1. Introduction

Permanent-magnet (PM) brushless machines are increasingly being used in various applications, such as electric vehicles, industrial servos and in wind power generation systems [9]. This increased popularity is due to their high torque, high power density, high efficiency, and low maintenance requirements as a consequence of the use of PM materials in the rotor. Surface-mounted permanentmagnet (SPM) motors have the advantages of low torque ripple and low cogging torque as compared with interior permanent-magnet (IPM) machines.

The overhang is defined in this paper as a configuration that the rotor length is longer than the stator length in the axial direction. In general, the overhang structure is used to enhance the air-gap flux and improve the power density while utilizing the free space caused by the stator end winding [29]. A 3-D finite-element method (FEM) is necessary to analyze the magnetic fields in the axial direction for a proper consideration of the overhang effects. Although FEM can precisely obtain the magnetic flux distribution and electromagnetic performances of electrical machines [22-24], it is time consuming and computationally expensive, especially at pre-design stages.

To reduce computational time for the motor design process, analytical methods are essential during the preliminary design of electric machines. Two types of analytical methods are usually used. The first one is based on the formal solution of Maxwell’s equations. It can calculate parameters and performances of electric machines with high accuracy [1-5, 8, 13, 15, 19-21, 27, 28, 30]. Also, this method can be used for analyzing practical electrical machines with the geometric complexity, such as synchronous reluctance motors [15], switched reluctance machines (SRM) [27], SPM machines [2, 4, 5, 30], slotless motors [19, 20], and PM actuator [13, 21]. The second method is an equivalent magnetic circuit model (EMCM). As an EMCM has the advantage of simplicity compared to the aforementioned method, it is widely used as an analytical method for various types of machines [6, 10-12, 14, 16-18, 25, 26, 31, 32], such as SPM machines [10, 12, 18], IPM machines [16, 31], flux-switching PM machines [32], SRM [25], axial flux PM motors [14, 17], and induction machines [26].

In the case of SPM motors, an EMCM required to predict the air-gap and magnet flux density analytically has been developed while taking into account the leakage flux around the magnets in the rotor [18]. However, overhang effects in SPM machines with rotor overhang are presently not included in the EMCM. In this paper, we propose an EMCM considering not only the leakage flux in the rotor but also the overhang effects for SPM machines with rotor overhang. A 3-D FEM analysis is used to verify the proposed model.

2. Analytical Model of SPM Machines with Overhang

Although the effects of the stator slots have been taken into account [10-12, 14, 16-18, 25, 26, 32], an EMCM which disregards slotting effects was developed in order to focus on the overhang effects and simplify the analytical model in this paper. It is assumed that the effects of saturation within the core are negligible, as the magnetic flux density within the core is low in general and because normally there is no significant saturation.

With the motor topology generalized as the linear translation type shown in Fig. 1, it is possible to derive an EMCM applicable to any specific topology. In order to enhance the air-gap flux and improve the power density, the overhang structure is used in the linear translation motor here, as shown in Fig. 2.

Fig. 1.Linear transfer motor topology

Fig. 2.Configuration of a motor with overhang

Qu et al. developed an EMCM for SPM machines that takes into account the air-gap leakage fluxes around the magnets [18]. The air-gap leakage fluxes consist of the magnet-to-magnet leakage flux and the magnet-to-rotor leakage flux. With the circular-arc straight-line permeance model [7], these leakage fluxes were modeled. However, this analytical model cannot be applied to SPM motors with overhang because the overhang results in asymmetry of the magnetic fields in the axial direction.

An EMCM based on the analytical model developed by Qu et al. considering overhang effects as well as air-gap leakage fluxes is proposed. Fig. 3 shows the proposed EMCM. The variables shown in Fig. 3 are defined as follows:

Fig. 3.Equivalent magnetic circuit for Fig. 1 and Fig. 2

Φg : the air-gap flux excited by one magnet pole Φr : the flux source of one magnet pole Rg : the reluctance corresponding to Φg Rmo : the reluctance corresponding to Φr Rr : the reluctance of the rotor yoke Rs : the reluctance of the stator yoke Rmm : the reluctance corresponding to the magnet-tomagnet leakage fluxRmr : the reluctance corresponding to the magnet-torotor leakage flux.

Rr and Rs can be ignored due to the assumption mentioned earlier. Fig. 3 can be simplified into Fig. 4 through the use of symmetry. In Fig. 4, Rm is calculated from Fig. 3 as

Fig. 4.Simplified circuit of Fig. 3

where

and

For the EMCM considering the overhang effect, the effective air-gap length should be defined. Flux lines between overhang structure and stator core are modeled with consideration for the distribution of magnetic flux density in the air gap. Fig. 5 shows modeling of flux lines in the air gap with a circular arc and a straight line. The magnetic flux distribution in the air gap which is calculated through FEM is shown in Fig. 6. The effective air-gap length is derived via the modeling of flux lines with a circular arc and a straight line as

Fig. 5.Modeling of the flux lines in the air gap with a circular arc and a straight line: (a) A straight-line permeance model; (b) A circular-arc permeance model

Fig. 6.Distribution of magnetic flux density by FEM

where z is the position in the overhang, as shown in Fig. 5, and g is the air-gap length.

The permeances in the non-overhang region is derived in [18] as

where μ0 is the permeability of air, μr is the magnet relative recoil permeability, Lst is the stator stack length, g is the air-gap length, wf is the width between two adjacent magnets, wm is the magnet width, hm is the magnet length, and min(·) is the minimum function.

With the effective air-gap length ge, the permeances of the infinitesimal stack length dz in the overhang can be derived. The permeances can be calculated by integrating the permeances of the infinitesimal stack length over the overhang length, as (9)-(12), where Loh is the rotor overhang length.

With the permeances in the non-overhang region and the overhang region, the permeances for the entire motor with overhang are derived as follows:

Using the equations above and the reciprocal relationship between the permeance and reluctance, the reluctances are calculated as follows:

According to the flux division, the air-gap flux and the flux from the magnet can be derived as

and

The air-gap flux density and the magnet flux density can be induced by

and

where

and

3. Results and Verification

For the verification of the proposed analytical model, the results from a FEM and the analytical results calculated from (23) and (24) are investigated. The investigations are conducted in various cases with different values of g, wf and Loh. Table 1 and Table 2 show the results for a motor employing ferrite magnets with Br = 0.4 T and rare earth magnets with Br = 1.07 T, respectively. Bm and Bg of FEM results is average value. Bm is calculated by averaging flux density in the middle of a magnet though rotor stack length. Bg is calculated by averaging flux density in the middle of the air gap though rotor stack length.

Table 1.At Lst =100.0 mm, hm = 4.0 mm, wm = 20.0 mm, μr = 1.05, and Br = 0.40. *: Difference = (3D FEM results − Analytical results) / 3D FEM results × 100.

Table 2.At Lst =100.0 mm, hm = 4.0 mm, wm = 20.0 mm, μr = 1.0384, and Br = 1.07. *: Difference = (3D FEM results − Analytical results) / 3D FEM results × 100.

As shown in Table 1 and Table 2, there is little difference between results of the motor employing the ferrite magnet and the rare earth magnet. The accuracy of the EMCM is affected by Loh. As the length of the overhang increase, the leakage flux at the end of the overhang is increased and the estimation of the magnetic flux path becomes difficult. This increases the difference between the analytical results and the FEM data. In cases in which Loh is less than 7 mm, the differences of Bm and Bg are less than 3% and 2%, respectively. The differences of Bm and Bg are respectively less than 5% and 3% when Loh is 10mm.

In the condition with same stator stack length, the total flux passing through air-gap increases when Loh increases. However, the average flux densitiesBm and Bg decline as the overhang increases, as shown in Table 1 and Table 2, this is because the area is more increased compared to the flux. In cases in which Lst is 100 mm, the total flux passing through the air gap increases by about 11% and 15% when Loh is 7 mm and 10 mm, respectively.

4. Conclusion

This paper is noteworthy in that the time for design of SPM machines can be reduced remarkably via the reduction of time for initial design by using the proposed EMCM, which is analytical method considering overhang effects.