Magnetic Field Uniformity of Helmholtz Coils and Cosine θ Coils

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1 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.

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).

2 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

3 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] where J = 1... N/2, and K = Eq. (4), 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. 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

4 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

5 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).

6 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

7 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. 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 = 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). Cosθ L/R=2.4 Cosθ L/R=3.2 Cosθ L/R=4.0 Cosθ L/R=4.8 Fig. 20: and modified cosine θ coil field fractional change along the x-axis (0.5 meter full range). Note it is the primary axis. Helmholtz

8 Fig. 21: 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: and modified cosine θ coil field fractional change along the z-axis (1.0 meter full range). Fig. 22: and modified cosine θ coil field fractional change along the y-axis (0.5 meter full range). Fig. 23: and modified cosine θ coil field fractional change along the y-axis (0.2 meter = 40% zoom-in). Fig. 25: 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

9 relative 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 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).

March 11. Physics 272. Spring Prof. Philip von Doetinchem

March 11. Physics 272. Spring Prof. Philip von Doetinchem Physics 272 March 11 Spring 2014 http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html Prof. Philip von Doetinchem philipvd@hawaii.edu Phys272 - Spring 14 - von Doetinchem - 32 Summary Magnetic