A Double Cosine Theta Coil Prototype
Elise Martin and Chris Crawford
October 28, 2010
1
Introduction
A cosine theta coil is a coil that produces a dipole field by arranging current density on a shell
in a cosine theta distribution. Inside the shell the field will be constant. The field outside the
coil can be cancelled to zero by adding a second coil with the right surface current. A physical
prototype of the shell coil is pictured in figure 2. The inner coil acts as a flux return and has
current running along the front face, down the coil and along the back face, then back, making
a circuit. This augments the field produced at the center by the shell coil and matches the
boundary conditions between regions. Theoretically the addition of the fields created by these
two coils should give a zero field external to the coils and a uniform, constant field inside. (See
C. B. Crawford, Yunchang Shin, “A method for designing coils with arbitrary fields”, 200905-24. Technical Note. http://www.pa.uky.edu/~crawford/pub/dsctc.pdf) This was
tested by programming the boundary conditions in COMSOL, which output the locations of the
equipotentials. The ROOT programming environment was then used to simulate discretized
current elements placed at the locations of the equipotentials, and then the Biot-Savart
formula (1.1) was used to find the B-fields at any location.(See figure 1) In addition, a
prototype outer coil was constructed from milled copper PCB. (See figure 2.) The copper
was milled using a computerized router along the equipotentials. The boards were then
soldered together at the edges with wire, current applied, and the resulting B-field mapped.
B=
2
µ0 I X ∆l × r
4π
r3
(1.1)
Simulated Coil
We can test the predictions of the electromagnetic theory by simulating a double square
cosine theta coil numerically. The COMSOL differential equation solver outputs the locations
of the equipotential lines that should give the required constant inner and zero outer Bfields. Both the outside shell and inside septum coils were modeled by plotting discretized
vectors along the equipotentials. The fields were then calculated in two ways. The first
was by creating polygon-shaped differential area elements between equipotential lines on the
1
x
z
x
O
y
Figure 1: Ideal coil. Top half of coil simulated for ROOT analysis. Equipotentials are seen.
Figure 2: Prototype coil. A prototype of the outer coil, built from milled copper PCB and
planes soldered together.
2
shell, integrating the Biot-Savart formula and doing a Taylor series expansion to obtain the
approximate B-field contribution from each differential area (see (2.1) and figure 3 (a)). The
second way was by creating discretized current elements along the equipotentials and again
expanding and integrating Biot-Savart to get the B-field(see (2.6) and figure 3 (b)).
Three variables were then varied to get the get the field to converge and to see how each
variable affected the field, the number of length divisions per wire segment, the number of
wires/length divisions for the inner coil, and the number of equipotentials on the shell. The
shell is 40cm long and centered on zero. The shell is 30 cm wide and 5 cm thick, leaving
room for the 20 cm wide by 40 cm long inner coil.
Figures 4 - 6 show the fields as these three variables are varied for the plane method.
Figure 7 shows variation for wire method. Figure 8 shows how the field changes depending
on how many terms in the expansion of the Biot-Savart integral are used. Zeroeth order is
just the first term of eqn 2.1. Second order is the first two lines (2.1-2.2). Fourth order is all
of the terms in eqn 2.1 (2.1-2.5). The figure shows that using many terms in the equation is
more accurate.
We can conclude from this simulation that using 20 length divisions on the shell and 600
inner wires makes a abetter field. We also see that using current planes or current wires
makes no difference, but the more equipotentials used, the closer to ideal the field is, since
the shell approaches a continuous current density.
1
1 r0 · w0
1 r0 · w0
∆l × w0 −
l0 × w 0 +
∆l × r0
2
12
4 r0
4 r02
!
w02
l02
5 (r0 · w0 )2 5 (r0 · l0 )2
+
+l0 × r0 − 2 − 2 +
2
2
8r0 8r0
8 (r02 )
8 (r02 )


2 2 !2
4
3 
r0 · w0
r
r0 · w0 
+l0 × r0
7 −
− 02
− 28 −
2
128
r0
r0
r02
r0 ·w0
2
2 !
− r2
w0
r0 · w0
0
−7 −
+3
+ (∆l × r0 − l0 × w0 )
2
32
r0
r02
2 2 !!
3
r0 · w0
w0
∆l × w0
5 −
−
2
160
r0
r02
µ0 I
B =
4πr03
l0 × r0 −
µ0 I
B=
4πr3
5 (r · dl)2 1 dl2
1+
−
8
r4
8 r2
3
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
!
dl × r
(2.6)
l1
r0
lt
w0 l
l1
Δl
l2
(a)
r
dl
(b)
Figure 3: (a) shows the differential area element used to solve for the B-field of discrete
planes. l1 and l2 are the equipotentials, lt is an arbitrary transverse vector in the area, r0
is the vector to the where the field is found. (b) shows the differential line element used to
find the B-field for differential wires. r is the vector to where the field is found
3
Data with Simulated Inner Coil
A prototype outer shell coil was constructed with the same dimensions as the simulation
and 1 Amp of current was passed through it. (See figure 2). Three dimensional B-field
measurements were then taken using a fluxgate magnetometer. All B-fields shown are in
Gauss, and all dimensions are in cm.
We look at a transverse plane near the center of the box. The plane is both inside and
extends outside the box. Figures 10-14 compare the measured fields to the calculated fields
and show the residuals for the comparisons.
We then look at a plane near the front of the prototype, where the wires to the power
supply are attached. Figures 15-16 show the comparisons for the front plane.
4
By (Gauss)
0.01
0
-0.01
-0.02
20 length divisions per segment
-0.03
10 length divisions per segment
9 length divisions per segment
8 length divisions per segment
5 length divisions per segment
-100
-50
0
50
100
z (cm)
Figure 4: The discretization along the current direction for each wire segment was varied.
There are four segments per wire and 100 wires per half shell. The inner coil was kept at 100
wires. It is shown that the field is limited by this variable by 20 divisions. 4π was subtracted
from the field inside the coil. Thus this plot shows the residual field. Field shown is in the
y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of
the coil.
3.1
Possible Sources of Error
The simulation of current planes using 100 equipotential lines shows that using only 100
equipotential divisions gives an inherent extra field of about .02 Gauss as compared to 1000
equipotential divisions.
Another possible source of error is that the soldered wires along the edges do not lie flat.
Wires were soldered between faces of the box that connect traces. If these are not perfectly
flat, each non-flat wire would create a small dipole. This field would go as
µ0
((m · r̂)r̂ − m)
(3.1)
4πr3
where m = IA and r is the vector between where the field is measured and where the
dipole is located at (rmeas − rdip ).
Figure 17 shows the field along a longitudinal line at the center of the box due to soldered
wires that have raised about 1mm. Only the joints along the horizontal edges of the coil will
contribute to a field in the horizontal direction. This field will be in the opposite direction
of the By field produced by the coil. This effect was approximated by simulating 30 of the
60 dipoles at the center of the edge, 15 at 6 cm left of center, and 15 at 6 cm right of center,
for an outside edge. For an inside edge, 30 dipoles were placed at the center, 15 at 5 cm
left of center, and 15 at 5 cm right of center. This was done for the four edges at the front
B=
5
By (gauss)
0
-0.005
-0.01
Entries
0
Mean x
0
Mean
y
0
100 equipotentials
RMS x
0
250 equipotentials
RMS
y
0
-0.015
500 equipotentials
-0.02
-200
1000 equipotentials
-150
-100
-50
0
50
100
150
200
z (cm)
Figure 5: The number of discretizations perpendicular to the current direction (number
of equipotentials) were varied. The number of wires/ length discretizations on the inner
coil kept at 100 and the number of length discretizations on the shell kept at 20 per wire
segment. It is shown that the greater the number of equipotentials, the closer the field is to
the expected result. 4π was subtracted from the field inside the coil. Thus this plot shows
the residual field. Field shown is in the y-direction, or across the face of the coil, while the
z-axis is along the longitudinal length of the coil.
face of the coil. The effects of this approximation, as seen in figure 17, show that there are
high field contributions at the longitudinal edges of the coil. These could account for the
abnormal field spikes at the same locations seen in the data, i.e. figure 14.
4
Conclusions
To find the tolerances of the field, begin with the equation of the B-field.
B=
µ0
(m · r̂)
4πr3
∂Bx
µ0
=
(18mx x + 6my y + 6mz z)
∂x
4πz 5
we know that the constraints in x,y, and z are
(4.1)
(4.2)
∇x = ∇y = 15cm
(4.3)
∇z = 2m
(4.4)
6
By (gauss)
200 inner wires
0.002
Entries
Mean x
Mean y
RMS x
RMS y
0.001
100 inner
0 wires
0 wires
600 inner
0
1000 0
inner wires
0
0
-0.001
-0.002
-200
-150
-100
-50
0
50
100
150
200
z (cm)
Figure 6: The number of discretizations both along the direction of the current and perpendicular to it were varied. The number of equipotentials on the shell was kept at 1000 per
half shell and the number of length discretizations kept at 20 per wire segment. The field
reaches its best value by about 600 wires, but 200 wires is also very close. 4π was subtracted
from the field inside the coil. Thus this plot shows the residual field. Field shown is in the
y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of
the coil.
and we know the field here is
Bx = 40mG
(4.5)
1 ∂Bx
< 10−6 /cm
B ∂x
(4.6)
mx < 470 A cm2
(4.7)
my < 1400 A cm2
(4.8)
mz < 106 A cm2
(4.9)
This gives a field tolerance of
and dipole tolerances of
It is concluded that the greater the number of equipotential discretizations used, i.e.
the more wires wound, the more closely the coil gives the desired field, since a continuous
current density is approximated. There are several errors present in the measurement of the
prototype coil. Some of this error can likely be explained by dipoles created by soldering
wires between the traces.
7
By (Gauss)
0
-0.005
Entries
0
Mean x
0
Mean y
0
RMS x
0
RMS y
0
250 wires
-0.01
-0.015
500 wires
100 wires
1000 wires
-0.02
-250
-200
-150
-100
-50
0
50
100
150
200 250
z (cm)
By (Gauss)
Figure 7: The discretization perpendicular to the current direction (the number of wires) was
varied for the continuous wires method. Very little difference is seen between this and the
same changes in the equivilant plots for the planes method (see figure 5). 4π was subtracted
from the field inside the coil. Thus this plot shows the residual field. Field shown is in the
y-direction, or across the face of the coil, while the z-axis is along the longitudinal length of
the coil.
0
-0.005
Entries
Mean x
Mean y
RMS x
RMS y
-0.01
0
0
0
0
0
-0.015
terms to 4th order
terms to 0th order
-0.02
-200
terms to 2nd order
-150
-100
-50
0
50
100
150
200
z (cm)
Figure 8: Shows By in z for zeroeth, second, and fourth order expansions of Biot-Savart for
finite area of current on the coil. 4π was subtracted from the field inside the coil. Thus this
plot shows the residual field. Field shown is in the y-direction, or across the face of the coil,
while the z-axis is along the longitudinal length of the coil.
8
By (Gauss)
14
inner coil
12
outer coil
total field
10
8
6
4
2
0
-2
-4
-50
-40
-30
-20
-10
0
10
20
30
40
50
z (cm)
By (Gauss)
Figure 9: Field from the prototype outer coil, field from the simulated inner coil, and the
sum of the two. This shows how the addition of the two feels gives the desired total field.
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-35
-30
-25
-20
-15
-10
-5
0
5
10
x (cm)
Figure 10: .Field along vertical line. Red is simulated field and blue is measured field.
9
By (Gauss)
0.2
0
-0.2
-0.4
-0.6
-0.8
-40
-30
-20
-10
0
10
20
30
40
y (cm)
Figure 11: .Field along horizontal line. Red is simulated field and blue is measured field.
10
By (Gauss)
0.2
0
-0.2
-0.4
-0.6
-0.8
-40
-30
-20
-10
0
10
20
30
By (Gauss)
y (cm)
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-40
-30
-20
-10
0
10
20
30
y (cm)
Figure 12: Field long several parallel horizontal lines. Red is simulated field and blue is
measured field. Black is residual.
11
By (Gauss)
0
-0.2
-0.4
-0.6
-0.8
-35
-30
-25
-20
-15
-10
-5
0
5
x (cm)
-30
-25
-20
-15
-10
-5
0
5
x (cm)
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-35
Figure 13: Field long several parallel vertical lines. Red is simulated field and blue is
measured field. Black is residual.
12
By (Gauss)
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
By residual (Gauss)
-0.8
-60
-40
-20
0
20
40
60
z (cm)
-60
-40
-20
0
20
40
60
z (cm)
0
-0.002
-0.004
-0.006
-0.008
-0.01
-0.012
Figure 14: Field long a line down the center longitudinal axis. Red is simulated field and
blue is measured field. Black is residual.
13
By (Gauss)
-0.3
-0.35
-0.4
-0.45
-0.5
-6
-4
-2
0
2
4
6
x (cm)
by+fieldx(-y+2.73,-x-20.6,-z+65.5)
by+fieldx(-y+2.73,-x-20.6,-z+65.5):(x+20.6)
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-8
-6
-4
-2
0
2
4
6
8
(x+20.6)
Figure 15: Fields along parallel vertical lines near the front of the prototype. Red is simulated
field and blue is measured field. Black is residual.
14
By (Gauss)
-0.25
-0.3
-0.35
-0.4
-0.45
-0.5
By residual (Gauss)
-8
-6
-4
-2
0
-4
-2
0
2
4
6
8
y (cm)
0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12
-0.14
-0.16
-0.18
-8
-6
2
4
6
8
y (cm)
Figure 16: Fields along parallel horizontal lines near the front of the prototype. Red is
simulated field and blue is measured field. Black is residual.
15
-3
solder_field(x,y,z)
solder_field(x,y,z)
0 ×10
-0.05
0
-0.0005
-0.1
-0.15
-0.001
-0.2
-0.25
-0.0015
-0.3
-0.002
-0.35
-60
-40
-20
0
20
40
60
x (cm)
-60
-40
-20
0
20
40
60
z (cm)
Figure 17: Horizontal fields along the vertical (left) and longitudinal (right) center lines.
The coil extends from -20 to +20 along z. In x, the inside of the coil is between -10 and
+10, and the actual coil takes up the space 5 cm beyond these.
16