Air Gap Flux Density Analytical Model for a Fractional-Slot Concentrated-Winding SM-PMSM

Novel ABC—Variable Based Closed-Form Analytical Model of Surface Mounted Permanent Magnet AC Machines With Sinusoidal and Non-Sinusoidal Back-EMF

“…where l is the stack length of the machine, xs w is the -phase winding function, pw n is the number of parallel windings per phase and v g is the airgap variation function that describes the variation of the airgap around the circumference of the machine, as explained in [32]. Expressions similar to (16) were presented previously but assumed that the airgap is spatially uniform [19].…”

“…where pm B is the airgap flux density due to permanent magnets only. Substituting expression (15) with s F set to zero into (32) By the same procedure, the band c-phases can be evaluated with appropriate phase-shifts. Note that in (31)- (33) the dependence on rm  is not expressed on the left-hand side for clarity.…”

Multi-Objective Optimization Design of Fractional Slot Concentrated Winding Permanent Magnet Synchronous Machines

“…These functions are the variation of the air-gap around the machine g v (φ sm , θ rm ), the variation of magnet thickness around the machine d mv (φ sm , θ rm ), the variation of the magnet remanence flux density around the machine B mv (φ sm , θ rm ), and the variation of the magnet relative permeability around the machine µ mv (φ sm , θ rm ). Analytical expressions for these four functions can be found in [15].…”

“…Note the functional dependency (r, φ sm , θ rm ) was omitted in (9) for clarity. Following the derivation shown in [15], it can be shown that the air-gap flux density at any radial position between r rb and r st is given by…”

“…where F m , R g and R m are the magnet MMF, air-gap quasi-reluctance and magnet quasi-reluctance, respectively. Expressions for these functions can be found in [15].…”