Axial Flux Permanent Magnet Motor Modeling using Magnetic Equivalent Circuit

Karim Abbaszadeh Electrical and Computer Engineering Dep. K.N. Toosi University of Technology

Tehran, Iran abbaszadeh@eetd.kntu.ac.ir

Seyede Sara Maroufian Electrical and Computer Engineering Dep.

K.N. Toosi University of Technology Tehran, Iran

sara.maroufian@ee.kntu.ac.ir

Abstract_ Several methods are introduced for precise modeling of electrical machines. Model accuracy and speed are the great concerns. Magnetic Equivalent Circuit Modeling of the Electrical Machines is one of the proposed methods and has been used for modeling the healthy case and a faulty condition example of an Axial Flux Permanent Magnet Motor in this paper. The faulty condition which has been modeled is the Demagnetization of the rotor PMs. The motor structure and feature are introduced, and then further analysis of Teeth Flux, Phase Flux and Back-EMF Voltage are proposed with according to magnetic circuit model. Detection of magnets demagnetization is established by Fourier analysis of the Back-EMF voltage induced in the Stator Windings of the proposed motor. The model then has been validated with the help of the experimental results.

Keywords: Axial Flux Permanent Magnet Motor, Magnetic Equivalent Circuit, PM Demagnetization.

I. INTRODUCTION

For a precise modeling of an Electrical Machine, Finite Element Method (FEM) is suggested; however the huge amount of calculations needed to be done is still a problem. This would be probably the main reason to perform less accurate but faster methods such as Magnetic Equivalent Circuit Method or briefly MEC [1]. Based on the magnetic properties of the material and considering the flux spatial paths, a magnetic circuit is established. The magnetic field characteristic can be technically obtained using electric circuit principles, e.g., Kirchhoff’s voltage law (KVL) and Kirchhoff’s current law (KCL) [2]. Magnetic Equivalent Circuit models provide reasonably accurate results and relatively fast computation time, when comparing to the time stepping finite element (TSFE) models [3].

The method has been used for modeling and fault detection of a variety of structures and machines. Sizov [3] used the method to model the stator and rotor failures in squirrel cage induction machines. Kano [4] presented a torque maximizing design of a double stator, axial flux PM machine using a simple nonlinear magnetic analysis. Hsie [2]

proposed a generalized equivalent magnetic circuit model to design a PM electric machine. In this paper, a magnetic circuit model of an axial flux permanent magnet motor is proposed. The procedure of establishing the magnetic circuit model is presented in section II. The motor is a 4-pole one, consisting 4 pieces of magnet on its rotor (designed and constructed in the machine lab. Fig.1). The proposed model is then revised to consider the probable demagnetization of the rotor magnet. The effect of PM demagnetization on the Back-EMF voltage is studied by Fourier analysis and a sample defected case is presented.

Figure.1. Axial Flux Permanent Magnet motor designed and constructed in the Machine Lab.

II. AXIAL FLUX PERMANENT MAGNET MACHINES MODELING

A. Permanent Magnet Motors

Axial Flux Permanent Magnet Motors have several unique features such as high efficiency, low rotor loss, small magnet thickness, compact construction and high torque at low speed [4]-[6]. Besides cogging torque which usually occurs in low speed and high load [7], is one of their drawbacks [7,8], which results in shaft vibration and noise. On the other hand rotor eccentricity is aggravated in AFPMs as a result of exposing the shaft with normal forces of great intensity [5]. PM Demagnetization is another drawback of PM machines [9]. PM demagnetization usually happens because of

opposite fluxes injected in the PM from external flux sources [9] or as a result of temperature rise.

Detection of the mentioned problems is an important subject and a variety of methods are introduced in papers [3]. The method speed and accuracy are the most important factors. The FEA method can directly calculate the flux patterns [2]-[4], but the entire process is computationally expensive; moreover changing the design parameters often requires the model to be reconstructed [2]. Magnetic equivalent circuit modeling is less time consuming in comparison with FEA method [3,6]. In addition fault conditions modeling is more practical with according to the model variation compatibility.

B. Magnetic Equivalent Circuit Modeling

Fig.2.a and Fig.2.b show the rotor and the stator of the axial flux permanent magnet motor. As mentioned previously it is a 4-pole motor with distributed stator winding. The motor feature and design parameters are listed in TABLE.I. The first step to model the motor is to consider the flux patterns so that the flux tubes which are the basic elements of the permeance network will be constructed. There are three types of permeances [1]:

1. Linear Permeances 2. Nonlinear Inheritance Permeances 3. Parametric Nonlinear Permeances

Figure.2. Motor Structure, a. Rotor, b. Stator

Parametric nonlinear permeances which are located in the air gap are the most important ones since they are responsible for energy conversion. In the proposed motor, the permeance of the flux tubes are calculated by a double integration on the surface which the flux lines enter the tube.

TABLE.I. Motor Parameters

Motor Parameter Value Pole Number 4

Phase Number 3 Speed 1500 rpm

Outer Diameter 158 mm Inner Diameter 88.5 mm PM Thickness 5 mm

Pole Pitch 118° electrical Stator Yoke Thickness 15 mm Rotor Yoke Thickness 17 mm

In order to determine the topology of the flux tubes, the flux path should be determined. With according to the motor structure and geometry, most of the flux tubes are considered as shown in Fig.3. With respect to the material magnetic permeability and by a 2D integration on the surface of the flux tube, its permeance is calculated. An example of such calculation for the stator tooth is presented:

(1)

The parameters mentioned in the above equation are noted in Fig.3.c.

Figure.3. Flux Tubes geometry used to calculate the permeance network elements. (a). Flux tube assigned to the Stator Teeth and Rotor PM segments.

(b). Flux tube assigned to the parts in the stator and rotor yokes. (c). Parameters for calculation of sample flux tube permeance.

Fig.4 shows a schematic view of the generated Permeance Network. In this figure 6 segments of a PM are depicted (each PM is divided into 10 segments). It should be noted that division of each PM to more segments will increase the model

(a)

(b)

(a) (b)

(c)

accuracy, although it is time consuming; so it is important to choose sufficient number of segments. Node potentials noted as are assigned to Magnetic potential levels in the rotor and stator and also the air gap between them.

Figure.4. Schematic view of the permeance network

Writing down the node potential equations of the machine magnetic circuit, one can derive the machine algebraic system of equations. Here examples of these algebraic equations are presented for four sample nodes of the machine, in different magnetic potential levels, located in the rotor yoke, the stator yoke, the air gap under the PMs, and the air gap above the stator teeth, respectively. The nodal equation for the first level of magnetic potential level, located in the rotor yoke is as follows:

, , ,, (2)

, , are the permeance above two segments of PM in the rotor yoke, the permeance between two poles in the rotor yoke and the PM segment permeance, respectively and

is the PM segment flux. Equation (2) defines the node magnetic potential for the first node of the magnetic potential level located in the rotor yoke. Here the first index of , shows the number of the magnetic level and the second index indicates the segment number (in this level varying from 1 to 40). With according to machine structure, the rotor part is divided into 40 nodes, that is, the potential levels of and consist of 40 nodes, each node corresponding to one of the magnet segments. The rest of the node potentials related to this potential level can be written in the same manner.

Dividing the stator part into 15 nodes (each corresponding to a stator tooth), the magnetic potential levels of and will contain 15 magnetic nodal equations. Equation (3) shows the nodal relation for the first node located in the stator yoke:

, 2 , , 0 (3)

In (3) is the stator tooth permeance. The same method is applied to magnetic levels of and . These magnetic potential levels are contracting with the permeances in the machine air gap. So the node potential equations will vary as the machine rotates. The calculation procedures of the nodal equations for a sample node in magnetic potential levels of and are as follows:

, , , ,, , ,

(4)

, 2 , , ,, , 0

(5)

is the leakage permeance between two PM segments of two adjacent PM poles, in the air gap. is the leakage permeance between two PM segments of one PM, in the air gap. is the leakage permeance between two stator teeth in the air gap. ∑ , defines the sum of the permeances connected to the node number 1 of the th magnetic potential level; the index shows the number of the permeance located in the air gap connected to the other magnetic potential level. The elements of the summation will change as the rotor rotates. To generate the nodal equations for these levels a time varying matrix is defined. In each step the matrix elements are calculated with according to the position of the rotor.

It is apparent that as a result of rotor rotation, only some specific elements of the permeance matrix are non-zero in each step. In fact based on the rotor position, only a few number of stator teeth will have contraction with a specific PM segment. Such relation and the value of the permeances are defined as follows [1]: , 0 2 21 cos2 1 cos 2 2 20

(6)

In (6), is the maximum permeance between a sample tooth of the stator and a sample PM segment, occurring while their axis are coincident. is the angular distance between stator tooth and PM segment. and are angular features of

the motor structure and are related to the statand segment width and diameter and theTherefore should be evaluated with accordinstructure.

Fig.5. shows the permeance variation ifor the 22th PM segment, calculated with acco

Figure.5. air gap permeances between stator teeth and th6 time steps(Horizontal axis: stator tooth n

After generating the permeance networkForce Transform Matrix and Flux Transformpotentials are easy to calculate. MagnTransform Matrix denoted as defines thetooth MMF vector and phase current vector [

Where and are the tooth magnetomand phase current vector, respectively. winding arrangement of the machine whTABLE.II. The table shows the location ofmachine slots.

TABLE.II. Winding arrangement

The Flux Transform Matrix denoted as connection between phase and tooth fluxes [1Φ Φ

The Back-EMF Voltage induced in thwinding is then computed in a step by step pposition will change gradually with a time stIn each step, the flux entering each stator tePMs are calculated; the flux matrix will turn

tor and rotor tooth e air gap length. ng to the machine

in 6 special steps, ording to (6).

he 22th PM segment in number)

k, Magnetomotive m Matrix [1], node netomotive Force e relation between [1] as follows:

(7)

motive force vector depends on the

hich is shown in f each coil in the

t

describes the 1].

(8)

the stator 3-Phase process. The rotor tep of 10 order. eth from the rotor

n these teeth fluxes

to phase flux. By means of mathemvoltage is calculated in each stepEMF voltage, calculated by magnePM on the rotor is divided into 10accuracy of the model although thexist as a result of this limited numfurther improve the model accsegments is suggested.

(a)

(b)

(c)

(d)

atic calculation the induced . Fig.6.a shows the Back-

etic equivalent model. Each 0 segments to increase the he voltage fluctuation still mber of PM segments. To

curacy, division to more

Figure.6. Back-EMF Voltage induced in the Stator Winding. (a). Simulated with MEC and (b). Experimental result (1/10). (c). Search Coil Voltage calculated by MEC. (d). Measured Search Coil voltage (with noise). (e).

Measured Search Coil Voltage (without noise)

The experimental result presented in Fig.6.b shows the validity of the magnetic equivalent circuit model. On the other hand, the search coil voltage calculated by means of MEC method, is shown in Fig.6.c. Again for validation, the experimental result of a sample search coil, wound on a stator tooth, is depicted in Fig.6.d. The search coil has been fitted around a stator tooth as shown in Fig.7. [7], [8]. The experimental result shown in Fig.6.d is together with the noise.

Figure.7. the machine stator together with the implemented search coil

For identification purposes it is better to transfer the Back-EMF voltage from time domain to frequency domain; this action is performed by Fourier Transform. Fig.8.a shows the result of this transformation for 1 turn rotation of the rotor. The harmonic components of the Back-EMF voltage are the result of the stator teeth number, motor pole numbers, PM segments and their combinations. The amplitude of the fundamental frequency is 29.4041 volts.

Figure.8. Back-EMF voltage in frequency domain

A closer view of the frequency spectrum of the Back-EMF voltage is shown in Fig.8.b. The main frequency which is 50 Hz is a result of motor structure and rotation speed. The effect of rotor PMs Segmentation is apparent in this spectrum. The harmonic component in the zero frequency (DC Component) is the result of the motor winding structure. In this fractional pitch winding, each phase on the stator consists of 5 coils, producing 4 poles, i.e. for each phase of the winding one pole has two overlapping coils; that is the pole arrangement is not symmetric. This would generate the DC component in the spectral analysis. Higher order harmonics which have significant amplitude in comparison with the others are the result of the stator teeth number and PM segmentation.

Fig.9 shows the Spectral components of a sample defected Magnet case. In this special situation which has been introduced to show the effectiveness of such analysis, one magnet segment has been removed, that is to generate zero flux.

Figure.9. Spectral distribution of the Back-EMF voltage for a sample defected case

As it can be seen in the above image, the defection has affected the harmonic frequencies and their amplitudes. The amplitude of the fundamental frequency is 28.8956 which shows a decrease of 1.7 percent in comparison with the healthy case. Any of the data extracted from this spectral analysis could be used for identification purposes. This data may contain the amplitude of the fundamental frequency, the (a)

(b)

(e)

DC component and the spectral distribution of the Back-EMF in different cases. Each type of magnet defection may affect a particular harmonic or shift the spectrum within some frequencies.

III. CONCLUSION

In this paper based on Magnetic Equivalent Circuit modeling, a Permeance Network was designed for an Axial Flux Permanent Magnet Motor. Magnet parts were divided into 10 segments to increase the model accuracy. The effect of PM demagnetization was studied with the help of Fourier analysis of the Back-EMF Voltage.

REFERENCES

[1] Vlado Ostovic, “Dynamics of Saturated Electric Machines”, University of Osijek.

[2] Min-Fu Hsieh, You-Chaiuan Hsu, “A Generalized Magnetic Circuit Modeling Approach for Design of Surface Permanent-Magnet Machines”, IEEE Transaction on Industrial Electronics. VOL. 59, NO.2, February 2012.

[3] Gennadi Y.Sizov, Chia-Chou Yeh, Nabeel A.O. Demerdash, “Magnetic Equivalent Circuit Modeling of Induction Machines under Stator and Rotor Fault Conditions”.

[4] Y.Kano, K.Tonogi, T. Kosaka, N.Matsui, “Torque-Maximizing Design of Double-Stator, Axial-Flux, PM Machines Using Simple Non-Linear Magnetic Analysis”, 2007 IEEE.

[5] Seyyed Mehdi Mirimani, Abolfazl Vahedi, Fabrizio Marigenetti, “Effect of Inclined Static Eccentricity Fault in Single Stator-Single Rotor Axial Flux Permanent Machines”, IEEE Transaction on Magnets, VOL. 48, NO.1, January 2012.

[6] P. R. Upadhyay, K.R. Rajagopal, B. P. Singh, “Design of a Compact Winding for an Axial-Flux Permanent-Magnet Brushless DC Motor Used in an Electric Two-Wheeler”, IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 4, JULY 2004

[7] S. Saied, K. Abbaszadeh, A. Tenconi, S. Vaschetto2, “New Approach to Cogging Torque Simulation Using Numerical Functions”, 2011 IEEE International Electric Machines & Drives Conference (IEMDC)

[8] Seyed Amin Saied, Karim Abbaszadeh, and Mehdi Fadaie, “Reduced Order Model of Developed Magnetic Equivalent Circuit in Electrical Machine Modeling”, IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 7, JULY 2010

[9] Yacine Amara, Sami Hlioui, Rachid Belfkira, Georges Barakat Mohamed Gabsi, “Comparison of Open Circuit Flux Control Capability of a Series Double Excitation Machine and a Parallel Double Excitation Machine”, IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 60, NO. 9, NOVEMBER 2011.

Appendix

Magneto motive Force Transform Matrix

18 0 18 0 18 0 36 18 18 18 18 36 0 0 018 36 0 18 0 18 0 18 0 18 0 36 18 18 180 36 18 18 18 18 36 0 18 0 18 0 18 18 18

Flux Transform Matrix

0.2 0.2 0.2 0 0.2 0.2 0.2 0.2 0.4 0.4 0.2 0.2 0.2 0.2 00.2 0.2 0.2 0.2 0 0.2 0.2 0.2 0 0.2 0.2 0.2 0.2 0.4 0.40.2 0.2 0.2 0.4 0.4 0.2 0.2 0.2 0.2 0 0.2 0.2 0.2 0 0.2