This article presents a nonlinear analytical (NA) solution based on subdomains for predicting magnetic field and performances of a spoke array vernier permanent magnet (SAVPM) machine which has spoke array type permanent magnet (PM) excitation. On one hand, equivalent current sheets replace tangentially magnetized PMs to make the magnetic vector potential in the region with PMs satisfy a Laplace's equation, which simplifies the problem and reduces the workload of numerical calculation. On the other hand, in contrast to the existing literature, regions of soft-magnetic material such as a tooth and a rotor pole are subdivided into small pieces, and are no longer a whole with a single permeability for considering local magnetic saturation. Poisson's and Laplace's equations with the given permeability of soft-magnetic material are analytically solved by separation technique of variables in seven subdomains, and nonlinear iteration is used to determine the solution of NA and update soft-magnetic material permeability. NA solves the magnetic field and performances of a SAVPM machine. Frozen permeability method (FPM) is used to determine PM flux linkage and d-axis inductance under load. Results from the NA model have been compared with those from FEA and experiments.