Differential Geometry Applied to ReBCO Coil- End Design
Thomas Nes1,2
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 0 Overview
• Objectives Coil-End Design * • Space Curves • Radius of Curvature / Example of Helix • Strip Surfaces • Development of hard-way bend free strip surfaces • Coil-End boundary conditions • Example Coil-Ends
• Stacking of tapes/cables • Calculation of the gap between tapes/cables
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 1 Objectives Coil-End Design
• Minimization of mechanical stress * • Minimize or compensate field error of the straight section • To ensure a tight fit to prevent movement during powering • To limit peak field enhancement in the coil-end for quench performance • To guarantee tight tolerances on the coil-end geometry for magnet-to-magnet consistency in the cable positioning. • To generate CAD data for designing and manufacturing purposes
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 2 Overview
• 4 Steps *
Draw Space Curve Calculate rotation vector
Draw tape surface Draw coil-pack thickness
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 3 Space Curves
• A space curve is a curve which “lives” in R3 * • In the following, only differentiable curves will be considered • Position can be parametrized by:
• Velocity:
• Acceleration:
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 4 Space Curves
• The arc length of a curve is given by: *
• A space curve can be expressed as a function of t or as a function of s. • One can switch between the two using:
• A curve having unit speed v=1 is said to be represented in terms of arc length.
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 5 Bezier Curves
• A type of parametrized curves are Bézier curves * • They are defined on an interval I=[0,1] and can be expressed as:
• We will use Bézier curves to draw our base curves • Great flexibility in drawing the shape • Derivatives can be calculated with relative ease using Casteljau’s algorithm (not going to be discussed) • Boundary conditions can be satisfied (stay tuned!)
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 6 The Frenet-Serret Frame
• The Frenet-Serret frame is an orthonormal frame used to * study curves • It consists of three vectors, the tangent vector T, the normal vector N and the binormal vector B • The Frenet-Serret frame is unique, i.e. for every curve the frame is uniquely defined • Associated with the frame are the curvature κ and the torsion τ of the curve
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 7 The Frenet-Serret Frame
• The tangent vector points in the same direction as v: *
• The normal vector is defined as
• Since T is a unit vector, T•T=1. Deriving with respect to s yields T•T’=0, and thus T is orthogonal to T’. Hence N is orthonormal to T. • The binormal vector is defined as
• This gives us the three orthonormal vectors of the frame
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 8 The Frenet-Serret Frame
• The curvature of a space curve is defined as κ=|T’| * and can be interpreted as the failure of a curve to remain straight • The torsion of a space curve is defined as τ=|B’| and can be interpreted as the failure of a curve to remain in- plane • The inverse of the curvature ρ=1/κ is called the radius of curvature. For ReBCO tape bend in-plane, this radius should be larger than the critical bending radius, otherwise loss of critical current Ic will occur
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 9 The Frenet-Serret Frame
• Using the definitions defined in the previous slides, one can * define the Frenet-Serret equations
• In matrix form:
• These equations tell us how the frame transforms when moving along the curve
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 10 The Frenet-Serret Frame
• In case of a curve parameterized by t one can show that * the following relations hold:
• The Frenet-Serret equations are then:
• The extra factor of v comes from the chain rule:
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 11 Rotation of Space Curves
• The transformation matrix is skew symmetric, i.e. *
• Using this property, we can rewrite the Frenet-Serret equations as:
• Here, ω is the angular velocity vector (also known as the Darboux vector), and is written as
• This gives another way to interpreted κ and τ: vτ and vκ are the magnitude of the rotation around T and B respectively
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 12 Radius of Curvature
• The magnitude of the angular velocity is given by: *
• Using we can find the bending radius:
• For ReBCO tapes, the bending radius should be larger than the critical bending radius, otherwise loss of critical current Ic occurs
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 13 Example: Bending radius of a helix
• The helix is an important example to study, * since CORC cable, as well as helical undulators are wound in a helical fashion • A helix is a curve which spirals on top of the surface of a cylinder and can be parameterized as:
• R is the radius of the cylinder it is wound on, and p the twist pitch.
Pic: advancedconductor.com
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 14 Example: Bending radius of a helix
• Using the curvature and the torsion are * then
• The bending radius of the helix is then:
• Consequence: A ReBCO helix can be wound on a cylinder with a radius smaller than the critical bending
radius without loss of Ic! Pic: Field Computation for Accelerator Magnets, S. Russenschuck
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 15 The Darboux frame
• The Darboux frame is used to study strips * • It consists of three orthonormal vectors t, n and b • The tangential vector t points along the tangential direction of the strip • The normal vector n points normal to the strip surface • The binormal vector b points along the width of the strip
Pic: Field Computation for Accelerator Magnets, S. Russenschuck
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 16 The Darboux frame
• The difference between the Frenet-Serret frame and * the Darboux frame is a rotation around the tangential vector:
• Differentiating this equation yields the equations for the Darboux frame:
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 17 Rotation of Strip Surfaces
• As this matrix is skew-symmetric, we can write it as *
• The angular velocity vector is then
• This gives another way to interpreted τ_r, κ_n and κ_g: vτ_r, vκ_g and vκ_n are the magnitude of the rotation around t, n and b respectively
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 18 Hard-way Bending
• Rotation around the normal vector n leads to a * difference in length of the edges of the strip surface. • ReBCO tape and cables do not like to be bent this way, and it is therefore known as the hard-way bend. • This condition is also known as the constant perimeter condition.
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 19 Ruled and Developable Surfaces
• By moving a line (called generator) with variable length g(t) * and direction g(t) along a line c(t) (called the directrix), one can trace out a strip surface S
• A developable surface is a special kind of ruled surface, it can be created by folding or bending a plane, but not by stretching or compression (no hard-way bending) • These are the properties of ReBCO tape, and thus it should be modelled as a developable surface
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 20 Ruled and Developable Surfaces
• One can show that a ruled surface is developable only if * dc/dt, g and dg/dt are in plane with each other, i.e. they are linearly dependent:
• One can show that the only function satisfying this requirement is
• This vector points in the same direction as the angular velocity vector • Conclusion: developing the strip surface using the angular velocity vector is the only way to develop a strip surface which is hard-way bend free.
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 21 Drawing of the strip surface
• The generators can be interpreted as the lines * around which the strip surface is bent (like folding lines on a piece of paper) • The length of the generators is
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 22 Coil-End Boundary Conditions
• At the beginning and end of the coil-end, the coil- * end meets a straight section, which has zero curvature and torsion. • Since we use Bézier splines, they start and end at t=0 and t=1 respectively. • Hence we have the condition that κ(0)= κ(1)=τ(0)= τ(1)=0 • We will use the properties of Bézier functions to adhere to these boundary conditions.
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 23 Bezier Curves
• The derivatives of a Bézier * functions at the beginning (t=0) and end (t=1) are: • The n-th derivative only depends on n control points, giving local control over the curve at the end points • Will be used to satisfy the boundary conditions, without changing the overall shape of the coil-end
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 24 Coil-End Boundary Conditions
• The condition κ(0)= κ(1)=0 implies *
• Looking at the table, we can see that this can be achieved by having P0, P1 and P2 on one line
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 25 Coil-End Boundary Conditions
• The condition |v x a|=0 gives an indeterminate * form of the torsion:
• It can be evaluated using L’Hôpital’s rule
• By evaluating the limit using L’Hôpital (twice), we can find its value:
• If τ=0, then vxȧ•ä=0, hence P0…P4 should be in the same plane
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 26 Coil-End Boundary Conditions
• The length of the generator *
at the end points should be g(0)=w, and thus τ/κ=0 • Since |v x a|=0 gives an indeterminate form
• We can use L’Hôpital again to find the limit, this yields:
• This can be made zero be setting P0…P5 to be in-plane
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 27 Effects of Control Points
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 28 Effects of Control Points
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 29 Effects of Control Points
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 30 Effects of Control Points
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 31 Effects of Control Points
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 32 Effects of Control Points
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 33 Cloverleaf Coil-End
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 34 Cos-θ Coil-End
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 35 Undulator Coil-End
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 36 GaToroid Layer Jump
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 37 Stacking Tapes
• We’ve found a way to describe one single tape, but * what about tape (or cable) stacks? • For a ReBCO tape, good face-to-face contact is desired, for mechanical stability and current sharing • Hence, it is natural to stack the next tape on top of the previous tape with identical normal vectors • This is trivial that this is possible for a racetrack coil, but what about twisted coils?
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 38 Stacking Tapes
• The parallel curve is parametrized by: *
• Differentiating yields:
• In terms of velocity:
• The magnitude of the parallel velocity is then
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 39 Stacking Tapes
• The tangential vector is then *
• The normal vectors of the original curve and the parallel curve must be the same for face-to-face contact:
• The binormal vector is then
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 40 Stacking Tapes
• In Matrix form: *
• Differentiating yields:
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 41 Stacking Tapes
• Instead of deriving to s, it is preferred to express the * frame with respect to s*. The derivative can be transformed with
• Using this, we can find the Darboux frame for the parallel curve:
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 42 Stacking Tapes
• In terms of parameter t, the frame can be expressed as *
with
• Note that the geodesic curvature κ_g is not necessarily zero, even though it was for the first curve!
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 43 Stacking Tapes
• The geodesic curvature is *
• For there to be no geodesic curvature of the parallel tape, the numerator has to be zero:
• This is the case when • κ and τ are constant (helix) • τ=0 (planar curve) • But generally it is not zero! • Thus, only planar windings and the helix can be wound with perfect face-to-face contact
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 44 Stacking Tapes
• If there is no natural way to stack the tapes, there is no * perfect way to draw the thickness of the coil-pack • Ignoring the hard-way bend and expanding in the normal direction is an option, but for the Cloverleaf it was deemed that this gives unnatural results, due to the coil-pack slipping under the xy plane. • In RAT, the coil pack is grown in the (N_r+N)/2 direction, where N_r equals the projection of the normal vector on the xy plane
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 45 Estimation of the gap
• Since the geodesic curvature came about by a rotation * around the tangent vector, we can also undo it by rotating around the tangent vector. • This leads to a gap between the tapes • Using the relations
• The angle of rotation is then:
• The gap is given by
• Combining these two relations yields:
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 46 Estimation of the gap
* • Using
the gap for the cloverleaf configuration is calculated. • It is in the order of nm, hence negligible.
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 47 Conclusion
* • A novel method using Bézier splines to develop ReBCO strips and cables has been developed, which has many advantages compared to previous methods • Great flexibility in shape • Properties of the curve can be calculated analytically everywhere • No hard-way bending, even at the boundaries • Various coil-end designs have been created using the Bézier method • It has been shown that no strips or cables can be laid together with perfect face-to- face contact, but without hard-way bending • The gap between two successive tapes has been estimated to be in the order of nm, and thus negligible.
6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 48 *
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6/4/2020 T.H. Nes – Differential Geometry Applied to ReBCO Coil-End Design 49