0018-9464 (c) 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TMAG.2017.2658543, IEEE Transactions on Magnetics 940 1 3-D Sub-Domain Analytical Model to Calculate Magnetic Flux Density in Induction Machines with Semi-Closed Slots under No- Load Condition Aida Mollaeian 1 , Eshaan Ghosh 2 , Himavarsha Dhulipati 3 , Jimi Tjong 4 and Narayan C. Kar 5 Centre for Hybrid Automotive Research and Green Energy, University of Windsor, ON, N9B 3P4 Canada 1 mollaei@uwindsor.ca, 2 ghoshe@uwindsor.ca, 3 dhulipah@uwindsor.ca and 5 nkar@uwindsor.ca In this paper, a novel 3-D sub-domain analytical model is developed to determine magnetic flux distribution in a single-cage induction machines (IMs) with skewed rotor bars under no-load condition in an effort to more detailed analysis of spatial harmonics. The proposed model along with optimization algorithm is as an alternative solution to finite element analysis in optimizing the geometry of IMs. The analytical method is based on the resolution of 3-D Laplace and Poisson’s equations in cylindrical coordinates using separation of variables method to calculate the magnetic vector potential for corresponding sub-domain. The proposed model includes the effect of slotting and tooth tips for both stator and rotor slots which is usually neglected in a 2-D analysis due to complexity of differential equations Also, the proposed 3-D model can be used for any slot-pole combination in addition to considering asymmetrical effect in the axial direction which is a source of noise, vibration and excessive losses in IMs. To evaluate the performance of the proposed 3-D analytical model, calculated magnetic field distribution is compared with the results obtained from the 3-D finite element analysis. Index Terms—3-D analytical solution, magnetic flux distribution, semi-closed slot, skew, sub-domain method. I. INTRODUCTION n order to reduce loss and vibrations occurring due to axial magnetic flux variation in skewed electric machines specially IMs, 3-D electromagnetic analysis is needed for more comprehensive analysis of spatial harmonics [1], [2]. Numerical methods such as finite element analysis (FEA) is found to be an efficient option in machine design field due to its accuracy. However, 3-D FEA analysis is a time consuming and inefficient approach in design optimization process [3]. Hence, the necessity of an exact 3-D analytical solution is crucial in order to reduce research, time and cost to obtain an optimal design for any industrial application. The most favorite methods by designers are determination of Maxwell’s equations using analytical technique and winding function method (WFM) to calculate inductances. They are fast and precise and are considered in optimization [3]-[4]. However, in WFM determination of instantaneous rotor position dependent inductances are challenging [4]. In [5], 2-D analytical solutions have been derived and proposed for non- skewed simple structures, solid and slotted rotor which mostly deals with eddy current calculation for rotor bars solving Maxwell’s equations. However, the accuracy of electromagnetic performance analysis is compromised over simplicity of developed differential equations, and axial asymmetrical effects are neglected. In this paper, a novel analytical method has been proposed to find a 3-D general solution for magnetic flux density distribution considering z- axis variation unlike previous methods, and magnetic flux densities are calculated in the air-gap, stator and rotor slots and compared to the ones with 3-D FEA. II. 3-D ANALYTICAL FIELD SOLUTION OF IM To solve Laplace and Poisson’s equations, the structure of three phase IM with any combination of rotor and stator slots can be divided into five major subdomains shown in Fig. 1: stator slot, stator slot opening, air-gap, rotor slot, rotor slot opening. Rotor and stator subdomains include 2Q r and 2Q s number of sub-regions, where Q s and Q r represent stator and rotor slot numbers. Therefore, in case of a single layer windings the total number of sub-domains is [2(Q r +Q s )+1]. The first step Fig. 1. Sketch of i th stator semi closed slot for a single layer concentric winding and i th rotor cage bar with domain indexing bar for stator slot, stator slot opening, rotor slot and rotor slot opening and air-gap subdomains. in solving Maxwell’s equations is to make the following assumptions to simplify the analytical solution: a) supply is balanced sinusoidal currents; b) stator and rotor yokes are nonconductive and are infinite permeable materials; c) electric conductivity and magnetic permeability of materials are constant and magnetic saturation is neglected for simplicity; d) current density in stator slot is considered constant; and e) slots are considered rectangular in shape. Magnetic vector potential (MVP), defined as A, is used to calculate the magnetic flux density at no-load condition by considering the machine’s geometrical configuration, as well as electrical and magnetic properties. As a consequence of axially uneven geometry, MVP has three components: A as radial , A as circumferential and A z as axial. Governing partial differential equations (PDEs) are expressed in each region as (1). Three terms in right hand side are expressed for existence of current source in air-gap, stator and rotor slot openings, stator and rotor slots accordingly. i r r i s i mz i mz i mz i mz z i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m j j z A z A A A A A A A z A A A A A A A z A A A A 0 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 ˆ 1 1 ˆ 2 ˆ 2 (1) where m is the index for sub-domains and i is number of slots. Relationship between MVPs and magnetic flux densities in all directions in cylindrical coordinates is as (2): I