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MI1-HDBK-218 1 June 1952 DEPARTMENT OF THE ARMY TM 700-5990-1
MILITARY STANDARDIZATION HANDBOOK APPLICATIONS OF ELECTRICAL RESOLVERS
m Downloaded from http://www.everyspec.com HEADQUARTERS, DEPARTMENT OF THE ARMY WAS1llNGTOS.25,D,C.,17 ~U@ 1.96%’ T.M 700-5990-1ispublishedfortl]eme of Illconcerned. IIYORDEROF THE SECRET+W OF ml~ AImY:
“G.H. DECKER, (+xwa[,UnitedStatexArmy, WJici,l]:~ Clli.fof.Staff. .J.C. Ii.iMBERT, ., Major (%wnd, UnitedStatesArmy, TILe.4djutmtGeneral,
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M1’L-ROBK-218 ● 1 June 1962
DEFENSE SUPPLY AGENCY WASRTNGT2N 25, D.C.
I MIL-RDBK Appliostionsof Electrical Reeolvers
1. This handbook hae been approved by the Department of Defense for uee by the Departmentsof the Arqy, the Navy, and the Air Force. I 2. Th accordancewith establishedprocedure,the Standardization Division has designatedOrdnance Corps, &reau of Naval Weapons, and Air Reeearch and Development Comnand, respectively,ae Aq-Navy-Air Force CuL+tOdimSof this handbook.
3. This handbook ie intended ae a guide to promote use of standards end etendardpractioes in the applicationof electrical resolversby the Departments of the Army, the Navy, end the Air Force.
b. Recommendedcorrections,additions,or deletioneshould be addressed to the StandardizationDivision, Defense Supply Agency, Washington 25, D.C.
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MCL-WDBK-218 ,0 1 June 1962 I
FOREWORD
,- This hemdbook is intended to help implementthe Department of Eefense standardizationprogrsm as it affects precision electricalreaolwn%. Re- solvers, their designationsewi terminologies,are defined in accordance ,. with militery specificati0n9. With eventual complianceby manufacturersto the proposed specifications,this handbook should serve as a guide to the standardizationof resolver applicationsand practices;to this end, the I fundamentalsand tlheoryof resolvers are briefly reviewed,providing engi- neers end designerswith a convenientreferenceto the mltiple capabilities of the resolver.
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MIL-RDBK-218 ● 1 June 1962
i CONTKNTS
1 DESCRIPTION 1 1.1 SCOFR OF RANDBOOK 1
1.2 DEFINITIONOF A PJ+SOLVER 1
1.3 CONSTRUCTIONOF A RESOLVKR 1 1.3.1 General. 1 1.3.2 Housing 1 1.3.3 Stator 1 1.3.4 Rotor 1.3.4.1 Ball Bearings 1.3.4.2 Lubricants 1.3.5 Distributionof Windings 1.3.6 CompensatorComponents
2 FUNDAMENTALS 5 2.1 GENERAL 5
2.2 RESOLVER EMPI.OYM?JIT 5 ,,
2.3 RESOLVER ZERO 7 2.3,1 Rotor-Exeited Resolvers 7 2.3.2 Stator-ExcitedResolvers 7
2.4 BASIC OPERATION 8 2.4.1 Two-WindingResolver 8 2.4.2 Three-WindingResolver 9 2.4.3 Four-WindingResolver 9
THEORY OF OFERATION 11 GENXRAL 1-1.
3.2 RESOLVER EQUIVALENT CIRCUIT 11 I FRINCIPALTYPES OF INFJTS 12 3:.1 Resolver Used for IiataTransmission 12 3.3.2 Resolver Used as a ComputingDevice 12 3.3.2.1 Sine-WaveVoltages in Phase 12 3.3.2.2 Sine-WaveVoltages Shifted 90 Degrees 12 3.3.2.3 Square-WaveVoltages 12 3.3.2.4 Sawtooth Voltages 12
3.4 STA’IX)RAND ROIOR R2SISTANCFS 12
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RESOLVER ERROR 13 ;:; .1 Function Error 13 3.5.2 InteraxisError 14 3.5.3 ProportionalVoltage Error 14
3.6 ROTOR ELECTRICALBAL4NCE 14
3.7 WINDING COM.FENSATION 14 3.7.1 Schematic Circuit 14 3.7.2 Equivalent Circuit 15 3.7.3 Phase Relations 15
CONSIDERATIONSOF F&50LVSR LoADING 17 %1 Resistive emd CapacitiveLoading 17
3.9 EFFECT OF TEMPERATORE 17
3.10 WINDIIKISWITROUT COMPENSATION 17 3.10.1 Phase Shift Iue to TemperatureVariations 17 3.10.2 TransformationRatio Xe to TemperatureVariations 17 b 3.11 TRANSFORMATIONRATIO 18 ● 3.12 TEMPERATURECOMPENSATION 18 3.12.1 Current Feedback 18 3.12.2 Effect of Loading 19
3.13 FREQUENCY RESPONSE 19 3.13.1 Stator-to-RotorResponse lg 3.13.2 Stator-to-CompensatorRes~nse 20 3.13.3 Carrier OperationsWith Winding Compensation 20 3.13.k Carrier Operatfons Without Winding Compensation 20
3.14 NULL VOLTAGE 20
3.15 HARMONIC DI.W)RTION 21
3.16 EFFECTS OF DIRECT CURRENT 21
3.17 !&?i?aTINGPROCEWRES 21 3.17.1 General 21 3.17.1.1 Mechanical.Positioningof Rotor a 3.17.1.2 Test Circuit 21 3.17.2 Tmnsform.atlonRatio 21 3,17.3 Ehction Error ‘ 2’2 3.17.4 Null Voltage 22 3.17.5 Phase Shift 22 3.~7.6 Frequency Response 22
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MIL-HDSK-218 1 June 1962
4 APPLICATIONS 24 4.1 GENZRAL 24
4.2 TYPICAL SYSTE~ 24 4.2.1 Automatic Control Syaterns 24 4,2.1.1 She Computation 24 4,2.1.2 Conversion0? Polar Coordinates 25 4.2.1.3 Vector Addition 25 ,. 4.2.1.4 Conversionof Rectwgul.arCoordinates 25 4.2.2 Analog Computers 25 4,2.2.1 ReciprocalOperation 27 4.2.2.2 Rotation of RectangularCoordinates 2a 4,2.3 Data Transmission 29 4.2.4 Phase Shifters 4.2.5 Radar Sweep Resolution :
5 TYPICAL SYNCHRO RESOLVERS 33 5.1 GEiW+AL 33
5.2 RESOLVERWITH ‘IWOEXCITED STATOR WINDINGS CONNEC’IRD 33 IN SERIES AND TWO RCTOR WINDINGS 5.2.1 Function 33 “o 5.2.2 Systems in Which Employed 33 5.3 RESOLVER WITH ‘IWOEXCITED ROTOR WINDINGS CONNECTED .33 IN SERIES AND TWO STATOR WINDINGS 5.3.1 Iluu?cion 35 5.3.2 Systems in Which Employed 35
5.4 RXSOLVER WITH ONE EXCITED STATOR WINDING, OM SHORTED 36 STATOR WINDING, A COMPENSATINGRESISTOR,AND TWO ROTOR WINDING-S 5.4.1 Fnnction 36 5.4.2 Systems in Which Employed 36 1. ?, 5.5 RESOL~ WITH TWO EXCITED S!l!ATORWINIXINGSAND ‘IWOROTOR WINDINGS d 5.5.1 FLlnction 1- 5.5.2 Systems in Which Employed 36 I 37 5.6 ~, RFSOLVER WITH TWO EXCITED S’IA!TORWIl,YDINr~, TWO COMPENSATINGWINDINGS, AND ‘IWOROTO,RW“~~l~ 37 1. 5.6.1 Punction ~ 5.6.2 Systems in Which Employed :; 0,. Downloaded from http://www.everyspec.com
MIL-RDBK-218 1 June 1962
5.7 RESOLVER WITH TWO EXCITED STATOR WIWDINGS, TWO COMPENSATINGRESISTOFS, AND ‘IWOROTOR WINDINGS 5.7..1 Function 5.7.2 Systems in Which Employed
6 PERFOF.MAWZSPECIFICATIONS 39 - 6.1 GEN-SBAL 39
6.2 CLASSIFICATION 6.2.1 Size ;; “ 6.2.2 Function 39 6.2.3 Input Impedance 39 6.2.4 me of Compensation 39 6.2.5 ExcitationFrequency 39 6.2.6 Modification 39 6.,2.7 Other Designation 39
6.3 MODE OF EXCITATION 39
6.4 DIRECTION OF ROTATION 4C 6.5 INFUT VOLTAGE 40 ● 6.6 INFIJICURRENT 40
6,7 INPOT KIWER
6.8 D-C RESISTANCE
6.9 TBANBFORM4TIONRATIO 6.9.1 Equality of the TransformationRatio
6.10 NOMINAL PRPJ3ES~ 6.10.1 Phase Shift Variation Caused By Rotation or Voltage Variation
6.11 INTERAXE ERROR 41
6.12 FUNDAMENTALNOLL VOLTAGE 41 “
6.I.3 TOTAL NULL VOLTAGE 41
6.14 FUNCTION ERROR 41
6.15 VOLTAGE ZERO SIUFT ,NULLSPACING 41
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Ml_L-HGRK-218 1 June 1962
6.16 FREQOENCYZERO ssmr OF NULL SPACING 42
6.17 FRICTION TORQUE 42
6.18 BRUSH CONTACT RESISTANCE 42 1. . 6.~9 TEMFERATURKRISE 42
6.20 42 ~: ‘ 6.21 WEIGRT 42
6.22 INSULATIONRESISTANCE 43 ~. 7 INSTALLATION 44 ?.1 GENERAL 44
7.2 PRECAOT!IOM 44
7.3 DIRE(?TMOUNTING PROM TERMINW BOARD END 4J+ ● 7.4 MOUNTING FROM SHAFT END, USING ADAPIER ASSEMSLY 44 7.5 MOIJiTCINGFROM TERMINAL ROARD END, USING ADAPI!ERASSEMSLY 46
7.6 MOUNTING FROM SRAFT END, USING AIMUBTA.BLECIAMPING DISK ASSEMBLY 46
7.7 MOUNTING FROM TEPJICNAT-BOARD END, USING ZEROING RING 47 t 7.8 MOUMTNG A GEAR ON A R2i30LvER5HAFT 47
7.9 ELWTRICAL CONNECWI.ON, 49
1 ~ 8 PREFERRED OR STANTMRD HARI?JAFJ? 50 8.1 GENERAL 50
8.2 CI&@HTG ASSEMSLY w ,: “ 8.3 CIAM?ING DISK ASSEMSLY 50
8.4 GEARED-mR ADAPTER ,- l:’ 8.5 ZEROING RIIW2
8.6 PINION WRENC~ 51 1- e,. ix
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MIL-RDBK-218 1 June 1962
FIGURES
Figure Page
1 1 ElectricalResolver,Assembled 2 1= 2 Major Componentsof an Electrical Resolver 3 Stator and Rotor Laminations 3 ; StandardizedFour-WindingResolver, Schematic Diagram 5 Vector Diagrsmm for Stsndardized Resolvers 1’ 2 Two-WindingResolver, Schematic Diagrzm ‘ : 7 Three-WindingResolver, Schematic Diagram 9 8 Four-WindingResolver, Schematic Diagram 10 9 Standard Resolver Equivalent Circuit 11 Lo 1 Typical Angular Error in Resolver Shaft Rotation 13 l-l Resolver with Winding-CompensatedStator, 15 Schematic Diagram 12 Equivalent Circuit for Resolver with CompensatingWinding 16 13 Test Circuit for Measuring TransformationRatio, 18 I Function Error, Null Voltage, and Phase Shift 14 Typical Resolver Frequency Response 19 15 Right-TriangleComputationswith ~ Three-WindingResolver 24 16 Vector Addition and CoordinateConversionwith a 26 I g Four-WindingResolver 1’7 ReciprocalFunctions with a Three-WindingResolver 27 18 Rotation of Rectsng.da.rCoordinates 28 19 Data Transmission 29 , 20 Phase Shifting with a Four-WindingResolver a Frequency Response for a Widebemd Resolver z 22 Radar Range Sweep Resolutionwith a Four-WindingResolver 32 23 Resolver with TWO Excited Stator Windings Connected in 34 ~ Series end Two Rotor WWiings Connected in Series 24 Resolver with Two Excited Rotor Windings Connected in 34 Series and Two Stator Windings 25 RepresentativeControl System for Prscision Bombing 35 1- = 26 Resolver with One Excited Stator Winding, One Shorted 36 Stator Winding, a CompensatingResistor, snd Two Rotor Windings 27 Resolver with Two Exciter Stator Windings and TWO 37 Rotor Windings ~. 1’ “ 28 Resolver with Two Excited Stator Windings, TWO CompensatingWindings, and Two Rotor Windifigs 29 Resolver with Two Excited Stator Windings, Two 38 CompensatingResistors,and Two Rotor Windings ! 30 Resolver Mounted Directly from Terminal Board End 45 31 Resolver Mounted with Adapter Aesersblyfrom Shaft End 32 Resolver Mounted with Adapter Assembly from Terminal 2 Board End
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MIL-HDBK-218 1 June 1962
Figure Page
33 Resolver Moumted with Clamping Disk from Shaft End 47 34 Resolver Mounted with Zeroing Ring from Terminal 48 Board End 35 Mounting of Gear on Resolver Shaft with Adapter 36 Clamp Assemblies Adjustable Disk Assembly 3 Geared-ToothAdapter Zeroing Ring % Pinion Wrenches 41 Socket Wrench 42 Shaft Hardware
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MIL-HDBK-218 1 June 1962
I SECTION 1 DESCRIPITON
1.1 Scope of Handbook. This handbook contains referencedata concern- ing the applicationof precision electricalresolversand is intended for use by engineers and designers. The physical and electricalcharacter- istics of resolvers are briefly described,nnd resolver fundsment.alsare discussed. Performancespecificationsare defined,but not restrictedto particularvalues. Preferred resolver configurationsare discussedwith reference to their applicationin various eq.~ipmentsor systems. A brief descriptionof installationmethods is included,based on stsndardhard- ware and synchro installationpractices.
1.2 Definition of a Resolver. An electricalresolver is a variable transformerwhose output is a trigonometricfunction of its input. It con- tains a pair of stationary (stator)windings electricallydisplaced 90 degrees, and a pair of rotating (rotor)windings, also displaced @ degrees and free to rotate within the stationarywindings, Either stator or rotor msy be excited, to serve as the primary. While all four windings are not necessarilyemployed in a particularapplication,they give the resolver multiple capabilities,in handling various types of input voltage, frequency, end waveform. The secondary or output voltages are trigonometricfunctions of the primsry voltages with a particulartremsfonnationratio determined by the angular displacementof the rotor with respect to the stator.
1.3 Constructionof a Resolver.
1.3.1 General. Pbysice.1.ly,electricalresolversare similar to syachros. They are classifiedaccording to size (diameter)in the ssme fashion, an& they may be mounted with most standsrd synchromounting hardware. Figure 1 shows sn assembled electricalresolver, end Figure 2 shows its major components.
1.3.2 A cylindricalfrsme with a standardizedmounting flsnge (Figure 1)%%%’the assembled resolver. External connectionscan be made to an insulatedterminal board on the rear of the housing. Internal con- nections of the rotor end stator are terminated at the terminal board (Fi@re 2), Miniature resolversoften have unterminatedlead wires brought out through the rear of the resolver,eliminatingthe need for a temninal block. A reference line is scribed on the face of the housing, for alignmentwith a similar line on the end.of the rotor shaft, to determine coarse electrical zero.
I 1.3.3 Stator. The stator of the resolver, shown in Figure 2, is a I cylindricalstructureof slotted laminationson which two coils are wound.
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hm.-HmMc-2lg 1 June 1962
Figare 1. [email protected], Assembled
These laminations (Figure 3) are stacked so that the slots which are fomned are either parellel to the rotor shaft or displaced in such a way that the front end of one slot may be fn a straight line with the back .endof the preceding slot. The displacementof the slots is called skew, end since the slot pitch is the angular disteme between slot centers,the winding’is said to be skewed one slot pitch,
1.3.4 Rotor. The rotor of the resolver, showt”inFigure 2, is composed Of a ahdt~tiaticms, two windings, end slip rings. The laminationsof the rator core (Figure 3) are etacked so that the slot formation differs from that of the stator. If the slots of the stator laminationsare skewed one slot pitch, the slots of the rotor laminationsare usually parallel to the rOtOr sh~t; conversely,if the stator slots are par~el to the rotor shaft, the rotor slots _ USuS.Ijyskewed. The stacked laminationsare heated and cemented under pressure and ~ rigidly mounted on the shaft.
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MIL-HDBK-218 1 June 1962
HOUSING STATOR - R~OR / I /REARiELL)TmM’ti*: !
,. $, .;’ ‘‘i 1
=4 \ ● kRUSHES
......
Figure 2. Major Componentsof an ElectricalResolver Q@
STATOR ROTOR
I Figure 3. Stator and Rotor Laminations ,0 -3- Downloaded from http://www.everyspec.com
MIL-SDBK-218 1 June 1962
Slip rings are mounted on the rear of the shaft opposite the mounting flange; insulated from the shaft, they are used to terminate the ends of the coils. Brushes riding on the slip rings (Fi@re 2) provide electrical continuity during rotation. The other end of the shaft is splined and threaded for connectionto dials or gears. An index line is scribed on the exposed end -1 of the shsf% for alignmentwith a similar line on the housing, to determine coarse electrical.zero.
1.3.4.1 Ball Bearings. In as assembled resolver, the rotor shei?tis mounted on ball benrings located on both ends of the rotor core. Lew- friction ball bearings of the rad.ieltype are preferred. Tne balls, re- tainers, races, and shields preferablyare made of corrosion-resistantsteel. The ball bearing at the rear of the core is mounted in a bell housing (Figure 2), and the ball bearing at the front of the core IS mounted in tilemain housing which encloses the resolver. C-rings are generallyused to secure the ball bearings in the housings.
1.3.4.2 Lubricants. Grease lubricantsmust conform to SpecificationMIL- G-3278; oil lubricantsmust conform to.SpecificationMIL-L-6c85.
1.3.5 Mstribution of Windings. Stator and rotor laminationsare stacked so that the slats formed are either parallel to the rotor shaft or skewed one slot pitch. If the slots of bath rotor and stator are skewed or both parallel to the rotor shaft, the resultantflux concentrationsof the rotor and stator coils tend to make the rotor slot-lock in certain positions, causing angylar errors. Efficiency increasesas the number of slots increases.
1.3.6 CompensatorComponents. Compensatorcomponents,which improve the ~~lar accuracy Of reS01ver5,MSy COM!iSt Of resistorsor additionalwind. ings in the stator sd rotor winding circuits. CompensatorMnungs me located inside the stator. Compensatingresistorsmay be mounted either in- side or outside the resolverhousing.’
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MIL-HDBK-218 1 June 1962
SECTION 2
FUNDMmTALS
I 2.1 General. An electricalresolver is similar to a synchro in that it has rotor and stator windings snd behaves like a variable transformer. Un- like a synchro,a resolver may employ one stator winding and one rotor wind- ing, one stator winding .md two rotor windings, or two stator windings and two rotor windings. Figurs 4 shows the schematicdiagram for a standardized four-windingresolver. The relationshipbetween the input and output voltages
S2 S4 LJ
R2 R-4
Figure 4. StandardizedFour-WindingResolver, Schematic Diagram
is best describedby vector diagrams, as shorn in Fi~re 5. These show the I stator and rotor in the resolver zero position (see paragraph 2.3). Angle representsthe displacementof the rotor when rotated in the positive or counterclockwisedirection (when facing the shaft extension end). Note in the operating equations in Figure 5 that the term for voltage across a winding indicatesthe two terminals sad the polarity or direction. Thus, E(S1-3) is 1“ the term for voltage across the stator winding between terminalsS1 and S3; the directionof the voltage vector is sl.wayswritten low (S1) to high (S3); I if the vector direction is reversedbecause of rotor displacement,the sign of the term IS negative in the equation.
2.2 ResolverEmployment. Electrical resolversoperate with one or two electricalinputs and one or two electricaloutputs, dependingupon whether ● all windings are used, and whether the stator or rotor windings are comected -5- Downloaded from http://www.everyspec.com
MIL-SDSK-218 1 June 1962
RI SI
R2-+R482+
R3 53
E SI-3 = ‘( ERI-3 COS 8 - ER2-4 SIN 8 )
= K(ER2-4COS8t ERl-3 SIN 6’) ‘S2-4
(A) ROTOR -EXCITED ● “
51 RI
53 R3
COS 6 + Es2-4 SIN 6 ) ‘R I-3 ‘ K(%.I-3
. K (E52-4 COSe - E51-3 SIN O ) ‘R2-4
(B) STATOR - EXCITED
~iSUre5; %CtOr Diagrams for StandardizedResolvere
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MIL-HDBK-218 1 JUne 1962
togetier in series. There is also a mechanical input (or feedback adjustment), through which the rotor is positioned relative to the stator, varying the electromagneticcoupling end the resultantvector magnitudes. By rotating the resolver rotor, a variety of trigonometricfunctionsmay be obtained for different vector sums and differenceson the output windings. Such em elec- trical.arrangementis ideal for solution of right-trianglerelationships,such as sine, cosine, and tangent functions. With perpendicularinputs end out- puts, the analogy to Cartesiem coordinatesis obvious. By applying propor- tional Cartesian coordinatevolteges to the stator windings, the output volt- age from one rotor winding in conjunctionwith the angular displacementof the rotor provides pol-arcoordinateinformation. Conversely,by applyingpro- portional polar coordinatevoltages to the rotor windings and rotating the shaft of the rotor to the proper angular displacement,the output voltages from”the stator windings provide Cartesian coordinateinformation.
2.3 Resolver Zero, T%e msgnitude of the output of a reeolver depends upon the orientationof the rotor windings with respect to the stator wind- ings. To express this fiction as sn angular displacementin the positive direction of rotation, resolver zero is required as a referencepoint. Re- solver zero is the condition in which the couplingbetween one stator winding (S1-3) end one rotor winding (R1-3)maximum, and the couplingbetween that stator winding (S1-3) end the other rotor winding (R2-4) is minimum.
2.3.1 Rotor-ExcitedResolvere. The basic equations for voltage vector6 in a.rotor-excitedresolver are as fOllows:
Es1-3 = K (EW-3 COe 8 - %2-4 sin t?)
%2-4 = K (ER2-4 coo o + %1-3 ‘in e ) where
K = transformationratio (definedin paragraph 6 .9), and I I 0 = angle of counterclockwisedisplacementfrom reeolver zero. 1- ~ Electrical zero for the resolver is so establishedthat, when 6 is zero, . with rotor winding R1-3excited and rotor winding R2-4 open, %1-3 will be at a maxinramand %2-4 will be zero. 1? winding R2-4 is excited end R1-3 is open, the outputs will be reversed.
2,3.2 Stator-ExcitedResolvers. The basic equationsfor voltage vectors in a etator-excitedresolver are as follows:
E~.3 = K (~1-3 cos 8 + Es2.4 sin 0 )
ER2.4 = K (%2.4 cos 0 - Es1.3 Sin 0 )
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with K and 9 as defined in 2.3.1. When 0 is zero, with stator winding s1-3 excited and stator winding S2-4 open, ER1.3 will be at a maximum and ER2-4 will be zero. If winding s2-4 is excited and S1-3 is open, the outputs will be reversed.
2.4 Basic Operation.
2.4.1 Two-WindingResolver. An electricalresolver employing one stator winding and one rotor winding Is shown in Figure 6. The tindingsof the resolver are arranged in the stator snd rotor slots in such a manner that the voltage induced in rotor winding R1-R3 is proportionalto the cosine of the angle of rotation and the voltage applied to stator winding S1-S3. Tne re- lationship between input voltage (Es1-3),output voltage (ERI-3), and shaft angle (0 ) is: 1 SI /’ ——. / 0 ES I-3 \/ — ‘R I-3 ‘$’~ / 1 / / S3 2 \ R3
Cos e ‘R1 -3 = ~%l-i
~igure6. !Pwo-WindingBesolver, Schematic Diagram ● -8- Downloaded from http://www.everyspec.com
143.L-~K-218 1 June 1962
2.4.2 Three-WindingResolver. A slightlymore complex resolver is that shown in Figure 7, employing two rotor windings in space quadrature. The voltsge induced in rotor winding R1-R3 is proportionalto the cosine of the sngle of rotation and the voltage applied to stator winding S1-S3. The voltage induced itirotor winding R2-R4 is proportionalto the sine of the m@e Of rOtatiOn.~d the ~Olts.geapplied to stator tinding s1-s3. The re- lationshipbetween input voltage (Es1-3),output ~oltages (ERI-zj~d ER2-4), end shaft angle (o ) is:
ER1-3 = EsL-3 cos O
%2-4 = ESL3 sin @
SI
Cos e ‘R I-3 ‘ES1-3 %1-3
R-3 1 1 X4 33 E R2-4=-E$I-3 SIN8
Fisure 7. Three-WindingResolver,Schematic Diagram
2.4.3 Four-WindingResolver. The most commonlyused resolver is that shown in Figure 5; it employs two rotor windings in space quadratureand two stator windings at right angles to each other. This permits complex com- mutations of both sine and cosine functions.?novided that two inuuts are used. For a stator-excitedoperation,the ~oilowing relationship-exists
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MIL-HDBK-218 1 June 1962
between the Input volt~es (Es1-3 and Es2.4), output voltages (E~.3 and ER2.4), and angle of rotation (o ):
ER1.3 = K (%31-3cos O + ES2-4 Sin e)
%2-4 = K (%2-4 cos @ - ~1-3 sin e)
SI
=K COS8 ‘R( -3 (%{-3 ● ESI.3 ——— — ‘ES2-4 SIN8 )
\ e ~ I \ R3 1 ~OR4 S3 R2 ~ ~2.4= K (ES2.4COS@
SIN6’ ) -ES I-3
Figme 8. Four-WindingResolver, Schematic Diagram
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MIL-HDBK-218 1 June 1962
SECTION 3
TREORY OF OPERATION
3,1 General. The output of a resolver Is determinedby the input volt- ages applied to the pri- windings and the position of the rotor with re- spect to the stator. Currents flowing in the primary windings set up mag- netic fields that are perpendicularto each other. The resultat magnetic field produced by the combinedprimary currents is the vector sum of the separateme@et ic fields. This resultantfield mqf be designatedby a vector having a magnitude proportioned.to the instantaneousmagnitudesof the primary currents and a directiondeterminedfrom the arc tsngent of the ratio of the two primary currents. The voltage induced in each secondary winding is proportionalto the componentof the resultantvector that lies in a plane perpendicularto the plane of the secondarywinding.
3.2 Resolver EquivalentCircuit. Figure 9 shows the equivalentcir- cuit for a resolverusing only one stator winding (S1-S3)and one rotor winding (R1-R3). It is similar to a transformerin that it has low core losses and high leakage inductance. ResistorsR1 and R2 representthe d-c resistanceof the stator and rotor windings respectively. When siialter- nating voltage is applied to the input (stator) winding,-. a magnetic flux is generated in the core. The greater part of this rlux is present In the stator nnd is representedby L1. L2 representsthe leakage inductanceof the rotor. L3 representsthe inductancecaused by the magnetic flux that links the stator with the rotor. Eddy currentsend magnetic hysteresis account for the energy lost in maintainingflux in the core. This 10ss is representedby en equivalentshunt resistance,R3. Cl representsan approxi- mation of the distributedcapacitancein the stafor. “3 r“ I DEAL TRANSFORMER
Figure 9. Stemiard Resolver EquivalentCircuit
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MIL-RIIBK-218 1 June 1962
3.3 PrincipalTypes of Inputs.
3.3.1 Resolver Used for Data Transmission. The principal type of input applied tO the pri~v win~ing (st~tor or rotor) of a data trsmsmission resolver is a sinusoidalinput at either 60 CPS”or 400 cps. The use of a single operating frequencymnkes the frequency response relativelyunim- portant and permits fairly accurate temperatureand load compensation.
3.3.2 R~. The principal types of in. puts applied to the primary windings (stator or rotor) of a computing re- solver are as follows: sine-wavevoltages in phase, sine-wavevoltsges shifted 90 degrees apart, sawtooth voltages, aud square-wavevoltages. Although other types of inputs can be used, the above inputs are the most consnon.
3.3.2.1 Sine-Wave Voltages in Phase. When two sine-wavevoltages of the same electricalphase are applied to the input windings of a resolver,the output voltages remain fixed in electricalphase shift and their magni- tudes vary in accordancewith the equations given in 2.3.1 and 2.3.2.
3.3.2.2 Sine-Wave Voltages Shifted 90 Eegrees. When two sine-wavevoltages, shifted 90 degrees apart, are applied to the input windings of a resolver, the amplitude of the output voltages remains constant,but the phase of the voltages varies continuouslywith changes in the augular position of the rotor. The phase shift between the two output voltages remnins fixed at 90 degrees.
3.3.2.3 Square-Wave Voltages. When two square-wavevoltages that are either in phase or shifted 90 degrees apart are applied to the input windings of a resolver, the output voltages can be determinedin the sane msnner as the sine-wavevoltages discussed in 3.3.2.1 and 3.3.2.2.
3.3.2.4 Sawtooth Voltages. When two sawtoothvoltages are applied to the input windings of a resolver,the output voltages vary in magnitude as sine and cosine functions of the rotor nugle position and the input voltage magnitudes. These outputs are especiallysuitable in radar sweep applications.
3.4 Stator and Rotor Resistance. The d-c resistancesof the stator and rotor differ from the a-c values dependingupon the smplitudeand fre- quency of the currents, These resistancesvary appreciab~ with tempera- ture. For example, if the d-c resistanceof one of the windings, either stator or rotor, is 125 ohms at room temperature(25” C), the ~esistnnceat 100° C is 170 ohms. This change is due to the chnnge in the resistanceof the copper wire with temperaturesnd is approximately0.4 percent per degree centigrade.
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3.5 Resolver Error. 3.5.1 Function Error. When the resolver stator wind.lngis excited, the rotor output voltaGe varies sinusoid.allywith the augular position of the rotor. Any deviationfrom this sinusoidalrelationshipconstitutesthe fvmtion error of the resoler. The major internal source of this error are the imperfectionsinherent in the resolvermechanicalparts which case small distortionsof the magnetic fields. This error is expressed as a percentage of the maximum output amplitude,aud may be stated mathematicallyas follows: E 0! Error = - sin@ E( @ =Y”)
where E @ = F&S value of the fidamental componentof output voltage at rotor angle 8 , sud
E( d = 90”) . RkS5value of the fundamentalcomponentof output at 6 = w“.
The clifference, when multipliedby 100, expressesfunction error in per- centage of the output at (?= 90°. For example, a resolverhaving a func- tional error.of N. 03 percent and a maximum output voltage of 20 volts develops an output voltage that deviatesno more than @.006 volt from any point on an idesl sine or cosine curve. A typical function error CUme is. shown in Figmre 10.
L 90 180 270 360
DEGREES OF ROTATION
Figue 10. ~lcsl Angular Error in Resolver Shaft Rotation
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3.5.2 InteraxisError. Theoretitally, in a four-windingresolver one rotor winding develops a null at exactly 90 mechanicaldegrees of shaft rotation from the ‘positionwhere the other winding develop$ a null. In actual prae,ticethis is not the case because the perpendicularityof the windings is not perfect and a small error exists. The interaxiaerror ia inserted in the ideil operating equationsto indicatethe efflectof this error as follows:
ER2-4 = K (ES2-4 sin ( 0 + A 0 )
- ESI--3cos(O+ A@) Therefore, the inters.xiaerror shifts the output of”ER2-4 90 degrees PLUS A ~ ERL-3 aad ERI-3 remains unchanged.
‘3.5.3 Proportional.Vol@ge Error. A proportionalvoltage error,is en- countered in a resolver that is used for data transmissionrather than as a computing device. This error is the deviation of the mechanical shaft agle from an ideal.angular position requiredto generate am ideal sine and cosine functio~l. For data tramsmiasion, one input winding is generally excited and the error is measured in minutes of arc. A comparisonbetween a resolver used for data transmissionan& a synchro used for data transmissionindicates that the resolver cperatea witiigreater accuracy. For example, the most accurate synchro transmittershave errors in the order of 6 to 8 minutes of arc while data transmitterreaolverahave errors of less than one minute.
3.6 ——Rotor ElectricalBalance. In some resolver application, the out- put voltages of both rotor windings’must be nearly equal.. This balance is usually achieved in the production of a resolver if the manufacturermain- tains the rigid standardsthat result in uaiform winding resistances,mag- net,iccharacteristicsand mechanicalparts. Any deviations in these char. acteristicsresult in unequal trsnsfomnationratioa between the rotor and stator windings.
3.7 W~nding Compensation. A technique commonlyused to improve the accuracy of resolvers,is the addition of an auxiliary or,compensatingwinding within the stator slots. This winding is inserted in such a manner that close ‘1 coupling is obtained between stator and compensatingwinding. With close I couplingthe frequency responsebetween stator and compensatingt.indingsis broader than that between atator and rotor windings. The voltage induced in the compensatingwinding is cipproximatelyequal to the bock emf of the stator winding. This back emf equals the input voltage minus all the series losses. Thus tinecompensatingwinding cam be used effectivelyto compensatefor the series losses, resistanceand inductance,in the stator windings.
3.7.1 Schematic Circuit. Figure 11 illustratesa resolver with a comp- ensated winding connectedto ah amplifier. The compensatin~winding voltage is employed as a negative feedback for the amplifier.through a summing network. The resistorsin the,summingnetwork are adjusted to provide ,unity gain from ● -i4- Downloaded from http://www.everyspec.com
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RI =, SI
1- * OUTPUT INPUT ?4 0+ In S3 c R3 F m = =
o Figare 11. Resolver with Winding-CompensatedStator, Schematic Diagrem the amplifier input to the resolver rotor output. When an input voltage is applied, the smplifier output drives the stator winding until the compen- sating voltage is equal to the input voltsge. As a result, the back emf of 1. the stator is equal to the input voltage regardlessof the influencestending to degrade the resolver’saccuracy. The rotor output voltsge is therefore equal in magnitude,and phase to the input voltage.
3.7.2 EquivalentCircuit. h equivalentcircuit for a compensatedre- t solver, having a stator winding, a compensatingwinding, and a rotor winding, is shown in Figure 12. It should be noted that this equivalent circuit con- tains all the characteristicsof the circuit for a noncompensatedresolver and also includes the compensatinginductanceL3, resistanceR4, end com- pensating transformer. Capacitor Cl signifiesthe distributedcapacitanceof the compe~satingwinding. Resistor R4 representsthe d-c resistanceof the 3 compensatingwinding. Inductor L3 representsthe leakage inductanceof the 1. compensatingwinding. 3.’7.3 Phase Relations. The leakage inductanceof the rotor is smaller I than that of the compensatingwinding. This differencein inductanceintro- duces a slight phase shift between the two windings. This phase shift ca be neglected in some applications;however, in critical resolver operation, it , c&n be compensatedfor by use of a resistor-capacitornetwork.
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3.8 —.Considerationsof Resolver Loading. Wnen consideringthe proyer loading for a resolver, reference should be made to the equivalent circuit. In this way, the loading may be calculatedthrough a few basic computationso the circuit pa.r.mneters. For most calculations,the shunt inductanceand the iron loss may be neglected. The out~ut impedanceof the resolver depends mainly on the leakage reactanceand the stator and rotor copper losses. Con- t siderationmust also be given to the variations of copper losses due to temp- erature and the variationsof leakage reactancewith rotor position. In . cases where both stator winclingsare recited, source impeh-,ce nmst be con- sidered. If only one stator is usei, the other stator should I]eshorted. r 3.8.1 Resistive— ______and Capaciti\.eLoading. t!henonly resistivekmding is used.,the outp~t voltage drops and the phase lazs. When only capacitive loading is used, the voltage increasesamd the phase leads. In both cases, however, the variations in w~tp~t voltages and phase may be easily calculated
3.9 Effect of-.—Temuerat’ure.!lemperaturevariationsaffect the operating char.acteri~~f~s—~fa resolverwhereas the other environmentalcharacteristics affect the life and durabilityof Lhe unit. Temperaturevariations cause a change in the value of winding resistanceaad inductance. ‘Thesepara..eter changes cause variations in the tramsformation ratio and phase shift which can be talculated from the parametervalues sho?m in the equivalent circuit. In an unloafiedresolver, the tenmeraturevariationsare due nainly to the copper losses in the stator windings.
3.10 Windings—. Without Compensation.
3.10.1 FtnaseShift Eue to—— TemperatureVariation. In a noncompensat,edre- solver, temperaturechanges affect the phase shift between input and output more directly than the amplitude. Assuming a no-load condition,this phase shift result.sprimarily fro.mthe copper 10Sses in the stator windifigs.There- fore, the phase shift will have the same coefficientas the temperatureco- efficient of the copper losses, which is approximatelyO.4 percent per degree centigrade. For example, when a resolverhas a net phase shift of 1.6”, a change in temperatureof 30“C will produce an additionalphase shift of apQrOXimtely O.2“; that is,
1.6 x 30 x o.ook = 0.192°.
3.10.2 ‘Ira.msformationRatio Jhe to TemperatureVariation. The trans- formation ratio is also varied by temperaturechaages in a noncompensated resolver. This variation is due to the .copgerlosses of the stator windings and is approximatelyproportionalto the cosine of the phase shift. In the example of the phase shift discussed in 3.10.1,the phase shift was given as O.l$P”. Therefore,the amplitude variation due to this phase variationbe- tween input and output is equal to the cosine of 0.192°, or, the cosine of 1.6” minus the cosine of 1.792” which is 0.02 percent.
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3.11 TransformationRatio. The transformationratio of a resolver is the ratio of the output voltage to the input voltage at maximum couplingbetween the rotor and the stator. This ratio can be determinedby a direct measure- ment of the input and output voltages,but a mOre accurate dete~inatiOn Cm be made by placing the resolver in a test circuit (Figure13) and computing the ratio of certain known impedancevalues in the output circuit (refer to . @ragraph 3.17.2).
R IOK)
TEST INPUT H / VTVM P. /
DIVIDING FI LTER NULL h HEAD DETECTOR I I* 1 =
Figura 13. Test Circuit for,MeasuringTransformationRatio, Function Xrror, Null Voltage, and Phase Shift
3.12 TemperatureConpensation.
3.12.1 Current Feedback. To decrease the effects of temperaturevari- ations, a current feedback loop is used in the ~mPlifier drive stage Of a resolver circuit. In this loop, a resistor develops the necessary feed- back current. If temperaturecompensationi;”””tobe obtained,the re- sistor should have a temperaturecoefficlent equivalentto that of the copper,wire used in the stator winding. Effectively,this feedback current produces a negative output reaistencewhich counteractsthe resistive variations of the resolver that occur during temperaturechamges.
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3.12.2 Effect of Loading. When a resolver is loaded with a finite re- sistance, i.e., fram 100,000 o’hmsto 1 megobm, the effect of this loading must be consideredon the temperaturecharacteristicsof the unit. Referring to the equivalent circuit for a noncompensatedresolver shorn in Figure 9, the copper resistance and the lowl resistanceact as a voltsze divider net- work. “Since the copper resistanceis a fumction of temperaturevariations, the loss in the divider varies with the temperaturecoefficientof cop?er. This loss can be readily calculatedfrom the values of the resistancesand I the temperaturecoefficientof co~per which is 0.4 percent per degree centigrade.
3.13 Frequency Response. Resolvers operate on carrier frequenciessuch as 60 and 00 cycles per second when used for data tra.ns.misd.on. When re- solvers are used as computingdevices the operating freq.ueuciesare con- siderablyhigher (10 kilocycles or greater). Therefore, computingresolvers should have a frequency response flat up to 10 kilocycles or greater, depending on the intended application.
3.13.1 Stator-to-RotorResponse. The frequency response curve for a typical resolver is shown in Figure 14. This response curve is similar to that for a transfom.erwhich has a high leakage reactance. At low-excitation frequencies,the resolver output amplitude decreasesbecause of the d-c resistance of the stator winding and the magnetizing inductancecaused by the ~metlc flu linking the statOr ad rOtOr wind%w. At this point the response is 3 decibels down. As the excitationfrequenciesincrease,the
+18- + 15- +12 +9 - +6 - +3 -
DB o , ~ I000 IOKC 100 KC -3 - FREQUENCY (CPS) -6 - -9 -12 – 15 - !8[
Figure 14. Typical Resolver Frequency Response
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output voltage zmplitude increasesto a constamt value for a wide frequency range. At some higher frequency,the distributedcapacitanceresonateswith the stator leakage inductanceto give a peak in the response. This occurs at about 30 kilocycles i-nthe curve and may result in a response rise of as nuch as 10 to 15 decibels. When the resolver is used with inputs consistingof irregular waves, the peaking of the response results in ringing. To COlll- pensate for this condition;a damping resistor is generallyused across the I rotor to increase the resonatingfrequency.
3.13.2 Stator-to-CompensatorResponse. In the compensatingtype of re- solver, the response for the stator-to-compensatorwindings is similar to that described in 3.13.1,except at very high frequencies. The leakage inductance of the compensatorwinding is lower $hm that of the ~otor, re- sulting in a much higher frequency response. In some resolver units, it is desirableto have a flat responsebetween stator and compensatorwindings to well over 100 kilocycles. To obtain this response in resolverswith a 1:1 ratio between stator emd compensatorwindings, correspondingterminals of these windings should be operated at equsl a-c potentials. This method of operationminimizes the capacity effects between windings e.rrdresults in a peaking effect beyond 100 kilocycles. I 3.13.3 Carrier Operations with Winding Compensation. Iiorms.Uy,the phase and amplitude response of resolverswith compensatorwindings is virtually ● independentof frequency,provided the frequencyvariation does not exceed aPprOxi~tely *1O percent. In cases where the frequency variation does ex- ceed *1O percent, the phase variation is approximately,fl ia t2 minutes, and the amplitude variation is only about O.O1.percent. These slight changes in amplitude and phase result primarily from the similaritybetween the fre- quency responses of the rotor and compensatorwindings.
3.13.4 Carrier OperationsWithout Winding Compensation. For carrier frequency operation at 60 or hoo cycles per second, the performanceof re- solvers without compensatorwindings,csnbe determinedfrom the equivalent schematicdiagram. The phase varies approximatelyinverselywith frequency. For example, if the frequency varies 10 percent above the carrier frequency, the phase also varies 10 percent, but in the.opposite direction. The ampli. tude variationsare proportionalto the cosine of the resolverphase shift. Therefore, it follows that resolverunits that are characterizedby ex- tremely 10V phase shift have negligible amplitude variations.
3.14 Null Voltage. ~en the position of the rotor of a resolver is fixed so that minimum coupling is obtainedbetween rotor and stator windings, a residual voltage exists at the resolver output. This output is proportional to the input and is caused by the eddy currents and core saturationsin the stator and rotor windings. To ensure that this residual or null voltage is kept at a minimum, the laminationsare insulatedbefore they are stacked.
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3.15 Hazmonic Distortion. Even under ide@. operating conditions,re- solvers generate some harmonicsbecause of the nonlinear characteristicsof qetic materials., In most resolvers,this distortion.of the input waveform is about.0.1 percent or less of the input voltage. This distortion is usucil.h I in the order of the third harmonic of the carrier frequency. Only when d-c - inputs sme used in the windings is there second harmonic distortion.
3.16 Effects of Direct Current. Direct current has a negligible effect on the operating characteristicsof resolvers,provided the current does not exceed the penk stator current of the unit. As mentioned previously, the use of direct current results in the introductionof second harmonic distortion;however, tinisdistortionusually does not exceed 0.1 percent of the input voltage.
3.CL7 Testing Procedures.
3.17”.1 General. Since a resolver is a precision instrument,etireme Care must be ex~ in testing these units. In order to perform the measure- ments in these tests, it is necessary to have test equipment of a greater accuracy than the unit umd.ertest. The usual measurementsinclude function error, null voltage, transformationratio, phase shift, and frecy~ency respmse.
3.17.1.1Mechanical Positioningof Rotor. Before attemptingto perform the vario.m re~er tests, a m=nnicsl means must be devised for rotating the rotor. A precision dividinghead is usually used for this purpose. This head, which is equipped with a fixture for supportingthe case of the re- solver, gositions the rotor with an accuracy of as great as *2 seconds, although .f30seconds is a.more typical case. Also.the couplingbetween the divider head and the shaft of the resolver does not substantiallyload the beariflgs.
3.17.1.2 Test Circuit. Figure 13 shows a test circuit for the measurement cf such resolver parametersas transforms.tionratio, phase shift and function error. A precision dividing head is coupled to the shaft of the rotor. Standard test voltages are applied to one stator winding, and output voltages are measured on one rotor winding. The filter in the output circuit is designed to pass only the frequency of the test or input voltage, eliminating harmonics. The variable capacitorsmd the voltage-dividingresistor may be decade-boxunits, for quick determinationof impedancevalues.
3.17.2 Trs.nsfomaationRatio. tie test circuit in Fi=mre 13 is suitable for measur~ng transformationratio when it is less than 1:1. With the Scale of the dividinghead aligned with the electricalzero psj.tion of the rotor, rotate the dividinghead 90”, I formaxhnun coupling of the stator and rotor windings. Adjust the voltage divider for maximum output (on the ~lfpl). AdJust C for minimum null, so that the voltage across the divider ● is In phase with the resolver output. The trs.nsformationratio (TR) can then -21- Downloaded from http://www.everyspec.com
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be obtained directly by comparingthe output reading (on the VTVM) with the known input voltage. A more accurate figdre can be obtained by computingthe ratio of impedances in the divider network. Thus,
Z
For a resolverwith a trsmsfommtion ratio equal to or greater than 1:1, a measuring device which indicates or compares the rms voltage of the funda- mentsJ frequency,and which does not alter the open circuit secondaryvolt- age by more than 0.1 percent, must be used.
3.17.3 Function Error. Function error is the daviation of the resolver output voltage from true sinusoidalcorrespondencewith the theoretical value of outpt voltage. It can be determinedin tem.nsof trs.nsform.ition ratio by measuring the ratio, as in paragraph 3.17.2, for various increments of rotation (USUSI.IYa minimum of every five degrees over an 180-degreearc). Thus, with TR1 the known transformationratio at mcuinnnncoupling,and TRn the ratio at the sagle ($ ) where error is being measured, the function error, in percent, may be computed as follows:
Error ~ (ain O ‘lTU- T%) x 100$ o ‘ml I
3.17.4 Null Voltage. Both fundamentaland totaL null voltsges may be measured directly,using the test circuit shown in Fi@re 13. With the specifiedtest voltage applied to the input, adjust C and R2 until a minimum ~ indicationis obtained on the null detector. This is the fundamentalcom- ponent of the null voltage. At this sama position, read the total null voltage with the VTVM connectedto the input side of the filter. I 3.17.5 Fhase Shift, Fhase shift can be computed as a function of the im- ~ pedsnce vahes in the test circuit shorn in Fi~re 13. With the rotor po- sitioned for maximum coupling, set the voltage-dividingresistorsand adjust C for a minimum null (in-phasevoltage). !Fheeagle of phaae shift, O , is then as follows: e=tan-l _xc_ R
3.17.6 Frequency Response. In order to measure the frequency response of a resolver,the rotor windings should be unloaded. The response should be measured from the electricalnull of one rotor winding while the other rotor is at a maximum couplingposition with the stator. A voltmetermust be used with an appropriate-fil~er so that it responds only to voltages of the
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fumiamentalfrequency of the input. An oscilloscopeshould be used to measure the large amplitude and phase shift variationsas the frequency is varied. The complete frequency response can be derived from the Lissajous figure on the oscilloscope.
●
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SECTION 4
APPLICATIONS
4.1 General. Resolvers are used extensivelyin automatic control sys- tems, @slog computers,angle data transmissionsystems, and plan position indicating radar systems. They are employed as phase shifters and data transmissiondevices, and csn also be used to resolve problems in trigonometry,
4.2 Typical Systems. 1
4.2.1 Automatic Control Systems. The simplest applicationof resolvers is found in automatic control systems. These systems use the resolver to perform sine and cosine computations,as well as vector additions.
4.2.1.1 Sine Computation. A voltage representingthe ~otenuse of a right triangle is applied to one stator winding, and the rotor is positioned to the ~gle @ as shown in Figure 15. The outputs of the rotor windings represent the coordinatesof a right triangle. This method of operation is used to convert polar coordinatesto rectangularcoordinates. SI ------2
RADIUS VECTOR
1 3 S3
s RADIuS VECTOR , A POLAR COORDINATE %1-3 FOR RADIUS ANGLE 8, ER2-4 AND ERI-3, AS SINE AND COSINE FUNCTIONS OF E , ARE THE EQUIVALENT RECTANGULAR C%-R%il NATES.
Fiare 15. Right-TriengleComputationswith a Three-Wind4ngResolver
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4.2.1.2 Conversionof Polar Coordinates. Assume that volts.ge%1-3 repre- sents one polar coordinate,the radius vector, and is applied.to the stator winding of the resolver shown in Figure 15; the rotor is positioned to angle o , which representsthe other polar coordinate. Output voltages ERP-L..- and represent the legs of a right trisngle, of which the radius vector is %-iypoten.se. Thus, the polar coordinatesof a point, P, cm be converted to the rectan~lar coordinates.
4.2.1.3 Vector Addition. A resolver is capable of adding vslues vector- ially. If the two stator windings are excited with voltages corresponding to the coordinatesof a right triangle, an output may be produced repre- senting the radius vector and the vector angle. This method of operation is used to convert rectangularcoordinatesto polar coordinates.
4.2.1.4 Conversionof Rectan@ar Coordinates. Assume that Es1-3 and %2-4 rePresent rect=8Uhr coordinatesamd are applied to the stator wind- ings of the resolver shown in Figure 16. The servo loop connectedto one rotor winding positions the rotor to an angle o ,
such that one output, ERp-4j is zerO. That is,
ER2-4 = ES2-4 CCS @ - Esl-3 sin o = o.
The other output, ER1-~, being 90 degrees avay, will be at a ~i~.
Expressed vectorially,the output
ERI..-j= Es1-3 cos O + Es2-4 sin o becomes
ER1-3 = .~(ESI-3)2 + (Es2-4)2 which is the radius vector, or equivalentpolar coordinate. 0 , of course, is the radius angle, or vector sngle.
4.2.2 Analog Computers. Resolvers are used in computer systems for reciprocaloperationssnd for the rotation of rectangularcoordinates.
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~. INPUT
SI
RA01u5 VECTOR. I S3 ) I
1“ R3 R2 —ER2.4 =o— R4 NI r———— I p ●’ I
I MOTOR, I ~F::RVO ‘ C9AcKb I ——— _ L__ I -+
Figure ~6. Vector,Addition and CoordinateConversionWLti a Four.Winung ReSOIVer
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4.2.2.1 Reciprocal Operation. In the reciprocaloperation shown in Figure 17, one output voltage, ER _ , is fed back through the high-gain amplifier. The ~gle .f the ~tor (~ ~ ~. li*ted t. produce the negative feedback v.l-t- age (ER1-~ = -El). The gain of the feedback loop is designed to varv as the cosine of”tl . Therefore,the gain of the closed loop varies as the.secant of 8, which is the reciprocalof the cosine of @ . The other rotor output be- comes equal to the tangent of d (c~~e~ ). This methOd of division obviates the need for a servo loop. Special considerationmust be given to the ampli- fier, since it mwt be capable of remaining stable over a wide range of loop-gain values.
FEEDBACK
VOLTAGE INPUT o AMPL El %1-3 ‘ ~@@ (EI)
I A ● ‘RI-3= ‘EI RI SI — e S3 h ~---
R2
E ~2-4 . E, SINe - ErTANe xCos 8
Fi@me 17. ReciprocalFunctions with a Three-WinClingResolver
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4.2.2.2 Rotation of RectangularCoordinates. A four-win&@ resolver~ be used for the rotation of rectangularcoordinates. As shown in Figure 18(A), with input voltages on the stator windings correspondingto X and Y coordinatesof a point, P, the output voltages from the rotor windings will correspondto X 1 and Y t coordinates,where the coordinateaxes have been rotated throu&b an angle 0 . This relationshipcem be seen in Figure 18(B). “rZ-ls’ ‘.1 I OR I
x Ou?pul INPUT ———\ —-_1 R3
/ S3 R2 — OUTyPUTe ‘4 7 w’(A)
X’ = OR + RM
(!$=CO’e)(!$=S,Ne)
x< = XCOS8+ YSIN8
Y’= ST- OT
(~ ’coSe)(+’SIN@)
I---X-4 Y’=YCOSL9 - x SIN o (B)
Fisure 18. Rotation of Rectangular Coordinates
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With angle 6 representingthe displacementof the rotor from resolver zero, the resultingequations of rotation are the same.as those for a standard four- winding resolver (paragraph2 .4.3). Actually, the input voltages on the stator WindingS fOrm a resultsntflux vector whose position, determinedby the arc- tengent of Y/X, is independentof the angular position of the rotor. And the ,’ rotor windings will at all times develop voltages which are proportionalto the sine or cosine of the engle the rotor makes with the flux vector.
,. . 4.2.3 2ata Transmission. In data transmissionapplications,more than one reSOIVermust be used, as showo in Figure 19. One of the prims.rytindings
!.
FIXEDSI INPUT
‘3 3
, L \
e SERVO ;\m& ;;;P:T I AMPL I POSITION RI R4 ‘l I
——_ ___ — I t“ .— _ __ ‘3
R2 R3 1. $x= P, Figure 19. Data Trsnsmission
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MIL-HIBK-218 1 June 1962 -. s .... ,.,: .*., -“:.2 of the first resolver’is supplied with a fixed excitationvoltage; the other winding is not excited. The shaft may be at any angle. The outputs appear- ing across both secondarytindlngs are rectangularcoordinates. These out- puts are suppliedto the secondarywindings of the second resolver for con- version to the original polar coordinates. The servo-drivenshaft of the second resolver repeats the rotor angle of the first resolver.
b.2,k Phase Shifter8. Figure 20 illustratesone of the many possible arrangementsof phase shifter circuits. When the primary windings are excited by two voltages of”equsl smplitudebut 9 degrees apart, the output voltages remain constant in emplitudebut vary in phase, dependingupon the angular position of the rotor. me output voltages, like the input volt- *e~, are $i?,*g”*es apart. Isolationamplifiers or resistiveloading in the output are%fien,‘necessaryto ensure the desired degree of precision in -’- phase aliiftinlg:<-
. ....
● ’
EI&
%1-3 = ‘S2-4 = ‘I
‘RI-3 ‘ ‘R2-4 = ‘1.
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4.2.5 Radar Sweep Resolution. Wideband resolversare used to resolve radar range sweepe in.plan position indicators. Additional informationcan be presented on a plan position indicatorthrough multiplexingtechniqueswith resolvers used in pLace of field coils. The response of a t~ical, wideband resolver is shown in Figure 21. The usable high frequency extends to approxi- mate~ lCO kilocycles. By using the compensatingwinding, the usable low frequency is limited only by the maximum current rating of the resolver and external circuit parameters. Figure 22 shows a typical applicationof a wide- bend resolver for radiarsweep resolution. The rotor of the resolver is con- nected, through gears, to the radar sntenna and a sawtooth voltage, Es1-3, is applied to the stator. When the antenna is positioned at O degree, the rotor is at resolver zero end the output sawtooth voltages,ER1-3 and ER2.4, applied to the horizontaland vertical deflectionplates of the display in-. dicator, are maximum and minimum respectively. As the antenna rotates in a counterclockwisedirection,ERI-3 decreases in amplitude while ER2-hj in- creases. This causes the radar sweep to move accordingly. When the rotor mgle ( 8 ) equals 45 degrees with respect to resolver zero, the output saw- tooth volteges are equal in amplitude end the sweep on the indicatorhas rotated 45 degrees from its original position. When O equals 93 degrees with respect to resolver zero, ER1-3 is minimum and ERP-4 is maximum. The radar sweep now indicates 90 degrees with respect to its original.position, Following this procedure, the amplitude of the output voltages will vary as the sine and cosine fumction of the rotor aagle causing the sweep to rotate 360 degrees for each revolution of the antenna.
2.0 Lo I I , , 1 , I I I 111111 /1 111111
-~ 1.0- -i o .E IK 10K FREOUENC. . . _, 11111 I
Figure 21. Frequency Response for a Widebaad Resolver
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MIL-FRJSK-218 1 June ‘1962
S2 S4
RI TO HORIZONTAL DEFLECTION PLATE Cos9 ‘R I-3 ‘ %.1-3 SA;W$:TH “
INPUT E SI-3 2 S3 R3
E =
Figure 22. Radar Range Sweep Resolutionwith a Four-WindingResolver ●
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MIL-HOBK-218 1 June 1962
SECTION 5
TYPICAL ELECTRICAL RESOLVERS
5.1 General. This section describes the function of some t~ical electricalresolvers in relation to the systems in which they are employed. The following resolver connectionsare discussed in subsequentparagraphs: a resolver with two excited.stator windings connected in series and two rotor windings connected in series; a resolver with two excited rotor windings COn. netted in series and two stator windings; a resolver with one excited stator winding, one shorted stator winding, a compensatingresistor, and.two rotor windingS; a resolver with two excited stator windings, two compensatingwind- ings, and two rotor windings; a resolver with two excited stator windings, two compensatingresistors,and two rotor windings; and a resolver with two excited stator windings and two rotor windings.
5.2 Resolver with Two Excited Stator Windings Connected in Series and Two Rotor Windings Connected in Series. The wiring connectionfor this type of resolver is shown in Figure 23.
5.2.1 Function. The function of the resolver connectionillustratedin Figure 23 ~ovide a variable amount of time-phase shift. The output voltage of the resolver is a direct function of the shaft position. There- fore, the output,may be shifted with respect to the input by simply rotating the resolver shaft.
5.2.2 Systems in Which Employed. Representativesystems in which this type of resolver is used include short rage navigation (SHORAN)and radar ranging equipments. In the SHORAN system a crystal-controlledoscillator generates a standard frequency of 186 kilocycles,from which three timing signals are derived. The periods of these signals correspondto SHORAN dis- tances of 1 mile (93kilocycles),10 miles (9.3kilocycles),end 100 miles (O.93kilocycles). The timing signals are generatedby three resolvers functioningas phase shifters. These three resolvers comprise a set and are linked together in 10-to-l ratios. Each output from the resolvers is variable in phase correspondingto the angular shift of the resolver rotor shaft. In the radar ranging system, the resolver is used to vary the phase of a fixed-phase input as a function of range. The fixed-phaseinput iS applied tO the stator windings. The rotor is mechanicallyconnected.to a rsnge driver unit which rotates the rotor shaft at a speed correspondingto a particular range. The output taken across the rotor is phase-shiftedin accordancewith the angular position of the rotor. The phase shift between input and output, therefore,is a function of renge. 5.3 Resolver with Two Excited Rotor Windings Connected in Series and ~o StatOr Windings. The tiring connectionfor this type resolver is ● shown in Figure 24. -33-
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MIL-HIBK-218 1 June 1962
S2 S4
lJ
SI
,s3 2
R2 f5R4
J
Fig.me 23. Resolver with Two Excited Stator Windings Connected in Series and Two Rotor Windings Connected in Series ● ’
I
S3 J R3 R2 RR4
Figure 24. Resolver with Two Excited Rotor Windings Connected in Series snd Two Stator Windings
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MIL-IDBK-218 1 June 1962
5.3.1 Function. The function of the resolver connectionillustratedin Figure 24 ~nvert rectangularcoordinatesto polar coordinates. Also, by short circuitingone of the rotor windings, the resolvermay be used to convert polar coordinatesto rectangularcoordinates.
5.3.2 Systems in Which Employed. Many aircraft, space vehicles, and naval vessels require informationgenerated in one coordinatesystem to be tras formed into another coordinatesystem. A representativesystem. employingthis resolver is the control system used in precision bombing, A polar axis bombsight is mounted in an aircraft so that the polar axis is vertical when the plane is in level flight. The sight acquires a point in space by means of an azimuth rotation,A (Figure 25), about the polar axis and then by em elevation rotation,E, about an axis perpendicularto the polar axis. The elevation axis is carried around by the polar axis. The problem is to maintain the sighting line on a target when the aircraft yaws, pitches, or rolls. It is assumed that yaw is always measursd about an axis parallel to the bombsight polar axis, and anY yaw indicationis fed directly into the azimuth servo. A pitch and roll ,gro measures the roll angle, R, about the aircraft’slongitudinalaxis nnd the pitch angle, P, about a I horizontaltransverse axis. This is equivalentto stating that the Ditch and roll aro is oriented so that roll is measured about the pitch line. The known position of the target in a vertical space frsme as measured ● from the aircraft position is specifiedby coordinatesX, Y, and Z. With pitch angle P and roll angle R known, the problem is to find sightingangles A and E of the sight. Fi@re 25 illustratesthe instrumentation. The known vertical space frame coordinatesX, Y, and Z are fed into the resolversP and R. ‘Fnistransformsthe data to coordinatesof the target as measured jn I the aircraft coordinatesystem. This new set of data is available as the out- put of the P ad R resolver chain. If this new data is fed to the sighting- angle resolver chain (A and E), it is then possible to position these re- solvers, as shown in Figure 25, so that the line of sight points to the target. Note that the slant range is one of the outputs.
FROM P, R GYRO LINE OF SIGHT
‘yz[~ MNTRANGE I 1 —[ I COORDINATES COORDINATES ; IN IN L.~ ~ERvo [.-j ~ERvo VERTICAL AIRCRAFT -SYSTEM SPACE FRAME
Fi~re 25. RepresentativeControl System for PrecisionBombing
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5.4 Resolver with One Excited Stator Winding, One Shorted Stator Winding, a CompensatingResistor, and Two Rotor Windings, me tiring con. nection for this t~e of resolver is shown in fi~re 26.
S4
—— ‘tIF-
Figure 26. Resolver with One Excited Stator Winding, One Shorted Stator Winding, a CompensatingResistor, ~d Two Rotor Windings
5.4.1 Function. The resolver illustratedin Figure 26 can be used in the fallowing applications: ~@-e cOmPutatiOn,sine and cosine computation, reciprocal operations,and radar sweep resolution.
5.4.2 Systems in Which Employed. This resolver is used in computing equipment resolving right triangular solutionsand in radar systems for the resolution of radar sweep voltages. It is especiallysuitable in pro- viding a rotating radial trace on a plan position indicator. In this application,Fhe rntor shaft is mechanicallyconnectedto the ra~r s.ntema. A range sweep voltage is applied to the stator. !I’heoutput from the rotor winding is applied to the deflectionplates of the PPI cathode-raytube, producing a radial trace synchronizedwith the sntenna. The compensating resistor in this applicationensures linearity of the radar sweeps. ‘1 5.5’ Resolver with Two Excited Stator Windings nnd Two Rotor Windin s. The wiring~ 5.5.1 Function. The resolver illustratedin Figure 27 can be used in the transm~f angular data and in vector additicm .amdsubtraction. It can also be used to provide a radial.‘sweepon a plsn position indicator, synchronizedwith a rotating antenna.
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MIL-RDRK-218 1 June 1962
S2 S4
I w“ i-
SI RI
S3 3 R3 R2 5R4 Figure 27. Resolver with Two Excited Stator Windings and Two 1 Rotor Windings
5.5.2 ~stems in Which Employed. This t~e of resolver is used exclusive- ly in antiaircraftfire control systems for producing azimuth marks and a radial sweep on the radar plan-positionindicators. ‘Resolversalso are used to add and subtrsct angles electricallysnd transmit augulsr informationbe- tween remote points. The operation of resolversused to produce a radial sweep on the plan-positionindicatoris similar to the operationsof the basic resolver system discussed in paragraph 5.4.2. When resolversare used for data transmission,one primmy winding of one resolver is excited. The two outputs from this resolver are used to excite the primaries snd drive the shaft of the second resolver (with a servo) so that it repeats the shaft sngle of the first.
5.6 Resolver with Two Excited Stator Winding, Two CompensatingWindings~ and Two Rotor Windings. The wiring.comection for this type of resolver is shown in figure 26.
s.6.~ Function. The functions of this type of resolver are similar to those of the resolver with two excited stator windings and two rotor tind- ings. The two compensatingwindings mske this type of resolver especially suitable in applicationswhere resolver performanceis susceptibleto en- vironmentalvariations and load and source impedancevariations.
5.6.2 Systems in Which Einployed. Refer to paragraph 5.5.2for a dis- cussion of a representativesystem in which this type of resolver is used.
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C2 C4 S2 w S4
SI c1 O R3 C3 ~ S3 R2 R4
Figure 28. Resolver with TWO Excited Stator Windings, Two CompensatingWindings, and TWO Rotor Windings I
5.-i Resolver with TWO Excited Stator Windin@, TWO compensating Resistors, and TWO Rotor Windings. The tiring connectionfor this type of resolver is shown in Figure 29. ● 5.7.1 Function. The functions of this type resolver are simller to the functions of the resolver with two excited stator tindings and two rotor wind- ings. The two compensatingresistorsmake this a temperature-compensated resolver. The effects of wide temperaturevariationsare minimizedbecause the coefficientof the resistors is equivalentto the coefficientof the copper used in the rotor and stator windings. 5.7.2 Systems in Which Employed. A representativesystem in which this type of resolver is used is discussed in paragraph 5.5.2,
S2 S4 I 10 u > SI RI
2“ 9 14
S33 R3 F] R2 R4 13 F@ure 29. Resolwr with Two Excited SitatorUindinge, Two CompensatingResistors,and.Two Rotor Windings o -3- Downloaded from http://www.everyspec.com
ICIL-HOBK-218 1 June 1962
I SECTION 6 FERFORM&NCESPECIFICATIONS
6.1 Generel. General specificationscover performancerequirements common to all typee of reeolvers. Detail specificationsinvoke the general specificationand completethe definitionof individualresolver types by stating parametermagnitudes, special requirementsand exceptions to the general specification. The performancespecificationsdefined in this section are applicableto all types of resolvers.
6.2 Classification. Resolvers are classifiedaccordingto size, input impedance,type of compensation,and excitationfrequency. A typicel .reCererice&signation is resolver,type 23R32N4e. These digitB and letters c~asslfy the *solver es followe.
6.2.1 Size. The first two digits (23) give the maximum diameter of the I housing,,i~nths of an inch.
6.2.2 I%nctlon. AU resolverstake the designation“R”
6.2.3 ~. me next tkm digits (32) designatethe”nomlnd impedance of t e input coil, in hundrede of ohms.
6.2A m of Compensation. The letter (N) designatesa resolverwith no compensation. Other code letters used are “R” for resistor compensated, “W” for winding compensated,and “B” for both resistor and wintig compensated.
6.2.5 ExcitationFrequency. The next .Uglt (k gives the frequency.of the excitationvoltage, cycles per second (cps1. Other stanti fre. uency desigaationeare 6 (60 cps), 8 (8OO cps), 10 (1000 cps), 100 ?1%0+30 CPS), and 200 (20,0f.xlCps):
6.2.6 Modification. The final.,lower-caseletter (a) indicatesthat the I%SOIVSr is the unmodifiedOriginal issue. The first modificationof this type will take the designation“b,” the second “c,” etc.
6,2.7 OtherDesignation. If the resolverhas.a fixed rated input voltage other than U5 volte, the referencedesignationwill be prefixedby the applicablevoltage rating,
6.3 hkdeof Excitation. Zhe mode of excitationindicateswhether the stator or rotor winding Ie energized. The energizedwimiing is always consideredthe primary winding of the resolver.
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MIL-~K-218 1 June 1962
6.4 Direction of Rotation. ‘The>ositive direction of rotation in standard resolvers is counterclockwise(facingthe shaft extension end of the resolver).
6.5 ~. A resolver must operate satisfactorilyover the range of input voltages requiredby the applicabledetail specification. This range of inputs includes the rm?ximum.,minimum, and test voltages. The test voltage is the input voltsge at which a resolver must meet all of the test requirementsof the applicabledetailed specification.
6.6 Input Current. The input current of a resolver is the ,current drawn by the prims~ windings when the specifiedtest voltsge is appl,ied I to the primary windings with the secondarywindings open-circuited. The input current value is specifiedby the applicabledetail specificationand must be measured‘witha millisnsneterhaving an accuracy of at least one percent.
6.7 Input Power. ‘I’heinput power of a resolver is the power consumed in the primary windings when the specifiedtest voltage is applied to the primary windings with the secondarywindings open-circuited. The input power value is specifiedby the applicabledetail specification.
6.8 D-C Resistance. The d-c resistsmceof the input and output wina- ings snd the compensatingresistorsand windings of a resolver must correspondto the vsLues specifiedby the applicabledetail specification.
6.9 TransformationRatio. The transformetion ratio, designatedas K, is the ratio of the fundamentalcomponent of the no-load output voltage, at maximum coupling,to the input voltage. The nominel value of the trens. formation ratio is obtained at the test voltage and is specifiedby tie applicabledetail specification.
6.9.1 Ewality Of the TransformationRatio. Equality of the trms- formstion ratio refers to the variation of the trensformation ratio for all the possible combinationsof stator sad rotor windings. These variations must not exceed the value specifiedby “theapplicabledetail specification.
6.10 Nominsl Phase Shift. Nominal phase shift is the time-phaseangle by which the secondary (output)voltage dlffer8 from the primary (excitation) voltage. The nominal phase shift value is specifiedby the applicabledetail specification.
6.10.1 Phase Shift Varhtion Caused by Rotation or Voltage Variation. Variations of phase shift mey occur as a function of the rotor angle or input voltage variations. Such variationsmust not exceed the vahe specifiedby the applicabledetail specification. ● -40- Downloaded from http://www.everyspec.com
MIL-SDSK-218 1 June 1962
6.11 InteraxisError. Intere.xiserror is the engular deviation of the null positions from space quadratureof all rotor,,stator, end rotor-stator winding combinations. When tested as describedherein, the interzxiserror must not exceed the value specifiedby the applicabledetail specification. To test the resolver, it is mounted in an e.ogularaccuracy tes~ stand, which can position the rotor within 15 seconds of arc. The resolver is energized with the specifiedtest voltage, and in the sequence indicated in Table 1. At each position in Table 1, fine angular adjustment is made to obtain minimum output. The algebraic differenceof these angles at positions 1 and 3, 2 ad. 4, 5 and 7, and 6 and 8 is the interaxiserror of the rotor. The algebraic Mfference of these nngles at positions 1 and 7, 2 and 8, 3 and 6, and 4 end 5 is the interaxis error of the stators. The algebraic difference of these angles at positions 1 and 5, 3 and 7, 2 and 6, ,md k and 8 is the interaxiserror of the resolver.
Table 1. InteraxisError Test
Null Null Excitation Shorted Cnltput Position Angle winding Winding Winding
1 0 S1-S3 s2-s4 R1-R3 2 18o S1-S3 S2-S4 P3.-R3 ● 3 w S1-S3 s2-s4 R2-R4 4 270 sl-s~ s2-s4 R%R4 o s2-sb S1-S3 R2-R4 : 180 S2-S4 S1-S3 R2-R4 7 w S2-S4 S1-s3 P3.-R3 8 270 S2-S4 S1-S3 R1-R3
6.12 Fundms.entslNull Voltege. The fundenmntalnull voltage is the minimum seconda~ voltage of the excitationfrequency,without harmonics, obtained for the null engles indicated in Table 1. The fundamentalnull voltage is specifiedby the applicabledetail specification.
6.13 Total Null Voltage. The total null voltage Is the minimum seconda~ voltage, includlngharmonics,obtained for the null angles listed in 6.11. ✌✎ The total.null.voltage is specifiedby the applicabledetail spscifIcation. 6.14 Function Error. Function error refers to the deviation of the actual output voltage from the theoretical.sinusoidal.output voltage expressed as a percentageof the maximum output. The error is normally tested at 5-degree intervals,between O an~ 18o degrees.
6.15 Voltage Zero Shift Null Spacing. After zeroing the resolver,the input voltage must be varied from maximum to test to one-half test to minimum. Variation of the angular deviationfram resolver zero is specified by the applicabledetail specification. ,0 -41- Downloaded from http://www.everyspec.com MIL-HDSK-218 1 June 1962
6.16 Frequency Zero Shift of Null Spacing. After zeroing the resolver, the frequency must be .veriedfrom 5 percent above to 5 percent below tlierated value. Variation of the emgu.1.ardeviationfrom resolver zero is specifiedby the applicabledetail specification.
6.17 Friction Torque, Friction torque, measured in ounce-inches,refers to the amount of rotationalforce required to overcome friction. Friction torque is determinedby measuring the rotational.force.required to rotate the stator housing about the rotor (with the resolver in a horizontalposition) at -55”c, 5?2’c,75”C, and 85°C. The stator must be rotated at least two times in both a clockwise and counterclockwisedirectionat a rate of four to six revolutionsper minute. me friction torque value is specifiedby the applicabledetail specification.
6.18 Brush Contact Resistance. The resistancedevelopedbetween the bruahea and collector rings of the rotor shaft. To measure this resistance, a Whetstone bridge, or another suitablemeasuring device, ie connected across rotor terminalsRI,-3 and R2-4. The ehaft is then rotated through a complete revolutionat the rate requiredby.the applicabledetail specification and the change in resistancenoted. This resistancechange should not exceed that allowed-bythe specification. ● 6.19 TemperatureRise. Temperaturerise is the Increase of the internal temperatureof a resolver above the smbient tenmeraturedue to dissipationof the-energizingpower. The rate of temperature,“fiseis specifiedby tie applicabledetail apecification.
6.20 Endurance. Resolvers are subjectedto 1200 hours of rotation (1150 * ~’~various attitudes and at various temperaturesas a test of endurance. Dining this test, they must operate satisfactorily,with the specifiedtest voltage applied, as follows:
Temperature Time Attitude
-25”c(-15”F) 62 to 64 hours Shaft horlzontal
85°c (185”F) 23 tO 25 hours ‘Shaftend up, 9Q0 23 tO 25 hours Shaft end,up, 45” 23 tO 25 hOurs Shaft end down, 45”
23 to 25 hours Shaft end down, $X3”
Ambient 1036to 104J+hO.~s Shaft horizontal.
6.ZL W*. Weight representsthe overall weight of the resolver and is expressed in ounces.
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MIL-HDBK-218 1 June 1962
1 6.22 InsulationResistance. The insulationresistanceis determined using a megohm bridge with 400 volts d-c applied between the windings and 1 the case. A minimum of 10 megohms is requiredbetween any winding and the case.
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SECTION 7
INSTALLATION
7.1 General. Installationof electricalresolvers involvesthree dis- crete operations -- mounting the resolver in .schassis or on a panel, Ming mechanical connectionto the rotor shaft, ad making electrical connectionto the stator and rotor windings. Mounting by clamp assemblies is possible; however, in many cases, various types of adapters must be used. The type of installationdepends upon the particularneed for zero awustment and upon accessibility,that is, whether it is more convenientto fasten the resolver fram the shaft end or the tenninsl end. Electrical connectionsare readily 1 accomplished,whether the resolveruses screw-t~e terminals,connectorpins, or lead wires:
7,2 Precautions. Resolvers are designed to withstand shock, vibra- tion, mud a wide range of atmosphericconditionswithout loss of accuracy. However, errors may be introducedas a result of faulty installation. Estab- lished ‘mouutingpractices, when followed,help to provide high standards of resolver performance. Special care nr-ultbe exercised when installingthe smaller feaolvers, such as size 8 or 11. Accuracy of performanceten~s to decrease with size. Thie is because the larger resolvers,with greater ● mechanical rigidity end closer relative tolerances,will.be less sensitive to variations in operating conditions.
7.3 Mrect Mounting From Terminsl Board End. Figure 30 shows a re- solver mounted directly on a panel from the terminalboard end. It is fastened to the panel tith three clamp assemblies (see Figure ’36)spaced 120 degrees apart about the resolver housing. (This method is used when no EU@SI. a%lui~ent iS required.) The shaft end of the resolver is first inserted in the hole in the panel, so that it is flush at point B. One end of each mounting clamp is placed against the rim of the housing, at point A. Each clamp assembly is then secured to the panel with a captive screw, which is inserted in a pretapped hole in the psnel.
.7,4 m~ting FIVSI shaft End, Using Adapter Assembly. Figure 31.shows a resolver mounted on a panel from the shaft end, using an adapter assembly (Figure ~) and three clamp assemblies. The shaft end of the resolver 1~ passed through the hole in the panel. !!headapter assembly is sligned with the four tapped holes in tie shaft end and secured with four mounting ecrews. The resolver is then withdrawn umtil the adapter is flush with the pauel. One end of each mounting clemp is placed against the flange of the adapter assembly. Each clamp assemb2y is then attached to the panel with a captive screw, which is inserted in a pretapped hole in the panel. Before the screws are tightened,a straight pinion wrench may be used to rotate tie adapter aseembly for angular adjustment. When the resolver is aligned, the three screws in the clamp assembliesare tightened, securingthe resolver at this angular position.
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MIL-RDBK-218 1 June 1962
CLAMP ASSEMBLY
\
Fi@re 30. Resolver Mouoted Dlrect2,yfrom Termlnd Board End
PINION WRENCH
.
!. 1.
Figure 31.. Resolver Mounted with Adapter Assembly from Shaft End
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MIL-HOSK-218 1 June 1962
7.5 Mounting From Terminal.Board End, Using Adapter Assembly. Figure 32 shows a resolvermounted on a panel from the terminal board end, using sn adapter assembly (Figure 38) and three clwp ‘assemblies. Tne adapter assembly is aligned with the four tapped holes in the shaft end of the resolver and securedwith four mounting screws. The adapter assembly is then inserted into the hole in the panel until Its flange is flush with the panel. One end of each mounting clamp is placed against the rim of the adapter.assembly. Each clamp assembly is then attached’to the panel with a captive screw, vkich is inserted in a pretappedhole in the panel, but not tightened. A straight pinion wrench may be used for angular adjustmentof the resolver. When the resolver is .sMgned, the.three screws in the clamp assembliesare tightened, securing the resolver at this sngdar position.
CLAMP ASSEMBLY
ADAPTER ASSEMBLY \
I u I PINION WRENCH
Figure 32. Resolver Mounted with Adapter Assembly from Terminal Board End
7.6 Mounti From Sh&ft End, Using Adjustable Clam ing Disk Assemb . Figure 33 ~~ adjustable clasping disk assembly (Figure 37). The shaft end of-the r~- solver is passed througl.the hole in the p~el until the flange of the housing is flush with the panel. The cl~ping disk assembly is tiigned with the four tapped holes in the shsft end. Four mounting screws are in- serted through the holes, but not tightened, so that the assembiy is flush with the shaft end of the resolver and the panel. A straight pinion wrench is then used for Sr@lar adjustmentof the resolver. When the resolver iS aligned, the four mounting screws are tightened, securin~the resolver at this SIV@ar pOSitiOn. ●
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MIL-lDIBK-218 1 June 1962
PINION WRENCH
~..,
> fxj i
!’ @ .,-, L.J ? 1- @ Q q!j ,,:’ @ @ .:5? ------
icLAMplNGE DISK ASSEMBLY
,. IWg-n-e33. Resolver Mounted with ClsnrpingMsk from Shaft End
7.7 MountingFrom Terminsl Board End, Using Zeroing Ring. Figure 34 shows a resolver mounted on a panel from the terminal board end, using a zeroing ring (Figure 39) and three CISMP assemblies. The zeroing ring is slipped over the shaft end of the resolver so that the tongue of the ring fits into the slot of the resolver flange. The shaft end.of the resolver is then inserted in the hole in the psnel until the ring is flush with the panel. One end”of each mounting clamp is then placed against the flemge of the resolver housing. Each clamp assembly is attached to the psnel with a captive screw, which is inserted in a pretapped hole in the panel, but not tightened. A 90-degreepinion wrench is used for angular adjustmentof the resolver. (Note that straight and ~-degree wrenches are actually interchangeable,de- pending upon accessibility;the 90-degreewrench is used here for illustration. only.) When the resolver is aligned, the three screws in the clsmp assemblies are tightened,securing the resolver at this an@lar position.
7.8 Mounting a Gear on a Resolver Shaft. Figure 35 shows how a gear is mounted on a resolver shaft by means of a drive washer and drive nut, placed on the resolver shaft. The drive washer is next placed on the shsft so that its drive nut is screwed tightly against the drive washer, which is deformed into the spline shaft, eliminatingbacklash. Figure 41 illustrates the use of the socket wrench to prevent the resolver sheft from turning during assembly. After the nut is tight, tabs on the washer are bent over the nut to secure it. This ssme procedure is followed when mounting en adjustable dial.
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MIL-HDBK-218 1 June 1962
CLAMP ASSEMBLY T
[
u u
Figure 3k. Resolver Mounted with Zeroing Ring from Terminal Board End
[
[ [
DRIVE WASH{R
SHAFT N Figure 35. Mounting of a Gear on Resolver Shaft with Adapter
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~-HDBK-218 1 June 1962
7.9 Electrical Comection. Provision is made for e%ternal electrical connectionsat the end opposite the shaft extension. Intern& stator and rotor tindings are braught out either to screw-typeterminals or connector pins, or as unterminatedlead wires. When lead wires are employed,they ame color coded in accordancewith the listing in Table II. Interconnection with screw-typeterminals is direct. For resolverswith connectorping, the leads from the chassis terminal board are first attached to a resolver connector,and then the connector is attached directLy to the terminal pins of the resolver. Table II. Lead Wire Color Code I Rotor Stator Conmensator Terminal Color Terminal Color l!erminal Color
m Red-white w. Red c1 Red-green tracer tracer
R2 Yellow-white S2 Yellow C2 Yellow-green tracer tracer
R3 Black-white S3 Black C3 Black-green tracer tracer
R4 Blue-white S4 Blue C4 Blue-green tracer tracer
Note: When sleeving is Used, a red sleeve is used for rotor leads, and a I black sleeve is used for stator leads.
I
1’
I
f
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SECTION 8
HWFERRED OR STANDARD HARIJJARE
8.1 General. Severalitems of hardware have been developedto facilitate resolver mouuting and the attaching of gears or dial.sto resolver ehafts. These hardware items are &=scrlbed In this section. Some of the items are stock materiel> end others are standard items that are not carried in stock.
8.2 Clamping Assembly. The clamping assembly (Figure 36) consists of a captive screw, j.ockwasher;end CIEUUP. ~ree clamptig assemblies,~s- placed from each other 120 degrees about the housing, mey be used to mount a resolver directly, or they w be used in conjunctionwith the adapter aseembly or zeroing ring.
L CAPTIVE SCREW p- LOCKWASHER CLAMP
~igure 36. Clemp Assemblies
8.3 clampIng Disk Assembly. The clamping tisk assemb~ (Figure 37) consists of a clemping diek, four captive screws, and four ‘lOckwasihers. Thisassembly secures the resolver against the,chassis end is made ror use on all standard resolvers. The four screws fit into threaded holes on tie Aeft end of the resolver.
8.4 Geared-ToothAdapter. The geared-toothadapter assembly (Figure 38)consists of en adapter plusfouk”captivescrew. end lockwasheis., This assembly Is used on resolvers of ell sizes. lhe four screws fit into the four tapped holes on the shaft end of the resolwr. When a,geared-tooth atipter’assembly is used, the resolver.is secured to the Cha&IiS by three clamping assemblies.
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I MIL-HDBK-218 ● 1 June 1962
I I
CAPTIVE SCREW [ LOCKWASHER U J$=+&=& i i’
Figure 37. AdjustableDisk Assembly
8.5 Zeroing Ring. The zeroing ring (Figura 39) is a flat-springsteel ring having one sector with 6 to 2U gear teeth. Adjacent to the gear sector is a ton~e which fits into a slot in the resolverhousing snd prevents the ring from turning around the resolver. When a zeroing ring is used, the re- solver is secured to the chassisby three clamping assemblies. Zeroing rings are made to fit resolversof all sizes.
8.6 Pinion llrenches.When resolversare mounted using SD adapter assembly, clsmping disk assembly,or zeroing ring, it is possible to adjust the resolverphysicallyfor electricalzero with either a straightor a $x)-degreepinion wrench. These wrenches are shown in Figure @. The chassis on which the resolver’is mounted must be provided with a hole located
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kEL-KDBK-218 I 1 June 1962 ●
CAPTIVE SC
. LOCKWASHER
.1
Fi~re ~, Geared-ToothAdapter
so that when the pinion wrench is inserted in the hole, the teeth on the wrench engage the teeth in the assembly attached to the resolver. Wlen the clamping arrangementis loosened,the resolvermay be accuratelyadjusted for electricalzero,
8.7 SocketWench. The socket wrench shown in Figdre kl holds the “/ aplined shaft by means of an internally splined socket while the shaft nut is tightened with the outer socket. Socket wrenches are not availablefor all shafts,
8.8 Shaft Nut. Gears or diels are usually mounted on the sheft of a resolver, and the shaft nut and the drive washer (see 8.9)are used to hold them in place. These items are shown in Figure 42. shaft nuts are ~de for resolver shafte of all sizes.
8.9 hive Washer. When a LUslor a gearh mounted on a resolver shaft, there must be no backlash between the shaft and the dial or ge~, Drive washers are splined to fit the shaft, and are shaped so that when the shaft nut is tightened,the teeth will dig into the shaft,. TWO tapered ● -52- Downloaded from http://www.everyspec.com
l.!IL-HDRK-218 ● 1 June 1962
drive dogs, showm in Figure 42, engage holes or slots in the dial or gear being mounted. The maximum width of the drive dog exceeds the hole diameter or slot width; when the shaft nut is tightened,the tapered drive dogs are wedged tightly into the holes or slots. The oversize dive dogs and splined center assure a.ntibacklashmounting. For locking the shaft nut, tabs are provided which may be bent around the nut. -–t- ● o!!TONGUE
TONGUE T
. 9
Figure 39. Zeroing Ring
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90-DEGREE WRENCH
Figure 40. Pinion Wrenches I Downloaded from http://www.everyspec.com
iICL-HDBK-218 ● 1 June 1962
I INTERNALLYSPLINEDSOCKET 7
Figure 41. Socket Wrench
INTERNAL SPLINE ●
TAPERED DRIVE DOG .
Figure 42. Shaft Kardware
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C!ARRI,ERl%Q,U??’NCY
COMP!ZNSATORWINDING An additionalwinfiinginserted within the stator slots of the resolver. ‘L’liswir.t- ing is stationary~,rithrespect tc the re- solver case and is used LO correct the transformationratio for various errors occurring during resolver operation.
Cltichciseor cOJntercloclmise., rotation, ?.eterm~.ned when facin~ tle c.ha?;exi,ensi,on end of the res,olver.
The deviationof the output voltages of a resmlver frcm the defln:~ngequations, designatedas’a percentage of the w.il:lum }-alueof the ideal sinusoidal.function.
l’hevariatic,q.of’transformationratios gf ● the two rotor .,
‘i’heponiti.m of tinerotar vimii:lgvhere minimlm voltage is iniinced‘O[:tweenstator and rotor windill~s,
,lllie ,le~o.{;[jof e,,er,2izin&the resolver, .! 1.s., either the s+ator or rotor winding is energi?.cd.
Thit.angular ,posit.ionof the rotcr at vhich tinefhndamenlalfrequeficycom- p:m~nt cf each secondaryouty~t voltage is a.:nir.imum ~:’nenone primary winding is, excited u.ndthe other is shorted.
?EAK STAT(RCowilri’ The maximum current rating of a resolver state,r winding.
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MIL-IiDSK-218 1 June 1962 I That frequency at which the rotor response penks because the distributedcapacitance in the rotor resonateswith the leakage inductance.
PHASE REFERENCE VOLTAGE The fundamentalcomponentof the second- ary voltage at the first position of ma.xinmmcoupling,when the resolver rotor ., is turned in a positive directionfrom resolver zero.
PRAs.iSHIFT The electricalangle by which the output voltage differs from the Wplt voltage.
PRIPWJ?Y‘MINDING The winding that receives the energizing power.
ROTOR ANGLE The angular displacementof the resolver rotor about the rotor1s exis from the re- solver zero position, measured as an increasingpositive angle in the ● counterclockwisedirection. ROTOR WINDII’JG The winding that rotates with respect to the resolver case.
SECONDARYWINDING The winding from which the transfomned voltage is t&en.
STATOR WINDING T’newinding that is stationarywith respect to the resolver case.
TEST VOLTAGE The input voltage at which the resolver 1’ will meet all the test requirements. TRANSFOPMATIONRATIO The ratio of the ma@itude of the output voltage, at 90 degrees from rssolver 1. zero, to the magnitude of the input voltage.
WINDING DESIGNATIONS The terminal designation (S, R, or C, for stator, rotor, or compensator)followed by an even or odd number, that is, S1 - S3 for one atator winding S‘2- S4 for the opposite stator winding R1 - R3 for one rotor winding R2 . R4 for the o>posite rotor winding
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BIBLICGPAPHY
1 Resolvers,Electrical,AC, MilitarySpecificationIMIL-R.I.4346 (ORD),,13 July, 1956
2 — Synchros, 60 and 400 Cycle, YtilitarySpecificationMIL-S-20708, 15 January, 1958 . PamphletOn Servomechanisms,Ordnance EngineeringDesign Han&Dook, 31 h~, 1957
TM u-674, Departmentsof the Army and the Air Force, August, 19Z
TN 9-6093-I,Departmentof the Army Technical M.nue.l,July, 19>6
T. O. 11J332-2-S-2, Departmentof the Air Force, 25 October, 1956
OP 1303; Bureau of Ordnance Publication,15 April, 1.958
AeronauticalRecommendedPractices,ARP 461,SocXety of Automative Engineers, Inc., New York, N. y., 18 peb~ary, 19~ ● Synchro and Rotary Compcmerlts,Clifton Precision Products Company, Clifton Heights, Pa.
—10 Resolver Handbook, Reeves InstrumentC~mpa.ny,New York, N. Y.
—11 .ResolverPkinu al,American Electronic, IP.c ., Culver Cit,y,Caj.if. g“ CoordinateTransformat.icnby Xeans of Wsolvers, anrlReso>\rer Character stics, Bartol Research Fouu.chtionof the Fran!din Institute of the State of perma., Swarthmore,Penna., 10 AOri], 1944 1 . g .—_The Microsolver, Engineeri~gReport NO. 2’j~B,Farra.ndOptical Co., Inc., 29 Jane, 1954 I
—14 ApplicationsFactors for ijl.ectrikal.Rasol~ Sidne>-Davis, Research Group, Arm Corparatio~~&-ch 1~>~
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MIL-HDBK-218 1 June 1962 I
INDEX
Paragraph Page
Adapter, geared tooth 8.4 50 Addition, vector 4.2.1..3 25 Analog computers 4.2.2. 25 Applications 4 24 Assemb~ Clamping 8.2 50 Clamping disk 8.3 50 Automatic control systems 4.2.1 24 Axes, perpendicularityof 6.11 41
Bmsh contact resistance 6.18 42
Capacitiveloading 3.8.1 17 Carrier operations Without winding compensation 3.13.4 20 With winding compensation 3.13.3 20 Clamping assembly 8.2 50 ,0 Clamping disk assembly 8.3 Classificationof resolvers 6.2 E Compensation Temperature 3.12 18 Winding 3.7 14 I Windings without 3.10 17 Compensatorcomponents 1.3.6 Computation,sine 4.2.1.1 2t Computers,analog 4.2.2 25 Computing device, resolverused as 3.3.2 I 12 Connections,electrical ;.; 49 Considerationsof resolverloading 17 Constructionof a resolver 1:3 Conversionof polar coordinates 4.2.1.2 2; , Conversionof recta.a,glarcoordinates 4.2.1.4 25 ,’ Current feedback 3.12.1 18 Current, input 6.6 40
Data transmission 4.2.3 29 Resolver used for 3.3.1 12 D-c resistmce 6.8 40 Definitionof a resolver 1.2 1 Diract current, effects of 3.~6 21 Distortionharmonic 3.15 21
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Faragraph Page
Distributionof windings L.3.5 4 Dividing Head, test use 3.17.1.2 21 Drive washer 8.9 5P
Effects of Direct current 3.16 .~l_ Loadin~ 3..12.2 l’jl , Temperature 3.9 17 Electrical.balnnce, rotor 3.6 14 Electrical connections 7.9 49 Employment,resolver Endurance .:::0 4; Equality of the transformationratio 6.9.L 40 Eq~f~alent circuit, remlver 3.2 11 Error Function 3.>.1 13 3.17.3 22 Interaxis 3.5.2 l!I 6.11 Ill Proportionalvoltage 3.5.3 1.4 Resolver 13 Excitation,mode of u 39
Feedback, current 3.12.1 18 Four-winding resolver 2.4.3 Frequency 6.4 4: Frequency response 3.13 19 Measurement 3.17.6 22 Frequency zero shift of null spacing 6.16 v< Friction torque 6.17 42 Function error 3.5.1 13 !ieasurement 3.17.3 ’22 Performancespecification 6.14 41 Fundamentalnull voltage 6.12 Iti Fund.wnentals 2 5 . Geared tooth adapter 8.4 53
Hardware, preferred or standard 8 Jo Harmonic distortion 3.15 21 Housing 1.3.2 I
Input Current 6.6 40 Power 40 Voltage ::; 40 Downloaded from http://www.everyspec.com
MIL-HDBK-218 1 June 1962
Paragraph Psge
Inputg, principal types of 3.3 12 Installation -1 44 General 44 Precautions ;:: 44 Insulationresistance 6.22 43 Interaxis error 3.5.2 14 Measurement 6.UL
Loading Effect of :.gi2 19 Reelstive and capacitive ., 17 Measurements,resolyer 3.17 21 Mechanicalpositioningof rotor 3.17.1.1 21 Mode of excitation 6.3 Mounting a gear on a resolver shaft 7.8 :? Mounting from shaft end Using adapter aesembly 7.4 44 Using a~ustable clamping disk assembly 7.6 46 Mounting from terminal.board end Direct Using adapter assembly :2 Using zeroing ring 47
Nominal phase shift 6.10 4Q Null spacing Frequency zero shift of 6.16 42 Voltage zero shift of 6.~5 41 Null voltage 3.14 20 Fundamental 6.12 41 Measurementof 3.17.4 22 Total 6.13 hl Nut, shaft 8,8 52
Operation Basic 8 Reciprocal %2.1 27 Theory of, 3 11 Performancespecifications 6 3? Phase Shift, measurement 3.17.5 22 Pinion wrenches 8.6 51 Polar coordlnatee,conversionof 4.2.1.2 Power, input 6.7 R Proportionalvolt~e error 3.5.3 14
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MIL-RDBK-218 1 June 1962
Paragraph Page
Radar weep resolution 4.2.5 31 Riviio,transformation 3.11 18 6.9 40 Reciprocal operation 4.2.2.1 27 Rectangular coordinates Conversion of 4.2.1.4 25 Rotation of 4.2.2.2 28 Resistance * Brush contact 6.18 42 D-c 6.8 4C Insulation 6.22 J+3 Resistances, stator and rotor 12 Resistive capacitiveloading % 17 Resolver Classification 6.2 39 Constrwtion 1.3 1 Definition 1.2 1 Employment 2.2, 5 Equivalent circuit 11 Error ::; 13 Four-winding 2.4.3 9 ● 3.13 19 Frequency response 3.17.6 22 Measurements 3.17 21 Rotor-excited 2.3.1 7 Stator-excited 2.3.2 Thee -vinding 2.4.2 i Two-winding 2.4.1 8 Used.as a computer 3.3.2 12 Used for data transmission 3.3.1 12 Zero 2.3 -f Resolver loading, considerationsof 3.8 1+ Resolver shaft, mounting gear on 7.8 b-( Response , Frequency 3.13 19 Stator to compensator 3.13.2 20 Stator to rotor 3.13.1 19 . Ring, zeroing 8.5 51 Rotation of rectangularcoordinates k.2.2.2 28 Rotor 1.3.4 Ball bearings 1.3.4.1 t Electricalbalance 3.6 14 Lubricants 1.3.k.2 4 Mechanicalpositioningof 3.17.1.1 21 Resistance 3.4 12 Rotor-excitedresolvers 2.3.1 7
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MIL-HDBK-218 ● 1 June 1962
Paragraph Page
Sawtooth voltages 3.3.2.4 12 Shaft nut 8.8 Shifters, phase k.2.L ;: ✎✌✚✌Sine computation 4.2.1.1 24 Sine-wave voltages in phase 3.3.2.1 12 Sine-wave voltages shifted 90 degrees 3.3.2.2 12 Socket wrench 8.7 5!! Specifications,perfomce 6 39 Square-wavevoltages 3.3.2.3 12 Stator 1.3.3 Stator-excitedresolvers 2.3.2 Stator resistance 3.4 Stator to compensatorresponse 3.13.2 Stator to rotor response 3.13.1 Sweep resolution,radar 4.2.5 Synchro resolvers,typical . Systems, automatic control i.2.l Systems, typical 4.2
Temperature Compensation 3.12 18 Effects of 17 Rise ::?9 42 Temperaturevariations Phase shift due to 3.10.1 17 Trnnsforqationratio due to 3.10.2 17 Test circuits 3.17.1.2 21 Testing procedures 3.17 21 Theoti of operation 3 11 Three-windingresolver 2.4.2 Torque, friction ‘ 6.17 Total null voltage 6.13 Transformationratio 2:? Eue to temperaturevariation 3.10.2 Equality of 6.9.~ Measurementof 3.17.2 21 Transmission,data 4.2.3 Two-windingresolver 2.4.1 Typical synchro resolvers Typical systems :.2
Vector addition k.2.1.3 25
Voltage Input 6.5 ho
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L-KtBK-218 i? June 1962
Paxagraph Page
Null \ 3.lb 20 Zero shift of null spacing “6.15, 41 {olte.ges In phase, sine-wave 3.3.2.1 12 ~“ Sawtooth 3.3.2.4 12 Shifted go degrees,“sine-weve 3.3,2.2 12 ● Square-wave 3.3.2.3 12
Washer, drive 8.9 Weight 6.21. z Winding compensation 14 Carrier operationswith ;::3.3 20 Carrier operationswithout 3.13.4 20 Equivalent circuit 3.7.2 15 Phase relations 3.7.3 ly Schematic circuit 3.7.1 14 Win&ings Distributionof 1.3.5 4 Without compensation 3.10 17 Wrenches; pinion 8.6 51 ● Wrench, socket 8.7 52 Zeroing ring 8.5 51 Zero, resolver 2.3 7
...... _ ... .
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P0S7A0RANo FEES PAID * — .- DOD 314 s: o. OFFICIAL BUSINESS I PENALTY FOR PR[VATE USE S3M 1.‘ Commander US Army Armament Research and Development Command Attn: DRDAR-TST-S Dover, NJ 07801 ,., . t I , ‘$ FOLD
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