Calibration of position sensors

Measuring therotor position in xy coordinates

In theprototype, there aretwo dierentialposition sensors at one end of the

rotor(end-A)andthree single-channelposition sensorsattheotherend ofthe

rotor(end-B),accordingtoFig.3.6. Assumingtheoriginofthecoordinate

sys-teminthecenterofthestatorandplacingthedierentialsensorsonthexandy

axesofthesystemresultsinthedirectmeasurementsoftherotorcenteratthe

end-A.Thesingle-channelsensors areequallyplacedin thestatorframeevery

120

◦

at theend-B (Fig. A.1). Therefore, obtainingthe rotorcenter

(x m , y m )

at theend-Brequires utilizingthe geometrypropertiesof thecircle(rotor

cir-cumference)circumscribedaroundthetriangleformedbythreemeasuredpoints

(

x km , y km

^).

Wedrawthestraightlinesfromthesensorstothecenterofthestator;they

arethesensinglines,indicatedbydashedlinesinFig.A.1. Thethreepointsof

theintersectionsbetweenthesensinglinesandtherotorcircumference(wecall

themmeasuredpoints)are

ro-tor. After some transformations, the measured center of the rotor in the xy

y x

s e n s o r - 1 s e n s o r - 2

s e n s o r - 3

b e a r i n g - 1 b e a r i n g - 3

b e a ri n g - 4

b e a ri n g - 2

( x m , y m ) ( x 1 m , y 1 m ) ( x 2 m , y 2 m )

( x 3 m , y 3 m )

FigureA.1: Single-channelsensorsareequallyplacedinthestatorframe,atthe

end-Boftherotor,atevery120

◦

. Thecoordinatesoftherotorcenter

(x m , y m )

canbeobtainedfromthegeometrypropertiesofthecircle.

x m = a 1 a 2 (y 2m − y 3m ) + a 2 (x 2m + x 1m ) − a 1 (x 1m + x 3m )

2 (a 2 − a 1 ) ,

^(A.12)

y m = a 2 (y 1m − y 3m ) − a 1 (y 2m + y 1m ) + (x 3m − x 2m )

2 (a 2 − a 1 ) ,

^(A.13)

wherethecoecients

a 1

^and

a 2

^are

a 1 = (y 1m − y 2m )

(x 1m − x 2m ) , a 2 = (y 3m − y 1m )

(x 3m − x 1m ) .

^(A.14)

Errorsin themeasuredrotor position

Letusexaminetheeectsoffourerrorsourcesonthemeasuredrotorposition.

Assumingthethreesingle-channelsensors,therstandsecondexaminederrors

aregainandoseterrors. Thethirdis anerrorinthepositionofthesensorin

thestatorframe(deviationfrom120

◦

equaldistributionofthesensors,

α

^angle

measuredin degrees). The fourth isthe inaccurate direction of the tip of the

sensor (angle

δ

^{betweenthe idealsensing line and theactual sensing line), as}

presentedinFig.A.2.

In orderto determine how these errorsinuence the calculatedposition of

therotor,wecomparetheobtainedair-gapsfromtheideallyplacedsensorswith

theair-gaps of thedeviated sensors. The inuence of dierent errortypes on

thecalculatedcenter oftherotorispresentedinFig.A.3.

When considering the accuracy of the measured rotor position, the least

importantbutnonlinearrelationexhibitstheerrorintheangle

δ

^{. Thecalculated}

y x

FigureA.2: Twotypesoferrorsin thepositionofthesensorsaredepicted,i.e.,

anerrorinthedirectionofthetipoftherstsensor

δ 1

^anddiscrepancyin120

◦

error in gain of 1st sensor [%]

relative error

error in offset of 1st sensor related to airgap [%]

relative error

error in α of 1st sensor [degree]

relative error

error in δ of 1st sensor [degree]

relative error

(D)

Figure A.3: Relative errorsin themeasured position of the rotor(in relation

to the average air-gap)are presented. Theresults correspondto the errorsin

measured

x m

^{for8dierentrotorpositionsat theend-B.}

xy position of the rotordepends mostly on the gain, then on the oset, and

nallyontheaccuracyofsensororientation

α

^{. Theerrorsinthegainandoset}

canbecompensatedbyrst-orderpolynomialapproximation.

Calculatingthe gain and oset

The gainand oset scaling could be based on themeasured air-gaps, the

ex-pectedair-gapsthatarecomputedaccordingtothedimensionsofthetestrotor

fromTableA.1(e.g.forfour dierentrotorpositions)andbyleastsquares

ap-proximation. Thefollowingprocedureis proposed forcalculatingthegainand

osetvalues:

1. By using the open-loop biasing currents, the center line of the rotor is

moved to four dierent positions

(x, y)

^:

(l bias , 0)

^,

(0, l bias )

^,

( − l bias , 0)

^,

(0, − l bias )

^{, where}

l bias

^{is the clearance, betweenthe safety bearingsand}

therotor.

2. Ateachposition,themeasuredair-gapsarestoredinthetable. Therotor

ismovedastomake

0.5k

^{circlesaccordingtotheclockdirectionand}

0.5k

circlesin thecounterclockdirection,where

k

^isintegerand

k ≥ 2

^.

3. Theexpectedidealair-gapsarecalculatedassumingnoerrorsinthe

mea-surementsandposition ofthesensors.

4. The measuredair-gaps and the corresponding calculatedideal ones

(ex-pected) arestored in twovectors

x

^and

y

^, respectively. Using theleast squares approximation,the gains andosets arecomputed foreach

sen-sorindividually. Theequationssuitablefortheimplementationarelisted

below.

(a) Let us dene themeasured position function fora singlesensor as:

x m = F (x) = θ 1 x + θ 0

^,wherethe

θ 1

^{, and}

θ 2

^{arethegainandoset.}

Then,wecancalculatetheJacobeanmatrix

J

^{ofthesensorfunction}

foreachmeasurement

k

^as

J =

(b) Next,wecanselecttheweightingmatrix

W

^{tobetheidentitymatrix}

and selecttheresidual vectoras

y

^{. Assumingthat the}

θ = [θ 1 , θ 0 ]

is identiable and that

J ^T J

^is non-singular,we canwrite (e.g. ac-cordingtoFranklinetal.,1998)theexplicitsolution

θ =

J ^T J −1

J ^T W y.

^(A.16)

(c) Thiscomputationisnecessaryforeachsensor.

Thepresentedprocedure isstraightforward. However,there canbecertain

diculties intheutilization ofthemethod andcalibrationoftheposition

sen-Table A.1: Dimensions of the test rotor according to naming convention of

Fig.3.6.

Selectedspecication Rotorend-A (2sensors)

/end-B(3sensors)

Rotordiametersatsensorsleeves[mm] 110/70

Diametersofsafetybearings[mm] 74.4/54.4

Rotordiametersatsafetybearings[mm] 75/55

•

^{Thesensorsmaybe}inaccuratelypositionedresultinginerrorsin

α

^and

δ

^.

•

^{Theresultingvectorsofmagneticforces(generatedbytheoppositeradial}

electromagnets) andthex andy coordinatesaremisaligned.

•

^Measureddisplacementmayexceedthemeasuringrange.

•

^{Thesafetybearingsandmagneticbearingsarepositionedina}non-concentric way. Thez-axesofsymmetryarebiased.

•

^{Owing to} non-uniform clearance and friction, the rotorcannot be posi-tioned accurately when using the open-loop bias currents. The radii of

theretainerbearingsarenotknownprecisely.

•

^{There areerrorsin} measurementsbecauseof arun-outof therotor (dis-crepancyin therotorshape).

As an alternative, it may be practical and easier to calibrate the sensors

ac-cording to the currentsin thecoils (orcontrol currentsorestimated constant

disturbances),intheclosed-loopsystem. Themethodutilizesthegravityforce

vectorthat is assumed to be directed between the x and y axes of the rotor

coordinates. Themagnetic forces, in steady-state,are proportionalto control

currentsanddisplacements,andthereforeosetsofthesensorsareadjustedto

minimizedierences betweencorrespondingmeasuredcurrentsofthebearings

that actin x andy axes (forslowlyrotatingrotor). Inthismethod, thegains

of theposition sensorsare assumedto beknown. Aftersuchacalibration, for

thezerodisplacementof therotor,therotorposition iscorrelatedwiththe

op-eratingpoint(onlytheforcecomponent,resultingfromthecurrentstiness, is

present). Next,weshouldmeasuretheclearancesundertheclosed-loopcontrol.

Finally, the rotorreference position would be given such that there are equal

clearancestothetouch-downbearings,inpositiveandnegativexy directions.

In document Design and implementation of FPGA-based LQ control of active magnetic bearings (sivua 146-150)