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Talk TECHNICAL ROTASYN Understanding Resolvers and Resolverto-Digital Conversion TECHNICAL TECHNICAL An introduction to resolvers and resolver-toAn introduction to resolvers and resolver-to-digidigital converters. resolvers work, resolAnconverters. introduction toHow resolvers resolver-to-digital How resolvers and work, resolver sigver signal format, and how to use tal format, converters. work,resolvers resolver signal and How how resolvers to use resolvers with comwith available resolver-to-digital nalcommercially format, and how to use resolvers with commercially available resolver-to-digital converters. converters. mercially available resolver-to-digital converters. Talk Fundamentally, then, all resolvers produce signals proportional to the sine and cosine of their rotor T A S Y N has a unique combinatiangle. O Since every angle O Tand A cosine S Y values, N on of sine a resolver provides absolute position information within one revolution (360°) of its rotor. This absolute (as opposed WHAT IS A RESOLVER? to incremental) position capability is one of the resolver’s main advantages over incremental enUnderstanding Resolvers and Resolver-to-Digital Conversion A resolver is a position sensor or transducer coders. Understanding Resolvers and Resolver-to-Digital Conversion which measures the instantaneous angular position of the rotating shaft to which it is attached. ELECTRICAL CHARACTERISTICS What isand a resolver? Electrical Resolvers their close cousins, synchros, have Electrically,Characteristics the Rotasyn, like a traditional resolver, What is asince resolver? Electrical Characteristics been in use before World War II in military is a transformer in which the coupling between A resolver is a position sensor or transducer which meaElectrically, the Rotasyn, like a traditional resolver, is a applications such as measuring and controlling the primary and the secondaries varies as the A resolver is a positionangular sensor or transducer meaElectrically,inthe Rotasyn, like a traditional resolver, is a sures the instantaneous position of thewhich rotating transformer which the coupling between the primary the angle of gun turrets on tanks and warships. sine and cosine of the rotor angle. Whereas a trasures the instantaneous angular position of the rotating transformer in which the coupling between the primary shaft to which it is attached. Resolvers and their close and the secondaries varies as the sine and cosine of the Resolvers are typically built like small motors with ditional resolver has its primary on the rotor and shaft to synchros, which it ishave attached. Resolvers andbefore their close and angle. the secondaries varies as the sine and has cosine of the cousins, been in use since World rotor Whereas a traditional resolver its pria rotor (attached to the shaft whose position is to its secondaries in the stator (necessitating brucousins, synchros, have been in use since before World rotor angle. Whereas a traditional resolver has its priII in militaryand applications as measuring and mary theslip rotorrings and its the stator (necesbeWar measured), a stator such (stationary part) sheson and orsecondaries a rotatingintransformer to War II in military applications suchonastanks measuring and mary on the rotor and secondaries in thetransformer stator (necescontrolling the angle gun turrets and warsitating and slipits rings or a rotating to which produces theofoutput signals. couplebrushes signals into the primary), the Rotasyn concontrolling the angle of gun turrets on tanks and warsitating brushes and slip rings or a rotating transformer ships. Resolvers are typically built like small couple signalsprimary into the primary), the Rotasyn contains in to The word resolver is a generic term formotors such with de- a tains both and secondary windings the ships. Resolvers are typically built like small motors with a couple signals into the primary), the Rotasyn contains rotorderived (attachedfrom to thethe shaftfact whose position is tomost be meaboth primary and secondary in thetostator and vices that at their bastator and uses a uniquewindings solid rotor directly rotor to the shaft ismechanical to be meabothaprimary and windings inthe thecoupling stator and a operate stator (stationary part)position which the uses unique solidsecondary rotor to directly vary besicsured), level(attached they by whose resolving theproduces vary the coupling between the primary andand the sured), and a stator (stationary part) which produces the uses a unique solid rotor to directly vary the coupling beoutput signals. tween the primary and the secondaries. angle of their rotor into its orthogonal or Cartesisecondaries. output signals. tween the primary and the secondaries. an (X and Y) components. From a geometric perLike all transformers, the Rotasyn (as well as a The word resolver is a generic term for such devices deLike all transformers, the Rotasyn (as well as a traditional spective, the relationship between the rotor angle traditional resolver) requires an AC carrier or reThe word resolver is a at generic termbasic for such devices deLike all transformers, the carrier Rotasynor(as well as signal a traditional rived from the fact that their most level they operresolver) requires an AC reference (some(θ) and its X and Y components is that of a right ference signal (sometimes also called the excitatirived from the fact that at their most basic level they operresolver) requires an AC carrier or reference signal (someate by resolving the mechanical angle of their rotor into times also called the excitation ) to be applied to its pritriangle:angle sensors are used, often built into on) to be applied to its primary. The amplitude of ate by resolving the mechanical angle of their rotorFrom into times The alsoamplitude called the of excitation ) to besignal applied to its modupriits orthogonal or Cartesian (X and Y) of components. mary. reference is then the driving motors. On the basis their physicala this reference signalthis is then modulated by the its orthogonal or Cartesian (X and Y) components. From a mary. The amplitude of this reference signal is then modugeometric perspective, relationship between the rotor lated thecosine sine and of theangle rotor angle to produce design, these angularthe transducers can be classisine by and ofcosine the rotor to produce the geometric perspective, the relationship between the rotor lated by the sine and cosine of the rotor angle to produce angle (θ)two andmain its X and Y components is that of a right the outputsignals signals on secondaries. fied into groups: output onthe thetwo two secondaries. angle (θ) and its X and Y components is that of a right the output signals on the two secondaries. triangle: R1 (Red/White) S1 (Red) triangle: θ R1 (Red/White) S1 (Red) θ 1 Secondary Primary sinθ 1 (Cosine) Secondary Primary sinθ (Cosine) θ θ cosθ cosθ Resolving an Angle into its Components Resolving Angleinto intoits itsComponents Components Resolving ananAngle Fundamentally, then, all resolvers produce signals proFundamentally, then,and all resolvers signals portional to the sine cosine of produce their rotor angle.proportional the sine cosinecombination of their rotorofangle. Since everytoangle hasand a unique sine and Since every has a provides unique combination of sineinforand cosine values,angle a resolver absolute position cosine values, a resolver provides absolute position information within one revolution (360°) of its rotor. This abmation(aswithin one revolution (360°)position of its rotor. This absolute opposed to incremental) capability is solute incremental) position capability is one of (as the opposed resolver’s tomain advantages over incremental one of the resolver’s main advantages over incremental encoders. R R R2 (Yellow/White) S3 (Black) R2 (Yellow/White) S3 (Black) S2 (Yellow) S2 (Yellow) Rotasyn Schematic Rotasyn Schematic Rotasyn Schematic Secondary (Sine) Secondary (Sine) S4 (Blue) S4 (Blue) By convention, the primary and secondary leads of all By convention, the primary and secondary leads of all resolvers are identified with the nomenclature and wire resolvers are identified with the nomenclature and wire colors shown above. For best performance, the reference colorsshould shownbe above. best performance, the reference signal a sineFor wave. signal should be a sine wave. In any transformer, there is a value which relates the out- Talk TECHNICAL ROTASYN By convention, the primary and secondary leads of all resolvers are identified with the nomenclature and wire colors shown above. For best performance, the reference signal should be a sine the primary. For resolvers, this quantity is called the transwave. the primary. For resolvers, this quantity is called the transformation ratio TR and isthis specified atisthe pointrelates of transmaxiIn any transformer, there isquantity a value which the primary. Fororresolvers, called the formation ratio or TR and is specified at thesecondary point of maxithe output voltage produced by the mum coupling between primary and secondary. For in- to formation ratio or TR and is specified at the point of maximum coupling between primary and secondary. For inthe primary. For resolvers, this quantity is called the transthat fed into the primary. For resolvers, this quandustrial resolvers, the de-facto standard transformation mum coupling between primary and secondary. For industrial resolvers, de-facto standard formation ratio or the TR and is specified at transformation theor point maxitity isis called the transformation ratio TR of and is ratio 0.5, which means that the maximum voltage produstrial resolvers, the de-facto standard transformation ratio is 0.5,atwhich means that theand maximum voltage promum coupling between primary secondary. For inspecified the point of maximum coupling betduced either secondary is half amplitude of theproratio is by 0.5, which means that the the maximum voltage duced either secondary is half the amplitude of the ween by primary and secondary. For industrial resoldustrial resolvers, the de-facto standard transformation reference signal. duced by either secondary is half the amplitude of the vers,isthe de-facto standard transformation ratio reference signal. ratio 0.5, which means that the maximum voltage pro-is reference signal. 0.5, which means that the maximum voltage proIfthe we define the reference voltage V as V , then the duced by either secondary is half the amplitude of the primary. For resolvers, this quantity is calledRthe trans(R1–R2) Ifduced we define the reference voltage V(R1–R2) as VR, then the by either secondary is half the amplitude voltages on the secondaries are given by the following signal. TR and isvoltage specified at theas point maxiformation ratio VR, of then the Ifreference we define theorreference V(R1–R2) voltages on the secondaries are given by the following of the reference signal. equations mum coupling between primary and secondary. For involtages on the secondaries are given by the following equations If we we define define the reference voltage V VR, the Ifdustrial the reference voltage V(R1–R2) as Vas (R1–R2) R, then resolvers, the de-facto standard transformation equations then the voltages on the secondaries are given by Sine Secondary: V ≡ V = V TR sin θ voltages on the secondaries are given following (S2–S4) S by the R voltage ratio is 0.5, which means that the maximum proSine Secondary: V(S2–S4) ≡ VS = VR TR sinθ the following equations Cosine Secondary: V(S1–S3) V =V VR TR TR sin cos equations duced by either secondaryV is half ≡ ofθθthe ≡the VCSamplitude = Sine Secondary: R Cosine Secondary: V(S2–S4) (S1–S3) ≡ VC = VR TR cosθ reference signal. Cosine Secondary: Vangle ≡V =rotor V TR cos θ where θSine is the mechanical of the assin shown Secondary: V(S1–S3) ≡V VRRTR TR θ Sine Secondary: V(S2–S4) º VS =CS = VR sinq (S2–S4) where θ is the mechanical angle of the rotor as shown as TR , cosq then If weCosine define the reference voltageº≡VC V(R1–R2) previously the Rotasyn schematic. Cosine V(S1–S3) VR Secondary: Vangle V θ the Rshown where θ Secondary: is in the mechanical the rotor asVcos (S1–S3) of V C = R TR previously thesecondaries Rotasyn schematic. voltages oninthe are given by the following previously inSignal the Rotasyn schematic. Resolver Format where mechanical angle of the rotor as shown equations whereθθisisthe the mechanical Resolver Signal Formatangle of the rotor as previously inSignal the Rotasyn schematic. Resolver previously inFormat the Rotasyn schematic. Ifshown we excite Rotasyn primary ) with the recom≡RV Sine the Secondary: V(S2–S4)(V S = VR TR sinθ If we excite the Rotasyn primary (VR) with the recommended sinusoidal reference signal as shown below, Secondary: V(S1–S3)(V ≡ Vwith cosθ the Resolver Signal Format Ifmended weCosine excite the Rotasyn primary recomC = Vthe R TR sinusoidal reference signalR)as shown below, the mended reference signal as shown FORMAT where θ sinusoidal is the mechanical angle of )the rotor asbelow, shownthe IfRESOLVER we excite theSIGNAL Rotasyn primary (V R with the recompreviously in the Rotasyn schematic. mended sinusoidal reference signal as shown below, the If we excite the Rotasyn primary (VR) with the reResolver Signal Format commended sinusoidal reference signal as shown below, the the secondary voltages sinusoidal If we excite Rotasyn primary (VR)are withalso the recomat the same frequency and nominally in below, phase the mended sinusoidal reference signal as shown For instance, with the rotor at 0° (called Electrical Zero or EZ and marked on the Rotasyn PC board), the amplitude of the sine secondary is 0 (since sin0° = 0) and the amplitude of the cosine tude of the cosine secondary will be at its of secondary will be at its maximum ofmaximum half the refetude of the cosine secondary will be at its maximum of half the reference (since 1): rence (since cos0° =cos0° 1): its=maximum tude of amplitude the cosine amplitude secondary will be at of half the reference amplitude (since cos0° = 1): half the reference amplitude (since cos0° = 1): tude of the cosine secondary will be at its maximum of half the reference amplitude (since cos0° = 1): tude of the cosine secondary will be at its maximum of half the reference amplitude (since cos0° = 1): Ref Ref Sin Ref Sin Cos Sin Cos Ref Cos Resolver Signals with Rotor at 0° (EZ) Sin Resolver Signals with with Rotor Rotorat at0° 0°(EZ) (EZ) Resolver Signals Cos Resolver Signals with at 0° voltages (EZ) With the rotor at 45°, theRotor secondary are the With the rotor at 45°, the secondary voltages are the are With but theonly rotor at 45°, the maximum secondary voltages same 70.7% of their since sin45° Resolver Signals with at 0° voltages (EZ) With the rotor at 45°, theRotor secondary are the= same but only 70.7% of theirof maximum since sin45° = the same but only 70.7% their maximum since Ref cos45°but = only 0.707: same 70.7% of their maximum since sin45° = cos45° 0.707: sin45° =rotor cos45° = 0.707: With the= at 45°, the secondary voltages are the Sin cos45° = 0.707: same but only 70.7% of their maximumCos since sin45° = cos45° = 0.707: Resolver Signals with Rotor at 0° (EZ) With the rotor at 45°, the secondary voltages are the same but only 70.7% of their maximum since sin45° = cos45° = 0.707: Ref Ref Sin Ref Sin Cos Sin Cos Ref Resolver Signals with Rotor at 45° Cos Sin Resolver Signals with Rotor at 45° Sinusoidal Reference Applied to Rotasyn Primary Sinusoidal Reference Applied to Rotasyn Primary Sinusoidal Reference to Rotasyn secondary voltages are Applied also sinusoidal at thePrimary same fre- secondary voltages are also sinusoidal at the same frequency and Reference nominally inApplied phase with the reference. Their Sinusoidal to Rotasyn secondary areinalso sinusoidal thePrimary sameTheir frequency andvoltages nominally phase with the at reference. amplitude is proportional to the amplitude of the referquency and nominally in phase with the reference. Their amplitude isvoltages proportional to the amplitude refer-fresecondary are also at of thethesame ence, the transformation ratiosinusoidal of the Rotasyn, and the amplitude is proportional to the amplitude of the reference, the transformation ratio ofwith theRotasyn Rotasyn, and the quency and nominally inApplied phase the Their Sinusoidal Reference to Primary sine or cosine of the mechanical angle ofreference. the rotor. Using Sinusoidal Reference Applied to Rotasyn Primary ence, the transformation ratio of the Rotasyn, and the sine or cosine of the mechanical angle of the rotor. Using amplitude is proportional to the amplitude of the referthe industry-standard 0.5also TR, we can lookatatthe thesame secondsecondary voltages are sinusoidal fresine orthe cosine of the mechanical angle of the rotor. Using the industry-standard 0.5 TR, can look at the secondence, transformation ratiowe ofwith thethe Rotasyn, and the quency and nominally in rotor phase reference. Their ary voltages for different angles as they would apwith the reference. Their amplitude is proportional the industry-standard 0.5 rotor TR, we can look at the secondary voltages forofdifferent asofthey would apsine or theof mechanical angle the rotor. Using amplitude proportional theangles amplitude oftransformathe referto the amplitude thetoreference, the pear oncosine anisoscilloscope. ary voltages for different rotor angles as they would appear on an oscilloscope. the 0.5 TR, we ator thecosine secondence, the transformation ratio ofcan thelook Rotasyn, and theof tionindustry-standard ratio of the Rotasyn, and the sine pear on an oscilloscope. For with the rotor 0° (called Electrical ary voltages rotor angles they ap-or sine or cosinefor ofdifferent the mechanical angleasof the would rotor. Using theinstance, mechanical angle ofatthe rotor. Using theZero indusFor instance, with the rotor at 0° (called Electrical Zero or EZ and on the Rotasyn board), the amplitude the industry-standard 0.5 TR,can wePC can look at the secondpear onmarked an oscilloscope. try-standard 0.5 TR, we look atElectrical the secondary Zero or For instance, with at 0°PC (called EZ and marked onthe therotor Rotasyn board), the amplitude ary voltages for different rotor angles as they would apof the sine secondary is 0 (since sin0° = 0) and the amplivoltages for different rotor angles as they would EZ and marked onthe therotor PC board), of the sine secondary isRotasyn 0 (since sin0° = Electrical 0) the andamplitude theZero ampliFor instance, with at 0° (called or pear on an oscilloscope. appear on an oscilloscope. of secondary 0 (sincePC sin0° = 0) the andamplitude the ampliEZthe andsine marked on theisRotasyn board), For instance, with the rotor at 0° (called Electrical Zero or of the sine secondary is 0 (since sin0° = 0) and the ampliEZ and marked on the Rotasyn PC board), the amplitude of the sine secondary is 0 (since sin0° = 0) and the ampli- Resolver Signals with Rotor at 45°Cos Resolver Signals with at 45°is at maximum and With the rotor at 90°, theRotor sin voltage With the rotor at 90°, the sin voltage isRef at maximum and the cosine Signals voltage is zero: Resolver with Rotor at voltage 45°isSin With the rotor at 90°, the sin voltage at maximum and With the rotor at 90°, the sin is at maxithe cosine voltage is zero: the cosine voltage is zero: mum and theatcosine is zero: With the rotor 90°, thevoltage sin voltage isCos at maximum and the cosine Signals voltage is zero: Resolver with Rotor at 45° With the rotor at 90°, the sin voltage is at maximum and the cosine voltage is zero: Ref Ref Sin Ref Sin Cos Sin Cos Ref Resolver Signals with Rotor at 90° Cos Sin Resolver Signals with Rotor at 90° Ref Cos Resolver Signals Resolver Signals with with Rotor Rotorat at90° 90°Sin Resolver Signals with Rotor at 90° Cos Resolver Signals with Rotor at 90° Talk Talk Talk TT EE CC HH NN I I CC AA LL TECHNICAL T ECHNICAL With the rotor at 135°, the amplitudes of the secondary R ROTASYN O T A S Y N With the rotor at 135°, the amplitudes of the secondary R O T A S Y N voltages are the same as at 45°, but the phase of the coWith the rotor at 135°, the amplitudes of the secondary R O T ATS EY C N H N I C A L voltages are the same as at 45°, but the phase of the covoltages are since the same as at 45°, but the phase of the cosine voltage reverses cos135° is –0.707: process of removing the carrier signal—leaving sine voltage since cos135° is –0.707: of the se- TheThe Withreverses the rotor at 135°, the amplitudes process of removing carrier signal – lea-just The process of removing the the carrier signal—leaving just sine voltage reverses since cos135° is –0.707: With the rotor at 135°, amplitudes secondary The process of removing carrier just Rthe O is T called A S signal—leaving Y N condary voltages arethe the same asof atthe 45°, but the the ving just the envelope – demodulation envelope—is called demodulation and is Iperformed T E C H N C A L demodulation and is performed thetheenvelope—is called voltages thecosine same as at 45°, reverses but the phase of the cocalled and isThe performed phase ofare the voltage since is performed by demodulation the Resolver-to-Digital (R/D) by and the envelope—is Resolver-to-Digital (R/D) converter. demoduby the Resolver-to-Digital (R/D) converter. The demodusine voltage reverses since cos135° is –0.707: by the Resolver-to-Digital (R/D) converter. Thecosine demoducos135° is –0.707: converter. The demodulated sine and reThesine process removing the carrier signal—leaving just lated andofcosine resolver signals are shown below: With the rotor at 135°, the amplitudes of the secondary R O Tbelow: Asignals S Y are N are lated and resolver shown below: solver signals are shown latedsine sine and cosine cosine resolver signals shown below: voltages are the same as at 45°, but the phase of the cosine voltage reverses since cos135° is –0.707: Ref Ref Ref Sin Sin Sin Cos Cos Ref Cos Resolver Signals with Rotor at 135° Sin Resolver Signals with Rotor at 135° Resolver Signals with Rotor at 135° Cos Other rotor angles may be shown similarly. Ref Other rotor angles may be shown similarly. Resolver Signals with Rotor at 135° Other rotor angles may be shown similarly. Sin signals apResolver with Rotor While it isSignals helpful to know how at the135° resolver Talk Talk the envelope—is called demodulation and is performed by the Resolver-to-Digital (R/D) converter. The demoduSine The of cosine removing the carrier signal—leaving just Sine latedprocess sine and resolver signals are shown below: Sine Cosine Cosine and is performed the envelope—is called demodulation Cosine by the Resolver-to-Digital (R/D) converter. The demoduSine lated sine and cosine resolver signals are shown below: Cosine Sine Cosine 0° 90° 180° 270° 360 0° Demodulated 90° Resolver Secondary 180° 270° 360 Signals Signals 0°Demodulated 90°Resolver Secondary 180° 270° 360 rotor may bethe shown similarly. While it isOther helpful toangles knowof how resolver signals appear as functions time since that isCos what one sees While it is helpful to know how the resolver signals apDemodulated Resolver Secondary Signals when oneoflooks them that with isanwhat oscilloscope, it is often Demodulated ResolverConversion Secondary Signals 0° 90° 180° 270° 360 pear as functions timeatwith since one sees Resolver-to-Digital Rotor at While itrotor isSignals helpful know how resolver pear asResolver functions of angles timetosince that isthe135° what onesignals sees apOther may be shown similarly. to work the envelope Demodulated Resolver Secondary Signals when onemore looksconvenient at them with an with oscilloscope, it is(amplitude often Resolver-to-Digital Conversion pear as functions of time since that is whatitone sees RESOLVER-TO-DIGITAL CONVERSION when one looks atangles themfrequency) with an oscilloscope, is respect often Other rotor may be shown similarly. The resolver-to-digital converter performs two basic funcat the reference of the signals with Resolver-to-Digital Conversion envelope (amplitude more convenient to work with the when one looks at them with an oscilloscope, it is often While it is helpful to know how the resolver sigmore convenient to workIf with the of envelope (amplitude tions: demodulation of the Conversion resolver format270° signals to re-360 Resolver-to-Digital 90° 180° to rotor position. the rotor the resolver is turned at The0° resolver-to-digital converter performs two basic at the reference frequency) of the signals with While it is helpful to work know how the envelope resolver signals nals appear as functions of the time sincerespect that is apwhat The The resolver-to-digital converter performs twofuncmore convenient to with (amplitude resolver-to-digital converter performs basic move the carrier, and angle determination totwo provide a funcat the reference frequency) of the signals with respect Demodulated Resolver Secondary Signals a rate such that it makes one complete revolution in the tions: demodulation of converter the resolver format signals toforreone one looks them with anrespect oscillobasic functions: demodulation of the resolver The resolver-to-digital performs two basic funcpear asreference functions of time since that is what one sees to rotor position. Ifwhen the rotor of theofat resolver is turned at at thesees frequency) the signals with digital representationofofthe the resolver rotor angle. The most popular tions: demodulation format signals to retime of 10If cycles of the excitation frequency (speeds to rotor scope, position. the rotor of convenient the resolver iswork turned at it is often more to with mat signals to remove the carrier, and angle detions: theConversion resolver format signals to re- a when one looks atIf them with of anthe oscilloscope, itturned isthe often thedemodulation carrier, andofangle determination toratiometric provide to rotor position. the rotor resolver is in at move a rate such makes complete revolution Resolver-to-Digital method performing functions is called to thisthat highitit are very one difficult and cause other problems move theofcarrier, and these angle determination provide a a rate such that makes one complete revolution in thein the envelope (amplitude atcomplete theenvelope reference fretermination to provide a digital representation move the carrier, and angle determination to provide aof more convenient to work with the (amplitude a rate such that it makes one revolution in the digital representation of the rotor angle. The most popular time of 10practice, cycles but of the excitation frequency (speeds . Since the resolver secondary signals tracking conversion are useful for understanding), the envelope representation ofthe therotor rotor angle. The most popular quency) ofofthe signals with respect to rotor posi- digital the resolver-to-digital rotor angle. The popular method of pertime of at 10thecycles the excitation frequency (speeds The converter performs two basic funcdigital ofmost angle. most popular frequency) of the signals with respect ofreference 10difficult cycles of thecause excitation frequency (speeds method ofrepresentation performing is The called ratiometric represent the sine andthese cosinefunctions of the rotor angle, the ratio this high time are very and other problems in of the secondary signals can be clearly seen. Shown betion. If the rotorIfof the resolver is turned forming these functions isfunctions called traofofperforming these is called ratiometric tions: demodulation of these the resolver format signals to rethis highto are very and cause other problems in inat method method performing functions isratiometric called ratiometric rotor position. the rotor the resolver is at turned this high aredifficult very difficult andofcause other problems of the conversion signal amplitudes is the tangent ofsecondary the rotor angle. tracking . Since the resolver signals low is such theuseful envelope of the sine secondary with practice, abut are forit understanding), the signal envelope rate that makes one complete revolution cking conversion. Since the resolver secondary conversion Since the resolver secondary signals move the carrier, and anglethe determination to provide asi.. Since resolver secondary signals tracking conversion practice,apractice, but useful for understanding), envelope rate are such that ituseful makes one complete the revolution in the tracking but are for understanding), the envelope Thus the rotor angle, θ , is the arc tangent of the sine sigrespect to rotor represent the sine and of the rotorofangle, the ratio in the time of 10position: cycles of the seen. excitation frequengnals represent thecosine sine and cosine the rotor of the secondary signals can be clearly Shown bedigital representation ofcosine the rotor angle. The mostthe popular the sine and cosine therotor rotor angle, the represent theby sine and ofofthe angle, ratioratio time ofsecondary 10signals cyclessignals of thebe excitation frequency (speeds of the secondary can clearly seen. Shown be-be- represent of the can be clearly seen. Shown nal signal divided the of cosine signal: cy (speedsof this high are very difficult and cause angle, theamplitudes ratio the amplitudes is theangle. of the is signal the tangent of the rotor low is thethis envelope the sine secondary signal with method of performing these functions is called ratiometric thesignal signal amplitudes amplitudes isisthe of thethe rotor angle. high are very difficult other problems in ofof the thetangent tangent rotor angle. low is the envelope theand sinecause secondary signal with low is the envelope of the sine secondary signal with other problems inof practice, but are useful for un- Thus tangent of angle, the rotor angle. Thus theofrotor angle, thethe rotor θSince ,,isis the arc tangent ofthe thesine sine sigrespect topractice, rotor position: . the resolver secondary signals tracking conversion Vs sin θ Thus rotor angle, θ the arc tangent of sigbut are useful for understanding), the envelope rotor the position: Thus θ,of is the the =arc tangent of the sine θ ≡ arctan arctan derstanding), envelope of the secondary sigθ, isthe therotor arc angle, tangent sine signal divided by sigrespect respect to rotortoposition: Vc cos θ divided bybysine the signal: represent the and cosine of the rotor angle, the ratio nal divided thecosine cosine of thecan secondary signals canShown be clearly seen.isShown be- nal nals be clearly seen. below the enthe cosineby signal: nal divided the cosinesignal: signal: of theratiometric signal amplitudes is the tangent of theanrotor angle. low is theofenvelope the sine secondary signal with The tracking converter performs implicit arc velope the sineofsecondary signal with respect VsVs sin sinθθ arc Thus the rotor angle, θ , is the tangent of the sine sigθ ≡ arctan = arctan θ ≡ arctan arctan respect to rotor position: to rotor position: q q q º calculation arctan sinon = arctan Vs tangent the ratio of the resolver signals by sin θ Vc cosθ = arctan Vc θ ≡ arctancos nal divided by thetocosine signal: Vcresolver. This cosposition θ cos Vs aVc forcing counter track the of the ratiometric tracking converter performs ananimplicit arcarc implicit arc tangent calculation is based on the trigonoTheThe ratiometric tracking converter performs implicit Vs sin θ = arctan The ratiometric tracking converter performs an implicit arc θ ≡ arctan The ratiometric tracking converter performs an tangent calculation on the ratio of the resolver signals by metriccalculation identity Vc θ of the resolver tangent on the cos ratio signals by tangent calculation the ratio of the implicit tangent calculation on the ratiosignals of This the by forcing a arc counter to on track the position of resolver the resolver. forcing a counter to track the position of the resolver. This resolver signals by forcing a counter to track the sin( θ − δ ) ≡ sin θ cos δ − cos θ sin δ The ratiometric tracking converter performs an forcing aarc counter tocalculation track the position of resolver. implicit tangent is based on the theimplicit trigono-arcThis implicit arccalculation tangent calculation is based on the trigonoposition of the resolver. This arc tangent tangent on the ratio ofisimplicit the resolver signals by metric identity 0° 90° 180° 270° 360 implicit arc tangent calculation based on the trigonoThis equation says that the sine of the difference between metric identity calculation is based onthe the trigonometric identity forcing a counter to track position of the resolver. This metric identity Modulation Envelope of Sine Secondary Signal two angles can byδcross multiplying the sin(θbe − δcalculated ) ≡ sinθ cos − cos θ sin δ trigonoimplicit arc tangent calculation is based on the sine and cosine of the two angles and subtracting the re− δ cosd ) ≡ sin-θcosq cos δsind − cos θ sinδ sin(q - dsin( ) º θsinq 0° 90° 180° 270° 360 metric identity sin(says θ − that δ ) ≡the sinsine θ cos cos θ sinδbetween This equation of δthe− difference 0° 90° 180°of Sine Secondary 270° 360 Modulation Signal360 angles can bethat calculated bysine crossofdifference multiplying the Thistwo equation says the sine between 0° 90° Envelope 180° 270° This equation sin( θ says − δthat ) ≡that sin cosof δof−the cos difference θthe sinδdifference This equation says theθthe sine the between sine and cosine ofcalculated the two angles and subtracting thethe reModulation Envelope of Sine Secondary Signal between twobeangles can be calculated by cross two angles can by cross multiplying 0° 90° of Sine180° 360 two angles can be calculated by cross multiplying the Modulation Envelope Secondary270° Signal This equation says that the sine of theof difference between multiplying the sine and cosine the two angles sine and cosine of the two angles and subtracting the resine and cosine of calculated the angles and subtracting Modulation two angles can be byFurther, cross multiplying Modulation Envelope Envelopeof ofSine SineSecondary SecondarySignal Signal and subtracting the two results. as longthe as the rethe and difference the two is relasine cosine ofbetween the two angles andangles subtracting the retively small (δ = θ ±30°), the approximation Talk T E C H N I C A L stored in the counter, is then incremented or decremented using a voltage controlled oscillator until this error is zero, R O T A S Y N at which point δ = θ (the digital angle output of the conTECHNICAL sin(θ − δ ) ≅ θ − δ verter is equal to the resolver angle). This incrementing sults. Further, as long as the difference between the two stored in the counter, thendigital incremented decremented and decrementing ofis the angle, δor , causes it to ROTASYN may also be used, further simplifying the equation. Thus, angles is relatively small (δ = θ ±30°), the approximation using a voltage controlled oscillator until this error is type zero,of track the resolver angle, θ , hence the name of this if the two angles are within 30° of each other, the differatconverter. which point δ = θ (the digital angle output of the conence using the The results (q - dbetween ) @ q - dthe angles are subtracted, demodulated by mulsin(θ − can δ ) ≅ be θ −calculated δ verter is equal to the resolver angle). This incrementing tiplying by the reference signal, and filtered to cross multiplication shown above. A more detailed description of tracking converter operaof the digital δ, causes may bebeused, further simplifying the the equation. mayalso also used, further simplifying equati-Thus, give and a DCdecrementing signal proportional to theangle, difference or it to tionthe is available from the converter Seeofthe resolver angle, θ, hence the manufacturers. name of this type Thus, if converter, the are twowithin angles areofwithin 30° ofthe each between the resolver angle, θ, and the digiIn the this equation is implemented usingerrortrack if on. the twoR/D angles 30° each other, differ2S80 and digital 2S90 series data sheets and applicaother, the difference between the angles can be tal angle, δ. The angle,product δ, stored in the converter. multiplying D/A converters the resolver ence between the angles can to bemultiply calculated using thesignals calculated using the cross multiplication shown counter, is notes then incremented or decremented tion from Analog Devices, the 19200 series data (proportional to sin θ andabove. cosθ) by the cosine and sine of cross multiplication shown above. usingAamore voltage controlled oscillator until this erdetailed description of tracking converter operasheets from ILC Data Devices Corp. (DDC), or the 168 the digital angle, δ, which is the output of the converter, ror is zero, In the R/D converter, this equation is implemenat which point δ = θ (the digital angle tion is available from the converter manufacturers. See In as the R/D converter, equation is implemented using and 268 series data sheets from Control Sciences Inc.the below. thisD/A tedshown using multiplying converters to multiply output of the converter is equal to the resolver 2S80 and information 2S90 series product data sheetsisand applicamultiplying D/A converters to multiplytothe resolver companies the resolver signals (proportional sinθ and signals angle).Contact This incrementing for andthese decrementing of given below. tion notes from Analog Devices, the 19200 series data The results are subtracted, demodulated by and multiplying (proportional tocosine sin θ and θ) by the cosine sine of by cosθ) by the andcos sine of the digital angle, the digital angle, δ, causes it to track the resolver Devices sheets from the ILC name Data Devices Corp.of(DDC), or the 168 the reference and filtered to give a DC signal proδ, which is thesignal, output ofisthe as shown angle, θ, hence of thisAnalog type converter. the digital angle, δ, which theconverter, output of the converter, Phone: +1 617.329.4700 and 268 series data sheets from Control Sciences Inc. portional to the difference or error between the resolver asbelow. shown below. Fax: +1is617.326.8703 Contact information for these companies given below. angle, θ, and the digital angle, δ. The digital angle, δ, The results are subtracted, demodulated by multiplying by Internet: www.analog.com Analog Devices R the Vreference signal, and filtered to give a DC signal proPhone: +1 617.329.4700 portional to the difference or error between the resolver Fax: +1 617.326.8703 angle, θ, and the digital angle, δ. The digital angle, δ, ILC Data Devices Corp. Internet: www.analog.com Phone: +1 516.567.5600 + sin θ cos δ VRV Synchronous Cos Multiplying Σ S Fax: +1 516.567.7358 Demodulator D/A Converter – Internet: www.ilcddc.com ILC Data Devices Corp. Phone: +1 516.567.5600 + sin θ cos δ Synchronous Cos Multiplying VS Σ cos θ sin δ Fax: +1 516.567.7358 Demodulator D/A Converter Sin Multiplying VC Control Sciences Inc. – D/A Converter Filter Internet: www.ilcddc.com Phone: +1 818.709.5510 Fax: +1 818.709.8546 Internet: δ cos θ sin δ Sin Multiplying VC Control Sciences Inc. www.controlsciences.com D/A Converter Filter Phone: +1 818.709.5510 Voltage Fax: +1 818.709.8546 Tracking Controlled Counter Internet: Oscillator δ www.controlsciences.com sults. Further, as long as the difference between the two angles is relatively small (δ = θ ±30°), the approximation (θ − δ ) (θ − δ ) Talk δ Tracking Counter Digital Angle Output of Converter Voltage Controlled Oscillator Digital Angle Output of Converter ©1997–1998 by Admotec Inc. TT02-0598 ©1997–1998 by Admotec Inc. TT02-0598 Advanced Motion Technology Admotec Inc. 85 Mechanic Street Lebanon NH 03766-1500 USA Tel: +1 603.448.7000 Advanced Motion Technology Fax: +1 802.448.7007 Admotec Inc. E-mail: [email protected] 85 Mechanic Street Lebanon NH 03766-1500 USA Tel: +1 603.448.7000 Fax: +1 802.448.7007 E-mail: [email protected] admotec δ Typical Converter TypicalTracking TrackingResolver-to-Digital Resolver-to-Digital Converter Typical Tracking Resolver-to-Digital Converter admotec Admotec Precision AG Kieselgasse 12 CH-8008 Zürich Telefon +41 44 422 22 75 Fax +41 44 422 22 76 www.admotec.com © 2016 by Admotec Inc. All rights reserved.