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In the rotary-magnetic refrigerator, a circular permanent-magnet array with side-openings (CPMAS) provides a magnetic field exceeding the remanence of each magnet. Here we show that the magnetic induction in the pole gap as well as inside the magnets of a CPMAS can be calculated using an analytical vectorial formula. The method is based on the representation of the field of a uniformly magnetized magnet by straight current-carrying wires on the surface of the magnet. The Biot-Savart law gives the vector field of a single wire of arbitrary orientation, position, and length; the sum of fields from each wire gives the total field. The magnetization hysteresis loop of the permanent magnet is included by varying the current density; this is then used to calculate modifications to the center field by demagnetization, including remanence inversion under the field generated by the CPMAS itself. The field analysis procedure described here can be used to optimize the structure of permanent-magnet arrays, especially for three-dimensional arrays of restricted or asymmetric geometry, for various applications.