Actuator dynamics

The bearing dynamics are examined using the voltage equation of the coilof

a single electromagnet. The voltage of the coil

u

^{equals to the voltage drop}

accordingtoOhm's Law

Ri

^{plusthechangeoftheux linkageintime.}

u = dψ

−5

Figure 3.10: Forcecharacteristicsof twopairedhorseshoeelectromagnetsasa

function ofthecontrol currentand displacementinthex andy planes.

0

Figure3.11: Calculateddynamicinductanceofoneelectromagnet,asafunction

of coil current anddisplacementof therotorfrom the central position toward

theelectromagnet.

−0.35 −0.2 0 0.2 0.4

Figure 3.12: Top two graphsdepict the control currentsas afunction of

dis-placements,measured(dots)andcomputedwith theRNM(solidlines),in the

radialbearings,aredepicted. Inthebottomgure,themeasuredmagneticforce

asafunctionofcontrolcurrentisshown.

−6 −4 −2 0 2 4 6

Figure3.13: Forcecharacteristicsofanaxialhorseshoeelectromagnet,forthree

biascurrents: (A)

i bias = 0.5 [A]

^,(B)

i bias = 2.5 [A]

^,(C)

i bias = 5 [A]

^.

Consideringthedisplacementoftherotorin

x

^{direction(towardthe} electromag-netwiththeinductance

L

^{),andusingthedynamic}inductance,dened earlier, thevoltageequationcan bewrittenas

u = L dyn di

Here, we introduce the ux linkage

ψ = Li

^{and the} velocity-induced voltage

k u x ˙

^(or motion-induced backelectromotiveforce). The inductanceof asingle horseshoe electromagnet (in the operating point, i.e.,

x = 0

^{, and when the}

magnetizationoftheironisneglected) canbecomputedusing theux density

in theair-gap(2.5)orreluctanceoftheair-gap(3.36)as

L = N S air dB di = N ²

2 < ^air = µ 0 N ² S air

2(l 0 − x) .

^(3.43)

Hence, using this result, comparing with Eq. (2.8), and (2.15), the

velocity-inducedvoltagecoecientmaybeapproximatedas

k u = i dL

dx = µ 0 N ² iS air

2(l 0 − x) ² ≈ 1

2 k i .

^(3.44)

Thesameresultmaybeobtainedfrom(2.5)anddirectlyfromthepartial

deriva-tiveoftheuxlinkage

ψ

^{withrespecttotheair-gap. Intheoperatingpoint,the}

k u

^{equalsthecurrentstiness ofthesingle}electromagnetthat ishalf ofthe

k i

(the currentstiness oftwopaired, opposite electromagnetsin the dierential

drivingmode). Ifamoreaccuratemodelfor

k u

^{isrequired,itcouldbeobtained}

bybuildingthelook-uptablefromthedynamicinductance.

Werecallthatweassumedacurrent-controlledAMBswiththebiasedcontrol

currents

i ref = max(i bias ± i c , 0)

^{. In the system, the input to the actuators}

consistsoffourradialreferencecurrentsandoneaxialone. Mostoften,ineach

actuator,anindependentlycontrolledcurrentloopisbuiltusingaproportional

controlleras

u ref = G P (i ref − i m ) ,

^(3.45)

where

u ref , G P

^,and

i m

^{arethereferencevoltage,the}proportionalgain,andthe measuredcoilcurrent

i

^,respectively. Whenanalyzingthedynamicsoftheinner currentclosed-loop,itistypical(e.g.Schweitzeretal.,2003,Lantto,1999,and

Larsonneur, 1990) to simplify the problem by neglecting the velocity-induced

voltage. Thisispossiblebecauseofitsrelativelyinsignicantmagnitude,

com-paredwiththevoltageofthecoil. Additionally,the

k u

^{introducesthederivative}

thathasastabilizingeect;itcounteractstheeectofthenegativeposition

sti-ness. Furthermore,itistypicalthattheresistanceofthecoilissmallandcould

beneglected. With allthese assumptions and utilizing (3.42), and (3.45), the

simplestapproximationoftheclosed-loopdynamicsbecomes

G cl (s) = i m

i ref ≈ G P

sL + G P

= 1

sτ cl + 1 ,

^(3.46)

where

τ cl

^istheclosed-looptimeconstant.

ThemoredetailedmodelincludesthePWMdelay. ThePWMdelaycanbe

approximatedusing a shiftin time in theLaplace domain and dierentseries

G f f

Figure3.14: Actuatorblockdiagram modelcomprisestheforceeld (withthe

current stiness

k i

^{and position stiness}

k x

^{), LR circuit, switched amplier}

modeledasthedelay,andinternalcurrentcontroller,where

G P

^and

G F

^indicate

theproportionalcontrollergainandfeedforwardcontrollergain,respectively.

e ^sξ ≈ 1 + sξ + (sξ) ²

2! + ... + (sξ) ⁿ

n! + ...,

^(3.47)

where

ξ

^{is the time shift in seconds. We can use (3.47) for the linear time}

invariantmodels(likestate-variableformandtransferfunctions). Forexample,

a second-order approximation of a delay is

e ^sξ ≈ 1/[s ² T _md ² /4 + sT md /2 + 1]

^.

In Matlab Simulink we can utilize a Transport Delay block, for PWM delay

toprovidemuch shortersimulationtimes comparedwiththeswitchingmodels

(Wurmsdobler,1997). Therst-orderMaclaurinapproximationcorrespondsto

therst-order(0/1)Padéapproximation(Franklinet al.,1998). Forthedelay

ξ = − T md /2

^{this rst-order}approximation,asthetransferfunction,yields

G del (s) ≈ e ^sξ ≈ exp 0/1 (sξ) = 1

1 − sξ = 1

s ^{T md} ₂ + 1 ,

^(3.48)

where

T md

^{istheaveragemodulationdelaythat equalshalf ofthemodulation}

periodowingtotheasymmetricregularsampling.

Finally, the slightly more detailed approximation of the current control

closed-loopdynamicsis

G cl (s) ≈ G P G del (s)

sL + R + G P G del (s) .

^(3.49)

Thevoltagerelationtogetherwiththeforcerelation(2.14),internalcurrent

controller and PWM delay form the complete actuator model, as presented

in Fig. 3.14, where the feedforward controller gain compensates the resistive

voltagedrop.

TheschematicofloadcapacitylimitationsoftheAMBactuatorispresented

in Fig. 3.15. Themaximum force of thebearing systemis determined by the

maximumcoil current

f max = F (i max )

^{. Themaximumconstantcoilcurrentis}

determinedbythevaluesoftheresistance

R

^{andavailableDClinkvoltage}

u DC

(

i DC,max = u DC /R

^{). However, usually this limit is high and thus} theoretical.

Inpractice,the maximumforce forthe continuousoperation islimitedby the

coiltemperaturelimit. Typically,thecoiltemperaturelimitisdescribedasthe

coilcurrent,at which amaximumcurrentdensityin acoilreachesthecertain

limit

J max = 4 − 6 A/mm ²

^{. Onebut adicult}opportunityfor moreaccurate

thermal analysis (Pöllänen et al., 2006). Another option could be to use the

fact that the static load capacity of the amplier is limited by the size and

geometryofthebearing(seeEq.2.17). Hence,tolimitthesaturationinuence

on system dynamics and linearity, we may select the limiting current to be

equaltothesaturationcurrent(seeEq.2.22). Apeaktransientcurrent,atlow

frequencies, is determined by the articially selected maximum peak current,

atwhichthecontroller,andcontrolelectronics(limitedmeasuringrangeofthe

currentsensorsandlimitednumberformat)canprovideastablesuspension. For

highfrequencies,thedynamicloadcapacityisdeterminedbyeitherthecurrent

control bandwidth (using Eq. 3.46,

ω cl ≈ G P /L

^{) orby the power bandwidth}

ω BW

^{,whicheverismore}restrictive.

A powerbandwidth resultsfrom thelimitedDC link voltage, thelow-pass

characteristicsofthecoiltransferfunctionappliedtosinusoidalsignals,andthe

maximumcurrent. It canbeestimatedbyusing theopen-loopplantscaled by

the

u DC

^{,whichis equaltothevoltagelimitedcurrentamplitudeas}

i u,max = u DC G pl = u DC

G del

(sL + R) ,

^(3.50)

wherethepeak-to-peakcurrentof

2 · i u,max

^{equalstothemaximumcurrent}

i max

^.

Thefrequencyatwhichthevoltageentersthesaturation

ω sat

^{maybedetermined}

bythe crossingofthe magnitudeBode plotsof thetransferfunction basedon

thecontrollerdynamics (3.49) andthevoltage-limitedcurrent-amplitude ratio

(3.50),specically

2u DC /i max · 1/(sL +R)

^{. Thepowerbandwidth}approximation

ω BW

^{canberead,directlyfromtheBodemagnitudeplot,}approximatelyatthe -3dBcrossingofthevoltage-limitedcurrent-amplituderatio(usedasatransfer

function). Another approximation of the dynamic force limitation that takes

theinductancevariationsinto account,asgivenby Lantto(1999),is basedon

therisetime

t rise

^{ofthecurrentinthecoilofthe}electromagnet

L dyn

^fromzero

to

i max

^{. Utilizingaresponseoftherst-order}approximationofthesystem,the powerbandwidthis

where

u DC

^{is theDC link voltage applied to thecoil. Replacing thedynamic}

inductance by the nominal one, assuming the

f max

^{at the}

i max = i sat

^{, using}

(3.43),and(2.22),thepowerbandwidthcanbepresentedin termsofthe

max-imum force amplitude

f max

^{(2.17) and the maximum amplier} performance

P max = i max u DC

^,as

Ifthecurrentcontrolbandwidthisabovethepowerbandwidth(

ω cl > ω BW

^),

the loadcapacity is limitedby theamplier performance and bythe DC link

voltage. Inconsequence, at thefrequenciesgreater thanthe

ω sat

^{, thebearing}

cannotproduceforceswithamplitudes determinedbythecontrollerdynamics.

The

ω sat

^{isdeterminedat thepointwhere thevoltagelimitcurvecrosseswith}

lo g f ( i )

Figure3.15: LimitsonAMB:(A)thestaticloadcapacity,(B)thedynamicload

capacityundercontinuousoperation,(C)thepeaktransientloadcapacity

themagnitude plotof thetransferfunction of thecontroller. Therough

rst-orderapproximationoftheactuatordynamicsbecomes

G cl (s) ≈ ω BW

s + ω BW

.

^(3.53)

Alternatively, forhigher-orderapproximationsoftheclosed-loopdynamics,we

canmodifyEq.(3.49),byutilizingadescribingfunctionmethod(Franklinetal.,

1998) and (Slotine and Li, 1991). The derivation of the describing function

Γ(G P i ref )

^{fortheactuatorsaturationispresentedinAppendixA.4. The}

quasi-linearizedactuator,forsinusoidalinputand output,yields

G cl (s) ≈ Γ(u ref )G P G del (s)

where

u ref

^{isassumed,here,to betheamplitudeofthesinusoidalinput(3.45).}

Thecontrolbandwidthcanbeestimatedasabandwidthof (3.54),where

u ref = 0.5 · G P i max

^(forsingle electromagnet). Forhigh amplitudes of acoilcurrent, theeect of thedynamic inductancemaybeincludedinto (3.54) by using(in

thecomputations),suchavaluefor

L

^{,thatgivesthesamerisetimeasthe}

L dyn

in Eq. (3.51). However,for high frequencies,the voltage is saturatedand coil

currents are bound in the regionwhere the inductanceis closeto its nominal

value.

TheexemplaryBodeplotsfordierentapproximationsofactuatordynamics

arepresentedin Fig.3.16. Themain benetof using (3.54)isthe inclusionof

boththesaturationandthemodulationdelayin onemodel.

Astothedynamicsoftwooppositehorseshoeelectromagnets,themaximum

loadcapacity at dierent frequenciesdepends on the maximumpossibleforce

slew rate, for

x = 0

^,

df /dt = k i (i) · di c (i)/dt

^{. As explained earlier, in the}

operatingpoint,thecurrentstinessofasingleelectromagnetequalshalfofthe

corresponding

k i

^{oftwoopposite}electromagnets,asdoestheforceslewrateand apeak-to-peakvalueoftheforce. Therateofchangeofthereferencecurrent,in

thesingleelectromagnet,correspondstotherateofchangeofthebiasedcontrol

current,fortwooppositeelectromagnets,whereeachofthemgeneratestheforce

−60

−40

−20 0

Magnitude (dB)

10 ⁰ 10 ² 10 ⁴ 10 ⁶

−180

−135

−90

−45 0

Phase (deg)

(A) (B) (C) (D)

Bode Diagram

Frequency (rad/sec)

Figure3.16: Bodeplotsoftheapproximatedactuatordynamics,for

ω cl > ω sat

^:

(A)basedon(3.49),(B)basedon(3.54),(C)basedon(3.53),(D)theDClink

voltagelimitfortheopen-loopEq.(3.50).

1. Ifthecontrolcurrentamplitudeislowerthanorequalto

i bias

^,therateof

changeofthecontrolcurrentisthesameastherateofchangeofthe

refer-encecurrents(bothcoilsareactive). Consequently,thevoltagesaturation

appearsapproximatelyat

ω sat

^{,and thefrequencyresponses ofthesingle}

electromagnet are the same as the frequency responses of two opposite

electromagnets(Fig.3.17).

2. If thecontrol currentamplitudeisgreaterthan

i bias

^{, theratesofchange}

of tworesultingreference currents(fromtheapplied

i c

^{)are greaterthan}

therateofchangeofthe

i c

^. Consequently,thevoltagesaturationappears earlierthanat

ω sat

(determinedforasingleelectromagnetwithoutreduced premagnetization current), and the frequency responses dier from the

predictedones(thederivedlinearmodelsaremoreuncertain). Theactual

ω sat

^{changes and depends on the control signalamplitude and bias. In}

addition,theintrinsicsystemnonlinearitiesbecomelargerforhighervalues

ofpositionandcurrentsignals(Fig.3.18).

Toput the second point in another way, for lowand medium frequenciesand

for amplitudes of the control current that are greater than the value of the

biascurrent,therateofchangeofamagneticforceiscontributedmostlybyone

electromagnet(forveryhighfrequencies,thecoilcurrentamplitudesarereduced

below

2 · i bias

^andbothelectromagnetsproduce force). Forthese reasons, the forceresponsemaydeviateconsiderablyfromthelinearizedone,attheoperating

point,modelprediction.

Figures 3.17and 3.18showcomparisonsof thenormalizedgains (resulting

from dierent force magnitudes) and phaseBodediagrams obtained from the

singleelectromagnet(A),twooppositeelectromagnets(B), anddierent

actu-atorapproximationsof thesystem with

ω cl > ω BW

^{. TheBodeplots (A)and}

0 1000 2000 3000 4000 5000 6000 0.4

0.6 0.8 1

magnitude [pu]

0 1000 2000 3000 4000 5000 6000

−80

−60

−40

−20 0

frequency [rad/s]

phase [degrees]

(A) (B) (C) (D) (E)

Figure 3.17: Frequency responses ofthe simulatedmodelsand dierent linear

approximations are presented. The

i max = 2 · i bias = 5 A

^{. The saturation}

occurred at

ω sat = 2800

^{rad/s, accordingto prediction (3.50). Thesaturation}

occurredat

ω sat = 2762

^{rad/s, accordingto}simulation.

(B) were obtained using the Simulink models, which included the force

non-linearities,inductancenonlinearities,modulationdelay,andvoltagesaturation.

Theresponses of theapproximatedmodels comprise(C) approximationbased

on(3.49),(D)approximationbasedon(3.54),and(E)approximationbasedon

(3.53). Forfrequencieslowerthan

ω sat

^{,thefrequencyresponsesoftheSimulink}

models aresimilarto theresponse ofEq. (3.49)(the agreementis notsoclose

forthesystemwiththehigh signalamplitudes), andthevoltagesdonot

satu-rate. Forthefrequencieshigherthan

ω sat

^{, theampliersenter}saturation,and thefrequencyresponsesoftheSimulinkmodelsapproachtheresponseof(3.54).

In thesimulations thesame current controllers (withthe samegains), with a

considerablyhighbandwidth

ω cl

^{(controldesignfor}

t r = 500µs

^{),wereused. The}

applied reference for single electromagnet

i ref = 0.5 · i max sin(ωt) + 0.5 · i max

^,

andtheappliedcontrolcurrentforoppositeelectromagnets

i c = i max sin(ωt)

^.

In document Design and implementation of FPGA-based LQ control of active magnetic bearings (sivua 69-77)