Improved Method for Calculating Magnetic Field of Surface-Mounted Permanent Magnet Machines Accounting for Slots and Eccentric Magnet Pole

Abstract

This paper presented an improved analytical method for calculating the open-circuit magnetic field in the surface-mounted permanent magnet machines accounting for slots and eccentric magnet pole. Magnetic field produced by radial and parallel permanent magnet is equivalent to that produced by surface current according to equivalent surface-current method of permanent magnet. The model is divided into two types of subdomains. The field solution of each subdomain is obtained by applying the interface and boundary conditions. The magnet field produced by equivalent surface current is superposed according to superposition principle of vector potential. The investigation shows harmonic contents of radial flux density can be reduced a lot by changing eccentric distance of eccentric magnet poles compared with conventional surface-mounted permanent-magnet machines with concentric magnet poles. The FE(finite element) results confirm the validity of the analytical results with the proposed model.

1. Introduction

Permanent-magnet machines have become more and more popular in the commercial, industrial and military products benefiting from higher power ratio to mass, torque ratio to volume, efficiency and lower vibration and noise over conventional electrically produced synchronous machines and asynchronous machines [1-3]. The magnetic field distribution in the air-gap is one of the most important issues in the permanent-magnet machines. It is foundation to other issues. At the present time, the numerical methods for magnetic field calculation, such as finite-elements method, provide accurate results concerning all kinds of magnetic sizes of permanent magnet machines, taking into account the saturation and without making any simplification of the geometry. But the numerical methods are very time-consuming, not suit to the initial design and optimization of the machines. Usually, the numerical methods are very good for the adjustment and validation of the design. Furthermore, the results which are obtained by numerical methods may be not accurate to calculate cogging torque and unbalanced magnetic force [4, 5] since it is sensitive to the FE meshes. Indeed, the motor performance can be obtained by the analytical methods of field computation based on the sufficient hypotheses [7].

Zhu [8] worked for both internal and external rotor motor topologies, and either radial or parallel magnetized magnets by accounting for the effects of both the magnets and the stator windings. B. Ackermann [9] proposed a two-step method, solving slotless field firstly and predicting the slotting effect based on slotless field, to reduce the computational complexity for each rotor position, although slotting effect still needs to be evaluated for all rotor positions. The method is very quick but is an approximate method. P. Kumar [10] developed an analytical model for the instantaneous air-gap field density with the assumption that the iron (both stator and rotor yoke) has finite permeability and the thickness of the stator yoke is finite. The slotting effect can be accounted for by the conformal mapping method which transforms the field solution in the corresponding slotless domain into the slotted domain. One typical example is the Carter’s factor which compensates the main flux for the slotting effect. F.W. Carter applied the conformal transformation to calculate magnetic field of electrically produced machines accounting for the slots [11] firstly. Cogging torque, acoustic noise and vibration spectra can not be analyzed by use of Carter’s coefficient.

It is very important to take into account slots in the analytical methods. Liu [11, 12] presents an analytical model suitable for analyzing permanent magnet motors with slotted stator core which calculate the air-gap flux density taking into account the effect of the interaction between the pole transitions and slot openings and solving the governing functions. In [7], the interface of magnet field between slots and air gap was obtained in form of Fourier series. Its high accuracy for the flux density distributions in both air-gap and magnets of the machines with different slot opening widths was confirmed by FE. Zhu [14] extended [7] to account for any pole and slot combinations, and an accurate analytical subdomain model with stator slotting effects was presented for computation of the open-circuit magnetic field in surface-mounted permanent magnet machines. Some mistakes were clarified at the same time. Wu [15] developed an improved analytical subdomain model for calculating the open-circuit magnetic field in surface-mounted permanent magnet machines accounting for the tooth-tips in the slots based on 2-D polar coordinate.

In [7], and [11-14], the field domain was divided into some types of subdomains, and calculated according to interface and boundary condition. The radial and circumferential components of magnet magnetization were expressed in the form of Fourier series to solve Possion equation in the concentric permanent magnet. But the interface between eccentric permanent magnet and air-gap is very difficult to be obtained. K. Boughrara [16] used two-dimensional field theory in polar coordinates to determine the flux density distribution, cogging torque, back EMF and electromagnetic torque in the slotted air gap of permanent-magnet motors with surface mounted magnet bars which are magnetized in shifting direction, but not like Halbach array magnetization. The sinusoidal waveform of the flux density distribution was obtained, but installation process of the permanent magnet bars is not easy to achieve.

Eccentric magnet pole is a typical example of eccentric magnet pole, and performance of motors can be optimized by changing eccentric distance of the eccentric magnet poles. Zhang[17] deduced expressions of Fourier transform coefficient for magnetomotive force of eccentric magnet pole. The main exciting force wave can be reduced through suitable selection of the eccentric distance. Xu [18] analyzed the influence of the eccentric magnet poles on the waveform of air-gap flux density and the motor performances, proposed a novel optimal designing method for the eccentric magnet poles with analytical expression. In [19], based on the magnetic field which was produced by a pair of windings on the air-gap, the expressions of the flux density produced by parallel-magnetized permanent magnet with different shapes were deduced with surface-current method. But slots were not accounted. At the present time, the analytical model of permanent-magnet machines accounting for eccentric magnet poles and slots has not been analyzed comprehensively.

In this paper, an improved analytical method accounting for slots and eccentric magnet pole is derived for calculating the magnetic field distribution of machine. In the derivation, magnetic field produced by radial and parallel eccentric permanent magnet is equivalent to that produced by surface current according to surface-current method of permanent magnet. The field domain is divided into two types of subdomains. The analytical field expressions of two subdomains produced by a pair of windings are obtained by the variable separation method. The coefficients in the field expressions are determined by applying the interface and boundary conditions. The magnet field produced by equivalent surface current is superposed according to superposition principle of vector potential. Compared with conventional surface-mounted permanent-magnet machines with concentric magnet poles, harmonic content of radial flux density can be reduced a lot by changing eccentric distance of eccentric magnet poles. The investigation shows the developed model has high accuracy to calculate the flux density of surface-mounted permanent magnet machines with eccentric magnet poles. The finite element (FE) results verify the validity of the analytical model.

 

2. Analytical Field Modeling

2.1 Equivalent surface current of magnet pole

In this paper, the analytical modeling is based on the following assumptions:

(1) Linear properties of permanent magnet; (2) Infinite permeable iron materials; (3) The relative permeability in the PM is equal to1; (4) Negligible end effect; (5) Simplified slot as shown in Fig. 1.

Fig. 1.Symbols and types of subdomains.

The two-dimensional conventional subdomain model is shown in Fig. 1. The magnet field is divided into two types of subdomains for the convenience of analysis: (1) subdomain of permanent magnet and air-gap (The first subdomain is limited by a circle characterised by a Rs radius); (2) subdomain of slots.

The permanent magnet with eccentric structure is shown in Fig. 2. The distance between point E and point O can be given by

Fig. 2.The eccentric structure of permanent magnet

where H is the eccentric distance, Rr is the radius of rotor, hmax is the maximum thickness of permanent magnet and ζ is the radian between OE and the center line of permanent magnet.

The radian between O1E and the center line of permanent magnet can be given by

where R2 is the radius of arc BC.

The radius of arc BC can be given by

The eccentric distance of the permanent magnet can be given by

where hmin is the minimum thickness of PM, αp is pole-arc to pole-pitch ratio and P is pole pairs.

The equivalent surface current is equal to the circumferential component of coercivity on the surface of magnet pole[19]. And the equivalent surface current of eccentric magnet pole is shown in Fig. 3(a) for parallel magnetization.

Fig. 3.Equivalent surface current of eccentric magnet pole. (a) Parallel magnetization; (b) Radial magnetization.

The current density of AB and CD can be given by

where Hcj is coercivity of permanent magnet.

The surface current density of side BC can be given by

The surface current density of side AD can be given by

The equivalent surface current of eccentric magnet pole is shown in Fig. 3(b) for radial magnetization.

The surface current density of side AB and CD can be given by

The surface current density of side BC can be given by

The surface current density of side AD is zero.

2.2 Magnet field produced by a pair of windings

Equivalent surface-current method is based on magnet field produced by a pair of windings. The current of windings can be given by

ic = JxΔl (x = 1, 2, 3)

where Δl is the length infinitesimal in the side AB, CD, AD and BC of magnet pole.

In this section, magnet field produced by a pair of windings is analyzed. Subdomain model with a pair of windings is shown in Fig. 4.

Fig. 4.Subdomain model with a pair of windings

2.2.1 Magnet field in the first type of subdomains

Since in the 2-D field, the vector potential has only z-axis component which satisfies:

α and β are the labels of winding. a and ζ present the position of the windings in the polar coordinate system. ic is current. The vector potential of point Q(r, θ) produced by α and β is given by

and

respectively.

According to (11) and (12), the sum of vector potential can be given by

where Am1 , Bm1 , Cm1 , Dm1 are coefficients to be determined, μ 0 is the permeability of the air, r is the radial of point Q, θ is the degree between point Q and center line of the windings, ρα and ρβ are the coordinates when the origins are α and β respectively.

If the origin is point O, ln ρα and ln ρβ can be expanded into infinite series about θ and r.

The radial and circumferential components of flux density can be obtained from the vector potential distribution by

While r < a , the flux density in the first subdomain can be given by

for the circumferential component.

In the outer surface of rotor, the circumferential component of the flux density is zero

Substituting (16) into (17), Bm1 and Dm1 can be given by

While r > a , substituting (18) and (19) into (13), the general solution of vector field in the first subdomain can be given by

where

While r > a , the flux density in the first subdomain can be given by

for the radial component, and

for the circumferential component, where

The vector potential produced by the equivalent surface current of jth magnet pole can be given by:

where Amj and Cmj are coefficients to be determined, and (j − 1)π / P is the angle between center line of first pole and that of jth pole.

2.2.2 Magnet field in the second type of subdomains

The governing function in the slots is:

The vector potential in the subdomain 2i is[15]

where Rs is the inner radial of the stator, Rsb is the radial of the slot bottom, θi is the angle between center line of the ith slot and center line of the windings as shown in Fig. 4, bsa is the slot opening width angle, Dn2i is coefficient to be determined, and

So the flux density in the second subdomain can be given by

for the radial component, and

for the circumferential component.

2.2.3 Interface condition between two types of subdomain

(a) The First Interface Condition

The first interface condition is that the circumferential component of the flux density in the inner surface of stator r=Rs is equal.

By evaluating (34) at the r=Rs interface, (34) simplifies down to

where

The circumferential component of the flux density along the stator bore outside the slot is zero since the stator core material is infinitely permeable. So Fourier series of the circumferential component of the flux density in the inner surface of stator can be given by

Where

Where

According to the vector potential distribution in the first subdomain, the circumferential component of the flux density in the inner surface of stator can be given as

According to (24), (37) and (42):

Combining (36), (38), (39) and (43), the following equations can be obtained:

(b) The Second Interface Condition

The second interface condition is that the vector potential of the ith slot opening is equal in the two types of the subdomains.

According to (20), the vector potential in the inner surface of stator can be given as

where

The equation (45) can be expanded into Fourier series along the stator inner surface of the ith slot:

Where

According to (30), the vector potential in the inner surface of stator can be obtained:

The vector potential in the inner surface of stator is equal in two subdomains.

Substituting (48) and (52) into (53), the following equation can be obtained:

Then

Substituting (49) into (55), the following equation can be obtained:

while n=1, 2, 3….

Combining (46), (47) and (53), the following equation can be obtained:

According to (44) and (57), the matrix format can be given as

Then the coefficients A1, C1 and D2i can be obtained according to (58).

2.2.4 The superposition principle of vector potential

The superposition principle can be applied to the vector potential by equivalent surface current in the surface-mounted permanent magnet machines.

In the Fig. 3, the vector potential produced by equivalent surface current of side AB and CD can be superposed:

where Δr is the length infinitesimal in the side AB and CD of PM and

The vector potential produced by equivalent surface current of side BC can be superposed:

where Δγ1 is the angle infinitesimal in the side BC of PM, and the origin is point O1,

The vector potential produced by equivalent surface current of side AD can be superposed:

where Δγ2 is the angle infinitesimal in the side AD of PM, and the origin is point O,

According to (15), (59), (61) and (64), the radial and circumferential components can be obtained.

 

3. Finite-Element Validation

The major parameters of two 30-pole/36-slot prototype machines which are used for validation are shown in Table 1. The minimum thickness of permanent magnet is 7mm in the prototype machine with eccentric magnet poles. And it is 12mm in the prototype machine with concentric magnet poles. The analytical prediction is compared with the linear FE prediction.

Table 1.Parameters of Prototype Machines (Unit: mm)

Fig. 5 show the results between analytical and FE predictions of flux density in the air-gap at r=198.5mm of motor with eccentric magnet poles. Fig. 6 show the results between analytical and FE predictions of flux density in the air-gap at r=198.5mm of motor with concentric magnet poles. As can be seen, the predicted flux density by subdomain model with equivalent surface-current method almost completely matches FE results.

Fig. 5.FE and analytically predicted flux density waveforms in the air-gap at r =198.5mm of motor with eccentric magnet poles: (a) Radial component; (b) circumferential component.

Fig. 6.FE and analytically predicted flux density waveforms in the air-gap at r = 198.5mm of motor with concentric magnet poles: (a) Radial component; (b) circumferential component.

Harmonic analysis of radial component of flux density with five pair of magnet poles in air-gap is shown in Fig. 7. Because five pair of magnet poles are one cycle to radial component of flux density in the air-gap. Then the 5th order spatial harmonic is fundamental harmonic. The same harmonic orders are 7th, 17th, 19th, 29th 31th et al. because of the influence of slots. The main different harmonic orders between Eccentric magnet poles and concentric magnet poles are the 15th and 25th order spatial harmonic. The harmonic content of radial component of flux density is 10.59% in the motor with eccentric magnet poles. The harmonic content of radial component of flux density is 23.17% in the motor with concentric magnet poles.

Fig. 7.Harmonic analysis of radial component of flux density at r = 198.5mm of motor. (a) Eccentric magnet poles; (b) Concentric magnet poles.

 

4. Conclusion

This paper presented an improved method for calculating the magnetic field in the surface-mounted permanent magnet machines accounting for slots and eccentric magnet pole. Magnetic field produced by radial and parallel eccentric permanent magnet is equivalent to that produced by surface current according to surface-current method of permanent magnet. The model is divided into two types of subdomains. The field solution of each subdomain is obtained by applying the interface and boundary conditions. The magnet field produced by equivalent surface current is superposed according to superposition principle of vector potential. The investigation shows harmonic contents of radial flux density can be reduced a lot by changing eccentric distance of eccentric magnet poles compared with conventional surface-mounted permanent-magnet machines with concentric magnet poles. The FE results confirm the validity of the analytical results with the proposed model.