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A superconducting coil includes (a) a plurality of windings of a coil comprising high-temperature superconductors and (b) an electrically conductive channel in which the high-temperature superconductors are mounted. The high-temperature superconductors can comprise at least one of the following: Ba2

A superconducting coil includes (a) a plurality of windings of a coil comprising high-temperature superconductors and (b) an electrically conductive channel in which the high-temperature superconductors are mounted. The high-temperature superconductors can comprise at least one of the following: Ba2Sr2Ca1Cu2O8 (2212), Ba2Sr2Ca2Cu3O10 (2223), and YBa2Cu3O7-x (123) superconductor.

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1. A superconducting coil, comprising: a plurality of windings of a coil comprising: a plurality of high-temperature superconductor strands; andan electrically conductive channel in which the high-temperature superconductor strands are mounted, with a plurality of but no more than nine high-temperat

1. A superconducting coil, comprising: a plurality of windings of a coil comprising: a plurality of high-temperature superconductor strands; andan electrically conductive channel in which the high-temperature superconductor strands are mounted, with a plurality of but no more than nine high-temperature superconductor strands mounted in a cross-section of the channel,wherein the coil has a cross-section that includes a plurality of the windings mutually attached in a unitary structure across the cross-section of the coil; anda filler matrix in which the windings of the high-temperature superconductor strands and electrically conductive channel are embedded in an array across the cross-section of the coil, wherein the array includes a plurality of rows and columns of electrically conductive channel sections, each containing the high-temperature superconductor strands and separated across the rows and columns of the cross-sectional array by the filler matrix. 2. The superconducting coil of claim 1, wherein the high-temperature-superconductor strands comprise at least one of the following: Ba2Sr2Ca1Cu2O8 (2212), Ba2Sr2Ca2Cu3O10 (2223), and YBa2Cu3O7-x (123) superconductor. 3. The superconducting coil of claim 1, further comprising solder in the channel, wherein the solder bonds the coil comprising the high-temperature-superconductor strands to the channel. 4. The superconducting coil of claim 3, wherein the channel comprises copper. 5. The superconducting coil of claim 1, wherein the coil includes at least 1,200 of the windings, and wherein the windings provide a current-carrying capacity of at least 2 million amps-turns in the coil. 6. The superconducting coil of claim 1, wherein the coil includes at least 1,500 of the windings, and wherein the windings provide a current-carrying capacity of at least 3 million amps-turns in the coil. 7. The superconducting coil of claim 1, wherein the filler matrix comprises epoxy. 8. The superconducting coil of claim 7, wherein the filler matrix further comprises fiber. 9. The superconducting coil of claim 8, wherein the fiber comprises glass or carbon. 10. The superconducting coil of claim 1, wherein the channel has a cross-section with a radial dimension no greater than about 2.4 mm. 11. The superconducting coil of claim 1, wherein no more than four high-temperature superconductor strands are mounted in the cross-section of the channel. 12. A magnet structure, comprising: at least one superconducting coil comprising a plurality of coil windings including: a plurality of superconductor strands comprising at least one of Ba2Sr2Ca1Cu2O8 (2212), Ba2Sr2Ca2Cu3O10 (2223), and YBa2Cu3O7-x (123);an electrically conductive channel in which the superconductor strands are mounted, with a plurality of but no more than nine superconductor strands mounted in a cross-section of the channel, wherein the superconducting coil has a cross-section that includes a plurality of the coil windings mutually attached in a unitary structure across the cross-section of the superconducting coil; anda filler matrix in which the windings of the high-temperature superconductor strands and electrically conductive channel are embedded in an array across the cross-section of the coil, wherein the array includes a plurality of rows and columns of electrically conductive channel sections, each containing the high-temperature superconductor strands and separated across the rows and columns of the cross-sectional array by the filler matrix; anda cryocooler configured and positioned to cool the superconducting coil. 13. The magnet structure of claim 12, wherein the cryocooler is a Gifford-McMahon cryocooler or a pulse-tube cryocooler. 14. The magnet structure of claim 12, further comprising at least one thermal coupling between the cryocooler and the superconducting coil. 15. The magnet structure of claim 14, wherein the thermal coupling is a superconducting current lead. 16. The magnet structure of claim 12, further comprising a matrix between the windings. 17. The magnet structure of claim 16, wherein the filler matrix comprises a glass-fiber/epoxy composite. 18. The magnet structure of claim 12, further comprising: a bobbin in which the superconducting coil is mounted; andradial-tension links coupled with the bobbin and applying outward radial tension on the bobbin at a plurality of positions. 19. A magnet structure, comprising: at least one superconducting coil comprising a plurality of windings of a coil including superconductor strands comprising at least one of Ba2Sr2Ca1Cu2O8 (2212), Ba2Sr2Ca2Cu3O10 (2223), and YBa2Cu3O7-x (123);an electrically conductive channel in which the coil comprising the superconductor strands is mounted, with no more than nine superconductor strands mounted in a cross-section of the channel;a cryocooler configured and positioned to cool the superconducting coil;a bobbin in which the superconducting coil is mounted;radial-tension links coupled with the bobbin and applying outward radial tension on the bobbin at a plurality of positions; anda pressurized bladder positioned between each coil and the bobbin to apply radial inward force on the coil. 20. The magnet structure of claim 12, wherein the structure includes at least two of the superconducting coils, the magnet structure further comprising a magnetic yoke that contains the superconducting coils. 21. The magnet structure of claim 20, wherein the magnetic yoke comprises a pair of poles, wherein the poles are on opposite sides of a median acceleration plane, and wherein the magnet structure further comprises an ion source configured to release ions into the median acceleration plane for acceleration. 22. The magnet structure of claim 21, wherein the superconducting coils have an outer radius no greater than 20 inches.