This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
1
Strain Distribution in REBCO Coated Conductors
Bent with the Constant-Perimeter Geometry
Xiaorong Wang, Diego Arbelaez, Shlomo Caspi, Soren O. Prestemon, GianLuca Sabbi, Tengming Shen
Abstract—Cable and magnet applications require bending
REBa2 Cu3 O7−δ (REBCO, RE = Rare Earth) tapes around a
former to carry high current or generate specific magnetic fields.
With a high aspect ratio, REBCO tapes favor the bending along
their broad surfaces (easy way) than their thin edges (hard way).
The easy-way bending forms can be effectively determined by
the constant-perimeter method that was developed in the 1970s
to fabricate accelerator magnets with flat thin conductors. The
method, however, does not consider the strain distribution in
the REBCO layer that can result from bending. Therefore, the
REBCO layer can be over-strained and damaged even if it is bent
in an easy way as determined by the constant-perimeter method.
To address this issue, we developed a numerical approach
to determine the strain in the REBCO layer using the local
curvatures of the tape neutral plane. Two orthogonal strain
components are determined: the axial component along the tape
length and the transverse component along the tape width. These
two components can be used to determine the conductor critical
current after bending. The approach is demonstrated with four
examples relevant for applications: a helical form for cables,
forms for canted cos θ dipole and quadrupole magnets, and a
form for the coil end design. The approach allows to optimize
the design of REBCO cables and magnets based on the constantperimeter geometry and to reduce the strain-induced critical
current degradation.
Index Terms—REBCO Coated conductors, rectifying developable surface, bending strain
I. I NTRODUCTION
ENDING REBa2 Cu3 O7−δ (REBCO, RE = Rare Earth)
coated conductors around a former is generally required
to make REBCO cables and magnets. The flat thin tape
geometry, however, makes the bending along the thin edge
of the REBCO tapes more difficult than along their broad
surface. Therefore bending REBCO tapes is more constrained
than bending round wires [1]. In addition, knowing the shape
of the bent REBCO tapes allows to properly support the tapes
against the electromagnetic force. In the 1970s, the flat thin
Nb3 Sn tapes faced similar issues for the development of beam
steering magnets and an effective solution called the constantperimeter method was developed [2–5].
The constant-perimeter method models the tape conductor
as a developable surface that can be unfolded to a plane
without stretching or tearing [4]. Thus, it approximates an
easy-way bend of the tape conductor that can facilitate the
cable and magnet applications. Besides the most straightforward example of single pancake coils, the method has
B
The authors are with Lawrence Berkeley National Laboratory, Berkeley,
CA 94720, U.S.A. E-mail: xrwang@lbl.gov
This work was supported by the Director, Office of Science, Office of High
Energy Physics and Office of Fusion Energy Sciences, of the U.S. Department
of Energy under Contract No. DE-AC02-05CH11231.
been successfully applied to the design of accelerator magnets
based on the high-temperature superconducting tapes, such
as a quadrupole magnet wound with Bi-2223 tapes [6, 7]
and cos θ dipole magnets made of REBCO tapes [8–10]. A
coil end design that is useful for saddle, block and cos θ
coils using thin tape conductors was also proposed in [11].
The constant-perimeter method was extended to model the
bending of Rutherford cables [12]. The method was also
implemented in ROXIE, a software package that is widely used
for accelerator magnet design [13]. In addition to accelerator
magnets, the method was also used for the design of fusion
magnets [14].
Although the constant-perimeter method enables an easy
way bend of REBCO tapes, it does not address the strain
distribution in the REBCO layer because the tape thickness
is not considered. For most commercial conductors available
today, the REBCO layer will strain during bending because it
is off the neutral plane of the tape. Excessive strain can permanently degrade the conductor critical current (Ic ) by damaging
the brittle REBCO layer. Therefore, understanding the strain
distribution in the REBCO layer is critical to minimize the Ic
degradation for REBCO cables and magnets [15, 16].
Few studies have been reported on the strain distribution
in the REBCO layer when the tape is bent in the constantperimeter geometry. van der Laan studied the strain along
the REBCO tape length when the tapes are helically wound
around a former [17]. A similar study was reported by Zhu et
al. based on a different analysis method [18]. Amemiya et
al. showed that significant edge-wise strain appears if the
REBCO tape in the winding deviates from the constantperimeter geometry [9]. Although these studies clarify the
strain distribution for specific cases, strain distribution for general bending cases based on the constant-perimeter geometry
remains to be addressed.
In this paper, we present a curvature-based approach to
determine the local strain distribution of the REBCO layer
bent with the constant-perimeter geometry. Four cases relevant for cable and magnet applications are analyzed: 1) a
helical winding, 2) a canted cos θ (CCT) dipole winding, 3) a
CCT quadrupole winding, and 4) a coil end design proposed
in [11]. The amplitude and distribution of two orthogonal
strain components, one along the conductor length and the
other along the conductor width, are determined. We show how
the strain depends on the cable and magnet design parameters.
With the presented method, one can optimize the design for
REBCO cables and magnets to minimize the strain-induced Ic
degradation.
Section II briefly reviews the constant-perimeter method and
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
2
how the shape of a bent tape is determined. In section III, we
present the method to determine the local strain distribution
based on the curvature of the neutral plane. The strain distributions for several practical cases are analyzed in section IV
as examples. Finally, we discuss the implications of the results
and limitation of the presented method.
II. A PPROXIMATE REBCO COATED CONDUCTORS WITH
RECTIFYING DEVELOPABLE SURFACES
We first review the constant-perimeter method based on
the concept of rectifying developable surface [4]. Our goal
is to determine the developable surface that approximates the
shape of the bent tape. The tape is neither extendable nor
stretchable, similar to a piece of paper. This property implies
the isometric bending that preserves the length of every arc
and the angle between every pair of intersecting curves on
the surface. Only a developable surface can be isometrically
mapped to a plane [19].
Fig. 1 illustrates a tape bending around a cylinder as an
example of the rectifying developable surface. After bending,
the center line of the tape becomes a space curve, X. For each
point along X, a trihedron of orthogonal unit vectors can be
established: the tangent vector v1 , the principal normal vector
v2 and the binormal vector v3 [19].
Fig. 1. (a) A REBCO tape on a plane before bending. The axial and
transverse directions are defined. (b) The tape around a cylinder as a rectifying
developable surface. The white dashed line is the center line of the tape
and the base curve X. Along X, a moving trihedron of unit vectors can be
established: the tangent vector v1 , normal vector v2 (pointing toward the axis
of the cylinder), and binormal vector v3 . r is the unit Darboux vector in the
v1 -v3 plane. The transport current flows along the axial direction. The 3D
interactive version of the figure can be viewed in the attached pdf file with
Acrobat Reader R .
The rectifying developable surface resulting from the space
curve X is given by the vector
Y = X + v r,
(1)
where the vector r and scalar v are to be determined based
on X, the base curve for the surface Y. The unit vector r is
given by the tangent and binormal vectors of X (Fig. 1), i.e.,
τ v1 + κv3
,
r= √
τ 2 + κ2
(2)
where τ is the torsion and κ is the curvature along the base
curve X. The torsion and curvature can be determined once
the base curve is given (Appendix A). The vector τ v1 + κv3
in (2) is also called the Darboux vector [19]. Equation (2)
shows that the developable surface is given by a generator vr
that is parallel to the Darboux vector along the base curve X.
Next, we determine the scalar v which is related to the tape
width. We use the property of isometric transformation (length
and angle are preserved) [20]. The unit Darboux vector r is
in the plane of v1 and v3 because τ and κ are scalars. Since
v1 and v3 are orthogonal unit vectors, the angle β (Fig. 1)
between r and v1 is given by
κ
(3)
β = tan−1 .
τ
From the isometric bending, we have
g
,
(4)
v=
sin β
where g ranges between ±w and 2w is the conductor width
(Fig. 1). The generator vr is now determined. With the base
curve X, we can now determine the rectifying developable
surface Y corresponding to a rectangular tape with zero
thickness according to (1).
III. D ETERMINE LOCAL ELASTIC STRAIN IN REBCO
LAYER BASED ON CURVATURE
The REBCO layer experiences strain when it is off the
neutral plane of the bent tape (Fig. 2). We treat the conductor
as a thin bending beam to estimate the local strain distribution
in the REBCO layer [15, 21, 22]. Elastic strains are considered
such that the deformations of the conductor and changes
of tape geometry are neglected. The Poisson effect is also
neglected.
Fig. 2. Sketch of the cross section for a typical commercial REBCO tape
(not to scale). The width of the tape is on the order of 1 mm. The thickness
of the tape is on the order of 10 µm. The tape has Cu stabilizers of the
same thickness on the top and bottom. The neutral plane of the tape lies at
the center of the substrate. The distance between the REBCO layer and the
neutral axis, d, is half of the substrate thickness.
The strain in the REBCO layer is determined by the local
curvature of the neutral plane κ and the distance between the
REBCO layer and the neutral plane d, i.e.,
= dκ =
d
,
ρ
(5)
where ρ is the radius of the curvature (ρ = κ−1 ). For a given
conductor architecture, d is assumed to be constant during
bending. The rectifying developable surface determined by
the base curve X (section II) is the neutral plane because by
definition, the developable surface preserves the length of any
two points on the surface and hence experiences zero strain.
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
3
We focus on two orthogonal strain components on the
REBCO layer. The first is an axial component along the
conductor length and parallel to the direction of transport
current (Fig. 1(a)). Its impact on the conductor critical current, in particular for the tensile strain, has been extensively
studied. The irreversible tensile strain limit, beyond which
the conductor Ic permanently degrades, ranges from 0.4%
to 0.7% for recent commercial conductors with Hastelloy
substrates [16, 22–24]. For the compressive axial strain, a
large intrinsic and reversible effect on conductor Ic has been
observed [25]. An axial compressive strain between 0.5% and
1.0% can reduce the conductor Ic for 20% [23, 26, 27].
The second component is the transverse strain along the
conductor width. An example of the transverse bend is given
by the cross sectional picture of the Conductor on Round Core
(CORC R ) cable (Fig. 8 in [26]). The transverse strain and its
impact on conductor Ic is less known compared to the axial
strain.
To determine these two strain components, we first determine the curvatures of the axial and transverse lines on the
neutral plane of the tape. The curvature κ of a curve on a
surface can be determined by the normal curvature κn and
geodesic curvature κg according to κ2 = κ2n + κ2g [19]. The
axial and transverse straight lines on a plane tape become the
geodesics on the developable surfaces after bending. By definition, the geodesics have zero κg . Therefore, the curvatures
for the axial and transverse lines on the neutral plane of a bent
tape are given by the their normal curvatures, i.e., κ = κn .
The normal curvature κn can be determined analytically
based on the principal curvatures κ1 and κ2 according to
Euler’s Theorem [19]. Developable surfaces feature κ1 = 0
and κ2 can be determined by the mean curvature based on
the first and second fundamental forms [19]. The analytic
approach is convenient for special cases, e.g., a circular
helix, but becomes less straightforward for more general space
curves.
Here we present a numerical approach to determine the
normal curvature κn . The axial and transverse lines on
the neutral plane are represented by discrete space curves
composed of vertexes. The κn of the discrete space curve
can be determined by the vertex osculating circles passing
three consecutive vertexes along the discrete space curve. We
choose a uniform mesh for the plane tape and determine the
corresponding coordinates for each vertex after bending. The
local curvature at each vertex can then be determined and gives
the distribution of the strains according to (5). More details
are explained in Appendix B.
IV. C ASE STUDIES RELEVANT FOR CABLE AND MAGNET
APPLICATIONS
Using the presented approach, we analyze the strain distribution in REBCO layer for several cases that are relevant for
cable and magnet applications. These cases are arranged with
increasing complexity of the strain distribution. For all cases,
the REBCO layer faces the former and is in compression after
bending.
A. Helically wound REBCO tapes
The configuration of a helical tape is used for CORC R [17,
26, 28, 29] and other round cables [30, 31]. The base curve
for this configuration is a circular helix
X = (r cos t, r sin t, p t),
(6)
where the radius r is the sum of the former radius and half of
the tape thickness; t is the parameter in radian and 2πp is the
pitch length. Fig. 3 shows the resulting rectifying developable
surface.
Fig. 3. A tape bent in a helical form around a cylindrical former. It is
generated from a circular helix given by (6).
The circular helix features a constant curvature according
to (20), i.e.,
r
κ= 2
.
(7)
r + p2
All the axial lines are in parallel with the base curve and they
have the same constant curvature as the base curve. If α is the
winding angle between the base curve and rotation axis of the
cylindrical former (Fig. 3), then p = r cot α and (7) becomes
κaxial =
sin2 α
.
r
(8)
Similarly, all the transverse lines have the same curvature
κtransverse =
cos2 α
.
r
(9)
The axial and transverse strain on the REBCO layer are
then given by
d sin2 α
,
r
d cos2 α
transverse =
.
r
axial =
(10a)
(10b)
The strain distribution calculated using the numerical method
presented in section III is consistent with (10).
Fig. 4 shows the axial strain in the REBCO layer as a
function of the substrate thickness for a 1 mm diameter former.
For a fixed winding angle, the axial strain decreases with
thinner substrates (smaller d). For a given substrate thickness,
smaller winding angle (longer pitch) reduces the axial strain
but increases the transverse strain. To limit the compressive
axial strain within 1% for a 1 mm diameter former, the substrate should be thinner than 20 µm for α = 45◦ . Alternatively,
one could wind tapes with a substrate thicker than 20 µm with
α < 30◦ .
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
4
α=30˚
-0.75
45˚
-1.00
60˚
-1.25
90˚
-1.50 0
10
20
30
40
Substrate thickness (micron)
50
Fig. 4. The axial strain in the REBCO layer as a function of substrate
thickness and winding angles for a 1 mm diameter former. According to (10),
the lines for α = 30◦ , 45◦ and 60◦ also give the transverse strains for
α = 60◦ , 45◦ and 30◦ , respectively.
B. A canted cos θ dipole winding
Canted cos θ (CCT) winding configuration is a concept
of generating multipole magnetic fields using helical current
paths for beam steering magnets. Originally proposed by
Meyer and Flasck in 1969 [32], the concept has recently gained
interest for various accelerator magnet applications [33–38].
The base curve for a CCT dipole winding can be given by
r sin t
X = r cos t, r sin t,
+ pt ,
(11)
tan α
where r is the radius of the conductor base curve; α is the tilt
angle between 0 and π/2. The pitch length is given by
2w + wrib
,
(12)
sin α
where 2w is the tape width and wrib is the rib width at t = 0.
More details, e.g., on the ribs, can be found in [36]. Fig. 5
shows a tape forming one turn of CCT dipole winding.
2πp =
Fig. 5. A tape forms a CCT dipole winding. It is generated from the base
curve given by (11). The peak axial strain occurs at the poles at t = 0.5π
and 1.5π. The attached pdf file gives the 3D interactive version of the figure
that can be viewed with Acrobat Reader R .
The winding scheme is similar to a pancake coil except
that the base curve approaches an ellipse rather than a circle.
Therefore, one expects that the CCT dipole winding leads
to mostly axial strain with minimum transverse strain in the
REBCO layer.
Fig. 6 shows the calculated strain distribution in the REBCO
layer using the method presented in section III. We consider
(a) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.03
-0.04
-0.05
-0.06
-0.07
-0.08
-0.09
-0.10
-0.11
-0.12
-0.13
(b) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
0
0.5π
Tape length in terms of parameter t
π
0.0e+00
-1.0e-03
-2.0e-03
-3.0e-03
-4.0e-03
-5.0e-03
-6.0e-03
-7.0e-03
-8.0e-03
-9.0e-03
-1.0e-02
0
0.5π
Tape length in terms of parameter t
Transverse strain (%)
-0.50
Normalized tape width
Axial strain (%)
-0.25
Normalized tape width
1.0 mm diameter former
Axial strain (%)
the following design parameters: r = 32 mm, α = 40◦ ,
p = 1.05 mm, 2w = 4 mm and d = 25 µm (a 50 µm thick
substrate). Due to the symmetry, half turn of the tape is shown
with t ranging from 0 to π.
0.00
π
Fig. 6. Strain distribution in the REBCO layer for a CCT dipole turn: (a)
axial, (b) transverse. r = 32 mm, α = 40◦ , p = 1.05 mm, 2w = 4 mm,
d = 25 µm.
The axial strain dominates, as expected. The peak axial
strains occur at the poles of the winding (t = π/2 and 3π/2)
with the smallest bending radii (Fig. 5). Negligible transverse
strain occurs along the base curve of the tape due to the
nonzero pitch and the resulting torsion of the base curve
(Fig. 6).
Longer pitches (larger p) will increase the torsion and hence
the transverse strain. For instance, doubling the p will increase
the magnitude of the peak transverse strain from 0.01% to
0.03% which is still negligible. Longer pitches also reduce
the effective current density for field generation, undesirable
for high-field CCT accelerator magnet applications. So we will
focus on shorter pitches (p . 2w) for the dipole case here and
the quadrupole case in the next section.
The peak axial strain can be approximated analytically when
p approaches zero to provide quick feedback for the CCT
dipole design based on the REBCO tapes. With zero p, the
base curve becomes an ellipse with no torsion. The ellipse has
a major axis of 2r/ sin α and a minor axis of 2r. Therefore,
the curvature along the ellipse satisfies
1
sin2 α
≤ κellipse ≤
(13)
r
r sin α
according to (21). Thus, the magnitude of the maximum axial
strain for a CCT dipole winding can be approximated as [33]
axial,max =
|d|
.
r sin α
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
(14)
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
5
1.00
30 µm
10 µm
5
TF
0.85
0.80
0.75
10 15 20 25 30 35 40 450.70
α (degree)
Fig. 7. The peak axial strain in the REBCO layer as a function of tilt angle
and conductor substrate thickness (2d) for a CCT dipole turn. r = 32 mm.
The secondary y-axis gives the transfer function (TF) in dashed line for the
dipole field normalized to that of α = 0.
Fig. 7 highlights the direct correlation between the peak
axial strain and magnet performance that is enabled by the
presented approach. For instance, even with a 50 µm thick
substrate, a high dipole transfer function of 0.95 can be
reached with the peak axial compressive strain around 0.3%
in the REBCO layer for r = 32 mm. This strain level is lower
than the 0.5% to 1% level where the conductor Ic can reduce
by 20% [23, 26, 27]. Larger magnet apertures will further
reduce the axial strain for the same α as indicated by (14).
However, thinner tapes are still preferred because they can
not only enable smaller α but also increase the engineering
current density of the winding pack, both benefiting the field
generation.
C. A canted cos θ quadrupole winding
The base curve for a CCT quadrupole winding [37, 39–41]
can be given by
r sin 2t
X = r cos t, r sin t,
+ pt ,
(15)
2 tan α
where the variables are defined as in the CCT dipole case.
Fig. 8 shows a tape forming one turn of CCT quadrupole
winding with four poles in a constant-perimeter geometry.
Note that the REBCO layer is in compression for the whole
winding.
Fig. 9 shows the calculated strain distribution in the REBCO
layer with the following design parameters: α = 40◦ , p = 1.05
mm, 2w = 4 mm and d = 25 µm (a 50 µm thick substrate).
We choose r = 32 mm for the direct comparison with the
dipole case.
As expected, the peak axial strain appears at four poles with
the smallest bending radius (t = 0.25π, 0.75π, 1.25π, 1.75π).
Fig. 8. A tape forms a CCT quadrupole winding. It is generated from the
base curve given in (15). The four poles are located at t = 0.25π, 0.75π,
1.25π and 1.75π. The 3D interactive version of the figure can be viewed in
the attached pdf file with Acrobat Reader R .
(a) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
(b) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.15
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
-0.14
-0.16
-0.18
-0.20
-0.22
-0.21
-0.20
-0.20
-0.21
-0.15
0
0.25π
0.5π
0.75π
Tape length in terms of parameter t
Axial strain (%)
50 µm
π
0.00
-0.10
-0.20
-0.30
-0.40
-0.50
0
0.5π
Tape length in terms of parameter t
Transverse strain (%)
0.90
Normalized tape width
0.95
Normalized tape width
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9
-1.0 0
Normalized dipole TF
Peak axial strain (%)
Based on (14), Fig. 7 plots the peak axial strain as a function
of the winding tilt angle α for a CCT dipole coil with r =
32 mm and three substrate thicknesses: 50, 30 and 10 µm.
Also shown is the dipole transfer function as a function of α
normalized to the value at α = 0 [37]. The strain value with
α = 40◦ and d = 25 µm from (14) is consistent with Fig. 6(a)
(within 7%).
-0.60
π
Fig. 9. Strain distribution in the REBCO layer for a CCT quadrupole turn:
(a) axial. Equi-strain lines of −0.15%, −0.20% and −0.21% are plotted,
(b) transverse. r = 32 mm, α = 40◦ , p = 1.05 mm, 2w = 4 mm and
d = 25 µm.
In addition, the transverse strain appears when the tape transits
between the poles, with the peak magnitude up to 0.6%
appearing at the tape edges. The maximum strain alternates
between the axial and transverse components (Fig. 9).
The magnitude of the peak axial strain can still be correlated
with the magnet design parameters and conductor substrate
thickness even though the distribution of the axial strain across
the conductor width is less uniform (Fig. 9(a)) compared to
the dipole case. With a zero pitch length, the maximum axial
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
6
strain can be approximated by
|d| p
1 + 3 cos2 α
(16)
r sin α
according to (22). Based on (16), we expect the ratio of the
peak axial strain between the quadrupole and dipole windings
to be 1.66 for α = 40◦ , consistent with Figs. 9(a) and 6(a).
Comparing (16) and (14), we see that, given the same r, d
and α, the peak axial strain in a quadrupole winding given
by (15) will be higher than but at most double that in a dipole
winding given by (11).
Fig. 10 plots the peak axial strain as a function of the
tilt angle α and the substrate thickness for CCT quadrupole
windings according to (16). Three substrate thicknesses are
considered: 50, 30 and 10 µm. Also shown is the normalized
quadrupole gradient as a function of the tilt angle. The strain
value from (16) is consistent with the peak axial strain in
Fig. 9(a) (within 8%) for α = 40◦ and d = 25 µm.
D. Saclay coil end
To use the flat thin conductors to form the coil ends for
cos θ or block magnet designs, Rochepault et al. at CEA
Saclay presented a base curve to determine the bending of the
conductors consistent with the constant-perimeter method [11].
The base curve is given by
π
π
X = r cos
+ φ0 cos t , r sin
+ φ0 cos t , δ sin t ,
2
2
(17)
where r is the radius of the conductor center line; φ0 is the
initial radial angle of the tape with respect to the vertical plane;
δ is the length of the coil end; and t is the parameter.
We consider two starting positions for the tape: a horizontal
and a radial position [11]. Fig. 11 shows an example coil end
with a 12 mm wide tape bending around a 50 mm diameter
former. The substrate thickness is 50 µm. Here φ0 = π/2 such
that the tape starts and ends on the same horizontal plane.
Fig. 12 shows the corresponding axial and transverse strain
distribution in the REBCO layer. Half turn of the tape is shown
with t ranging from 0 to π/2 due to the symmetry. The equistrain lines illustrate the pattern of the axial strain distribution
in the REBCO layer. Similar to the CCT quadrupole case,
both axial and transverse bending modes appear in the Saclay
coil end winding scheme. As opposed to the CCT winding
(b) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.02
-0.08
-0.04
-0.12
-0.06
-0.08
-0.08
-0.10
-0.12
-0.14
-0.15
0
0.25π
0.5π
Tape length in terms of parameter t
Axial strain (%)
Normalized tape width
(a) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.16
-0.18
0.00
-0.05
-0.10
-0.15
-0.20
0
0.25π
0.5π
Tape length in terms of parameter t
Transverse strain (%)
Fig. 10. The peak axial strain in the REBCO layer as a function of tilt
angle and conductor substrate thickness (2d) for a CCT quadrupole turn. r =
32 mm. The secondary y-axis gives the transfer function for the quadrupole
gradient normalized to that of α = 0.
Fig. 11. A tape forms a coil end on a 50 mm aperture mandrel based on [11].
The bending form is generated from the base curve given by (17). The tape
is 12 mm wide. r = 31 mm (the radius of the aperture plus half width of
the tape), φ0 = π/2, δ = 50 mm, d = 25 µm. The 3D interactive version
of the figure can be viewed in the attached pdf file with Acrobat Reader R .
Normalized tape width
0.0
1.00
-0.1
10 µm
0.95
-0.2
-0.3
0.90
30 µm
-0.4
0.85
-0.5
TF
-0.6
50 µm
0.80
-0.7
-0.8
0.75
-0.9
-1.0 0 5 10 15 20 25 30 35 40 450.70
α (degree)
Normalized quadrupole TF
Peak axial strain (%)
axial,max =
-0.25
Fig. 12. Strain distribution in the REBCO layer for a Saclay coil end turn
bent on a 50 mm aperture mandrel with the horizontal start position: (a) axial,
(b) transverse. The tape is 12 mm wide. r = 31 mm, φ0 = π/2, δ = 50
mm, d = 25 µm.
cases, the peak axial strain at t = π/2 is no longer uniform
across the tape width because of the non-negligible torsion of
the base curve at t = π/2. The peak axial strain appears at
the center of the outer edge of the tape (t = π/2, normalized
width = −0.5). The axial strain also increases at both ends of
the inner edge (t = 0 and π/2, normalized width = 0.5). This
is related to the boundary conditions for the Saclay coil end
design with φ0 = π/2 such that they match the conductors
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
7
0.00
-0.10
-0.2
-0.15
-0.20
-0.35
Fig. 13. The same coil end design as in Fig. 11 except for φ0 = π/4. The
3D interactive version of the figure can be viewed in the attached pdf file
with Acrobat Reader R .
Tilting the tape from the horizontal to a radial starting
position leads to tighter bending at t = π/2 and less torsion
of the tape. As a result, the magnitude of the peak axial strain
increases from −0.18% (Fig. 12(a)) to −0.40% (Fig. 14(a)).
The magnitude of the peak transverse strain decreases from
−0.25% to −0.14% compared to the horizontal starting position.
A third initial position as discussed in [11] is that the
tape tilts by an additional angle with respect to the radial
positioning angle φ0 toward π/2 − φ0 , i.e., from the radial
to the horizontal position. In this case, the magnitude of the
peak axial strain will decrease while the peak transverse strain
will increase, as demonstrated here by Figs. 12 and 14. The
detailed strain distribution can be determined by the approach
presented here.
The design parameter δ affects mostly the axial strain at
t = π/2 with negligible impact on the transverse strain: longer
coil end (larger δ) makes a tighter bend at t = π/2 and hence
a higher axial strain. Using the case shown in Fig. 14 as an
example, the magnitude of the peak axial strain increases 20%
from 0.40% to 0.60% when δ increases 20% from 50 mm to 75
mm. One can again perform the detailed design optimization
on the strain distribution in REBCO layer with the presented
approach.
V. D ISCUSSION
The REBCO tapes can be bent into various easy-way
bending forms based on the constant-perimeter method. We
used this method to describe the helical winding form that
is relevant to the round REBCO wires. Two bending forms
were also demonstrated for the canted cos θ accelerator magnet
concept. Although examples of bending single tapes are shown
-0.25
-0.30
-0.35
0
0.25π
0.5π
Tape length in terms of parameter t
-0.40
0.00
-0.02
-0.04
-0.06
-0.08
-0.10
Transverse strain (%)
(b) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.1
Axial strain (%)
-0.05
Normalized tape width
(a) 0.5
0.4
0.3
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
Normalized tape width
from the magnet straight section. This boundary condition
also results in zero transverse strain at the flat ends of the
tape. The transverse strain starts increasing in the first and
last quarter of the tape where the transition toward the axial
folding completes, a pattern similar to the CCT quadrupole
case.
Fig. 13 shows the second case where the tape starting at
π/4 radial position as opposed to the horizontal position to
study the impact on the strain distribution.
-0.12
0
0.25π
0.5π
Tape length in terms of parameter t
-0.14
Fig. 14. Strain distribution in the REBCO layer for a Saclay coil end turn
bent on a 50 mm aperture mandrel at the 45◦ radial position: (a) axial, (b)
transverse. The tape is 12 mm wide. r = 31 mm, φ0 = π/4, δ = 50 mm,
d = 25 µm.
here, the method is applicable to a stack of tapes and Roebel
cables [42].
The constant-perimeter method describes an easy-way bend
for the REBCO tapes; but it does not consider the strain
distribution in the REBCO layer. We attempted to address this
issue with a numerical approach to determine the distribution
of the axial and transverse strains in the REBCO layer based
on the local curvature of the bent tape. The approach is
consistent with earlier analysis on the axial strain for the
helical tape case [17, 18] (Appendix C). Our approach can
be applied to more general bending forms given the local
curvature of the neutral plane.
The most effective way to reduce the bending strain in the
REBCO layer is to place it on the neutral plane of the tape
as demonstrated previously [43, 44]. For helical tapes around
a former, the conductor Ic degraded less when the REBCO
layer was closer to the tape neutral plane through additional Cu
lamination [17] or customized conductor architecture [31]. For
a conductor layout shown in Fig. 2, a thinner substrate places
the REBCO layer closer to the neutral plane and reduces the
minimum bending diameter for round REBCO wires [28, 29].
For conductors with the REBCO layer that is off the tape
neutral plane, the winding strain also depends on the design
parameters of REBCO cables and magnets, e.g., the former
radius and winding angle with respect to the former rotation
axis. Formers with small radii and/or small winding angles
are desired as they can lead to compact round REBCO wires
and compact magnets with higher fields or gradients. However,
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
8
they also result in higher strains in the REBCO layer (Figs. 7
and 10 in section IV). This design optimization issue can
be addressed with our method that correlates cable/magnet
designs and strain distribution in the REBCO layer. The presented approach can help ensure proper strain margin to avoid
conductor degradation during cable fabrication and magnet
winding.
Given the strain dependence of Ic , the presented approach
allows to determine the expected Ic of the REBCO tape after
bending [45–48]. Considering the Ic dependence on both strain
and magnetic field, one can determine the expected cable
and magnet performance in order to assess if the fabrication
process has introduced any conductor degradation [49].
We assumed that the conductor Ic depends primarily on the
axial strain component in the REBCO layer. If the assumption
holds, small winding angles (longer pitches) are favored for
a helical tape configuration as they reduces the axial strain,
as shown by (10). This assumption is supported by the
experimental results on helically wound tapes in tension [18]
and compression [50].
The transverse strains are expected in the bending forms
relevant for cable and magnet applications but their impact
on the conductor Ic is largely unknown. For REBCO tapes
fabricated with the metal-organic chemical-vapor deposition
route, both axial and transverse strains lead to the highest
strain sensitivity of the Ic through the impact on the critical
temperature of REBCO [51–53]. The transverse and axial
strain in the helical configuration can be varied with the
winding angle (section IV-A). Thus, comparing the Ic of tapes
wound with different angles may help to clarify the impact of
each strain component on the Ic of REBCO layer.
Although several practical cable and magnet schemes were
presented here, it can be less trivial to obtain the explicit
base curve as required by the constant-perimeter method for
some winding cases, e.g., the continuous ramp between the
two layers of pancake or racetrack coils where the strain in
the tapes can limit the critical current [15, 16, 54]. This can
be a challenge to apply the constant-perimeter method. With
the approach presented here, however, it is now possible to
compare different base curves and determine the optimal one
that minimizes the bending strain in the REBCO tapes. The
approach also gives the shape of the bent tape which can help
the mandrel design to provide proper mechanical support to
the tape.
VI. C ONCLUSION
We present a curvature-based approach to determine the
local axial and transverse strain in the REBCO layer after
being bent with the constant-perimeter geometry. The presented approach allows to determine the strain distribution in
the REBCO layer during the cable and coil winding process
against the key design parameters. It offers an effective and
simple tool for REBCO cable and magnet designers. Four
cases relevant for cable and magnet applications were given
as an example for the application of the presented method: the
helical winding, the CCT dipole and quadrupole configuration,
and the Saclay coil end design. Based on the presented method,
one can determine the expected conductor Ic after bending.
The transverse strain is not negligible for some cable and
magnet winding schemes and its impact on conductor Ic needs
to be clarified. The presented method can be further improved
by considering the Poisson effect as a next step through, e.g.,
finite-element analysis.
ACKNOWLEDGMENTS
We thank the following colleagues for the useful discussion:
D. R. Dietderich and S. A. Gourlay of LBNL, D. van der Laan
and J. Weiss of Advanced Conductor Technologies LLC, and
E. Rochepault of CERN. We thank the referees for suggesting
many improvements.
A PPENDIX A
C URVATURE AND TORSION OF A SPACE CURVE WITH
ARBITRARY SPEED
Suppose X(t) = (x1 (t), x2 (t), x3 (t)) is a regular space
curve with the parameter t and arbitrary speed. The unit
tangent vector v1 is defined as
v1 =
X0
,
|X0 |
(18)
where X0 = dX(t)/dt. The unit binormal vector v3 is given
by
X0 × X00
v3 =
.
(19)
|X0 × X00 |
The unit normal vector v2 is given by v2 = v3 × v1 [55].
The curvature is given by
κ=
|X0 × X00 |
.
|X0 |3
(20)
When p = 0 in the base curves for CCT dipole (11) or
quadrupole (15), we have
√
a2 + 1
κdipole (t) =
,
(21)
r(a2 cos2 t + 1)3/2
q
3 a2 sin2 (2 t) + a2 + 1
,
(22)
κquadrupole (t) =
3/2
r (a2 cos2 (2 t) + 1)
where a = cot α.
The torsion is given by
τ=
|X0 X00 X000 |
(X0 × X00 ) · (X0 × X00 )
.
(23)
More details can be found in [19].
A PPENDIX B
L OCAL CURVATURE BASED ON VERTEX OSCULATING
CIRCLES
To determine the local curvature, we first discretize the
axial and transverse lines on the neutral plane of the flat tape
(Fig. 15(a)). Except for the vertexes on the tape boundaries,
we can define a vertex osculating circle for each vertex along
the axial and transverse lines [56]. For example, the vertex
osculating circle at vertex γk is the circle through γk and its
two neighbors γk−1 and γk+1 (Fig. 15(b)).
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
9
The axial strain in REBCO layer per pitch length is then
lREBCO − lneutral axis
lneutral axis
s
(r − d)2 + p2
=
−1
r 2 + p2
s
d2 − 2rd
−1
= 1+ 2
r + p2
=
(27)
2
Since | dr2−2rd
+p2 | < 1, (27) can be expanded with Taylor series
as
s
1+
d2 − 2rd
1 d2 − 2rd
rd
−1
≈
1+
−1 ≈ − 2
, (28)
r 2 + p2
2 r 2 + p2
r + p2
2
by neglecting the term 2(r2d+p2 ) . Equation (28) is equivalent
to (10a).
R EFERENCES
Fig. 15. (a) The mesh along the axial and transverse lines of the flat tape
before bending. The black dots are the vertexes. (b) The local curvature of
the bent tape is determined by the osculating circles passing three consecutive
vertexes. (c) The osculating circle at vertex γk .
After bending, the axial and transverse lines become space
curves. The coordinate of each vertex can be determined based
on (1). The curvature at vertex γk of the discrete space curve
can be determined based on the sine rule, i.e.,
κk =
2 sin ϕk
,
||γk+1 − γk−1 ||
(24)
where ϕk is the turning angle at vertex γk (Fig. 15(c)). Define
Tk−1 as the unit vector given by γk−1 and γk and Tk as
the unit vector given by γk and γk+1 (Fig. 15(c)), then ϕk =
cos−1 (Tk · Tk−1 ).
A PPENDIX C
A XIAL STRAIN AS DETERMINED BY THE LENGTH CHANGE
Here we show for the helical winding, the axial strain
defined as the length change over a pitch length [17] is
equivalent to the local strain.
The length of the neutral axis is
lneutral axis = 2π
p
r 2 + p2 .
(25)
The length of the REBCO layer in compression is
lREBCO = 2π
p
(r − d)2 + p2 .
(26)
[1] G. A. Levin and P. N. Barnes, “The integration of
YBaCu3 O6+X coated conductors into magnets and rotating machinery,” AIP Conference Proceedings, vol. 824,
no. 1, pp. 433–439, 2006.
[2] B. Colyer, “The geometry of constant perimeter dipole
windings,” Rutherford Laboratory, Tech. Rep. RL-73143, 1973.
[3] H. I. Rosten, “The constant perimeter end,” Rutherford
Laboratory, Tech. Rep. RL-73-096, 1973.
[4] T. C. Randle, “A few mathematical notes on constant
perimeter layers in coil windings,” Rutherford Laboratory, Tech. Rep., December 1974, RL-74-165, and
references therein.
[5] M. N. Wilson, R. B. Hopes, and R. L. Roberts, “A pressure impregnated superconducting dipole,” in Proceedings, 5th International Conference on Magnet Technology
(MT-5): Rome, Italy, April 21-25, 1975, 1975, pp. 511–
518.
[6] G. Dugan, “Developable constant perimeter surfaces:
application to the end design of a tape-wound quadrupole
saddle coil,” Cornell University, Tech. Rep., 1998, cBN
98-1.
[7] M. Manlief, H. Picard, G. Snitchler, A. Ghosh, M. Harrison, W. Sampson, P. Wanderer, G. Dugan, J. Rogers,
A. Temnykh, J. Brandt, P. Limon, and R. Sood, “The performance of Bi-Sr-Ca-Cu-O superconducting quadrupole
coils,” IEEE Trans. Appl. Supercond., vol. 9, no. 2, pp.
293–296, 1999.
[8] K. Takahashi, N. Amemiya, T. Nakamura, Y. Mori,
T. Ogitsu, M. Yoshimoto, I. Watanabe, and T. Yoshiyuki,
“Magnetic field design of coil-dominated magnets wound
with coated conductors,” IEEE Trans. Appl. Supercond.,
vol. 22, no. 3, p. 4901705, 2012.
[9] N. Amemiya, H. Miyahara, T. Ogitsu, and T. Kurusu,
“Design of a cosine-theta dipole magnet wound with
coated conductors considering their deformation at coil
ends during winding process,” Physics Procedia, vol. 67,
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
10
pp. 776–780, 2015, Proceedings of the 25th International Cryogenic Engineering Conference and International Cryogenic Materials Conference 2014.
[10] K. Koyanagi, S. Takayama, H. Miyazaki, T. Tosaka,
K. Tasaki, T. Kurusu, and Y. Ishii, “Development of
saddle-shaped coils for accelerator magnets wound with
YBCO-coated conductors,” IEEE Trans. Appl. Supercond., vol. 25, no. 3, p. 4003104, June 2015.
[11] E. Rochepault, G. Aubert, and P. Vedrine, “Threedimensional magnetic optimization of accelerator magnets using an analytic strip model,” Journal of Applied
Physics, vol. 116, no. 2, p. 023910, 2014.
[12] J. M. Cook, “Strain energy minimization in SSC magnet
winding,” IEEE Transactions on Magnetics, vol. 27,
no. 2, pp. 1976–1980, March 1991.
[13] S. Russenschuck, Field Computation for Accelerator
Magnets: Analytical and Numerical Methods for Electromagnetic Design and Optimization. Weinheim: John
Wiley & Sons, 2010, ch. 19.
[14] P. L. Walstrom, “Twisted coil geometry in plasma confinement devices,” Journal of Fusion Energy, vol. 6,
no. 3, pp. 265–273, 1987.
[15] W. D. Markiewicz and C. A. Swenson, “Winding strain
analysis for YBCO coated conductors,” Superconductor
Science and Technology, vol. 23, no. 4, p. 045017, 2010.
[16] C. Barth, P. Komorowski, R. Tediosi, R. Herzog,
R. Schneider, and C. Senatore, “A size-constrained 3-T
REBCO insert coil for a 21-T LTS magnet: Mechanical
investigations, conductor selection, coil design, and first
coil tests,” IEEE Trans. Appl. Supercond., vol. 26, no. 4,
p. 4600609, 2016.
[17] D. C. van der Laan, “YBa2 Cu3 O7−δ coated conductor
cabling for low ac-loss and high-field magnet applications,” Superconductor Science and Technology, vol. 22,
no. 6, p. 065013, 2009.
[18] Y. Zhu, P. Yi, M. Sun, X. Yang, Y. Zhang, and Y. Zhao,
“The study of critical current for YBCO tape in distorted
bending mode,” Journal of Superconductivity and Novel
Magnetism, vol. 28, no. 12, pp. 3519–3523, 2015.
[19] E. Kreyszig, Differential geometry. New York: Dover
Publications, Inc., 1991.
[20] P. Bo and W. Wang, “Geodesic-controlled developable
surfaces for modeling paper bending,” Computer Graphics Forum, vol. 26, no. 3, pp. 365–374, 2007.
[21] D. C. van der Laan, J. W. Ekin, J. F. Douglas, C. C.
Clickner, T. C. Stauffer, and L. F. Goodrich, “Effect
of strain, magnetic field and field angle on the critical
current density of YBa2 Cu3 O7−δ coated conductors,”
Superconductor Science and Technology, vol. 23, no. 7,
p. 072001, 2010.
[22] S. Otten, A. Kario, A. Kling, and W. Goldacker, “Bending properties of different REBCO coated conductor
tapes and Roebel cables at T =77 K,” Superconductor
Science and Technology, vol. 29, no. 12, p. 125003, 2016.
[23] C. Zhou, K. A. Yagotintsev, P. Gao, T. J. Haugan, D. C.
van der Laan, and A. Nijhuis, “Critical current of various
REBCO tapes under uniaxial strain,” IEEE Trans. Appl.
Supercond., vol. 26, no. 4, p. 8401304, 2016.
[24] H.-S. Shin and M. J. Dedicatoria, “Intrinsic strain effect
on critical current in Cu-stabilized GdBCO coated conductor tapes with different substrates,” Superconductor
Science and Technology, vol. 26, no. 5, p. 055005, 2013.
[25] D. C. van der Laan and J. W. Ekin, “Large intrinsic effect
of axial strain on the critical current of high-temperature
superconductors for electric power applications,” Applied
Physics Letter, vol. 90, no. 5, p. 052506, 2007.
[26] D. C. van der Laan, X. F. Lu, and L. F. Goodrich,
“Compact GdBa2 Cu3 O7−δ coated conductor cables for
electric power transmission and magnet applications,”
Superconductor Science and Technology, vol. 24, no. 4,
p. 042001, 2011.
[27] M. Sugano, K. Shikimachi, N. Hirano, and S. Nagaya,
“The reversible strain effect on critical current over
a wide range of temperatures and magnetic fields for
YBCO coated conductors,” Superconductor Science and
Technology, vol. 23, no. 8, p. 085013, 2010.
[28] D. C. van der Laan, J. D. Weiss, P. Noyes, U. P. Trociewitz, A. Godeke, D. Abraimov, and D. C. Larbalestier,
“Record current density of 344 A mm−2 at 4.2 K and
17 T in corc accelerator magnet cables,” Superconductor
Science and Technology, vol. 29, no. 5, p. 055009, 2016.
[29] J. D. Weiss, T. Mulder, H. J. ten Kate, and D. C.
van der Laan, “Introduction of CORC R wires: highly
flexible, round high-temperature superconducting wires
for magnet and power transmission applications,” Superconductor Science and Technology, vol. 30, no. 1, p.
014002, 2017.
[30] W. Luo, S. Kar, A. Xu, X. F. Li, A. Ben Yahia,
and V. Selvamanickam, “Fabrication and electromagnetic
characterization of ultra-small diameter REBCO wires,”
IEEE Trans. Appl. Supercond., vol. 27, no. 4, p. 6602705,
June 2017.
[31] S. Kar, W. Luo, and V. Selvamanickam, “Ultra-small
diameter round REBCO wire with robust mechanical
properties,” IEEE Trans. Appl. Supercond., vol. 27, no. 4,
p. 6603204, June 2017.
[32] D. Meyer and R. Flasck, “A new configuration for a
dipole magnet for use in high energy physics applications,” Nuclear Instruments and Methods, vol. 80, no. 2,
pp. 339–341, 1970.
[33] C. L. Goodzeit, M. J. Ball, and R. B. Meinke, “The
double-helix dipole — a novel approach to accelerator
magnet design,” IEEE Trans. Appl. Supercond., vol. 13,
no. 2, pp. 1365–1368, June 2003.
[34] A. Devred, B. Baudouy, D. E. Baynham, T. Boutboul,
S. Canfer, M. Chorowski, P. Fabbricatore, S. Farinon,
H. Félice, P. Fessia, J. Fydrych, V. Granata, M. Greco,
J. Greenhalgh, D. Leroy, P. Loverige, M. Matkowski,
G. Michalski, F. Michel, L. R. Oberli, A. den Ouden,
D. Pedrini, S. Pietrowicz, J. Polinski, V. Previtali,
L. Quettier, D. Richter, J. M. Rifflet, J. Rochford,
F. Rondeaux, S. Sanz, C. Scheuerlein, N. Schwerg,
S. Sgobba, M. Sorbi, F. Toral-Fernandez, R. van
Weelderen, P. Védrine, and G. Volpini, “Overview and
status of the Next European Dipole Joint Research Activity,” Superconductor Science and Technology, vol. 19,
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.
This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication.
The final version of record is available at
http://dx.doi.org/10.1109/TASC.2017.2766132
11
no. 3, p. S67, 2006.
[35] H. Witte, T. Yokoi, S. L. Sheehy, K. Peach, S. Pattalwar,
T. Jones, J. Strachan, and N. Bliss, “The advantages and
challenges of helical coils for small accelerators —a case
study,” IEEE Transactions on Applied Superconductivity,
vol. 22, no. 2, p. 4100110, April 2012.
[36] S. Caspi, F. Borgnolutti, L. Brouwer, D. Cheng,
D. R. Dietderich, H. Felice, A. Godeke, R. Hafalia,
M. Martchevskii, S. Prestemon, E. Rochepault, C. Swenson, and X. Wang, “Canted-cosine-theta magnet (CCT)
— a concept for high field accelerator magnets,” IEEE
Transactions on Applied Superconductivity, vol. 24,
no. 3, p. 4001804, June 2014, and references therein.
[37] L. N. Brouwer, “Canted-cosine-theta superconducting
accelerator magnets for high energy physics and ion
beam cancer therapy,” Ph.D. dissertation, University of
California, Berkeley, 2015.
[38] L. Quéval and R. Gottkehaskamp, “Analytical field calculation of modulated double helical coils,” IEEE Trans.
Appl. Supercond., vol. 25, no. 6, p. 4901307, 2015.
[39] R. B. Meinke, C. L. Goodzeit, and M. J. Ball, “Modulated double-helix quadrupole magnets,” IEEE Trans.
Appl. Supercond., vol. 13, no. 2, pp. 1369–1372, June
2003.
[40] S. Caspi, F. Trillaud, A. Godeke, D. R. Dietderich, P. Ferracin, G. Sabbi, C. Giloux, J. G. Perez, and M. Karppinen, “Test of a NbTi superconducting quadrupole magnet based on alternating helical windings,” IEEE Trans.
Appl. Supercond., vol. 19, no. 3, pp. 1195–1198, June
2009.
[41] F. Bosi, P. Fabbricatore, S. Farinon, U. Gambardella,
R. Musenich, R. Marabotto, and E. Paoloni, “Compact
superconducting high gradient quadrupole magnets for
the interaction regions of high luminosity colliders,”
IEEE Trans. Appl. Supercond., vol. 23, no. 3, p. 4001004,
June 2013.
[42] C. Lorin, M. Durante, P. Fazilleau, G. Kirby, and
L. Rossi, “Development of a Roebel-cable-based cos ϑ
dipole: Design and windability of magnet ends,” IEEE
Trans. Appl. Supercond., vol. 26, no. 3, p. 4003105, April
2016.
[43] D. Verebelyi, E. Harley, J. Scudiere, A. Otto, U. Schoop,
C. Thieme, M. Rupich, and A. Malozemoff, “Practical
neutral-axis conductor geometries for coated conductor
composite wire,” Superconductor Science and Technology, vol. 16, pp. 1158–1161, 2003.
[44] M. Sugano, S. Machiya, M. Sato, T. Koganezawa,
K. Shikimachi, N. Hirano, and S. Nagaya, “Bending
strain analysis considering a shift of the neutral axis for
YBCO coated conductors with and without a Cu stabilizing layer,” Superconductor Science and Technology,
vol. 24, no. 7, p. 075019, 2011.
[45] D. C. van der Laan and J. W. Ekin, “Dependence of the
critical current of yba2 cu3 o7−δ coated conductors on inplane bending,” Superconductor Science and Technology,
vol. 21, no. 11, p. 115002, 2008.
[46] M. Takayasu, L. Chiesa, L. Bromberg, and J. V. Minervini, “HTS twisted stacked-tape cable conductor,” Su-
perconductor Science and Technology, vol. 25, no. 1, p.
014011, 2012.
[47] N. C. Allen, L. Chiesa, and M. Takayasu, “Structural
modeling of HTS tapes and cables,” Cryogenics, vol. 80,
pp. 405–418, 2016.
[48] N. Bykovsky, D. Uglietti, R. Wesche, and P. Bruzzone,
“Design optimization of round strands made by twisted
stacks of HTS tapes,” IEEE Trans. Appl. Supercond.,
vol. 26, no. 2, p. 4201207, 2016.
[49] X. Wang, D. R. Dietderich, A. Godeke, S. A. Gourlay,
M. Marchevsky, S. O. Prestemon, and G. L. Sabbi,
“Performance correlation between YBa2 Cu3 O7−δ coils
and short samples for coil technology development,”
Superconductor Science and Technology, vol. 29, no. 6,
p. 065007, 2016.
[50] M. Vojenčiak, A. Kario, B. Ringsdorf, R. Nast, D. C.
van der Laan, J. Scheiter, A. Jung, B. Runtsch,
F. Gömöry, and W. Goldacker, “Magnetization ac loss
reduction in HTS CORC R cables made of striated coated
conductors,” Superconductor Science and Technology,
vol. 28, no. 10, p. 104006, 2015.
[51] D. C. van der Laan, D. Abraimov, A. A. Polyanskii,
D. C. Larbalestier, J. F. Douglas, R. Semerad, and
M. Bauer, “Anisotropic in-plane reversible strain effect
in Y0.5 Gd0.5 Ba2 Cu3 O7−δ coated conductors,” Superconductor Science and Technology, vol. 24, no. 11, p.
115010, 2011.
[52] M. Sugano, S. Machiya, H. Oguro, M. Sato, T. Koganezawa, T. Watanabe, K. Shikimachi, N. Hirano,
S. Nagaya, T. Izumi, and T. Saitoh, “The effect of
the 2D internal strain state on the critical current in
GdBCO coated conductors,” Superconductor Science and
Technology, vol. 25, no. 5, p. 054014, 2012.
[53] S. Awaji, T. Suzuki, H. Oguro, K. Watanabe, and
K. Matsumoto, “Strain-controlled critical temperature
in REBa2 Cu3 Oy -coated conductors,” Scientific Reports,
vol. 5, p. 11156, 2015.
[54] X. Wang, S. Caspi, D. W. Cheng, D. R. Dietderich, H. Felice, P. Ferracin, A. Godeke, J. M. Joseph, J. Lizarazo,
S. O. Prestemon, and G. Sabbi, “Measurements on subscale Y-Ba-Cu-O racetrack coils at 77 K and self-field,”
IEEE Trans. Appl. Supercond., vol. 20, pp. 368–372,
2010.
[55] A. Gray, E. Abbena, and S. Salamon, Modern Differential
Geometry of Curves and Surfaces with Mathematica,
3rd ed. Chapman & Hall/CRC, 2006, ch. 7.
[56] A. I. Bobenko, “Geometry II: Discrete differential geometry,” December 2015, http://page.math.tuberlin.de/ bobenko/Lehre/Skripte/DDG Lectures.pdf.
Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.