CBN 98-1 Developable constant perimeter surfaces: Application to the end design of a tape-wound quadrupole saddle coil

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Nickolas Beasley
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1 CBN 98-1 Developale constant peimete sufaces: Application to the end design of a tape-wound quadupole saddle coil G. Dugan Laoatoy of Nuclea Studies Conell Univesity Ithaca, NY Intoduction Constant peimete sufaces ae geneally thought to e desiale fo the winding sufaces fo saddle coils fo supeconducting dipoles o quadupoles. Winding multiple layes on such sufaces is facilitated, since the winding of each laye can poceed in eithe diection without any tendency of the tuns to slip to a "shote" path due to the winding tension Ȧ suface is said to e "uled" if it is geneated y moving a staight line continuously in space. A "developale" uled suface is a suface that can e olled on a plane, touching along the entie suface as it olls. Such a suface has a constant tangent plane fo the whole length of each uling. Paallel geodesic loops (in a diection pependicula to the ulings) on closed developale uled sufaces all have the same length; such sufaces ae thus "constant peimete" sufaces. Winding of tape conductos on such developale constant peimete sufaces should povide fo minimum stain to the tape. The tangent plane of the developale suface is fixed, so the nomal to this, the diection of cuvatue of the suface, is constant ove the whole width of the tape. Consequently, all the ending involved is in the "easy" diection (i.e., nomal to the suface of the tape). The developale suface (the tape) can e unolled to a flat suface without twisting o distotion.. Constuction of a developale constant peimete suface A developale constant peimete suface may e constucted fo any given space cuve y following the pesciption outlined in efeence 1. The asic idea is the following. We define a space cuve (β ) fom which we will develop the constant peimete suface. In the case of a tape conducto winding, this cuve would coespond to the path followed y one edge of the tape. At each point along the space cuve, we define a tiplet of othogonal unit vectos (tangent, nomal, and inomal), using the following elations: t =, n = k, = t n (1) in which and = d ds = d dβ ds dβ, = d ds () 1 Januay 14, 1998

2 The cuvatue is ds dβ = dx dβ + dy dβ + dz dβ (3) dx k = + dy + dz ds ds ds (4) As shown in efeence 1, the equation of a uled, developale constant peimete suface, geneated fom the space cuve (β ), is Ω(β,v) = (β ) + v p(β ) (5) in which p = + κ t (6) and κ is the atio of the space cuve tosion τ to its cuvatue k: κ = τ k ; = τn The tosion τ is otained fom last equation. Fo the application to a quadupole saddle coil, we define the efeence space cuve (β ) fo one edge of the tape, which estalishes the geomety of the coil end. z Space cuve (7) O a y D Σ x Fig. 1 Illustating the space cuve coesponding to one edge of the tape Januay 14, 1998

3 The space cuve is taken to e an ellipse, with majo and mino axes a and, on the suface of a cylinde of adius. The cylinde axis is paallel to the z-axis, and its cente is located at the head of the vecto in the x-y plane. This is illustated in Fig. 1. Fig. shows the space cuve on the cylinde's suface. z a β s Σ Fig. The space cuve on the cylindical suface The coodinates of the space cuve on the cylinde ae given in tems of the paametic angle β y s = cosβ z = asinβ (8) in which s is the ac length on the cylinde. Fig. 3 shows the pojection of the space cuve onto the x-y plane. The stategy in specifying the space cuve is the following. Fo a space cuve defined as an ellipse on a fixed adius () cylinde, centeed on the oigin, we would have =0 and D= in Fig. 3. In ode to accommodate coil locks in which the winding suface in the staight pat of the coil is paallel to the x-axis, athe than adial (as in a "keystoned" coil), we allow the cente of the cylinde to move as the space cuve tansits aound the ellipse; that is, we allow the vecto to e a function of β. The vecto (β ) is paameteized as (β ) = î β sin + ĵ β cos To insue quadupole symmety, the adius vecto of the efeence cylinde must always point fom the head of to the point Σ, which lies on the intesection of the cylinde with the 45 o line. Hence we have the constaint D = + (β ) () whee (9) 3 Januay 14, 1998

4 D = d(β ) î + ĵ ( ) is a vecto whose diection lies along the 45 o line. The quantity d(β) is given y solving Eqs. (9), (), and (11) simultaneously; the esult is d(β ) = cosβ ( 1 + cos[ β] ) In Fig.3, the vecto ρ is the pojection of onto the x-y plane. Fom the geomety of Fig. 3, we have ρ = v + (13) in which v is a vecto of length. y (11) (1) Σ Line along which tavels s φ v D ρ π/4 O x Fig. 3 Pojection of the space cuve onto the x-y plane, and associated defining geomety The elation etween, v, and the angle φ = s is illustated in Fig. 4 4 Januay 14, 1998

5 Σ φ v s Fom Fig. 4, we have So, Eq. (13) ecomes The vecto Fig. 4 elation etween v and v = cosφ ( ˆk )sinφ (14) ρ = cos s ( ˆk )sin s + is given in Eq. (). Sustituting into Eq. (15) gives ρ(β ) = D ( (β ))cos cosβ ˆk D ( ( (β )))sin cosβ + (β ) (16) Eqs. (9), (11), (1) and (16) give the complete specification of the space cuve in tems of the ellipse paametes a and, the cylinde adius, and the offset. The values fo a,, and ae detemined y the geomety of the coil's staight section. Fig. 5 shows the x-y plane pojection at β=0, the point whee the end meets the staight section. The ectangula lock, a distance x 0 fom the oigin, and of height y 0, epesents the coss section of the tape conducto coil lock in the coil's staight section. The tape width is T. (15) 5 Januay 14, 1998

6 y D p φ T y 0 π/4 α x 0 Fig. 5 Pojection on the x-y plane and β=0; geomety to define,, and. x As discussed aove, the instantaneous diection of the developale constant peimete suface is given y the vecto p defined in Eq. (6). We equie that, at the points at which the space cuve desciing the end of the coil meets the staight section (at β=0, and at β=π), the diection of the winding suface must e the same as that in the staight section (i.e., it must only have a component in the x-diection, as shown in Fig. 5). Since the tangent vecto t points along the z-axis at this point, we equie that κ(0)=0, and that must only have an x-component at β=0. When z(β) has the fom given in Eq. (8), the equiement that κ(0)=0 is satisfied if 6 Januay 14, 1998

7 d ρ(β ) = 0 and d3 ρ(β ) dβ β =0 dβ 3 = 0 β =0 This condition is satisfied fo all even functions of β with continuous deivatives. Since ρ(β ) as given in Eq. (16) is such a function, this equiement is satisfied y constuction. The equiement that have only an x-component povides the following constaint: in which ( + [ ])cosφ + ( + )sinφ + ( + [ ]cosφ + [ + ]sinφ )= 0 (17) (18) φ = (19) Fom the geomety of Fig. 5, we have the additional elations: in which, fom Eq. (5), and D = d(0) = + + cos π φ x 0 cos 1 = + + x 0 sinφ + ( y 0 )cosφ () d(0) = + (1) = x 0 + (y 0 ) () Solving (18), () and () simultaneously gives, and. We ae still fee to choose ε, the ellipticity of the ellipse desciing the end cuve, to fix a=ε. 3. Application to the end design of a tape-wound quadupole saddle coil The paametes fo the developale constant peimete suface fo such a coil ae given in Tale 1. Coil paamete Value x 0 35 mm y 0 15 mm T 3 mm ε mm mm mm φ ad 7 Januay 14, 1998

8 Tale 1: Paametes fo the constant peimete suface of a tape-wound quadupole saddle coil Fig. 6 shows the pojection of the space cuve, fom which the winding suface is developed, in the x-y plane (the uppe cuve). The space cuve pojection is indicated. The ectangula locks epesent the tape coil locks in the staight section of the coil, the cuve aove the space cuve is the pojection of the uppe edge of the tape as it goes aound the coil end. The quate-cicle is a 35 mm adius ac. 50 Coil lock Space cuve Coil lock 50 Fig. 6 Pojection of the space cuve and elements of the tape onto the x-y plane. Fig. 7 shows the space cuve in thee dimensions, with the tiplet of othogonal unit vectos shown at β=0, π/, and π. 8 Januay 14, 1998

9 y 0 x 0 t z n n t t 0 n - Fig. 7 The space cuve in thee dimensions, with the tiplet of othogonal unit vectos shown at β=0, π/, and π. 9 Januay 14, 1998

10 Fig. 8 Constant peimete tape winding suface (thin stip) and the tape edge guiding suface Fig. 8 shows the constant peimete suface (the thin stip), as well as a suface geneated fom the space cuves y allowing the angle α in Fig. 5 to vay fom 0 to tan 1 y 0 x 0. This suface epesents the winding suface which guides the edges of the tape elements compising the coil lock. Januay 14, 1998

11 0 Fig. 9 End suface of the coil winding fom 0 0 Fig. 9 shows the complete end suface fo the coil winding fom. The ottom suface is extended in z using a cylinde of 35 mm adius, centeed on the oigin. The top suface is fomed y extuding the uppe edge of the constant peimete suface in z. This suface is joined to a staight section. The suface shown in Fig. 9 is duplicated at the othe end of the staight section, and the whole foms the complete coil winding fom fo one saddle coil of the quadupole. 4. Acknowledgments I'd like to thank Jim Welch fo helpful suggestions, fo uncoveing efeence 1, which foms the asis of this wok, and fo tanslating the sufaces geneated y Mathematica into CAD files. 5. efeence 1. T. C. andle, "A Few Mathematical Notes on Constant Peimete Layes in Coil Windings", uthefod La intenal note L (Dec., 1974) 11 Januay 14, 1998

ME 210 Applied Mathematics for Mechanical Engineers

ME 210 Applied Mathematics for Mechanical Engineers Tangent and Ac Length of a Cuve The tangent to a cuve C at a point A on it is defined as the limiting position of the staight line L though A and B, as B appoaches A along the cuve as illustated in the