Biplanar homogeneous field electromagnets and method for making same

A method for making a biplanar, symmetrical electromagnet having K coils, where K is an even integer, comprising the steps of:

• a) producing K-1 equations of the form 0 = ∑ k = 1 K ⁢ I k ⁢ R k - ( n + 1 ) ⁢ sin ⁢ ⁢ θ k ⁢ P n + 1 1 ⁡ ( cos ⁢ ⁢ θ k )
• wherein each equation has a different value of n, wherein each value of n is a positive, even integer less than 2K, and wherein Ik denotes the Ampere-tums and Rk and θk denote the spherical coordinates of a kth coil;
• b) solving the equations produced in step (a) for values of Ik, Rk, and θk, wherein the coils are constrained to two parallel planes and the coils are symmetrical;
• and
• c) building coils corresponding to values of Rk and θk found in step (b) and capable of carrying Ampere-turns proportionate to values of Ik found in step (b).

A biplanar, symmetrical 8-coil electromagnet comprising:

• a) a first small coil;
• b) a first medium coil;
• c) a first large coil;
• d) a first extra large coil, wherein the first small coil, the first medium coil, the first large coil, and the first extra large coil are located in a first plane;
• e) a second small coil;
• f) a second medium coil;
• g) a second large coil;
• and
• h) a second extra large coil, wherein the second small coil, the second medium coil, the second large coil, and the second extra large coil are located in a second plane;
• wherein:
• 1) the first plane and the second plane are parallel and spaced apart by a distance 2z;
• 2) all 8 coils are coaxial about an axis;
• 3) the first small coil encloses a first small filamentary loop with a radius equal to 0.294922z, and the second small coil encloses a second small filamentary loop with a radius equal to 0.294922z;
• 4) the first medium coil encloses a first medium filamentary loop with a radius equal to 0.590474z, and the second medium coil encloses a second medium filamentary loop with a radius equal to 0.590474z;
• 5) the first large coil encloses a first large filamentary loop with a radius equal to 1.031283z, and the second large coil encloses a second large filamentary loop with a radius equal to 1.031283z;
• 6) the first extra large coil encloses a first extra large filamentary loop with a radius equal to 2.632601z, and the second extra large coil encloses a second extra large filamentary loop with a radius equal to 2.632601z;
• and
• 7) the first filamentary loops are coplanar with the first plane, and the second filamentary loops are coplanar with the second plane;
• whereby the electromagnet provides a homogeneous magnetic field between the first plane and second plane near the axis when energized with appropriate currents.

A biplanar, symmetrical 10-coil electromagnet comprising:

• a) a first extra small coil
• b) a first small coil;
• c) a first medium coil;
• d) a first large coil;
• e) a first extra large coil, wherein the first extra small coil, the first small coil, the first medium coil, the first large coil, and the first extra large coil are located in a first plane;
• f) a second extra small coil;
• g) a second small coil;
• h) a second medium coil;
• i) a second large coil;
• and
• j) a second extra large coil, wherein the second extra small coil, the second small coil, the second medium coil, the second large coil, and the second extra large coil are located in a second plane;
• wherein:
• 1) the first plane and the second plane are parallel and spaced apart by a distance 2z;
• 2) all 10 coils are coaxial about an axis;
• 3) the first extra small coil encloses a first extra small filamentary loop with a radius equal to 0.236199z, and the second extra small coil encloses a second extra small filamentary loop with a radius equal to 0.236199z;
• 4) the first small coil encloses a first small filamentary loop with a radius equal to 0.457269z, and the second small coil encloses a second small filamentary loop with a radius equal to 0.457269z;
• 5) the first medium coil encloses a first medium filamentary loop with a radius equal to 0.734885z, and the second medium coil encloses a second medium filamentary loop with a radius equal to 0.734885z;
• 6) the first large coil encloses a first large filamentary loop with a radius equal to 1.164685z, and the second large coil encloses a second large filamentary loop with a radius equal to 1.164685z;
• 7) the first extra large coil encloses a first extra large filamentary loop with a radius equal to 2.745650z, and the second extra large coil encloses a second extra large filamentary loop with a radius equal to 2.745650z;
• and
• 8) the first filamentary loops are coplanar with the first plane, and the second filamentary loops are coplanar with the second plane;
• whereby the electromagnet provides a homogeneous magnetic field between the first plane and second plane near the axis when energized with appropriate currents.