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A ' Microprocessor-Based Lookup Bearing Table NASA Technical Paper 1838 A'Microprocessor-Based Table Lookup Approach . for Magnetic Bearing Linearization . -. Nelson J. Groom and, James B. Miller .- MAY 1981 NASA .. TECH LIBRARY KAFB, NM NASA Technical Paper 1838 A Microprocessor-BasedTable Lookup Approach for. Magnetic BearingLinearization Nelson J. Groom and James B. Miller LarrgleyResearch Ceuter Hatnpton,Virgilria National Aeronautics and Space Administration Scientific and Technical Information Branch 1981 SUMMARY An approach forproduci ng a linear transfer chlar acteristic between force command and force output of a magnetic bearing actuator without flux biasing is pres.ented. The approach is mlcroprocessor based and uses a table 'lookup to generatedrive signals for the magnetiCbearing power driver. An experimental test setup used to demonstrate the feasibility of the approach is described, and testresults are presented. The testsetup contains bearing elements simi- lar to those used in a laboratory model annular momentum control device (AMCD) . INTRODUCTION ,. This paper describes an approach for producing a lineartransfer charac- teristic between the force command and force output of a magneticbearing actuator. The approach, which is microprocessor based and uses a table lookup to generate drive signals for the magneticbearing actuator power driver, was investigated for application to a laboratory model annular momentum control device (AKD) . The laboratory model (described in ref. 1 ) was built to inves- tigate potential problem areas in implementing the AMCD concept and is being used aspart ofan AKD hardware technology development program. The basic AMCD concept is that of a rotating annular rim,suspended by a minimum of three magneticbearing suspension stations and driven by a noncontacting elec- tromagneticspin motor. A detaileddiscussion of therationale for the AMCD configuration andof some of its potential applications is presented.in reference 2. As described in reference 1 , themagnetic-bearing linearization technique used in the original laboratory model AMCD magnetic.suspension was to differen- tially control sets of magneticbearing elements about a permanent-magnet bias flux. Preliminarytests indicated that this approach (permanent-magnet flux biasing)presented a problem from a control system standpoint(ref. 3). For a given equivalent permanent-magnet stiffness, a minimum-bearing servo band- width is requiredfor stability. The existence of structural modes in thearea of the laboratory model bearing servo crossover restricted the amountof bear- ingservo damping that could be achieved. Because of the limitations of permanent-magnet fluxbiasing encountered with the laboratory modelAKD, a decision wasmade to explore an alternate approach tothe design of the magneticsuspension system (ref. 4). As a result, a new magnetic suspension system for the laboratory model has been designed, fabricated, and tested(ref. 5). Thenew system uses a zerobias flux approach forthe magneticbearing actuators. Analog multiplier and squareroot modules produce a direct solution to the ideal magnetic actuator force equation to provide a linearforce-current characteristic. (For furtherdiscussion of mag- neticbearing actuator control approaches, see ref. 6.) The accuracy of this approach is limited by theaccuracy with which theideal force equation approxi- mates the actual characteristics of the actuator and by theaccuracy of the analog components. This paper presents a zerobias flux linearization approach which is digital and which uses a lookup table constructed from measured actua- torcharacteristics. An experimental test setup that was used to develop this approach is described, and test resultsare presented. s YMBOLS Dimensional quantities are presented in both SI Units and U.S. Customary Units. Measurementswere made in U.S. Customary Units. force produced by bottanelectromagnet force command for magneticbearing actuator value of force command associated with mth line segment force produced by topelectromagnet microcanputersystem sample rate = Fc(m+l ) - Fc(m) for all m, where m < N = Fc - Fc (m) , where Fc (m) 6 Fc < Fc (m+l ) displacement of suspended element with respect to centered position in magnetic bearing actuator gaps gap ofbottom electromagnet magneticbearing actuator gap with suspendedelement centered gapof top element current in bottom electromagnet current command for magneticbearing actuator stored value of current command associated with mth line segment current in top electromagnet electromagnetconstant number of line segments slope of mth line segment from (Fc (m) ,Ic (m) ) to (Fc (m+l ) ,Ic (m+l ) ) Abbreviations: AD analog-to-digitalconverter 2 AMCDannular momentum controldevice D/A digital-to-analogconverter dc directcurrent dc em f electromotiveemf force APPROACH Magnetic Bearing Control Approach The magnetic bearing control approachis one which uses zero bias flux. Figure 1, a schematic representation of a magnetic bearing element ispair, Suspended element Figure 1.- Magnetic bearing actuator. presented in order to describe this approach. Included in the figureare top and bottom electromagnets, with currents IT and IB, respectively; a portion of the suspended element whichis centered between the electromagnets; and a position sensor which measures the displacementG of the suspended element with respect to the centered positionGo. In the zero bias flux control approach, one electromagnet at a time is controlled; top the electromagnet is controlled for an upward force and the bottom electromagnetis controlled for 3 a downward force (sinceeach electromagnet can physically produceonly a uni- directional force). Figure 1 indicates that if up is taken as the positive direction, the electromagnet gapsare and Gg=Go+G (2) Assuming negligible fringing and ignoring nonlinearcore effects, the force pro- duced by each electromagnet is given by (ref.4) and The composite force-current characteristic of a zero bias flux actuator with the suspended element centeredis shown in figure2. Y Uppermagnet charactsristic * L'T Current Figure 2.- Composite force-current characteristic of a zero bias flux magnetic actuator. 4 . , Linearization.. Approach .. .. The linearization approach is microprocessor based anduses: a lookup table. The most straightforwardapproach for producing a linear transfer characteristic betweenforce command and force output wouldbe to solveequations (3) and (4) for theelectromagnet current required to producethe desired force. That is, and where Fc is the commanded force.Implementing the solutions of equations (5) and (6) digitallycould produce a more camputationallyaccurate result than the analogsolution but would not account for deviationof the actual bearingsfrom the ideal model. Thissolution would also require thatconsiderable computing power be dedicated to a relatively minorportion of the total controlsystem. The tablelookup approach, which employs a one-to-onecorrespondence between force command Fc andoutput current command IC permits theoutput/ inputfunction to conform,within the limits ofquantization error, to the actual bearingcharacteristics. However, this formof table requires con- siderable memory space.The minimummemory requirement is obtained by select- ing the minimum set of straightline segments which approximate the output/ inputrelationship within the desired tolerance. The major disadvantageof thischoice of line segments is thatconsiderable time may be spentsearching for the appropriate line segment. Since in this application computational time is more critical than memory size,an approximation was selected whichrequires more memory but eliminates thesearch time. The minimum set of equallyspaced line segments N are selec- ted so that N = 2", andthe N + 1 pointsdefining the line segments are sepa- rated by equal force command steps AF~. If therange of Fc is scaled to a B bit binary word (B > n) , the n most significant bits of, Fc uniquely identifythe N linesegments. In actual computation these bits identify'the 5 lookup table data associated with themth force commanddata point Fc(m) such that F, (m) 2 F, Fc (m+l ) . The B - n least significant bits of F, repre- sent the difference 6Fc between F, and F, (m) . (See fig. 3. ) line seqment m I Figure 3.- Table lookup approach. The data contained in the lookup table consistof the slope SLOPE(m) of each line segment m between (Fc(m) ,Ic (m)) and (Fc (m+l) ,Ic (m+l) ) and of the current command Ic(m) corresponding to F,(m). The location of the appropri- ate data is obtained directly from the force command word by masking and shift- ing. Calculation of output current command IC at G = 0 requires a single multiplication of 6Fc by SLOPE(m) and addition of this result to Ic(m). Since actual variation of electromagnet current gapwith was observed to be linear, as indicated by equations (5) and (6), calculation of IC for other values of G is accomplished by multiplying IC by the appropriate bearing gap. For a more detailed descriptionof the table lookup algorithm, see appendix A. 6 HARDWARe TESTS Description of Test System The test system consisted of a magnetic bearing test fixture connected to a microcomputersystem as shown infigure 4. Thissystem was used to obtain the data required to develop the lookup table and to obtain data on the perfor- mance ofthe proposed approach. A descriptionof the current driver shown in figure 4 is givenin appendix B. Figure 4.- Magneticbearing test system. Magneticbearing test fixture.- The magneticbearing test fixture shown in figure 5 consistsof a magneticbearing
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