This paper presents a review of the existing analytical approximations of the magnetic field of a sector winding quadrupole, evaluating to which extent these formulae are usable as a first estimation of the resultants of the electromagnetic forces and related mechanical stresses produced. The distribution of the forces has been investigated to better understand the stress provided to the structure containing the windings and the compression effect induced on the coil mid plane. Therefore, the study has been carried out setting the magnet current at its critical value, analyzing forces and stresses as functions of the geometrical layout and of the magnetic gradient as well. In the last part the effect of an iron yoke on the magnetic field and forces is presented

Fessia, Paolo

Regis, Federico

Todesco, Ezio

We first present a review of the existing analytical equations for electromagnetic forces and stress in a sector coil quadrupole, evaluating their extent of application in function of the coil geometrical layout. We analyze the distribution of stress provided on the coil retaining structure as well as on the coil mid plane, this one related to the degradation of critical current in the superconductor. We focuse on the maximum compressive stress dependence on the magnetic gradient at short sample conditions, considering two possible low temperature superconductors: Nb-Ti and Nb$_{3}$Sn. In the last part the effect of an iron yoke on the magnetic field and forces is presented, proposing a correction of the critical current density expression when an infinite permeable iron yoke is used

Fessia, P

Regis, F

Todesco, E

CERN Document Server

Parametric Analysis of Forces and Stresses in Superconducting Quadrupole Sector Windings

IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007 1269Parametric Analysis of Forces and Stresses inSuperconducting Quadrupole Sector WindingsP. Fessia, F. Regis, and E. TodescoAbstract—This paper presents a review of the existing ana-lytical approximations of the magnetic field of a sector windingquadrupole, evaluating to which extent these formulae are usableas a first estimation of the resultants of the electromagnetic forcesand related mechanical stresses produced. The distribution ofthe forces has been investigated to better understand the stressprovided to the structure containing the windings and the com-pression effect induced on the coil mid plane. Therefore, the studyhas been carried out setting the magnet current at its criticalvalue, analyzing forces and stresses as functions of the geometricallayout and of the magnetic gradient as well. In the last part theeffect of an iron yoke on the magnetic field and forces is presented.Index Terms—Current density, stress, superconducting magnets.I. INTRODUCTIONI N superconducting quadrupoles, e. m. forces and associatedstresses of the order of 100 MPa are generated by the inter-action of the cable current with the magnetic field. Two mainissues have to be analysed: first, a mechanical structure mustbe envisaged to contain these forces and limit the cable move-ments during the magnet ramp. Secondly, the stresses inside thecoil must not exceed the limits beyond which insulation can startcreeping and the superconductor properties are degraded. Thissecond aspect is critical for the , which according to somecable measurements cannot tolerate compressive stresses largerthan 150 MPa [1].In this paper we aim at analyzing how forces and stressesdepend on the quadrupole aperture, on the width of the coil,and on the superconducting material. We used a simplified coillayout made up of a sector of inner radius , radial width andan angular extension (thus canceling the first allowedfield harmonic), fed by a uniform current density (see Fig. 1).This simple geometry has the advantage of being closer to areal coil rather than the classical , whilst still allowing ananalytical approach.In [2], it has been shown that a similar lay-out well represents,from e. m. point of view, several quadrupoles based on the shellgeometry that have been built in the last 30 years.The main steps carried out in this analysis have been:Manuscript received August 28, 2006. This work was supported by the Euro-pean Community-Research Infrastructure Activity under the FP6 “Structuringthe European Research Area” programme (CARE, contract number RII3-CT-2003-506395). The work of F. Regis was supported by ASP (Associazione per loSviluppo Tecnologico e Scientifico del Piemonte) and Compagnia di San Paolo,Turin, Italy.The authors are with the Accelerator Technology Department, CERN, Geneva23, 1211 CH Switzerland (e-mail: paolo.fessia@cern.ch).Digital Object Identifier 10.1109/TASC.2007.898523Fig. 1. Quadrupole sector coil layout.• Estimate analytically using the formalism developed in[3]–[5] the electromagnetic forces and the induced stressesin the sector coil as a function of and for a given . Asimilar approach has been developed for the dipoles usinga model in [6], [7]. In particular, we will focus on thedefinition of the position and of the value of the maximumcompressive stress on the coil mid plane.• Substitute with the critical current density relative to thatcoil lay-out and to that superconductor (either Nb-Ti or), and evaluate the peak stress as a function of theobtained gradient.• Further develop the formulae for the critical current den-sity in presence of an iron yoke, revising the peak stressbehavior in this new condition.The study is based on an analytical approximationcross-checked with a two-dimensional finite element modelcoded in ANSYS of one quadrupole octant (see Fig. 1). Themagnetic model of the winding built in ANSYS is surroundedby air.II. FORCES AND STRESSES FOR A UNIFORM CURRENT DENSITYA. Lorentz ForcesAs first step, the profiles of the magnetic field in the aperture,in the coil, and outside the coil have been computed using an an-alytical approach [3]–[5] and the numerical model. The resultsshow that• There is a good agreement between the numerical and theanalytical estimate of the magnetic field both in the aper-ture and outside the coil.• The agreement is worse inside the coil, the error being de-pendent on the angular dimension of the sector coil andincreasing as approaches 45 .Despite of the aforementioned discrepancy, the analysis of themagnetic energy reveals that the analytical approximation1051-8223/$25.00 © 2007 IEEE1270 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007Fig. 2. Azimuthal stress distribution (compression) on coil mid plane  (r)for a coil with inner radius r of 28 mm and a width w of 30 mm, for a currentdensity j = 1000 A=mm .agrees with the numerical one within 4%. A similar result isfound for the forces integrated over the coil.One can conclude that notwithstanding the discrepancy be-tween numerical and analytical results relative to the field dis-tribution inside the coil, the analytical approach provides a verygood estimation of the coil magnetic energy and of the coilforces. However, the analytical formulation cannot be used todetermine the coil peak field (and therefore the critical current)since it does not provide adequate estimation of the magneticfield inside the coil. This estimate will be carried out using anumerical method as outlined in [2].B. Stresses on Coil Mid Plane and Collar Contact EdgeThe equations for the azimuthal compressive stress on thecoil mid plane and of the radial compressive stress onthe external profile of the coil can be evaluated throughbalancing the forces acting on an infinitesimally small coil ele-ment not considering the shear effect. Integrating the differentialequations given by the force balance, we can get:(1)(2)where is a function involving geometrical parameters. In theseequations, stresses are in Pa, lengths in m, current density in. In this analytical approximation higher order terms in ,, and shear stresses are neglected. In Fig. 2 as providedby (1) is compared with the numerical results of a model withan isotropic material, showing a very good agreement, with amaximum error of 5%, for an inner coil of 28 mm radius and30 mm width. A similar agreement is found also for the radialstress.C. Dependence on Material PropertiesThe previous analysis shows that the analytical approach is agood approximation for the stresses distribution in an isotropicmaterial. In order to verify the applicability in case of real su-perconducting windings, where some anisotropy is present dueFig. 3. Dependence of the azimuthal stress distribution on coil mid plane (r)on the material anisotropy for a case with inner radius r = 14mm, coil widthw = 40 mm, and current density j = 1000 A=mm .to the structure of the cable and insulation, simulations havebeen carried out setting an azimuthal Young’s modulus ofabout 13 GPa and increasing the modulus in the radial direc-tion. Notwithstanding a relevant difference in the inner radiusvalues (see Fig. 3), only minor discrepancies are found for themaximum stress on the mid plane, that is the quantity of interest.A weak dependence of the position of the maximum stress onthe material anisotropy is also observed. We conclude that theanalytical approximation is also suitable for anisotropic mate-rials as measured for the superconducting coils.III. LORENTZ FORCES AND STRESSES AT CRITICALCURRENT DENSITYIn the previous equations, stresses are proportional to thesquare of the current density . Indeed, in a superconductingquadrupole is limited by the critical current density , i.e. thecurrent corresponding to a peak field on the critical surface. Inthis section we will compute stresses when the current densityis set at its critical value. In order to estimate the critical current,following the approach of [2] we define the lay-out parameters(3)(4)For a sector coil of one has(5)and is well fit by an empirical expression(6)where , and are constants related to the 30 lay-out.The Nb-Ti critical surface is fitted by a linear relation(7)where the equation is written for the engineering current density, i.e. the current divided by the area of the insulated conductor.FESSIA et al.: PARAMETRIC ANALYSIS OF FORCES AND STRESSES IN QUADRUPOLE SECTOR WINDINGS 1271TABLE IPARAMETERS OF THE CRITICAL SURFACE FITS FOR Nb-Ti AND Nb SnThe fitting parameter is the slope of the critical surface, andis the critical field according to the fit (see Table I), under-estimating the actual one by around 10%. The filling ratio is adilution factor taking into account the presence of copper, voidsand insulation in the coil; is equal to 1 for a coil made only ofsuperconductor and in the real case is in a range between 0.23and 0.33. Introducing the critical surface fit in (3)–(7), we ob-tain the expression for the critical current of a sector winding ofinner radius and width made of Nb-Ti conductor.(8)and the critical gradient can be obtained by multi-plying by (5).For the we use the fit proposed in [2](9)which is accurate within 5% with respect to the Summer’s ap-proximation [8] between 11 and 17 T at 1.9 K, and has the ad-vantage of allowing an explicit solution for the critical current(10)with given by (6), and similarly for the critical gradient. Thefitting parameters and for both materials at 4.2 and 1.9 Kare listed in Table I.We then evaluate from (1) the position along the mid planewhere the compressive stress is the maximum in the coil; wethen substitute the critical current expressions (8) or (10). Weiterate this computation for different inner radii and differentcoil widths, evaluating both the maximum stress and the criticalgradient obtained by each configuration. Quadrupole aperturesbetween and have been analysed;coil widths have been chosen between 5 mm and the valuecorresponding to a saturation of the critical gradient toits maximum value.Results are shown in Figs. 4 and 5 for Nb-Ti andrespectively. In both cases, we also evaluated a few lay-outsthrough a numerical simulation to verify the agreement with theanalytical approach, which is shown to be very good.The stress increases for larger apertures: in the largest anal-ysed case the stress is just below 90 MPa forthe Nb-Ti and about 150 MPa for , which is the max-imum acceptable stress in magnet design. The dependence onthe coil thickness is more involved: for the Nb-Ti in largeFig. 4. Maximum azimuthal compressive stress on coil mid plane for differentcoil layouts considering a Nb-Ti cable (constant aperture radius). T = 1:9 K.Fig. 5. Maximum azimuthal compressive stress on coil mid plane for differentcoil layouts considering a Nb Sn cable (constant aperture radius). T = 4:2 K.apertures the peak stress does not increase mono-tonically with (which is a function of ): a maximum isfound for a given , after having reached that point the stressdecreases as increases (since increases faster then ).This behavior appears also for the , but at larger aper-tures .IV. IRON EFFECTAn iron yoke has the main function of closing the magneticcircuit, thus increasing the magnetic field produced for the samecurrent density. This means that also the magnet critical currentis reduced because the load line hits the critical surface athigher field. The gradient and the peak field are assumed to belinear functions of the current , i.e. we neglect saturation. Atfirst order one can expect that the gradient and the peak fieldwill have a similar relative increase when an iron yoke is added.In Fig. 6 we plot the difference between the relative increase inthe peak field and the relative increase of the gradientin the aperture , normalized to , for different ra-tios and collar thickness. The increment of the gradientin the aperture has been analytically derived using the for-mulae of a sector coil with constant current density and ironscreen, whereas the increment of the field has been com-puted through the numerical code ANSYS. It can be stated that• The 1st order approximation is correctwithin 10% in a large part of the domain of interest1272 IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 17, NO. 2, JUNE 2007Fig. 6. Relative difference (in %) between the relative increment of B and Gversus w=r .Fig. 7. Evolution of  for a collar width w = 20 mm, Nb-Ti case.Solid lines refer to the same coil width, varying only the iron radius.Fig. 8. Evolution of  for a collar width w = 20mm, Nb Sn case.Solid lines refer to the same coil width, varying only the iron radius.. Since this difference has to be applied on anincrement, representing therefore a small error.• The relative difference mainly depends on the ratio ,and is practically independent on the distance between thecoil and the iron (i.e. the collar width).The curve in Fig. 6 has been empirically fitted with(11)with , , and . Having an analyticalestimate of (given by the analytical model), and an empir-ical fit giving , we can estimate through (8) and (10) thenew critical current density with the iron . This new cur-rent density can then be used together with the formulae of theforces to evaluate the maximum stress on the mid-plane. Resultsare shown in Figs. 7 and 8, for two apertures. The increment ingradient is followed by a variation of the peak stress that bringsthe new structure to the same maximum stress condition as acoil generating the same gradient in air.V. CONCLUSIONSWe analysed a simplified model of superconductingquadrupole, namely a 30 sector coil with uniform currentdensity. We outlined a semi-analytical approach that allow topredict the stress distribution along the mid-plane with goodprecision, and that holds also for more realistic non-isotropicmaterials. We used deduced in [2] in the previous derivedequations, to show that the stress increases for larger quadrupoleapertures and that the dependence on the coil thickness is moreinvolved. For the Nb-Ti the stress is below 100 MPa for aper-tures radii smaller than 60 mm; for the the stress inaperture radii smaller than 60 mm is below 150 MPa, whichis taken as guideline for before degradation of thesuperconducting properties of the material.Introducing an iron screen, both the critical gradient and thepeak stress increases to reach the same level as it would havein an ironless larger coil creating the same gradient. So furtherstudies can be accomplished on coil in air (easier to treat ana-lytically) since the peak stress depends only on .REFERENCES[1] E. Barzi et al., “Sensitivity of Nb3Sn Rutherford-type cables to trans-verse pressure,” in ASC’04, IEEE Trans. Appl. Supercond., June 2005,vol. 15, no. 2, p. 1541.[2] E. Todesco and L. Rossi, “An estimate of the maximum gradient in su-perconducting quadrupoles,” IEEE Trans. Appl. Supercond., this con-ference, submitted for publication.[3] R. Meuser, Magnetic Field for a Thick cos n Winding EngineeringNote M5254, Lawrence Berkeley National Lab., 1978.[4] R. Meuser, Stresses in a Thick cos n Winding Engineering NoteM5256, Lawrence Berkeley National Lab., 1978.[5] A. Asner, Cylindrical Aperture Multipoles With Constant CurrentDensity Sector Windings CERN Internal Report, SI/Note MAE/69-15,(7-10-1969).[6] S. Caspi and P. Ferracin, “Limits of Nb3Sn accelerator magnets,” inParticle Accelerator Conference, 2005, pp. 107–111.[7] S. Caspi, P. Ferracin, and S. Gourlay, “Graded high field Nb3Sn dipolemagnets,” in 19th Magnet Technology Conference, IEEE Trans. Appl.Supercond., 2006, pp. 354–357.[8] L. Summers et al., “A model for the prediction of Nb Sn critical cur-rent as a function of field, temperature, strain and radiation damage,”IEEE Trans. Appl. Supercond, vol. 27, pp. 2041–2044, 1991.[9] F. M. Asner, High Field Superconducting Magnets. : Oxford SciencePublications, 1999, pp. 132–141, 155–159.

We first present a review of the existing analytical equations for electromagnetic forces and stress in a sector coil quadrupole, evaluating their extent of application in function of the coil geometrical layout. We analyze the distribution of stress provided on the coil retaining structure as well as on the coil mid plane, this one related to the degradation of critical current in the superconductor. We focus on the maximum compressive stress dependence on the magnetic gradient at short sample conditions, considering two possible low temperature superconductors: Nb-Ti and Nb$_{3}$Sn. In the last part the effect of an iron yoke on the magnetic field and forces is presented, proposing a correction of the critical current density expression when an infinite permeable iron yoke is used

Parametric Analysis of Forces and Stresses in Superconducting Quadrupole Sector Windings

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