A Self-Sensing Homopolar Magnetic Bearing: Analysis and Experimental Results

Transcription

1 A Self-Sensing Hmplar Magnetic Bearing: Analysis and Experimental Results Perry Tsa Seth R. Sanders Gabriel Risk Department f Electrical Engineering and Cmputer Science University f Califrnia, Berkeley Abstract One f the chief bstacles t cnstructing a reliable, lw-cst, practical magnetic bearing is designing a suitable psitin sensr. A hmplar magnetic bearing that uses the same cils t stabilize the rtr and sense its psitin is intrduced. A prttype bearing has been built and successfully used t stabilize a rtr. The electrmagnetic analysis f the self-sensing functin, the implementatin f the prttype design, and experimental results verifying the theretical analysis f the self-sensing scheme are presented in this paper. 1 Intrductin Sensrs are a critical element in all active magnetic bearings, but unfrtunately, they are cstly and frequently represent the weakest pint f the system with respect t reliability. As part f an effrt t design an ecnmical flywheel energy strage system, a self-sensing hmplar bearing has been designed and develped. In a self-sensing bearing, the same cils used fr stabilizing the rtr are als used t sense its psitin. This eliminates prblems with aligning and lcating sensrs in tightly cnfined and pssibly envirnmentally hstile areas, reduces cst, and imprves reliability by reducing part cunt and cmplexity. An inductive sensing scheme similar in sme respects t that described here is discussed in [1]; hwever the sensr in [1] is nt integrated int a self-sensing bearing and uses a sinusidal carrier rather than a PWM wavefrm. In additin, the scheme in [1] addresses nly sensing alng ne axis. In the present paper, the basic principles f peratin fr a hmplar magnetic bearing, the electrmagnetic analysis f the self-sensing scheme, the design f the prttype bearing, and experimental results f the prttype bearing are presented. Principles f Operatin The magnetic flux paths f the prttype bearing are essentially the same as thse in a cnventinal hmplar bearing [, 3]; hwever, in ur design rtr lsses are minimized by using a sltless statr and a tridal winding scheme. A cutaway view f the bearing is shwn in Fig. 1, shwing the tw statr stacks and the magnet used t prvide dc bias flux. Fur cils are Figure 1: Cutaway view f hmplar magnetic bearing. wrapped tridally arund each f the statr stacks, with each cil ccupying ne quadrant. There are tw phases n each statr, each cnsisting f tw cils in ppsite quadrants, cnnected in series with their magnetic flux in ppsitin. Figure shws a cutaway view with the ac and dc flux paths superimpsed. Fr clarity, nly the ac cntrl flux path thrugh the upper statr is shwn; a similar pattern exists in the lwer statr. Frce is applied t the rtr when the ac flux reinfrces the dc permanent magnet bias flux n ne side f the rtr and reduces the field n the ppsite side f the rtr. Varying the currents in phases AC and BD cntrls the directin and magnitude f the ac cntrl flux, and thus the frce n the rtr. Since frce is prprtinal t the square f flux density, the bias flux reduces the pwer needed t perate the bearing. 3 Electrmagnetic Analysis f Self- Sensing Operatin The sensing functin f the magnetic bearing is analyzed, and it is shwn here that psitin measurement in bth axes can be accmplished by sampling vltage at the midpint f nly ne f the tw phases. Thrughut the analysis, variables subscripted with a capital A,B,C, r D indicate values pertaining t cil A, cil B, cil C r cil D, respectively. Subscripts AC r BD indicate values pertaining t the phase AC r phase BD, where phase AC cnsists f cils A and C cnnected in series and wund in ppsitin, and phase BD cnsists f cils B and D als cnnected in series and wund in ppsitin. Definitins fr several f the variables are indicated in Fig. 3 The distance between the rtr and the inner diameter f the statr at angle, and its crrespnding inverse, can be apprx-

2 Figure : 3D cutaway view shwing flux paths. Figure 3: Schematics f statr and cils. Fr clarity, phase BD and the ac cntrl flux it induces have been mitted in the diagram n the left. imated t first rder as: g( ) ß g x cs y sin (1) 1 ß 1 1 x + cs y + sin () g( ) g g g where g indicates the nminal gap. Fr ease f analysis, the MMF distributin in the statr is apprximated as sinusidal: MMF statr = MMF AC + MMF BD (3) = N I AC sin N I BD cs (4) where N is the number f turns per cil. T calculate the MMF f the rtr, we first slve fr B( ), the magnetic flux density thrugh the gap at angle : B( ) =μ H( ) =μ MMF statr ( ) MMF rtr g( ) We then cmbine (), (4), and (5), and then slve Maxwell s magnetic flux cntinuity equatin by integrating ver the surface f the rtr t find the rtr MMF: 0 = ) MMF rtr = Z ß 0 (5) B( ) d (6) 1 4g (NI AC y NI BD x) (7) Nw t relate the displacements t the inductance, we slve fr the flux linkage in each winding. First the flux thrugh a

3 crss-sectin f the statr at is defined as Φ( ) (cnsult Fig. 3): Φ( ) =Φ +Z Z Φ radial d =Φ + hrb( ) d (8) Here h represents the height f the statr, and R its inner radius. The flux linkages fr cil A and cil C are then calculated by taking the definite integral f the flux ver the length f the winding: Λ A = N ß Z ß 4 Φ( ) d (9) ß Z 4 3ß 4 N Λ C = ß Φ( ) d (10) 5ß 4 The term N ß, where N is the ttal number f turns in the cil, is the turns density. The cnstant Φ represents the flux at, and is canceled ut in the definite integral. The resulting flux linkages are: ψ μ p hrn p! y Λ A = [ I AC ßg g + 1 x (11) 4 g ψ p!# y x + I BD + 1 y g 4 g ψ μ p hrn p! y Λ C = [ I AC ßg g 1 x (1) 4 g ψ p!# y x + I BD 1 y g 4 g ) Λ A μ hrn Λ C = [ x I ßg AC + y I BD ] (13) Therefre, the difference in the flux linkages f the tw cils in phase AC is prprtinal t the psitin f the rtr and the current in phases AC and BD. Fr nw, we neglect the resistive drps t btain: v a v c = d(λ A Λ C ) μ hrn =» x di AC ßg + y di BD (14) (15) under the assumptin that x and y change slwly. T first rder, the ttal inductance f phase AC, Λ A +Λ C ß p μ hrn I AC = L AC I AC (16) ßg is cnstant with respect t x and y. A similar analysis can be dne t btain L BD = L AC. We then substitute diac = v AC L AC and dibd = vbd L BD v A v C = 1 and and L BD int (14) arriving at: 4 p g [ x v AC + y v BD ] (17) Figure 4: System diagram f prttype. Thus, it has been shwn that t first rder the difference in vltage between cil A and cil C is linear with respect t the vltage applied at the terminals f phase AC and phase BD and bth displacements x and y. An analgus result hlds fr phase BD and v B v D. v B v D = p 1 [ x v AC + y v BD ] (18) 4 g Sampling the midpint vltage f phase AC, fr example, allws ne t deduce the individual cil vltages v A and v C. Nte that since v A v C cntains infrmatin n bth x and y it is pssible t sense bth displacements x and y by sampling the vltage at nly the midpint f phase AC, r equivalently, by sampling at nly the midpint f phase BD. There are a variety f PWM drive schemes that can facilitate this, but the mst straightfrward methd and the methd used in ur prttype senses x by sampling when v BD =0and senses y by sampling when v AC =0. The analysis shwn here is nt limited t bearings with ur winding cnfiguratin r hmplar bearings. Fllwing the analysis thrugh fr bearings with a typical active magnetic bearing winding cnfiguratin wuld lead t similar results. The nly restrictin is that the ppsing cils need t be cntrlled such that their currents are equal. This requirement is autmatically met in bearings where ppsing cils are cnnected in series. 4 Prttype Implementatin 4.1 System Overview A prttype self-sensing bearing was cnstructed and its selfsensing functin has been successfully demnstrated and used t stabilize a 1.5 kg rtr. Fig. 4 shws the cmpnents f the system. The diagram shws bth phases f the radial bearing, where phase AC cntrls mtin alng the y- axis, and phase BD cntrls mtin alng the x-axis. In rder t remve the cmmn mde signal, an analg differential

4 Figure 6: Mdel f bearing with resistances. Figure 5: Example PWM wavefrm. amplifier was used t cmpare v AC;mid and v AC;ref t create v AC;diff. The amplifier als scaled and biased the signal int the crrect range fr the A/D cnverter, which sampled v AC;diff. A prprtinal-derivative (PD) cntrller and ther cntrl lgic was implemented using a field prgrammable gate array (FPGA). As nted earlier, vltages nly need t be sampled at ne f the tw midpints, thus nly ne A/D channel, and ne analg frnt end is necessary. Hwever, as is shwn in ur diagram, a secnd sensing channel was implemented in ur prttype fr testing purpses and t verify ur mdel. 4. Errr canceling sampling scheme Example PWM wavefrms fr v AC and v BD as implemented in ur prttype are shwn in Fig. 5. Wavefrm v AC is shwn at 65% duty cycle and v BD is shwn at 50% duty cycle. Nte that v AC is cmmanded t be zer at times kt +t 1 and kt +t fr all integers k, independent f duty cycle. Likewise, v BD is always cmmanded t be zer at times kt + t 3 and kt + t 4. Frm inspectin f (17), it may seem that the easiest methd f sensing psitin y wuld be t sample v AC;diff at t 1, and then use the assumptin that V s = v A + v C and the fact that v AC;ref = V s = t arrive at v A v C =v AC;diff. Hwever, the assumptin that V s = v A + v C ignres resistive drp f the cils, which can be significant since these cils are als used t actuate the bearing. An intuitive way f understanding hw the sensing is affected by the resistance f the cils is t think f the bearing mdel develped earlier as a black bx with tw leads frm the terminals f each phase and ne lead frm the midpint f each phase (see Fig. 6). Placing a resistr in series with each terminal lead will nt affect the inductance relatinships inside the bx, therefre (17) and (18) still hld. Hwever, in this mdel, when currents are applied, the vltages the bearing mdel sees are v AC i AC (R A + R C ) and v BD i BD (R B + R D ), which changes (17) t: v A v C = 1 4 p g [ x (v AC i AC (R A + R C )) (19) + y (v BD i BD (R B + R D ))] Clearly, an unwanted current dependence has appeared. Hwever, these currents are assciated with stabilizing the rtr and therefre change slwly in cmparisn t the sampling frequency. T remve this current dependence in the prttype implementatin, v AC;diff is sampled at times t 1 and t. Examining Fig. 6 yields: v AC;diff = 1 (v C v A )+ 1 (R C R A )i AC (0) Thus: v AC;diff (t ) v AC;diff (t 1 ) (1) = 1 [v A(t 1 ) v C (t 1 )] 1 [v A(t ) v C (t )] + 1 (R C R A )(i AC (t ) i AC (t 1 )) T see the current dependence f [v A (t 1 ) v C (t 1 )] [v A (t ) v C (t )], (19) is evaluated at times t 1 and t, which leads t: [v A (t 1 ) v C (t 1 )] [v A (t ) v C (t )] () = p 1 f x [( i AC (t 1 )+i AC (t ))(R A + R C )] 4 g + y [V s +[( i BD (t 1 )+i BD (t ))(R B + R D )]]g Using the assumptin that the currents change slwly in cmparisn t the sampling perid implies i AC (t 1 ) ß i AC (t ) and i BD (t 1 ) ß i BD (t ). Then, expressin () simplifies t: [v A (t 1 ) v C (t 1 )] [v A (t ) v C (t )] = p 1 V s y (3) g Substituting back int (1) leads t: v AC;diff (t ) v AC;diff (t 1 ) (4) = 1 4 p g V s y + 1 (R C R A )(i AC (t ) i AC (t 1 )) Using the assumptin f a slwly changing current again yields: v AC;diff (t ) v AC;diff (t 1 )= 1 4 p y g V s (5)

5 Figure 7: Pht f self-sensing hmplar bearing. Analgusly, the abve analysis can easily be mdified t shw that: v AC;diff (t 3 ) v AC;diff (t 4 )= p 1 x V s (6) 4 g Therefre, it has been shwn that ur self-sensing scheme cancels ut errrs frm resistive drps due t bth phase currents i AC and i BD when sensing psitin alng bth the x and y axes. By sampling v AC;diff twice and subtracting, nt nly were the errrs in () due t the ttal phase resistance remved, but the errrs in (4) due t imbalances f the cil resistances in each phase were als canceled ut. Design values Rtr diameter 5.3 cm Stack height 1cm Nminal gap 1 mm (radial) Laminatins M19 Steel (14 mil) # Turns/Cil 10 # Cils/Phase PWM switch frequency 0 khz Calculated design values dc bias field strength 0.8 T Max. frce 500 lbf Inductance 30μH Sensing gain :6V=mmat V s =15V Measured values Inductance f each phase 6μH Resistance f each phase 0:091Ω Sensing gain 0:75V=mmat V s =15V Table 1: Self-sensing hmplar bearing design summary. 5 Experimental Results A pht f the prttype appears in Fig. 7., and sme parameters fr the prttype are presented in Table 1. The prttype was cnstructed using tw statr stacks and tw rtr stacks built up frm M19 steel laminatins. The rtr stacks were attached t a slid steel bdy. A ferrite permanent magnet prvided the dc bias flux. The cils were cnstructed frm slid cpper pieces bent arund the statr stacks. The measured inductance was significantly higher than that predicted by (16) due t leakage inductance. A simple digital cntrller was implemented with an FPGA. Sme scillscpe traces f signals in the system are shwn in Figs. 8, 9, 10. Fig. 8 shws v AC at its minimum, n-lad, 50% duty cycle. The ntches cut int the wavefrm at t 1 and t are apprximately :5μs lng. In Fig. 9, the tp trace displays a lgic utput frm the cntrller that indicates whether psitin data fr the x-axis r y- axis is being sampled. The lwer trace is v AC;diff. Ntice hw v AC;diff is ffset frm its nminal vltage at times t 3 and t 4, which crrespnds t a large x-axis displacement. The y- axis displacement is much smaller, therefre at times t 1 and t v AC;diff stays much clser t its nminal vltage. Fig. 10 shws a clseup f v AC;diff at time t 3. The 1μs time perid t s is when the A/D samples v AC;diff. Figure 8: Pht f v AC wavefrm n scillscpe. Data frm the self-sensing utput is graphed in Fig. 11. As shwn in the graph, the utputs f the sensing system, v AC;diff (t 4 ) v AC;diff (t 3 ) and v BD;diff (t 4 ) v BD;diff (t 3 ), are linear with respect t x. As is evident frm the figure, the respnse was similar regardless f which phase was sampled, whether it be phase AC at times t 4 and t 3 r phase BD at times t 4 and t 3. This agrees with equatins (17) and (18) f the analysis, which predicted that the magnitude f the sampled vltage wuld be linearly prprtinal t the displacement, and the same n bth phases. In additin, n measurable crss-axis cupling was bserved, i.e. mtin alng the y-axis did nt influence the x-axis signal, and vice-versa. These bservatins all agree with ur analysis. The gain f the sensing scheme as shwn in Fig. 11 is ap-

6 Sensr Perfrmance Sensr utput vltage (mv) Figure 9: Tp trace is lgic utput indicating whether psitin data fr the x-axis r y-axis is being sampled. Lwer trace is f v AC;diff X axis displacement (mm) Va Vc Vb Vd Figure 11: Graph f sensr utput versus rtr displacement. 6 Cnclusin A self-sensing hmplar magnetic bearing has been cnstructed and successfully used t stabilize a rtr radially. An electrmagnetic analysis f the self-sensing scheme has been presented, and measurements cnfirming the analysis and successful peratin f the sensr have been described. References [1] T. Haga S. Mriyama, K. Watanabe. Inductive sensing system fr active magnetic suspensin cntrl. In 6th Internatinal Sympsium n Magnetic Bearings, Cambridge, MA, [] G. Schweitzer, H. Bleuler, and A. Traxler. Active Magnetic Bearings. vdf Hchschulverlag, [3] C. Srtre, P. E. Allaire, E.H. Maslen, R.R. Humphris, and P.A.Studer. Permanent magnet biased magnetic bearings-design, cnstructin, and testing. In nd Internatinal Sympsium n Magnetic Bearings, Tky, Japan, Figure 10: Clse up f v AC;diff at time t 3 with sampling time t s marked. prximately 0:75V=mm, and was btained with V s = 15V. The gain predicted by Eqn. (6) is :6V=mm. The discrepancy is due in large part t the large leakage inductances which reduce the effective vltage n the windings. Fr example, the measured inductance was mre than twice the cmputed inductance, accunting fr at least a factr f tw. Other cntributing factrs may be additinal lead inductance, spatial winding harmnics, and three-dimensinal effects.