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When designing permanent magnet synchronous machines, the voltage waveforms and mechanical forces are important in many applications. Many analysts use the finite-element (FE) method to determine fields in the air gap of machines. These fields are then used to calculate vector forces, voltages, or other quantities of interest. Although the FE method has been used for decades, it requires significant computational resources, especially when transient analyses are required. This limits its application to a tool for analysis rather than a large design space, iterative optimization. To address this, an enhanced field reconstruction technique (EFRT) has been developed. Using the EFRT, the fields in the air gap of a machine are computed using a minimum number of FE evaluations. Using the results, a set of basis functions are established that enable the calculation of fields under arbitrary excitation and rotor position. The EFRT greatly reduces the computational effort required to compute the vector fields inside machines and compares favorably with the FE method and hardware results.