AXISYWMETRIC WINDING DESIGN UNDER ...

IEEE TRANSACTIONS ON MAGNETICS, VOL. MAG-21, NO. 6, NOVEMBER 1985

A PROCEDURE

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WINDING DESIGN UNDER PARAMETRIC CONSTRAINTS: TBE RFX P0LOIDb.L TRANSFORMER

FORAXISYWMETRIC

AN APPLICATION TO

Massimo Guarnieri and Andrea Stella Abstract - The problem of winding design to perform a specified field synthesis is usually carried out as analytic determination of current distribution within a givenregion.Thisapproachmaybeus,efulfor conceptual or preliminary design, but presents severe technical difficulties from a construction point of view. A construction oriented procedure for axisymmetric air core windings for fusion devices has been developed, to account for parametric constraints such as current per turn, current density, number of turns and diagnostic accessibility. Starting from a preliminary tentative design, presenting su'itable electrical parameters, each coil position is automaticallyadjusted,withinspecifiedlimits,to achieve the required field synthesis. A package has RFX been implemented and succesfully applied to the [ l ] Poloidal Field Transformer design and the main results are reported in the paper together with a number of numerical examples. INTRODUCTION The most common approach to two or three dimensional field computations nowadays is represented by Finite Element or Difference methods. This approach consists of three steps: automatic grid generation, iterative' field equation computation and [2], Due to graphicrepresentationoftheresults theirgenerality,thesemethodsarecomplex,and hardly ever suitable for inclusion as routines in a morecomplexprocedureperformingautomaticfield synthesis. For axisymmetric\and linear configurations, a much faster and easier approach can conveniently be used.Thefieldcomponentscanbecomputedasthe superimpositionofcurrentfilamentcontributions, obtained through the Legendre complete elliptic integrals [3]. The most common approach to magnetostatic field synthesis is based upon the linearity between current filaments and field component.contributionsleadingto a Fredholm-type integral equation, where current density distribution, within a fixedwindingcrosssection,isunknown. These equations belong to the class of the ill-posed problems, whose solution can be correctly obtained by minimizationof a properfunction e.g.the copper weight [ 4 , 5 ] . The method is fast and convenient, but raises some problems as far as technical feasibility For instance the solution can require a is concerned numberofturnsnotsuitabletothepower ypply capabilities. Manufacturing problems can also arise If very uneven coil shapes are obtained.

Winding specifications The winding has to comply with tight specifications which can be summarized as follows: a)to

store the magnetic flux as required to ionize thegasandtoinduceandsustaintheplasma current inside the vacuum vessel; b) to present the correct inductance and resistance valuestofitthecircuitandpowersupply capabilities; c)to comply with layout constraints imposed by the load assembly and other machine components, and to leave accessibility for diagnostics; d) to produce negligible field over the plasma regiontopreventequilibriumproblemsandto facilitatedischargebreakdown(theprescribed equilibrium "vertical" field being generated bya . specific different set of coils). Thefirstthreeconditionscangenerallybe easily met by means of an ordinary,package able to compute the magnetic field and the flux produced by a given current distribution within a set of axisymmetric coils placed around the toroidal vacuum vessel. It should be pointed out that minimization of the power requirement is achieved if condition a) is met with a good mutual coupling between winding and vacuum vessel and that condition 6 ) may require such a low number of turns as to represent a serious constraint in complying with condition d). This is because the specified maximum stray field inside the vacuum vessel is usually some orders of magnitude lower than the highestmagneticfieldintheregioninsidethe central solenoid.

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PROBLEM

OUTLINE

Problemsofthiskindareusuallyencountered when designing the poloidal transformer for toroidal fusionmachinessuchasTokamaks,RFPsorsimilar devices. The magnetizing winding consists of a number of coaxial turns, in general grouped ina set of coils of different size, mirrorwise placed with respect to a 1 asanexample.The midplane,asshowninFig. resulting geometry is generally rather complex and, especiallyforaircoremachines,evenrelatively small coildisplacementscanproduceunacceptable stray fields in the plasma region. TheauthorsarewithIstitutoGasIonizzati Associazione CNR-EURATOM Via Gradenigo, 6/A PADOVA Italy.

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Fig. 1 Pictorial view o f a typical magnetizing winding arrangement. The vacuum vessel is placed in a region completely enclosed within the coils. Problem approach Trial and error coil positioning can meet condition d ) only in a rough approximation, which is usually not enough for a pr-oper design. Nevertheless such a tentative winding configuration, presenting the appropriatenumberofturns,coilsandcurrents, represents the starting point for a further automatic

0018-9464/85~1100-2449%01.0001985 IEEE Authorized licensed use limited to: University of Science and Technology of China. Downloaded on July 12,2010 at 06:35:05 UTC from IEEE Xplore. Restrictions apply.

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integral quantity and the solution of equation ( 2 ) is optimization performed by a new package able to adjust each coil position, within given boundaries, to reduce much less sensitive and leads possibly to a much lower accuracy in terms of local field B. This was found the stray fields below specified limits: the field particularly important in our case, the condition on synthesis performed not is through current the configuration accuracy being given in terms of redistribution but through rigid coil displacements. field. In otherwordsthecoilcross-sectionandcurrent remain unchanged while its barycenter can be displaced Physical model in the R direction (an increase or. reduction of the coil radius) and in theZ direction (rigid translation Fromapracticalpointofvieweachcoilis along the main axis) as shown in Fig. 2, where a so represented by a proper number of current filaments cylindrical reference system is assumed. that the magnetic field can be computed in any point as the sum of contributions resulting from well known formulaeinvolvingtheLegendrecompleteelliptic integrals of first and second kind[ 3 ] . For small AP; the variation of each field contribution A B i can be i t " coil cross section linearized as hPi -ARiu,*AZiu, AB.(AP.,Q) 1 1

=

-*

Api

8Pi

i Fig. 2. Cylindrical coordinates system used in the model. Q is any point of the W region where the is the coil synthesis performed, is APi displacement. ANALYTICAL

where pi represents the curvilinear coordinate and the derivatives are computed as average values over an appropriate and fixed APmax. In this way (1) becomes a linear equation where the known quantity is the error field Be= Bs-Bo, the unknowns are the displacements AP; and the coefficients are the field derivatives .described above. It can be solved as a least square problem, finding the set of APi displacements which minimizes the form

BACKGROUND

Magnetic field approach The unknown quantities are the coil displacements AP;= A R ~ - u R + A Z i - u from Z their former positions and can be determined from an equation which also includes the magnetic field variations, written in the following form:

N

COILS

n

U

N

Q representsageneralpointinsidetheregion W where the synthesis is intended to be performed, AB; is the field variation produced in by Qthe displacement AP; ofthei-thcoil ; Bo 1 s the original field produced in Q by the whole winding, B, Q and N isthe is the desired synthesis field in numberofcoils.Ifsomecoildisplacementisnot allowed the sum is done over -=N'N. In application our vectors all are two-dimensional axisymmetry the for this of particularcase,butthemethodissuitablefor solving more general problems, in any geometry.

2

1.5

Fig. 3. Schematicrepresentation of thecoilsand The Q points are evenly distributed plasma region over the whole cross section where the stray field has to be minimized.

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NUMERICAL

METHOD

Forcomputerimplementationtheintegralhas been substituted by a sum over a number M of points Qj distributed inside W, s o that the problem is give'n Similar equations can also be written in terms of the matrix form: flux function 'JJ instead of field B:

Magnetic flux approach

N

CP

being the flux linked with the axisymmetric loop Q, the other quantities and indexes passing through maintaining the same meaning already given (1). in Apart from the fact that B is a vector and 'f! is a scalar (1) and ( 2 ) are mathematically equivalent. Nevertheless for our problem the first approach seems tobemoreappropriate:thisisbecause 'JJ is an

where [dB/dp] represents a MxN matrix, [AP] and [Be] represent two Nxl and Mxl matrixes of vectors respectively. Its least square solution can now be found in terms of the Euchlidean norm if M is chosen greater'than N. In thiswaythesolutionofthe Pi problem is represented by a set of displacements which give the new coil positions selected to produce j points. the minimum mean square error over Qthe

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Additional remarks Some comment is worthwhile on the choice of the Qj points distribution inside the W region where the meansquareerrorfieldiscomputed.Onecouldbe tempted to distribute these points along the boundary of W, on the assumption that a minimum mean square error field along the boundary would be sufficient to guarantee similar conditions also inside W, as would happen for a harmonic function. Actually the amplitude of the field B is not a harmonic function, unlike the magnetic flux or the vector potential, as can easily Q' be shown, It isthusconvenienttochoosethe points evenly distributed inside W. The number M points, required for an accurate computation, dependson the magnetic field gradient V - B in W: the smoother the field the fewer' Qj points are required.

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Thewholeprocedureprovedtoworkproperly, converging in a few iterations towards geometrically regular solutions and producing very low stray fields inside W. The possibility of prevepting coils being displaced some indirection can used beto "orientate" the configuration so that several alternatives can be derived, starting from the same tentative manual winding positioning. Finallyitshouldbe . pointedoutthatthe limitations on the range of coil displacements APi (andthecomputedadjustementsaregenerally smaller) allow tight control the of winding arrangement. For this reason not only currents and numberofturnsremainunchanged,butalsothe parametersofintegral kind, suchasresistances, inductances and flux, are little affected. RFX

POLOIDAL

TRANSFORMER

Practical computation The package has been used to design the RFX air The solution of the problem proves to be mpre and core transformer. RFX is a Reversed Field Pinch (RFP) o f simultaneous allowed more unstable as the number experiment presently under construction in Padua displacements N is increased, producing increasingly (Italy) with the financial support of EURATOM, ENEA irregular winding positions. Such a result is and CNR. It is designed to operate up to a plasma consistent with the evidence that, by increasing the current of 2 MA within a toroidal vacuum vessel of2 m degrees of freedom, a larger and larger number of coil and 0.5 m majorandminorradiirespectivelyand distributions meeting the required field synthesis can represents the leading experiment of this line be found. To better direct the coil rearrangement two alternative Tokamak. to transformer RFX The precautions have therefore been taken: specifications are given in Table I. 1) each displacement is limited within a given range ( A P i 5 A Pi max) where the linear Table I RFX basic magnetizing vinding and power ( 3 ) can be considered well approximation assumed in supplyspecifications:theparameters havebeen verified; chosen to comply with the plasma performance. -2 ) coil displacements are computed one by one in sequence over the whole winding and iterations are Maximum magnetizing flux 15 Wb performed a convenient number of times. Maximum stray field 5 mT At every step the new value of the mean square Number of turns 25 x 8 errorfieldBeiscomputedandcomparedwiththe Winding inductance 57 mH previous one. New iterations ,are performed only if the Maximum winding resistance 17 m i 2 mean square field error actually decreases so that No load power supply voltage 1700 V more and pore accurate configurations are obtained at Load power supply voltage 1350 V every step until a relative minimum is achieved. It Maximum power supply current 50 kA shouldbeobservedthat,aroundtheminimum,the field derivatives are dB/ap;=O, and hence minor coil For Fusion experiments the region of field displacementsfromthecalculatedpositionswould produce negligible field errors; therefore manufacture synthesis W corresponds to the vacuum vessel where the plasma must be produced. tolerances and small mistakes in the coil positioning canbeaffordedwi'thoutjeopardizingthemagnetic configuration.

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Use of the package

1

0.8 The optimization is automatically performed: the user is only required to modify the boundaries of the coildisplacementif,inhisopinion,theobtained configuration does not meet the expected performance. 0.4

Magnetic lields in mT

0 1.2

0

0.8

1.6

2

2.4

2.8

3.2

3.6

4

1.2

0

Magnetic fields in mT

Magnetic lieldr in mT

0.8 0.4 0.4

0

0 0.8

Pig. 4. Windingconfigurationand contours after a first tentative the stray field obtained is well above the specified level.

Mgnetic field coil positioning;

2.81.2

2.4 1.6

2

Fig. 5. Two winding configurations obtained starting from the same reference layout: both are magnetically acceptable and subtantially differ from a mechanical pointof view only.


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For RFX the Qj points were chosen evenly distributed over concentric circles, from the boundary (that is the wall of the vacuum vessel) inwards as schematically represented in Fig. 3. According to condition d) above the .ideal configuration would present no stray field at all in 0 : in practice an the plasma region W, that is B, upperlimitofupto5mTwasconsideredstill acceptable, as specified in Table I. Comments on the results The stray field obtained, on a trial and error basis, with manual coil positioning was well beyond such a limit, as shown in Fig. 4 wherethefie'ld contoursarerepresented:theotherspecifications [l], listed in column A of Table 11, were satisfactorily met. It was thus taken as the starting point configuration to be used as an input to the automatic field synthesis package under a number of Two of the several different space constraints. solutionswhichhavebeenobtainedareshownin Fig. 5-B and 5-C and their parameters are listed in the columns B and C of Table 11. It i s evident that alltheelectricalspecificationsaremetandthe integral parameters are only slightly modified after optimisation.Thesealternativesolutionshavebeen developed in order to try to minimise the mechanical problems arising from the high electromagnetic forces [6]. In fact they substantially differ from a mechanical point of view only. The results prove that the package is capable of finding several solutions, equivalentfromanelectromagneticpointofview, with stray fields in the plasma region around 1 mT, which is more than one order of magnitude lower than the former configuration.

Fig. 7. Magneticfieldamplitudetruncatedabove 50 mT to show the stray field in the plasma region. CONCLUSIONS

Thepackagehasbeenexhaustivelytested.The results were compared with similar software developed at Culham (U.K.) [7] and a very good agreement was found. Fig. 6 shows a 3-D representation of the field magnitude in a region covering the central coils and the vacuum vessel (for the winding configuration of Fig. 5-C). The plasma region appears completely flat in the scale shown and the stray field can only be appreciated if a truncated representation is used as Table I1 - RFX computed winding parameters: A corresponds to a configuration obtained with a first in Fig. 7. The procedure can be easily generalized to othergeometries,forexampletoperformmagnetic tentative and manual coil positioning, B and C show field synthesis in linear systems: moreover the same two alternative solutions derived from A. concepts can be easily extended to perform magnetic field synthesis by finding the proper amplitude of ,A B C currents in a set of fixed conductors (instead of movingcurrentfilamentsofagivenamplitude)as 14.9114.9514.90 Wb Flux required, for instance, for plasma position control. 14 1.0 0.9 mT Stray field 200 200 200 Number of turns ACKNOWLEDGEMENT 4.5 T 4.5 Field in the 4.5 bore mH 58.2 57.5 57.6 Inductance We wish to thank T.A. Mace and T.J. Martin of mJZ 16.1 16.2 Resistance 15.8 CulhamLaboratoryfortheirinvaluablesupportin kA 50 50 50 Current performing comparisons and tests on the package. REFERENCES

[ 11 G. Rostagni et al. "The RFX Project: A

1.3

I I

PLASMA REGION-

.,'

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1

Fig. 6 . A 3-D view of the magnetic field amplitudeon the midplane. The plasma region is completely flat in the scale shown.

Design Review", in Proc. of the 13th SOFT, Varese 1984. [2] M.V.K. Chari, et a1 "Three-Dimensional Vector Potential Analvsis for Machine Field Problems", in IEEE Transactions on Magnetics, Vol. MAG-18, n02, March 1982. [3] E. Durand, Magnetostatique, Masson et C., Paris, 1968. [4] E. Coccorese, R. Martone, "A Design Procedure for Air Core Transformer in Toroidal Axisymmetrical Geometry", in Proc. of 10th SOFT, Padova; 1978. [5] of Magnetic .~ R. Sikora, R. Polka,"Synthesis Fields", in IEEE Transactions on Magnetics, Vol. MAG-18 n02, March '82. [6] M. Guarnieri, C. Modena, B.A. Schrefler, A, Stella "Electromagnetic and Mechanical Design ofRFXMagnetizingWinding",inProc.ofthe Sixth Topical Meetingon the Technology of Fusion Energy, S. Francisco, 1985. [ 71 T.J. Martin "The Interactive Design of Magnetic Fields for Controlled Thermonuclear Research", in Proc. of Compumag Conference, Oxford, U.K., 1976.
