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196 IEEE TRANSACTIONS ON EDUCATION, VOL. E-29, NO. 4, NOVEMBER 1986

Design of a Magnetic Levitation Control System AnUndergraduate Project

T. H. WONG

Abstract-A magnetic levitation control system is built as a class- controller amplifler electromagnetroom demonstration device for systems and control courses. System operating Rlinearization and phase-lead compensation techniques are used to con- pointtrol the unstable nonlinear system. Col

INTRODUCTIONpho to resils tor lih sorcA CLASSROOM demonstration device is always very ligght source

helpful in teaching engineering courses particularlyfor automatic control. The magnetic levitation control Fig. 1. Magnetic levitation control system.system is considered an interesting and impressive devicefor this purpose. In addition, the system is so simple and d2xsmall that is very convenient to be carried from class to m 2 = mg - f (3)class.

There are, generally, two approaches for the design of wheremagnetic levitation. One way is by using the eddy currentmagnetic repulsive force [1]. Another way is by using f electromagnetic forceelectromagnetic attractive force [2]-[4]. This design proj- i coil currentect is based on the second method because it is more ef- x distance between electromagnet and ball bearingficient than the first method in energy consumption [4]. C constantThe magnetic levitation system is an unstable nonlinear e voltage across the coilsystem. The following is a detailed description of how to R coil resistancedesign the controller to stabilize the system. L coil inductance

SYSTEm ANALYSIS m mass of ball bearingg gravitational acceleration.In Fig. 1, a ball bearing of mass m is placed underneath

the electromagnet at distance x. The current flowing into The system dynamic equations are nonlinear. Before wethe electromagnetic coil will generate electromagnetic can apply the linear control theoty, the system dynamicforce to attract the ball bearing. The net force between the equations are linearized at an operating point. The linear-electromagnetic force and gravitational force will induce ized equations describing the variations from the operat-an up or down motion of the ball bearing. The photores- ing point are obtained by using only the linear terms fromistor senses the variation of the position of the ball bearing the Taylor series expansion. If the variables of the oper-by the amount of shadow casted on its surface and feeds ating point are expressed with subscript "0" and the var-back this signal to the control circuit and amplifier to iables at the neighborhood of the operating point are rep-regulate the input current i. The ball bearing is kept in a resented with subscript "1," then linearized equations aredynamic balance around its equilibrium point. 200 2Ci2The system's dynamic equations can be obtained as (see fi= 2 i1 3-2-I (4)

Appendix) x0 x0i 2 ~~~~~~~~~~~~~~~~~~di,

f () M(1) el= Ri1 + L d- (5)di d 2x,

e=Ri+L (2) mdF2L = f (6)

Manuscript received July 11, 1985; revised December 19, 1985. whereThe author is with the Department of Mechanical Engineering, Tulane li 2

University, New Orleans, LA 70118. fo = mg = C lIEEE Log Number 8610515. \Xo/

0018-9359/86/1100-0196$01.00 1986 IEEE

WONG: MAGNETIC LEVITATION CONTROL SYSTEM 197

disturbance 70E, electromagnet

controller __________bal l-bearingl_r+ IE(S)l T(S) IF (S) X1(S)1I 7m

90 /00

8

Fig. 2. Block diagram.Fig. 3. System's dimensions (mm).

Laplace transformation of (4)-(6) yields ImF1(S) = kLI,(S) -- Xl(S)] (7)E1(S) = (R + LS)11(S) (8)

mS2X1(S) = -F1(S) (9)where --

-58 -49.5 49.5 Re

k = 2C .x2o

The block diagram of the magnetic levitation system isshown in Fig. 2. The characteristic equation of the controlsystem can be obtained. Fig. 4. System root loci (constant amplifier gain).

Q(S) = xoLmS3 + xoRmS2 -kioLS -kioR + Gc(S)kXoB = 0. (10) TABLE I

From (10), we know that the system is unstable without xO 0.008 ma proper controller. m 0.068 kg

R 28 QL 0.483 H

COMPENSATION TECHNIQUE iO 0.76 AC 7.39 x i0-5 N .m2/A2In Fig. 3, the electromagnet is made of a 3600-turn coil k 1.756 N/A

of gage 22 insulation wire closely wound around a low B 1.14 x 103 V/mcarbon steel cylinder of diameter d = 25 mm and length1 = 100 mm. The other measured parameters are listed inTable I. First, considering Gc(S) is a proportional controller (i.e.,

Equation (10) can be rearranged as constant amplifier gain), the root loci of the system isQ(S) = 1 +GC(S)kXOB 0 shown in Fig. 4.

V(S) = I + LGS3c+R S2 kxB LS k Fig. 4 indicates that system can never be stabilized byxOLMS3 xORmS2 - kioLS - kioR simply adjusting the amplifier gain only. It is required to(11) shift the loci to the left of the S plane so that the system

can be stabilized by selecting the proper amplifier gain.and Since a lead network can serve this purpose, a phase-lead

kB controller is considered for Gc(s). The technique of theGC(S) ML phase-lead compensation design in this case is to place

Q(S) = 1 + =0. the zero of the compensator in between 0 and -49.5, andki0 Fki0 close to -49.5, on the, real axis of the S plane, while themx0 L> S,

198 IEEE TRANSACTIONS ON EDUCATION, VOL. E-29, NO. 4, NOVEMBER 1986

Im

K' - 4.702 x 106s2 /

K'= 3.05 x 10 /(K = 50)

{ t 72g)j;tZ ~~~~~K'=1.421 x 106-400 -58 40 S1 49.5 Re

-4.5\

n 100

Fig. 5. System root loci (with phase-lead compensation).

+42 V

110 1000II h

-42 v_outpu)lt disturbance

detection input

Fig. 6. Electronic circuit of the magnetic levitation system.

K' (S + 40) /-'9015Q(S) = 1 + (S + 49.5)(S - 49.5)(S + 58)(S + 400) 33t

(15)~ ~~~~0

K' = 60990 K, K: amplifier gain. 233< 3 3 1 ]0The root locus of (15) for K' changes from O OD is

shown in Fig. 5. In Fig. 5, for a stable system the mini-mum system gain is K' = 1.421 x 106 or K = 23 at point -40 20AFSi and the maximum system gain is K' = 4.702 x 106 or _l ______ 0 |K = 77 at point S2 We select the medium amplifier gainof K = 50. The control circuit for the magnetic levitation Fig. 7. Design layout (mm).system is shown in Fig. 6.

a feedback signal to control the position of the ball bear-SYSTEM DESIGN AND EXPERIMENTS ing. Since the photoresistor is very sensitive to the inten-

The layout of the control demonstration device is shown sity of light on its surface, a black tube is used to coverin Fig. 7. The deviation from the equilibrium position of the photoresistor and to block the disturbance effect fromthe ball bearing casts a shadow on the photoresistor which outside light sources. In designing the system, the leadwill generate the varying voltage corresponding to the compensator is achieved by using a simple RC circuit aschanging position. Then, the changing voltage is used as in Fig. 8.

WONG: MAGNETIC LEVITATION CONTROL SYSTEM 199

c frequency response. The demonstration device wasbrought to the class twice for the course. First, a dem-

+ + onstration was conducted at the beginning of the semesterFrom R2 To to explain the concept of feedback control systems. Stu-

phtresistor Vl 2RV amplifierphotorRt2 2 inpVt dents reacted with a great deal of curiosity and interest.0 O The second demonstration was given after the students

Fig. 8. Lead compensator. had learned the root locus compensation technique. In thisdemonstration, the complete magnetic levitation controlsystem from system modeling, analysis, and synthesis wasused as a live control system design example. The stu-dents' responses were enthusiastic. Many questions wereasked about the technical details in the design of the con-trol system.

In demonstrations, in order to enhance the effect, afunction generator was used to supply a sinusoidal wave-forn of different frequencies at the disturbance input ofthe circuit. Then, the ball bearing would oscillate at thefrequencies accordingly.

Students' reaction to this course is very encouraging;that can be observed from 58 percent of senior studentschoosing this course as their one of the two career elec-

Fig.9.Actual levitated system. tives out of six courses offered by the department this se-Fig. 9.Actual levitated system.metr mester.x

(mm) CONCLUSION

8.5 The magnetic levitation control system has proven suc-cessful for classroom demonstration of feedback control.

7.0 l/_,_,_,_,_,_ ._It is very effective in teaching a control course using this0.08 0.16 Tlme (Second) demonstration device as a live control system design ex-

ample. It is also possible to include this device as part ofa control lab for compensator design with a different op-erating point or a different size of the ball bearing.

Fig. 10. System step response. This paper presents the design details on how to use thelinear control theory to control a nonlinear unstable mag-

The most difficult part for this design work is to select netic levitation system.RI and R2 such that the steady state value of V2 has theproper bias voltage to the power transistor which, in turn, APPENDIXwill supply the current for the coil to generate the electro- DERIVATION OF SYSTEM EQUATIONSmagnetic force approximately equivalent to the weight ofthe ball bearing at the operating point. Several experi- The magnetic forcefexerted on the ball bearing is givenmental trials are necessary. by the well known equation [21

Fig. 9 is the picture of display of the controlled system. '2 dLIn Fig. 10, the system's dynamic time response is ob- f= --- (Al)tained from HP 7015B X-Y recorder by applying a step 2 dxdisturbance to the system. If the disturbance exceeds the The inductance L has its largest value when the ball bear-linearized range, the system will become unstable and loss ing is next to the coil and decreases to a constant value LIof control. In the lab experiments, we found that the limit as the ball bearing is removed to x = oo. For the presentof allowable disturbance to the system is about + 1.0 mm purpose we assume that this dependence isaway from xo. Loxo

DEMONSTRATIONS L L + x (A2)The magnetic levitation system is intended to be used where Lo is a constant and xo is the operating levitation

as a classroom demonstration device for my control and gap. Substituting (A2) into (Al) we getautomation course which also includes six control labs 2 2consisting of instrumentations, dc-motor system parame- f = 2 LoXo (' = C . (A3)ter identification, velocity control, position control, and 2 0 (D

200 IEEE TRANSACTIONS ON EDUCATION, VOL. E-29, NO. 4. NOVEMBER 1986

The differential equation of the circuit is [61 L. 0. Kehinde, "Analysis of limit cycle oscillations in a magneticsuspension system using the describing function method," Int. J. Eng.

d(Li) Sci., vol. 22, no. 4, pp. 419-437, 1984.e = Ri + (A4) 17] J. J. D'azzo and C. H. Houpis. Linear Control System Analysis anddt Design, Conventional and Modern. New York: McGraw-Hill, 1975.

Since x i always ept aroud the cloe neighbrhood of [8] H. H. Woodson and J. R. Melcher, Electromechanical Dvnamnics:Since x is always kept around the close neighborhood of Part I Discrete Systems. New York: Wiley. 1968.xo, L can be considered as a constant that will greatly sim- 19] G. H. Hostetter, C. J. Savant. Jr., and R. T. Stefani, Design ofFeed-plify the system model without losing much accuracy. back Control Systems. Ncw York: Holt, Rinehart and Winston,Therefore, (A4) can be written as 1982.110] R. H. Cannon, Dynamics ofPhysical Systems. New York: McGraw-

di Hill 1967.e =Ri+L-. (AS)dt

The equation of motion of the mechanical system isd 2x

m 2=mg-J (A6)The simplified system dynamic equations are (A3), (A5),and (A6).

REFERENCES _ _ T. H. Wong received the B.S.M.E. degree fromREFERENCES Tatung Institute of Technology, Taiwan, in 1967,

11] E. R. Laithwaite, 'Electromagnetic levitation," Proc. IEE, vol. 112, the S.M. degree from the Massachusetts Instituteno. 12, pp. 2361-2375, 1965. of Technology, Cambridge, in 1974, and the

[2] B. V. Jayawant and D. P. Rea, "New electromagnetic suspension and Ph.D. degree from the State University of Newits stabilisation," Proc. IEE, vol. 115, no. 4, pp. 549-554, 1968. York at Buffalo, in 1983.

13] B. V. Jayawant, P. K. Sinha, A. R. Wheeler, R. J. Whorlow, and J. From 1967 to 1968 hc served in the ChineseWillsher, "Developement of I-ton magnetically suspended vehicle army as an R.O.T.C. Officer. From 1968 to 1972using controlled dc electromagnets," Proc. IEE, vol. 123, no. 9, pp. he was with Toung-Yuan Electrical Company and941-948,e1976. m Tatung Company, Taiwan. He joined the Me-

[4] F. Matsumura and S. Yamada, 'A control method of suspension con- chanical Engineering Faculty of Tatung Institutetrol system by magnetic attractive force," Trans. I.E.E.J., (Japa- of Technology Taiwan from 1974 to 1980, and taught courses primarily innese), vol. 94-B, no. 11, pp. 33-40, 1974. the systems and controls areas. In 1983 he became an Assistant Professor

15] R. Frazier, P. Gillinson, and G. Oberback, Magnetic and Electrical at Tulane University, New Orleans, LA. His research interests include sys-Suspension. Cambridge, MA: MIT Press, 1974. tem dynamics and control theory.