Magnetic Field Uniformity of Helmholtz Coils and Cosine θ Coils

Magnetic Field Uniformity of Helmholtz Coils and Cosine θ Coils
Ting-Fai Lam
The Department of Physics, The Chinese University of Hong Kong
ABSTRACT
The magnetic field uniformities of the Helmholtz coil, the cosine θ coil and the
modified cosine θ coil have been studied, because a uniform magnetic field is essential
to many experiments, including the SNS nEDM Experiment. Computer simulations
have been used to calculate the magnetic fields of different coils. It was found that the
cosine θ coil in general produces more uniform field than the Helmholtz coil, hence is a
better candidate for experiments which requires higher magnetic field uniformity.
I. Background and Introduction
The Helmholtz Coils
A Helmholtz coil is a device with two
identical circular loops of electrical wires,
parallel to each other, with the two centers
located on the same imaginary perpendicular
axis. An identical electric current runs
through the loops in a parallel direction,
generating a magnetic field in the central
axis direction.
As shown in Fig. 1, a Helmholtz coil is often
considered as the combination of two rings
of current with negligible winding thickness.
Thus, the only parameters in a Helmholtz
coil are the separation of rings, h, and their
radius.
Such configuration has a zero first-derivative
of the on-axis field strength with respect to
z-position, at the central point between the
rings. Derived from the Biot-Savart Law, it
can be shown that the radius of the rings
should be adjusted to equal the separation h,
so as to attain the best field uniformity,
defined by a zero second-derivative of field
at the coil center.
The overlapping cylinders
It has been noted that the overlapping of two
current-carrying cylinders can create a
constant magnetic field region. In the
overlapping volume of two parallel infinitely
long cylinders, each carrying a constant
current density J running in opposite
direction, it can be derived that the magnetic
field is a constant vector (Fig. 2).
Fig. 1: Helmholtz coil schematic drawing.

Fig. 2: Constant B-field generated by two
overlapping current-carrying cylinders.
The proof is as follow. Consider the volume
in the cylinder carrying a current density J,
whose axis coincides with the z-axis, the B-
field inside should be:
Fig. 3: Overlapping cylinders for small d.
Hence, it is also possible to represent such
configuration with only one cylinder,
carrying a surface current whose density σ(θ)
is proportional to cos θ, as shown below:
... Eq. (1).
Similarly, inside another cylinder with
current density –J, whose axis is parallel but
displaced towards –y direction by d, the B-
field is:
... Eq. (2).
Fig. 4: The “idealized” cosine θ setup.
A vector addition will show:
The Cosine θ coils
... Eq. (3).
In the limit when axial displacement d
approaches 0, only two “new moon shaped”
channels would still contain a current. The
amount of current in the infinitesimal
volume is proportional to cos θ, which is
defined as:
Fig. 5: Cosine θ coils schematic drawing. [1]
The cosine θ coil in Fig. 5 can be considered

as a practical approximation to the
arrangement in Fig. 4. In order to reproduce
the density distribution σ(θ) = σ 0 cos θ, the
wires are wound on the coil such that the x-
axis projections of all windings are
uniformly distributed. The angle θ of the
windings can be expressed as: [2] ... Eq. (4),
where J = 1 ... N/2, and K = 0.
The cosine θ coil replaces the continuous
current densities with a total of N discrete
wires, and also the infinitely long cylinder
with a finite L/R ratio, where L is the
cylindrical length of the coil and R is the
radius. The effects of the two approximations
will be investigated.
Modified cosine θ coils
A simple cosine θ coil has a field gradient. In
order to produce more uniform magnetic
field, it is possible to combine two coils with
different number of turns, N, to partially
cancel the gradient. [3] The combination of a
main coil and a secondary coil, having
different number N, and running the currents
in opposite directions, will be called a
“modified cosine θ coil”.
The uniformity of field is reflected by the
fractional change, which is independent of
the actual value of the current running in the
coil. The modified cosine θ coils introduce
the following two parameters, namely N 2 in
the secondary coil, and the current ratio I 2 /I,
where I and I 2 are the currents in the main
and secondary coils.
Magnetic field uniformity and fractional
change
Magnetic field uniformity is represented by
the fractional change of field relative to the
field at the origin. This is a function of
position, explicitly:
... Eq. (5).
The closer this value is to zero, the more
uniform the field is. Similar definitions hold
for y, z field components, and y, z position
dependence.
II. Method
Computer simulations
A Fortran program and a Mathematica
program were written to numerically
integrate the Biot-Savart Law, hence
calculate the resultant B-field generated by a
coil.
Because the Mathematica program allows
flexible numerical plotting, it was used to
investigate the field dependence on position.
Other than absolute values of magnetic field,
the program was used to calculate the
fractional change of field as a function of
position, defined as in Eq. (5).
Comparison of various coils
Field dependence on position cannot be
compared in all 3 dimensions, owing to the
different orientations of the Helmholtz and
the cosine θ coils.
To compare the cosine θ coil with the
Helmholtz coil, one “primary” axis is
identified. Even if the field in the coil
volume is not “unidirectional”, it is clear that
the Helmholtz coil generates field primarily
towards its z-axis, while the cosine θ coil
generates towards z-axis. Such axis will be
addressed the “primary” axis of each coil. In
other words, the B z (z) for Helmholtz coil and
B x (x) for cosine θ coil will be considered in

every comparison plot.
along the x-axis (1.0 meter full range).
The cosine θ coil radius R is always set to be
h/2, where h is both the separation and radius
of Helmholtz coil. The magnetic field is
generated in x direction for Helmholtz coil,
while in z direction for cosine θ coil. This
setting would allow the lengths of the
primary axes to be the equal for both coils.
For the cosine θ coil, the two parameters N
and L/R ratio are varied to see the effects on
the field uniformity. Also the Helmholtz coil,
both the simple and modified cosine θ coils
are compared.
Fig. 7: Helmholtz coil field fractional change
along the x-axis (0.2 meter = 20% zoom-in).
III. Results and Discussion
Field generated by a Helmholtz coil
Fig. 6 to 9 show the B z fractional change
along the 3 axes for the Helmholtz coil. The
first pair of plots are the dependence along
the x-axis, with the first showing the whole
0.5-meter range, while the second showing
only 0.2-meter zoom-in.
The next pair are for z dependence. Because
x and y directions are symmetric for the coil,
the y-axis plots are omitted.
Fig. 8: Helmholtz coil field fractional change
along the z-axis (0.5 meter full range). Note
it is the “primary” axis.
The specifications are:
h = 0.5 meters,
where h is both the radius and separation.
Fig. 9: Helmholtz coil field fractional change
along the z-axis (0.2 meter = 40% zoom-in).
Note it is the “primary” axis.
Fig. 6: Helmholtz coil field fractional change
From the above plots, it is found that the
fractional change takes a “saddle-shaped”
positional dependence. As expected, it has a

stationary point at the center, which is also
the global maximum.
Some characteristics are the fractional
change plots do not change in shape upon
zooming in. The uniformity of the primary
axis is relatively worse than the other (x- and
y-) axes.
Field generated by a cosine θ coil
Fig. 10 to 15 show the B x fractional change
along the 3 axes for the cosine θ coil. The
first pair of plots are the dependence along
the x-axis, with full range and zoom-in
respectively. The remaining 2 pairs are for y
and z dependence.
Fig. 11: Cosine θ coil field fractional change
along the x-axis (0.2 meter = 40% zoom-in).
Note it is the “primary” axis.
The specifications are:
R = 0.25 meters, L = 1 meter, L/R = 4.0,
where R is the radius and L is the cylindrical
length.
N = 22.
Fig. 12: Cosine θ coil field fractional change
along the y-axis (0.5 meter full range).
Fig. 10: Cosine θ coil field fractional change
along the x-axis (0.5 meter full range). Note
it is the “primary” axis.
Fig. 13: Cosine θ coil field fractional change
along the y-axis (0.2 meter = 40% zoom-in).

sections, explicitly stated as follow.
For Helmholtz coil:
h = 0.5 meters.
B z (z) is observed.
For all cosine θ coils:
R = 0.25 meters, L = 1 meter, L/R = 4.0.
B x (x) is observed.
Fig. 14: Cosine θ coil field fractional change
along the z-axis (1.0 meter full range).
Helmholtz
Cosθ N=40
Cosθ N=22
Cosθ N=12
Cosθ N=8
Fig. 16: Taking N as the parameter, field
fractional change plot of various coils along
the primary axis (0.5 meter full range).
Fig. 15: Cosine θ coil field fractional change
along the z-axis (0.2 meter = 20% zoom-in).
On the primary x-axis, the fractional change
takes an “m-shaped” positional dependence,
with the center being a local minimum.
Both the y and z dependence has a saddle
shape. Same as the Helmholtz coil, plot of z
dependence retains a similar shape upon
zooming. The uniformity along y axis, on the
other hand, clearly worsens only near the
edges.
Comparison for various N
N is taken as a parameter here. The primary
component field fractional change along the
primary axis for various cosine θ coils, and
the Helmholtz coil, are plotted together in
Fig. 16 and 17.
The specifications are the same as above
Cosθ N=8
Cosθ N=12
Cosθ N=22
Cosθ N=40
Helmholtz
Fig. 17: Taking N as the parameter, field
fractional change plot of various coils along
the primary axis (0.2 meter = 40% zoom-in).
Since the field uniformity is better when the
fractional change is close to zero, N = 40 has
the best uniformity while N = 8 has the
worst. It verifies the expectation that a
greater N produces better uniformity,
because it better compensates the
approximation to the continuous current
density (Fig. 4) by more lines of current.
For the particular length specifications used

here, N = 40 approximately corresponds to
the Helmholtz coil uniformity considering
the full range, while N = 22 approximately
corresponds to that in the zoom-in.
Comparison for various L/R ratios
L/R ratio is taken as a parameter here. The
primary component field fractional change
along the primary axis for various cosine θ
coils, and the Helmholtz coil, are plotted
together in Fig. 18 and 19.
The specifications are the same as above
sections, explicitly stated as follow.
For Helmholtz coil:
h = 0.5 meters.
B z (z) is observed.
For all cosine θ coils:
R = 0.25 meters, N = 22.
B x (x) is observed.
Cosθ L/R=2.4
Cosθ L/R=3.2
Cosθ L/R=4.0
Fig. 19: Taking L/R ratio as the parameter,
field fractional change plot of various coils
along the primary axis (0.2 meter = 40%
zoom-in).
From the plots above, L/R = 4.8 has the best
field uniformity, while L/R = 2.4 has the
worst. It can be justified in terms of
approximation to the ideal case, where L/R is
infinity. Referring to Fig. 5, it is also noted
that the “edge current” is less effective when
L/R is large.
Modified cosine θ coils
Cosine θ coil can be modified by combining
another cosine θ coil with different N, which
runs in opposite directions. The number N 2
and the current ratio I 2 /I for the secondary
current can be determined by trial-and-error.
The specifications for the Fig. 20 to 25 are:
For both coil:
R = 0.25 meters, L = 1 meter, L/R = 4.0.
For main coil:
N = 22.
For secondary coil:
N 2 = 8, I 2 /I = –0.75.
Cosθ L/R=4.8
Helmholtz
Fig. 18: Taking L/R ratio as the parameter,
field fractional change plot of various coils
along the primary axis (0.5 meter full range).
Modified
Simple
Cosθ L/R=2.4
Cosθ L/R=3.2
Cosθ L/R=4.0
Cosθ L/R=4.8
Fig. 20: Simple and modified cosine θ coil
field fractional change along the x-axis (0.5
meter full range). Note it is the “primary”
axis.
Helmholtz

Modified
Simple
Simple
Modified
Fig. 21: Simple and modified cosine θ coil
field fractional change along the x-axis (0.2
meter = 40% zoom-in). Note it is the
“primary” axis.
Fig. 24: Simple and modified cosine θ coil
field fractional change along the z-axis (1.0
meter full range).
Modified
Simple
Modified
Simple
Fig. 22: Simple and modified cosine θ coil
field fractional change along the y-axis (0.5
meter full range).
Modified
Simple
Fig. 23: Simple and modified cosine θ coil
field fractional change along the y-axis (0.2
meter = 40% zoom-in).
Fig. 25: Simple and modified cosine θ coil
field fractional change along the z-axis (0.2
meter = 20% zoom-in).
The plots show that a secondary coil can
effectively reduce the field gradient in the
central region of the coil. For the 0.2-meter
zoom-in region, the uniformity is improved
by more than 25 times and about 7 times, for
the x-axis and the y-axis respectively.
The uniformity of a modified cosine θ coil is
not as good for the edge regions of each axis.
Also the uniformity on z-axis is totally
unaffected, as seen from Fig. 24 and 25.
IV. Conclusions
To conclude, the magnetic field uniformities
of the Helmholtz coil, various simple cosine
θ coil and modified cosine θ coil, which are
represented by the fractional change of field

elative to the coil origin, have been
observed and compared through simulation.
It was found that the cosine θ coil is a better
candidate than the Helmholtz coil to
generate a uniform magnetic field, given
suitable number of turns N, length-to-radius
ratio L/R, and secondary coil modification if
possible.
V. Acknowledgments
I gratefully acknowledge the help and
beaconing advice from Jen-Chieh Peng and
Ping-Han Chu. This work is completed
under the REU program, which is supported
by National Science Foundation Grant PHY-
0647885.
VI. Footnotes, Endnotes and
References
[1] B. Plaster, talk presented at the SNS
nEDM Experiment Collaboration Meeting
(October 2006).
[2] S. Balascuta and R. Alarcon, talk
presented at the SNS nEDM Experiment
Collaboration Meeting (June 2009).
[3] R. Schmid, talk presented at the SNS
nEDM Experiment Collaboration Meeting
(March 2006).