Actuators

Inthis study, theactuatorscomprisethe switchingampliers and the

electro-magnets. Theamplier convertsthe control signals,namely control currents,

into the electrical currentsin thecoils. These currents produce themagnetic

eldin theelectromagnet,which inturnproducesthemagneticforce.

2.3.1 Electromagnets

Formulti-degreeof freedomsuspensionsystem,weuse radialand axialAMBs

x y

i = i b i a s + i c , x

i b i a s + i c , y

i b i a s - i c , y

c u

i b i a s - i c , x

Figure2.4: Geometryofthestudied radialbearingwithits coilsand currents,

where

i c,x

i c,y

i bias

^and

χ

^are^control ^currents^associated^with ^x ^and^y ^axes,

biascurrentandforceactingangle,respectively.

polestatorarrangedinto fourhorseshoeelectromagnets. Theyarewoundand

connected such that wehave NSNS conguration and four total coil currents

perradialbearing. By usingthe current biasing, theoppositeelectromagnets

arepairedinorder toprovidetwoperpendicularforcecomponents,specically

in the direction of x and y axes, as in Fig. 2.4 and Fig. 2.5. With such an

arrangement,thereis,notmuchuxcouplinginthestator,whichwouldresult

intheforcecouplingbetweenthexandydirections. Asanalternative,anNNSS

congurationof thefour horseshoeelectromagnetswould lowerrotatinglosses

butincreasetheforcecoupling(Antila,1998),increasetheironsaturation,and

decreasethemaximalattainablemagneticforce.

Ingeneral,in thebearingswithhorseshoeelectromagnets,thecurrent

con-trol is simple,but the minimal loadcapacityis lower(i.e., lessecientuse of

iron) compared with the bearings with independently controlled coils. As an

illustration, in the eight-pole radialAMB with horseshoe electromagnets, the

force (e.g. in x direction) is generated only by thesingle electromagnet (two

poles). However,in thebearingswith independentlycontrolledcoils,theforce

isgeneratedbythefourelectromagnets(fourpoles). Usually,theincreasedload

capacity isnotworththeexpense. It requires twice asmanypowerampliers

asthetraditionalsolution.

In the traditional statorconguration (Fig. 2.4), the control of the forces

canbeperformedwith onlytwocontrolcurrents

i c,x

i c,y

^. ^A^constant

premag-netizationcurrent, alsocalled abias current

i bias ≤ 0.5 · i max

^is ^applied ^to ^all

ofthecoils,wherethetotalcoilcurrentis

0 ≤ i ≤ i max

^. ^The^reason^for^such^a

controlscheme,calledadierentialdrivingmode,isanonlinearrelationofthe

attractivemagneticforcewithrespectto thecurrentanddisplacement.

ThemagneticforcephenomenacanbeexplainedwiththehelpofMaxwell's

equation, namely the currents and changing electric elds produce magnetic

elds

∇ × H = J + ∂ D

∂t ,

^(2.1)

where

H

J

^and

D

^are^the^magnetic^eld ^strength,^current^density,^and^electric

N S S N

Figure2.5: Ontheleft,themainuxpathsarepresented. Ontheright,theload

capacityof theeight-pole radialbearing,with statorarrangedinto four

horse-shoe electromagnets (NSNS conguration), is indicated. The tip of maximal

forcevectorformstheperimeter oftherectangle.

ux density. Assuming no time dependence and applying the Kelvin-Stokes

theorem,weobtaintheintegralformofAmpère's CircuitalLaw

I

Theleft-handside integralof Eq.(2.2)iscarriedoutalongtheclosed uxline

c

^, ^which ^denes ^the circumference of the area

S

^. ^We ^consider ^the ^magnetic

circuitof Fig.1.1. The ux travelsthrough twomedia: ironand air, and the

uxdensity

B = const

^is ^equalⁱⁿ ^both^media. ^The^magnetic^circuit^equations

are

relativepermeabilityofiron. Furthermore,assuming

µ Fe 1

^themagnetization ofironcanbeneglected,andthemagneticuxdensityintheair-gapbecomes

B = µ 0 N i

( _µ ^l ^Fe _Fe + 2l air ) ≈ µ 0 N i 2l air

.

^(2.5)

Now, having the ux density, we can compute the eld energy stored in the

air-gap. Assuminglinearmagneticcircuit,thestoredmagneticenergy

W fe

^and

co-energy

W ce

^are ^equal. ^For^a^single^horseshoeelectromagnet,asin Fig.1.1, andhomogeneouseld intheair-gap,themagneticenergyequals

W ce =

where

S air

^is ^the cross-section area of a stator tooth tip. According to the principleofthe virtualwork,presentedbyKrause andWasynczuk (1989),the

force equalsthepartial derivativeof the magneticco-energy

W ce

^with ^respect

to the virtual displacement

l

^. ^The ^derivation ^of ^this ^relation ^is ^repeated ⁱⁿ

Appendix A.1. Using theprevioussolutionfor

W ce

^, ^the^magnetic ^force

f

^can

bewritten as

f = ∂W ce

∂l = B ² S air cos χ µ 0

,

^(2.7)

where,

χ

^is^the^force^acting^angle^(half^the^angle^between^the^poles^of^an

electro-magnetequalsto

π

8

^). Substitutingmagneticuxdensity(2.5)thatneglectsthe eect of magnetization of iron gives theapproximationof theattractiveforce

generatedbythesingleelectromagnet

f = µ 0 N ² i ² S air cos χ

4l ² _air .

^(2.8)

Thesinglehorseshoeelectromagnetgeneratesonlyattractiveforceinone

direc-tion.

Considering the force that can act in opposite directions along the x ory

axis,generatedbythepairoftwooppositehorseshoeelectromagnets,yields

f x = f 1,x − f 2,x = µ 0 N ² S air cos χ

where

(x, y)

^represents^thecoordinatesoftherotorcenterrelativetothebearing center, respectively. Bearing in mind the nonlinear, attractive nature of the

magneticforce,it isconvenienttolimitthecoilcurrents

i 1 , i 2

^and ^introduce^a

biaslinearization,e.g for

f x

i 1 = max(i bias + i c , 0),

^(2.11)

where

l 0

^and

i bias

^are^the^nominal^air-gap^and ^the^bias^current,^which ^is^equal

toorsmallerthanthehalf ofthemaximalcoilcurrent(

i bias ≤ 0.5 · i max

^). ^The

force relation, linearized about the operating point (

x = 0, i c = 0

^), ^for ^two

coupledhorseshoeelectromagnets,becomes

f x = k i i c,x + k x x,

^(2.14)

k i = ∂f

∂i c | x=0, i c =0 = µ 0 N ² S air i bias cos χ

l ₀ ² ,

^(2.15)

k x = ∂f

∂x | x=0, i c =0 = µ 0 N ² S air i ² _bias cos χ

l ₀ ³ ,

^(2.16)

where

k i

^and

k x

^denote ^the^current^stiness ^(actuator^gain)^and ^position

sti-ness(perceived by arotoras anegative open-loop stiness). In addition, the

introducedbiaslinearizationprovidesalargerforceslewrate(maximum

possi-blerateofchangeofamagneticforce),whichintheoperatingpointisdoubled,

in regard to the single magnet. Some authors even suggest to ignore the

ef-fectofsaturation(KnospeandCollins,1996),whenahighbiascurrentisused.

Forsingleelectromagnet,thecurrentstinessand position stiness(linearized

about the operating point) are equal to half of the values computed for two

coupledhorseshoeelectromagnets.

Themaximumforce,which thesingleelectromagnetcanattainisassumed

at thesaturated siliconiron statorcore

B sat ≈ 1.6

^T,^for ^the^rotor ^remaining

in the central position. Assuming the saturation occurring at the saturation

current

i c + i bias = i sat

^and ^using ^Eq. ^(2.7), ^the ^maximal ^force ⁱⁿ ^terms ^of

bearingdimensionsyields

f max = B _sat ² S air cos χ

µ 0 ≈ L br d r · 37 N

cm ² ,

^(2.17)

where

µ 0 = 4π · 10 ⁻⁷ NA ⁻² , L br

^and

d r

^are^the^bearing^core^length^and^the^rotor

diameter. Fortheeight-poleactuator,theloadcapacityvarieswithdirectionas

shownin Fig.2.5.

2.3.2 Ampliers

In the actuator, the coils of the electromagnets are fed by the power

ampli-ers. The ampliers operate within the current control scheme, where the

control current isbiased, asexplained earlier, producingthe referencecurrent

(

i ref = i bias ± i c

^). ^Then, ^a ^total ^coil ^current

i

^is ^generated ^according ^to ^the

referencesignalbythepowercircuitusinginternalcurrentcontrolandcurrent

feedback as in Fig. 2.6. There are twotypes of power ampliers: linear and

switchedpowerampliers. Inthesimplecase,thelinearamplieriscontrolled

bythecurrentcontroller,whichisjustaproportionalgainandthecurrent

feed-back. Such alinearamplier producesripple-freecurrent,but hashigh power

losses. Therefore, almostall applicationsuse switched powerampliers. With

thistypeofampliers,abettereciencyisachieved,butwehaveto dealwith

electromagnetic compatibility (EMC) issues. Another, but a small

disadvan-tageof the switching ampliers is the ripple producedin the current and the

remagnetizationofthemagneticcircuit,which itcauses. This ripple decreases

withanincreasedcarrierfrequencythatleadstoashorterswitchingperiod.

As the power circuit, we used a three-phase line-frequency diode rectier

to produce a DC link voltage and an H-bridge switching amplier (depicted

in Fig.2.7) to build thecoil current. The switching pattern isproducedwith

carrier-basedPWM (twocarriersignals) and asymmetricregular sampling(at

double carrier frequency) analogically to Holtz (1992) and Zhang and Karrer

(1995). ThisPWM resultsin theunipolarswitchingscheme,describedby

Mo-C u r r e n t c o n t r o l

a n d P W M P o w e r c i r c u i t

C u r r e n t m e a s u r e m e n t c i r c u i t i r e f

i m

A m p l i f i e r

i

Figure 2.6: Schemeofcurrentamplierwithanradiallyappliedelectromagnet

u D C

R c o i l L c o i l

S 1 D 1 S 2 D 2

u L L

u m d

Figure2.7: PowercircuitwithH-bridgeswitchingamplier

orbetweenzeroandnegativeDClinkvoltage. Indetail,thetrianglecarrier

sig-nal

u tri

^(and^its^inverse)^are^compared^with^the^reference^voltage

u ref

^produced

bythelocalcurrentcontroller. Thisresultsinthefollowingswitching patterns

fortheS1and S2transistorsoftheH-bridgeamplier asinFig.2.7:

S1: When

u ref > u tri

^, ^the^S1^is^on.

S2: When

u ref > − u tri

^,^the^S2^is^on.

The PWM signalsand currentsresulting from this scheme, when asinusoidal

u ref

^is^applied,^are^shownⁱⁿ^Fig.^2.8^and^Fig.^2.9,respectively.

In document Design and implementation of FPGA-based LQ control of active magnetic bearings (sivua 39-44)

x y

i = i b i a s + i c , x

i b i a s + i c , y

i b i a s - i c , y

c u

i b i a s - i c , x

i c,x

i c,y

i bias

χ

i c,x

i c,y

i bias ≤ 0.5 · i max

0 ≤ i ≤ i max

∇ × H = J + ∂ D

∂t ,

H

J

D

N S S N

I

c

S

B = const

µ Fe 1

B = µ 0 N i

( µ l Fe Fe + 2l air ) ≈ µ 0 N i 2l air

.

W fe

W ce

W ce =

S air

W ce

l

W ce

f

f = ∂W ce

∂l = B 2 S air cos χ µ 0

,

χ

π

8

f = µ 0 N 2 i 2 S air cos χ

4l 2 air .

f x = f 1,x − f 2,x = µ 0 N 2 S air cos χ

(x, y)

i 1 , i 2

f x

i 1 = max(i bias + i c , 0),

l 0

i bias

i bias ≤ 0.5 · i max

x = 0, i c = 0

f x = k i i c,x + k x x,

k i = ∂f

∂i c | x=0, i c =0 = µ 0 N 2 S air i bias cos χ

l 0 2 ,

k x = ∂f

∂x | x=0, i c =0 = µ 0 N 2 S air i 2 bias cos χ

l 0 3 ,

k i

k x

B sat ≈ 1.6

i c + i bias = i sat

f max = B sat 2 S air cos χ

µ 0 ≈ L br d r · 37 N

cm 2 ,

µ 0 = 4π · 10 −7 NA −2 , L br

d r

i ref = i bias ± i c

i

Mo-C u r r e n t c o n t r o l

a n d P W M P o w e r c i r c u i t

C u r r e n t m e a s u r e m e n t c i r c u i t i r e f

i m

A m p l i f i e r

i

u D C

R c o i l L c o i l

( _µ ^l ^Fe _Fe + 2l air ) ≈ µ 0 N i 2l air

∂l = B ² S air cos χ µ 0

f = µ 0 N ² i ² S air cos χ

4l ² _air .

f x = f 1,x − f 2,x = µ 0 N ² S air cos χ

∂i c | x=0, i c =0 = µ 0 N ² S air i bias cos χ

l ₀ ² ,

∂x | x=0, i c =0 = µ 0 N ² S air i ² _bias cos χ

l ₀ ³ ,

f max = B _sat ² S air cos χ

cm ² ,

µ 0 = 4π · 10 ⁻⁷ NA ⁻² , L br