MAGNETIC FIELD ANALYSIS OF AN INDUCTION MACHINE WITH ... · phase shift. That is caused by spatial winding distribution on the stator surface. Dual three-phase machine is a six-phase

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International Journal of Mechanical Engineering and Technology (IJMET) Volume 9, Issue 11, November 2018, pp. 789–801, Article ID: IJMET_09_11_080 Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=11 ISSN Print: 0976-6340 and ISSN Online: 0976-6359

© IAEME Publication Scopus Indexed

MAGNETIC FIELD ANALYSIS OF AN

INDUCTION MACHINE WITH MULTIPHASE

STATOR WINDING THROUGH FINITE

ELEMENT METHOD

Oleh Mykhailiuk

Department of Renewable Energy and Transportation Electrical Systems Faculty of Power Engineering and Electrical Mechanics

Vinnitsa National Technical University, Vinnytsia, Ukraine

ABSTRACT

Induction machines are the most frequently used type of electrical machines for

industrial and domestic applications. Therefore, increasing machine performance and

reduction of energy losses are substantial problems. The fundamental operation principle

of the induction machine is based on attraction and repulsion between rotor and stator

magnetic fields. Thus, in some cases winding configuration can cause significant

influence on machine operation and efficiency. The aim of this work is to model and

describe magnetic field of the induction machine with three-phase and multiphase

windings in order to identify appropriate winding configuration.

Numerical computations were performed for two winding configurations with the use

of finite element method magnetics (FEMM). First configuration consists of three-phase

single-layer stator winding. Second configuration consists of multiphase single-layer

winding. For both cases field line maps were obtained and numerical values were

presented in co-ordinate system, shown in figures and compared. Results show that

multiphase winding provides better field distribution in the stator core and in the air gap.

Keywords: induction machine, finite element method, magnetic field distribution, multiphase winding.

Cite this Article Oleh Mykhailiuk, Magnetic Field Analysis of an Induction Machine with Multiphase Stator Winding Through Finite Element Method, International Journal of Mechanical Engineering and Technology, 9(11), 2018, pp. 789–801. http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=11

1. INTRODUCTION

Electric machines are one of the most fundamental motion generation mechanisms in both industrial and household products. As such, they work often as noise and vibration generating sources as well. Although there are various types of electric machines, induction machines are the most popular among them. Three-phase induction machines are commonly used in industrial and domestic applications because of efficiency and cost effectiveness [1]. Moreover, the most

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important feature which makes the induction machine most viable alternative to DC system is its cost per KVA which is approximately one fifth of that of DC system and its suitability in hostile environment. Thus in the present situation induction machines consume large amount of electric power and an increase in the efficiency will reduce the consumption of the electric power which in turn further reduces costs [2–4].

Due to the importance of induction machines, new strategies and configurations of design are being sought that are capable of improving efficiency [5–10]. In order to increase effectiveness of electric systems, the idea of multiphase was conceived. Multiphase (more than three) induction machines are more reliable compared to the traditional three-phase machines. This is true because the loss of one or two stator phases does not necessarily stop the machine from working as in three-phase machines [11]. Multiphase induction machines are becoming popular and have been being studied for many years as well as conventional three-phase induction machines or induction machines with lesser phases. Compared to conventional three-phase machines, multiphase machines are credited with having better fault tolerance, higher efficiency, lower current ripple, less torque pulsation, reliability and facility to split certain amount of power into multiple phases [12–14]. These desirable features make multiphase machine a promising candidate for high power applications [15].

Among multiphase machines, those with multiple three-phase windings, such as six-, nine-, and twelve-phase machines are often considered for various applications. Compared with multiphase machines with prime phase numbers, machines with multiple three-phase windings can benefit from the well-established three-phase technology [16]. Multiphase machines have been proposed for some specific advantages, can be better exploited justifying the higher complexity in contrast to the three-phase solutions. Some of the most suitable applications are the high current ones, where the main advantage of multiphase technology is determined by splitting the controlled current on more inverter legs, reducing the single switch current stress compared to the three-phase solutions [17, 18].

In this paper, finite element method was used to investigate magnetic field distribution for the machine with multiphase winding and to obtain numerical values of stator field for comparison three-phase and multiphase windings.

2. MATERIALS & EXPERIMENTAL PROCEDURES

2.1. Multiphase winding technology

Multiphase stator winding consists of coils that can be connected in series to form one circuit or parallel to form several identical circuits. The coils are placed into the stator slots. Depending on design requirements, winding parameters can be calculated with the use of (1–4) according to M. Hadžiselimović et al. [19].

The number of stator slots per pole defined as:

pQ

Q =2 p

(1)

Where Q is the total number of stator slots, 2p is the number of poles.

For multiphase machine number of slots per pole and phase is needed:

Qq =

2 p m, (2)

where m is the number of phases.

Stator slots are normally distributed around the inner surface of the stator. Therefore, each slot is shifted in space for particular angle:

Magnetic Field Analysis of an Induction Machine with Multiphase Stator Winding Through Finite Element Method


q2 π

α =Q

. (3)

Considering the number of pole pairs, electrical angle between stator slots:

2 πα = p

Q, (4)

where p is the number of pole pairs.

Fig. 1 shows the single-layer three-phase winding with two pole pairs. Each phase consists of 4 coils which are connected in series.

Figure 1. Three-phase single-layer winding.

Each of 24 slots is shifted by 15° mechanical. Considering that each coil is placed in neighbouring slots, the phase zone is divided by 2 opposite parts. The star of electromotive force phasors for each slot is shown in Fig. 2.

Figure 2. The 24 slot emf phasor diagram.

The phasor diagram contains 12 phasors, one for two equally energized coils. Diagram is presented in electrical degrees. The angle between two phasors is 30°. The phasors are numbered around the circle with two full turns. Each phasor represents two slots with the same emf direction. Phase A pole zone represented by 4 phasors, two positive 1;13, 2;14 and two negative 7;19 and 8;20. In that case the geometrical emf sum is less than algebraic sum because of 30° phase shift. That is caused by spatial winding distribution on the stator surface.

Dual three-phase machine is a six-phase induction machine having two sets of three-phase windings spatially shifted by 30 electrical degrees with isolated neutral points or star-delta connections [20]. Detailed circuit diagram of the winding is shown in Fig. 3. Among the groups of multiphase machines, the six-phase has received more attention due to its simplicity in converting a three-phase machine to six-phase machine. This is achieved by splitting the phase belt of three-phase machine into the two parts I and II. As a result there are two sets of three-

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phase stator windings likewise the original three-phase with set II spanning 30° from set I having a common magnetic structure. In fact, any higher phase order in multiple of three is easily realizable using the conventional three-phase machine [11]. The schematic structure of multiphase winding is presented in Fig. 3. Likewise in Fig. 2 multiphase winding placed in 24 slots. But in this case there are two three-phase systems A, B, C and A1, B1, C1. The emf star diagram is shown in Fig. 4.

Figure 3. Two-pole multiphase winding.

The multiphase winding is divided into two parts. Each part has to be fed from separate power source. However, there is the way to use one power source by the means of interconnecting multiphase winding into the star and delta parts. For small harmonic contents, both of them must generate the same magnetomotive force. But the currents in the delta part are smaller by the concatenation factor 1.73. Therefore, the number of turns in the delta part must be increased by this factor (6). Consequently, the conductor area must be decreased by this factor (7), so that the same slot geometry and the same core stack could be used. The fact that two types of windings must be manufactured is the only disadvantage of the multiphase winding. If the number of turns is very low, the ratio of the number of turns between star and delta parts cannot be exactly adjusted [21].

The number of slots between the star and delta parts can be calculated:

x3 0 Q

Q =2 π p

, (5)

Where Q is the total number of stator slots, p is the number of pole pairs

Number of turns in delta-connected part can be calculated

xN = N 3 , (6)

Where N is the number of turns in star-connected part.

Considering increased number of turns in delta part, conductor area can be defined

xS

S =3

, (7)

where S is the star part conductor area.



Figure 4. The phasor diagram of multiphase winding.

Six-phase winding is constructed by splitting the 60° phase belt into two 30° spatially shifted parts. A, B, C part is placed in 1;13, 7;19, 3;15, 9;21, 5;17, 11;23 slots and A1, B1, C1 part is placed in 2;14, 8;20, 4;16, 10;22, 6;18, 12;24 slots. Each coil is placed only in specific slots without distribution, so the winding distribution factor increases from 0.966 for three-phase to 1 for six-phase winding.

In conventional three-phase machine if one of the phases is lost, the rotatory field also disappears and machine stops. The multiphase machine offers the improvement of the system reliability which is of great interest in specialized applications. Independently of the number of phases the multiphase machine has, it only needs two degrees of freedom to generate a rotatory field. Consequently, if one phase is lost, machine continues operating although at different rating values [16].

2.2. Finite element method

FEMM is a finite element software package for solving low-frequency electromagnetic problems on two-dimensional planar and axisymmetric domains. It allows the simulation of linear and nonlinear magnetostatic problems, linear and nonlinear time harmonic magnetic problems, linear electrostatic problems, and steady-state heat flow problems. It can be used in a variety of electromagnetic problems, but has a few limitations when applied to machines with moving parts [22]. By the use of FEMM, magnetic field distribution can be easily computed. The major benefit of FEMM is that it is free of cost and easily available. Easiness in control is additional advantage. Geometrical models can be formed unswervingly in FEMM, but drawing is uncomfortable. Auto-CAD designing is another possibility to draw a geometrical model. Designed models in Auto-CAD must be saved in .dxf format because FEMM allows only .dxf files. So, after being designed, Auto-CAD models imported in FEMM [23, 24].

FEMM is used in many fields of science and engineering to obtain approximate solutions for the partial differential equations to be solved. Its basic concept is that although the behavior of a function may be complex when viewed over a large region, a simple approximation may be sufficed for a small subregion. In practice, it utilizes a variational problem that involves an integral of the differential equation over the problem domain by dividing it into a large number of non-overlapping subregions, each with a simple geometry. Over each subregion the solution of the partial differential equation is approximated by a polynomial function. These polynomials have to be pieced together so that where the edges of adjoining elements overlap the field representations must agree to maintain continuity of the field. Then the variational integral is evaluated as a sum of contributions from each element resulting in an algebraic system with a finite size than the original infinite-dimensional partial differential equation. When enough small

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regions are used, the approximate solution closely matches the exact one. The advantage of breaking the domain down into a number of small elements is the problem transformation from a small but too complex into a big but relatively easy to solve [25]. One-quarter part of the machine cross-section geometry is shown in Fig. 5.

Figure 5 Discretization of the induction machine geometry.

Linear triangular elements are used for discretizing cross-sectional area of the induction machine such that the mesh coordinates during computing process can be accessed rapidly [26, 27]. In this approach, the FEM meshes of the cross section of the machine are divided into two parts, which are the stator and the rotor, each including a part of the air gap. Meshes of the two parts are generated separately. The air gap is divided into two layers, one of which is to the rotor, and the other is to the stator. This means that the stator and the rotor meshes are generated once at the beginning and then can be used without any further change throughout the computing [28–30].

Electromagnetic problem analysis starts with Maxwell’s equations [25]:

ρE - = 0

ε∇ ⋅

, (8)

where E is the electric field intensity, ρ is the charge density, ε is the electrical permittivity.

B = 0∇ ⋅ , (9)

where B is the magnetic flux density.

B × E + = 0

t

∂∇

∂ , (10)

EB - μ J + ε = 0

t

∂ ∇ ×

∂ , (11)

where μ is the magnetic permeability.

Ohm’s law can be written with J and E quantities [25]:

J = σ E , (12)

where J is the current density, σ is the medium conductivity.

For the time-invariant cases (10) and (11) are simplified into:

E = 0∇ × , (13)

B - μ J = 0∇ × . (14)



In a magnetostatic problem FEMM goes about finding a field that satisfies (9) and (14) via the magnetic vector potential approach. Flux density is written in terms of the vector potential A [31] is represented as follows:

B = A∇ × , (15)

1A = J

μB

∇ × ∇ ×

. (16)

If linear isotropic magnetic material is used, (16) simplifies to:

2A + μJ = 0∇ . (17)

Electric scalar potential V is used for solving electrostatic problems:

E = - V∇ , (18)

2 ρV + = 0

ε∇

. (19)

Low frequency time harmonic magnetic problems are also solved when the field is oscillating in one fixed frequency. In this case the equation is:

eff

1α + σ V - J + jωσα = 0

μ B

∇ × ∇ × ∇

$

. (20)

where ω is the oscillation frequency, J$ is the phasor transformation of the applied current sources, effμ is the effective magnetic permeability, α is the complex amplitude of the phasor transformation [25].

2.3. Numerical model

Three-phase squirrel-cage induction machine was used for simulation. Machine rated values for operation in normal condition are given in Table 1.

Table 1 Machine rated values.

Parameter Value Parameter Value phase number 3 frequency 50 Hz phase voltage 220 V rotor speed 3000 rpm

rated power 20 kW number of poles 2

Besides electrical rated values, main geometrical dimensions are given in Table 2. Such parameters as number of rotor and stator slots, stator and rotor diameters, air gap and core length were determined for building machine geometry.

Table 2 Machine main dimensions.

Parameter Value Parameter Value

number of stator slots 48 number of rotor slots 38 slots per pole 24 rotor outer diameter 199 mm

inner stator diameter 200 mm rotor length 157 mm Stator length 157 mm air gap 0.5 mm

The final geometry used for simulation is shown in Fig. 6. Also, mesh was applied by FEMM preprocessor for breaking machine geometry into a number of small triangular elements. Number of obtained nodes is 164890 and number of elements is 329584. After preparing geometry in

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preprocessor and applying mesh, processor and postprocessor were used for solving the problem and obtaining results.

Figure 6 Machine geometry with mesh applied.

In the first case stator contained single-layer two-pole distributed winding. Second configuration presents multiphase stator winding (Fig. 7). It consists of two three-phase systems A, B, C, and A1, B1, C1. Each pole zone contains two coils of one system and two coils of another system. Each system is spatially shifted by 30° and fed separately. The machine was fed by three-phase symmetrical current system at no load condition.

1

11

1 18

1818

1810

1010

102

22

2 3

33

3 4

44

4 5

55

5 6

66

6 7

77

7 8

88

8 9

99

9 14

1414

1412

1212

1211

1111

11 13

1313

13 16

1616

1615

1515

15 17

1717

17 22

2222

2220

2020

2019

1919

19 21

2121

21 24

2424

2423

2323

23

A XB

Y

CZ A1

B1

C1 X1 Y1Z1

26

2626

2625

2525

25 28

2828

2827

2727

27 30

3030

3029

2929

29 32

3232

3231

3131

31 34

3434

3433

3333

33 36

3636

3635

3535

35 38

3838

3837

3737

37 40

4040

4039

3939

39 42

4242

4241

4141

41 44

4444

4443

4343

43 46

4646

4645

4545

45 48

4848

4847

4747

47

Figure 7 Multiphase stator winding.

3. RESULTS

Fig. 8 shows graphical representation of magnetic field lines in rotor and stator cores. Not all teeth are magnetized equally. One part is under high field density, but another part is magnetized at lower level.



Figure 8 Magnetic field distribution map.

Magnetic field distribution graph across the air gap between stator and rotor is shown in Fig. 9. Field is distributed sinusoidally with minimal and maximal ratings. Table 3 contains stator teeth induction ratings for one pole zone.

Figure 9 Magnetic induction distribution across the air gap.

Table 3 Stator teeth induction ratings.

Tooth number B, [T] Tooth number B, [T] Tooth number B, [T]

1 0.18 9 1.4 17 0.04 2 0.42 10 1.42 18 0.03 3 0.81 11 1.24 19 0.035 4 1.4 12 0.88 20 0.055

5 1.52 13 0.54 21 0.09 6 1.53 14 0.29 22 0.23 7 1.51 15 0.15 23 0.06 8 1.45 16 0.08 24 0.05

4. DISCUSSION

It can be seen from Fig. 9 and Table 3 that maximal values of the magnetic induction belong to 2 - 13 teeth and minimal values belong to 14 - 24 teeth. Tooth field magnitude is higher than in the air gap. The magnitude of the tooth induction represents the winding configuration in stator slots.

Fig. 10 shows simulation results for multiphase winding. Compared with Fig. 8 more field density lines are presented in stator teeth. That is because of improved winding configuration and better magnetic field distribution.

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Figure 10 Magnetic field distribution with multiphase winding.

Fig. 11 presents magnetic induction distribution graph across the air gap. Compared with Fig. 9 magnetic induction is higher in teeth and air gap zones.

Figure 11 Air gap induction distribution for multiphase winding.

Table 4 contains teeth induction ratings for multiphase winding. Unlike in Table 3 there are higher induction ratings in 17 – 24 teeth. Also, in 1–10 teeth there are higher induction values too. However magnetic field decreased in 11–15 teeth.

Table 4 Stator teeth induction ratings for multiphase winding.

Tooth number B, [T] Tooth number B, [T] Tooth number B, [T] 1 0.34 9 1.44 17 0.16 2 0.76 10 1.5 18 0.32

3 1.66 11 1.2 19 0.1 4 1.49 12 0.58 20 0.06 5 1.62 13 0.2 21 0.24 6 1.72 14 0.08 22 0.52

7 1.59 15 0.054 23 0.16 8 1.51 16 0.08 24 0.08

Stator teeth induction differences for three-phase and multiphase windings are given in Table 5.



Table 5. Stator teeth induction differences for three-phase and multiphase windings.

Tooth number ΔB, [T] Tooth number ΔB, [T] Tooth number ΔB, [T] 1 0.16 9 0.04 17 0.12 2 0.34 10 0.08 18 0.29 3 0.85 11 -0.04 19 0.065

4 0.09 12 -0.3 20 0.005 5 0.1 13 -0.34 21 0.15 6 0.19 14 -0.21 22 0.29 7 0.08 15 -0.096 23 0.1

8 0.06 16 0.001 24 0.03

Performed calculations revealed that the multiphase winding provides higher induction ratings. Although multiphase winding induction values are periodically changing along the stator surface likewise for the three-phase winding. So, multiphase winding is the optimal solution for the induction machines because of better magnetic properties.

5. CONCLUSION

In this paper, magnetic field investigation of the induction machine with multiphase winding was conducted by using finite element method. FEMM software was used to investigate magnetic field distribution in the stator core. The simulation results show that common single layer stator winding has irregular field distribution. In that case not all stator teeth are involved into energy transfer process. Multiphase winding allows to improve field distribution in the air gap by means of dividing coils into two separate groups spatially shifted by 30° and fed from two individual power sources. The same result can be achieved with the use of series or parallel connection of the star and delta winding parts.

The simulation results show that multiphase winding provides better field distribution in the stator core and in the air gap. Field density line map shows that more magnetic field lines intersect the core parts. Air gap induction distribution graph obtained to display the air gap field modification. Teeth induction values give the numerical significance of magnetic field improvement. The obtained results were compared for both types of windings. The analysis of computation results indicates that it is expedient to use multiphase windings for induction machines.

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MAGNETIC FIELD ANALYSIS OF AN INDUCTION MACHINE WITH ... · phase shift. That is caused by spatial winding distribution on the stator surface. Dual three-phase machine is a six-phase