∞, op- timal H∞ control – min||T(s)||2≤1. The robust controller design problem can be formulated as follows: “Given a state space realization of a plant P(s) (14) find a stabilizing feedback control law u2=K(s) y2 (Fig.6) such that the norm of closed-loop transfer function matrix (15) is small”[5].

ICf€f …ff L …fc L \ > w L …cc L ^tfw L …cf L (15) Equations describing the designed controllers are given below:

†v‡ L

fˆ,‰Š·fX^yŒ2^•|Žff,‰f·fX^yŒ2^•|•‰‰,•‘·fX^yŒ2^’|‰‰‰,ŠŠ·fX^yŒ2|ˆf,“•·fX^yŒ X,c·fX^yŒ2^”|fˆ,ˆ•·fX^yŒ2^•|Žff,“ˆ·fX^yŒ2^•|•‰‰,ŽŠ·fX^yŒ2^’|‰‰‰,‘‰·fX^yŒ2|ˆf,“‘·fX^yŒ

(16)

†s•–L

f,ˆ“·fX^”2^”|X,“•·fX^yŒ2^•|fŠ,‰“·fX^yŒ2^•|‘Ž,ŽX·fX^yŒ2^’|cc,‘X·fX^yŒ2|‘,Š•·fX^yŒ X,f2^—|X,•Š·fX^”2^˜|X,XXŽ·fX^yŒ2^”|X,•Š·fX^yŒ2^•|fŠ,Šc·fX^yŒ2^•|‘Ž,‘‘·fX^yŒ2^’|cc,cŽ·fX^yŒ2|‘,Š“·fX^yŒ

(17)

†s•–L

f,ˆ·fX^”2^”|X,“Š·fX^yŒ2^•|f‰,ˆ•·fX^yŒ2^•|‘‘,fˆ·fX^yŒ2^’|cf,‰f·fX^yŒ2|‘,‰‰·fX^yŒ X,f2^—|X,•Š·fX^”2^˜|X,XXŽ·fX^yŒ2^”|X,•Ž·fX^yŒ2^•|fŠ,XŽ·fX^yŒ2^•|‘‘,fc·fX^yŒ2^’|cf,Ž‰·fX^yŒ2|‘,‰Ž·fX^yŒ

(18) Considered transfer function models of the robust controllers have much larger orders than the motor model order. These controllers can be difficult to labora- tory implementation or completely unfulfillable. For that reason, order reduction is required.

There are three methods to obtain a reduced order stable model:

• truncation method;

• residuum method;

• Hankel`s norm method.

Fig. 8. Designed system with robust controller frequency Bode

characteristic

Amplitude

Fig. 9. Designed robust controller system step response

5. SUMMARY

For many years in Military University of Technology are performed the investigation of magnetic bearings. In the paper are presented the results of the bearingless motor simulation analysis.

Mathematical and simulation bearingless motor model was derived, as well as robust control method designing was presented. Time and frequency characte- ristics of advanced control system was shown, too.

Positive effect of research has opened new perspec- tives for the development work associated with the construction of this type electrical machines.

The work was financed in part from the government support of scientific research for years 2009 ÷ 2012, under grant No. O N509 032736.

References

1. Henzel M., P. Mazurek P.: The analysis of the control system of the active magnetic bearing. In: “Electrody- namic and mechatronic systems”, 3rd International Students Conference on Electrodynamics and Mechatronics (SCE III). Opole 2011.IEEE Catalog number: CFP1174M-PRT, p. 53 – 58.

2. Henzel M.: The robust controller for the bearingless electric motor with permanent magnets”. In: The 7th Inter- national Conference "Mechatronic Systems and Materials", Kowno, Lithuania (abstract) and CD version) 2011, p. 266 – 267.

3. Chiba A., Fukao T., Ichikawa O., Oschima, M. Takemoto M., and Dorrel D.: Magnetic bearings and bearingless drives. Elsevier’s Science Technology Rights Department in Oxford, UK, 2005, p. 127 – 158, 201 – 219.

4. Zhou K., Doyle J.C.: Essentials of robust control”. Prentice Hall, Inc., 1998.

5. Henzel M., Falkowski K., Żokowski M.: The analysis of the control system for the bearingless induction electric motor. “Journal of Vibroengineering” 2012. Vol. 14, Iss. 1, p. 16 – 21.