IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 40, NO. 1, JANUARY/FEBRUARY 2004 121 Analysis and Modeling of Air-Gap and Zigzag Leakage Fluxes in a Surface-Mounted Permanent-Magnet Machine Ronghai Qu, Member, IEEE, and Thomas A. Lipo, Fellow, IEEE Abstract—In this paper the magnetic characteristics of surface- mounted permanent-magnet machines are analyzed and modeled. The air-gap and zigzag leakage fluxes are analytically expressed in terms of the magnetic material properties and the motor dimen- sions. Both factors are essential quantities for the accurate predic- tion of the flux distribution within the machine and of the machine torque. Therefore, they are desired for the purpose of machine de- sign and optimization. In order to evaluate the validity of the proposed models, the fi- nite-element method (FEM) analysis is used. The results show that the errors between the FEM results and analytical predictions are less than 7% for the nonsaturated tooth flux and less than 17% for the saturated case. Finally, the models are applied to a novel per- manent-magnet machine design. Index Terms—Air-gap leakage flux, permanent-magnet (PM) machine, PM machine design, zigzag leakage flux. I. INTRODUCTION A CCOMPANIED by the fast development of perma- nent-magnet (PM) materials and the availability of modern power electronics, PM machine topologies have evolved quite quickly [1]–[3]. They have a wide variety of applications in the area of variable-speed drives, servo drives, appliance and industrial automation. Therefore, design work on PM machines has becoming increasingly important. It is well known that leakage fluxes have a substantial effect on the air-gap fluxes interacting with the armature current to produce torque. It is an essential quantity for the accurate pre- diction of machine torque and the average flux densities within the air gap and the magnet. Therefore, it is desirable to analyti- cally express leakage flux factors in terms of the magnetic ma- terial properties and the machine dimensions. The air-gap leakage flux model for interior PM machines has been developed [4]. For surface-mounted PM machines, one portion of the air-gap leakage flux, magnet-to-magnet leakage, Paper IPCSD 03–107, presented at the 2002 Industry Applications Society Annual Meeting, Pittsburgh, PA, October 13–18, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Ma- chines Committee of the IEEE Industry Applications Society. Manuscript sub- mitted for review September 16, 2002 and released for publication September 22, 2003. R. Qu is with the Global Research Center, General Electric Company, Niskayuna, NY 12309 USA (e-mail: ronghaiqu@ieee.org). T. A. Lipo is with the Department of Electrical and Computer Engi- neering, University of Wisconsin, Madison, WI 53706-1691 USA (e-mail: lipo@engr.wisc.edu). Digital Object Identifier 10.1109/TIA.2003.821790 Fig. 1. Simple linear translational motor topology. has been well modeled using the circular-arc straight-line per- meance model [5]–[8]. The other portion of the air-gap leakage flux, magnet-to-rotor-iron flux, which can be as large as the magnet-to-magnet leakage flux, has not been included in the air-gap leakage flux model. A more accurate air-gap leakage flux model including both leakage flux paths is highly desired and required for the purpose of machine design and optimiza- tion. To derive such a model is one of two focuses in this paper. In addition to the air-gap leakage flux, zigzag leakage flux is another main part of the leakage flux. The zigzag leakage can be composed of three portions: one part of the zigzag leakage is short circuited by one stator tooth; the second part links only part of the windings of one phase; the third part traveling from tooth to tooth does not link any coil. Although the last two portions are well described in the literature [9], an analytical model for the first portion has not been found. In fact, the zigzag leakage flux short circuited by one stator tooth has a dominant effect on PM machines, especially for the semi-closed slots. To model this part of the zigzag leakage flux will be the other emphasis of this paper. The new models of the air-gap and zigzag leakage fluxes are proven by finite-element method (FEM) analyses to be within an error of 7% and 17% for the nonsaturated tooth flux and the saturated case, respectively. The two models should be benefi- cial for designing and optimizing of machines. II. ANALYTICAL MODEL OF AIR-GAP LEAKAGE FLUX A. Assumptions and Equivalent Circuit The magnetic circuit approach is used to analyze the model machine with a simplified motor topology. This approach takes the leakage flux into account and yields analytical expressions for the average flux densities within the air gap and magnets. 0093-9994/04$20.00 © 2004 IEEE