Field Analysis of Induction Machine Using Two Different Models

Gheorghe Madescu1, Marius Biriescu2, Marian Greconici2, Sorin Muşuroi2, Marţian Moţ1

1 Romanian Academy – Timişoara Branch, 300223 Timişoara, B.dul M. Viteazul nr. 24, Romania, e-mail: gmadescu@d109lin.utt.ro 2 Politehnica University of Timişoara, 300223 Timişoara, B.dul V. Pârvan nr.2, Romania.

Abstract- The paper presents a comparison between two

models used for nonlinear magnetic field analysis of three phase induction machine in no-load conditions. The firs one is a numerical model based on Finite Element Method (FEM) and the second one is an analytical anisotropic model. In the paper are presented some numerical comparative results concerning space distributions of the magnetic field and some advantages or disadvantages of these models.

Keywords- Magnetic field, Numerical method, Analytical method

I. INTRODUCTION

The FEM is now considered as one of the most powerful tools for calculating the electromagnetic field in complicated structures, like those of induction motors with great number of slots in stator and rotor [1, 2, 3]. The advantage of the FEM is the possibility to take into account the saturation of the magnetic material in every point of the machine cross-section. The difficulty of the FEM is to take account the movement between the rotor and stator. However, there were developed such programs that compute step by step the transient magnetic field from stand-still until the no-load steady state operation. But in this case it is necessary a very long computation time. In order to reduce the computing time, different programs and formulations of the field problem were elaborated [4, 5] for the computation of the induction motors. The magnetic field of the induction motor in steady-state condition can also be computed by using analytical methods [6, 7, 8] that is very attractive for the designers and have some advantages in comparison with numerical methods.

The aim of the paper is to present some comparative results concerning space distribution of the magnetic field in different areas of the machine cross section, obtained with FEM and with an analytical model (AIM) presented in a previous paper [9]. The paper also presents some advantages and disadvantages of both models.

II. THE NUMERICAL MODEL

This model is based on FEM. In order to analyze the magnetic field in some areas of the magnetic core and in the air-gap of the induction motors with cage rotor, two programs (MESH and SOLV) were developed by the authors. For any stator and rotor diameters, pole numbers, slot numbers, or slot dimensions of the motor, MESH generate a 2D fixed-structure mesh. For a great accuracy, the density of the mesh points is bigger in the teeth heads. The area of each stator slot

pitch includes fixed number of 91 finite elements (figure 1) and for each rotor slot pitch, a number of 47 elements, respectively. In such a manner the mesh density increases if the slot number is bigger. Finally, MESH provides all the data about the mesh: node coordinates, area of each finite element (∆e), the number and position of the finite elements.

Figure 1. The mesh points.

It is assumed that the current density vector (J) and the

magnetic vector potential (A) have only the Z-axis components. In consequence, the flux density (B) has only components on the XY – plane (the cross section of the machine). Because of geometric and electromagnetic symmetries, the computing domain is reduced to a sector including one pole-half. In this case boundary conditions of both type Neumann and Dirichlet was imposed.

For i-node of the mesh, it can be write the following linear equation:

1 1 1 1

4 03

i i i im m m me e e e e e e

i ii ij j ik ke e e e

A K K A K A J= = = =

+ + − ∆∑ ∑ ∑ ∑ = (1)

with:

( ) ( )21eii k j j ke eK x x y y = − + − µ ∆

2

(2)

( )( ) ( )( )1eij k j i k j k k ie eK x x x x y y y y = − − + − − µ ∆

( ) ( ) ( )(1eik k j j i j k i je eK x x x x y y y = − − + − − µ ∆

)y

Figure 2. The influence of cµ on the convergence process.

where: (xi, yi) are the i-node coordinates; the subscripts (i, j, k) denotes the three nodes of each element (e); µe is the magnetic permeability obtained from the standard magnetization characteristic and depending on the flux density on the respective element (e). It is known that the flux density, the magnetic permeability and the current density are constant on each element surface, but different to each other.

The sums of the equations are performed for all the elements (mi) around the i-node.

The number of some equations that can be writing is equal with the number of the nodes and the number of the unknown quantities (Ai). This system of linear equations allows determining the magnetic potential vectors in all the mesh-nodes and finally, the flux densities on all the mesh-elements. The program SOLV deals with this side of the model and includes an iterative algorithm in order to solve the equation system.

Figure 2 show the convergence process on the element with number 797, for cµ=0.2 and cµ=0.1. It can be see a great influence of this coefficient on the convergence rate.

III. THE ANALYTICAL MODEL At each iteration all the values of magnetic vector potential are determined and used to find in the same iteration the components of the flux density:

In the analytical model is used the anisotropic (orthotropic) field model presented in a previous paper [9] and complete validated by test on no load and on load, for several three-phase induction motors. In this model the cross section of the induction machine is divided into five circular layers (D1 - D5, figure 3). The both stator and rotor slotting layer are homogenized and become anisotropic domains, with distinct magnetic permeabilities µr and µθ along the radial and tangential directions. This multilayer homogeneous model leads, as a consequence, to the analytical solution of the field differential equations. Finally, the magnetic potential expressions for the five layers are:

( )

( ) ( )

( )

( ) ( )

12

1 12 2

12

1 12 2

ee ex k j ie

e ei k j j i ke e

ee ey k j ie

e ei k j j ie e

AB x x Ay

k

x x A x x A

AB y y Ax

y y A y y A

∂= = − +

∂ ∆

+ − + −∆ ∆

∂= − = − +

∂ ∆

+ − + −∆ ∆

(3)

( ) braprhrgrA pp <<⋅+⋅= − θ);sin()θ,( 111111

( ) drcprhrgrA pp <<⋅+⋅= − θ);sin()θ,( 133311

( ) freprhrgrA pp <<⋅+⋅= − θ);sin()θ,( 155511

(7)

and the resulting magnetic flux density on each mesh-element:

( ) ( )2 2e e ex yB B B= + . (4)

( ) crbprkrhrgrA pp <<⋅+⋅+⋅= α−α θ);sin()θ,( 12

22222121

( ) erdprkrhrgrA pp <<⋅+⋅+⋅= α−α θ);sin()θ,( 12

44444141

From magnetizing curve, these values allow to determine the new magnetic permeability on the elements.

In order to obtain a good convergence rate, for the next cycle of computation each permeability and each magnetic vector potential were changed according to:

where gi and hi are constants [9] and k2 and k4 are depending on the stator and rotor current sources:

( ) ( ) ( ) ( )( )1 1n n nc+ +µµ = µ + µ −µ n

)n

(5)

( ) ( ) ( ) ( )(1 1n n nAA A c A A+ += + − . (6)

4 44 42

4 1 44r

mr

k Jpθ

θ

µ µ= −

µ − µ

2 22 22

2 1 24r

mr

k Jpθ

θ

µ µ= +

µ − µ.

(8)

Many values of the coefficients cµ and cA were tested in order to reduce the number of iteration in the calculus algorithm. It has been proved that cµ=0.1 and cA=1.3 were adequate values for a wide power range of the machines.

In the last relation, Jm2 represent the equivalent rotor current density that has a demagnetizing effect which justifies the ⊕ sign in the second equation of the relation (8).

Figure 4. The computing domain and the geometric dimensions.

Figure 3. The multilayer homogeneous model.

Figure 5. Magnetic vector potential distribution in the stator yoke

along the θ - coordinate.

The stator and rotor current density Jm2 and Jm4 are related

to the reactive stator and rotor phase currents 1rI and 2'

rI by

the expressions (for the three phase motor):

4 12 26 2 1

m w1 1rJ W K Ie d

=π −

'2 12 2

6 2 1m w1 2rJ W K I

c b=

π −.

(9)

This analytical model, even using complicate expression of the coefficients, allows to drastically reducing the computing time for solving field problems.

IV COMPARATIVE RESULTS The resulting flux density distribution in the stator yoke

along the θ - coordinate, for r=0.072 m was represented in figure 6. On the element surface the flux density value is constant but different to each other. It can be see that the yoke flux density calculated with FEM is bigger in the slot axes (marked on the figure), whereas AIM assume a slotless stator with continuous distribution of the flux density along the pole pitch.

The both models above described, were comparatively used to calculate the magnetic field in the cross section of a 4 kW, two-pole three-phase induction motor with cage rotor, in steady-state no-load condition, at nominal voltage.

The dimensions of the computing domain (one pole-half) are presented in figure 4.

In figure 5 the space distribution of the magnetic vector potential in the stator yoke along the half of the pole pitch (θ - coordinate) and r=0.069 m was presented. The AIM model provides a sinusoidal distribution because of relation (7) and because of the homogenized slotting layer, whereas FEM shows an irregular distribution because of the stator slots (the slots positions were marked on the figure).

Figure 7 present the flux density distribution in stator yoke area along the r – coordinate direction, for θ=constant. It is easy to find that curve 4 is obtained with AIM which provide a continuously distribution of flux density on the axis

2θ = π , when “r” increase between 0.066 m and 0.09 m.

- curve 2, with intermediate values of axis angle corresponding to θ6 < θ < 2π .

Figure 6. Flux density distribution in stator yoke along θ - coordinate.

The air-gap flux density distributions are shown in figure 8 as a function of θ, for both AIM and FEM models. According to AIM (7), the air-gap flux density has a sinusoidal distribution along the pole pitch, with magnitude depending on the saturation level of magnetic material (teeth and yokes).

Figure 8. Air-gap flux density distribution a long a half of the pole pitch.

Figure 7. Flux density distribution in stator yoke along radial direction.

In FEM, the field distribution is depending on the time.

From this reason, in figure 8, two distributions were calculated: at the time 1 2t = π ω when the current in the first phase is the maximum, respectively at the time 2 2 3t = π ω when the current in second phase is zero. These distributions obtained with FEM shows that the air-gap magnetic field has a high content of space harmonics that produces different parasitic phenomenon.

V. DISCUSSIONS ABOUT THE MODELS

Each of the two models has some advantages or disadvantages. The choice of the model is depending on the complexity of the computing domain and on the aim or preference of the researcher.

With FEM, very different curves were obtained depending on the θ - axis. At θ=constant, in the radial direction of the yoke there are 5 finite element with different values of flux density. If θ - axis is overlapped with slots axes (example θ=θ6 in fig. 4), the flux density values are higher than the case in which θ - axis is overlapped with teeth axes ( 2θ = π ). The three curves in figure 7 correspond to the three values of θ - axis, as follow:

If it is desire the analysis of the magnetic field space distribution in the cross section of the induction machine, indisputable, the FEM is a good choice. This model provide all the field distortions determinate by the non-homogeneous areas (slots, slot-openings) and in consequence allows to compute both the space and time harmonics of the air-gap flux density in order to estimate the parasitic torques, magnetic noise, vibrations, additional loses and heating, or other effects. - curve 1, at θ=θ6, with higher values of the flux density;

On the other hand, if the aim is to compute just the fundamental wave of the main field path accounting for the saturation effect of the ferromagnetic material and the

- curve 3, at 2θ = π , with lower values of the flux density;

reaction field of the cage rotor, then the AIM is the better choice. Such an analytical model with six-layer system was recently presented in [10]. These analytical approaches, like AIM, were preferable to use if it is possible because has some advantages:

- It requires lower computation time [10]. Although modern computers are much faster than a few decades ago, this reason remains valid today. The AIM provides a field parameterized solution that includes some mathematical functions and geometrical dimensions. If these dimensions must be changed, a simple calculation of new solution will suffice. In the FEM, this requires re-meshing of the computing domain for each new dimension.

- Gives greater insight into the problem [10]. The reason for the greater insight is that dependencies can be seen simply by inspection. The analytical field solution provides transparency, something that is completely absent in FEM.

- Allows connecting it very easy to the circuit model in order to obtain a coupled field-circuit model.

- Allows stating an integrated criterion for optimal synthesis and analysis of the induction machines.

VI. CONCLUSIONS

The paper presents two different models used for 2D-field analysis of induction machine. Using these both models, the magnetic field of a particular 4 kW three-phase induction motor, in steady-state no-load conditions was calculated. The obtained results allows to see the difference between the discreetly solution and the continuously solution of the field problem. It is known that some designers prefer the analytical models for computing and optimization of usual electrical machines. These examples emphasize the advantages or disadvantages of both models in order to choose the appropriate model for some field problem.

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