1 Romanian Academy Timisoara Branch, Timisoara, Romania
Tel: +40 256 491823, Fax: +40 256 491816, E-mail: ileana_torac@yahoo.com
2 Dpt. of Electrical Engineering, “Politehnica” University of Timisoara, Romania
Tel: +40 256 403457, E-mail: alin.argeseanu@et.upt.ro
3 student, “Politehnica” University of Timisoara, Romania
E-mail: Krisztina_leban@yahoo.com.au
Abstract
I. Torac 1 , A. Argeseanu 2 , K. Leban 3
The paper deals with the optimization of the reluctance machine with axiallylaminated rotor. An analytical method was used for the magnetizing inductances computation. The rotor data have the most important influence on the machine parameters. Therefore, for the machine optimization, that of the rotor has the most important contribution.
Keywords : reluctance machine, axially-laminated rotor, genetic algorithms, optimization method
1. Introduction
As it is known, the performance indexes of the synchronous reluctance motor with axially laminated rotor (such as torque per-unit volume, power factor, the maximum value of the torque) are essentially dependent on the magnetic dissymmetry of its rotor, respectively on the saliency ratio, L d
/L q
[1], [2], [3]. Figure 1 shows an example of rotor shape for the two-pole machine.
1. lamination; 2. insulation sheet 3. clamping plates 4. non-magnetic stainless steel bolt
Figure 1: Example of rotor shape for a two-pole machine

The optimal design of an electrical machine supposes the analyze of a very high number of machine data in order to find the solution corresponding to the required performances.
The variable data are all the geometric dimensions (for the ferromagnetic core: the inner and the outer diameter of the stator and that of the rotor, the stack length, the stator and rotor slot dimensions, the windings data: the turns per phase number, the conductor section, the number of currents ways, the number of the elementary conductors, the pole pair number) and the material proprieties. Therefore, the search domain for the required solution is a very large one. So, for the machine calculus, a computing program having a low running time value is very useful.
Obviously, an improved version of the computing program includes the optimization subroutine.
The stator of the reluctance machine with axially-laminated rotor is similarly to that of the induction machine. So, the stator optimization has no specific problems.
Moreover, the axially-laminated rotor influences essentially the saliency ratio and consequently, the rotor data are the most important to obtain the required performances. For this reason, the machine optimization supposes especially the rotor optimization.
Obviously, the rotor dissymmetry depends on all his non-magnetic domains: the insulation sheets of the axially laminated area (which is defined by the number n, and width of the insulation sheet,
 iz
) and clamping plates (defined by the width,
 c
).
It is not possible to estimate the influence of every non-magnetic rotor area on the saliency ratio as function of a single independent variable. Usually, the width of the clamping plate is obtained from mechanical strength conditions, and lower values for
 c
cannot be accepted.
Therefore, one can assume that the rotor dissymmetry depends only on the axially laminated area, (which is defined by the number n, and width of the insulation sheet,
 iz
).
In this case, the low number of variable for the optimization problem is a significant advantage. For instance, the analytical used method for the machine inductance estimation has as input data 24 variables (excepting the ferromagnetic material data), but only 4 variables are rotor data.
In order to find the appropriate values of rotor data, which can improve the motor performance, the “classical” optimal design supposes to analyze combinations of rotor data (width and number of laminations respectively of insulation sheets, and width of the clamping plates).
2. The inductance estimation
2.1.Simplifying hypothesis
One agrees that in all the orthogonal sections of the machine the magnetic field is the same; one neglects the magnetic field, which closes through the front area.

One considers the m.m.f. with sinusoidal distribution on the inner surface of the stator.
The influence of stator slots is taken into account by the Carter factor; the influence of the insulation sheets on the equivalent air gap length and that of the clamping plates is computed as in [2].
One considers the non-linearity of the steel laminations only for the d-axis field computation.
One considers that the q-axis field between two adjacent steel lamination of the rotor is plan-parallel one.
2.2 The inductances computation
One considers that the machine data (geometrical dimensions, B-H curve of the ferromagnetic material) and the stator m.m.f. distribution on the inner stator surface are known.
In order to estimate the d-axis flux density distribution in the air gap, B  d
(x), one divides the motor air gap into sections where the flux density is considered the same, having the value corresponding to the point in the middle of the section. The coordinate x defines the position of the point relating to the d-axis.
Figure 2 : The approximate field line for a two-pole machine
For the considered point in the air-gap, one gives as initial value the B  d
(x) that corresponds to the case
 fe

. With this value one computes the m.m.f. for each section of the magnetic circuit (stator tooth, stator yoke, air-gap, rotor lamination).
The computed resultant m.m.f., considering
 fe
finite, becomes greater than
 initially considered, and then a new value is given for the air gap flux density. The iterative process continues until the difference between the computed value of the m.m.f. and the initial admitted value is smaller than the fixed error.

In the air gap area corresponding to the rotor part composed of lamination and insulation sheets, one computes the flux density for a number of points equal to the laminations number, n (because the lamination width is usually less than 1 mm).
In the air gap area corresponding to the clamping plates, one computes the flux density for n c
points, n c
being less than n. So one obtains the d-axis flux density distribution along the pole pitch, B  d
(x).
The amplitude of the fundamental of the air gap flux density, corresponding to the d axis and the magnetic flux corresponding to B  d1
are:
B
 d 1

4




  n


1
B



 sin x
 f

  sin x
 i





 k n
 c

1
B
 k


 sin x ki

  sin x kf









(1)
 d 1


8
2 l



 n


1
B


 sin x
 f

  sin x

 i



 k n c 

1
B
 k

 sin x ki

  sin x kf





 (2)
The magnetizing inductance L md
, respectively the synchronous inductance L d
, are:
L md

2
 m ( Nk b
)
2

 d 1 (3)
L d

L md

L

(4)
Applying the magnetic circuit law for every rotor lamination, respectively the magnetic flux law for every insulation sheet as in [2] one can estimate the q-axis air gap flux density distribution along the pole pitch, B  q
(x). The amplitude of the fundamental of the q-axis flux density in the air gap is:
B
 q 1
 
0
( b q

 b qc
)

(5)
The term b q
 depends on the geometrical dimensions of the rotor area composed of laminations and insulation sheets and b qc
depends on the rotor dimensions and on the clamping plates width [2].
Then one obtains the relations for the magnetizing inductance L mq
respectively for the synchronous inductance L q
:

L mq

4


0
2 l
 m ( Nk b
) 2 ( b q

 b qc
)
L q

L mq

L

(6)
(7)
Using the analytical method mentioned above it was developed a program which estimates the air gap flux density and the inductance considering known the motor data and the stator phase current. The running time is significantly shorter in comparison to the finite element method based program. Therefore, this program is very useful in the early stages of motor design, in order to minimize time for finding optimal configuration.
The analytical method for the inductance estimation was validated [2].
3. The rotor optimisation strategies
Usually, in order to obtain the required machine parameters, one analyzes a large number of rotor data.
For the axially-laminated rotor optimisation, one considers as variables data the width and the number of insulation sheets.
In the special case of the axially-laminated rotor optimization problem, it was possible to reduce the input variable so that, using an analytical method based program, the time for finding the best solution is small enough, and the optimization method does not increase the running time.
Usually, the electric machine optimization using the genetic algorithm method, needs a set of seven-ten design variables. These are selected according with their significance in the general machine design.
In this particular case of the reluctance machine with axially-laminated rotor, the original analytical proposed method allows to reduce significantly the design variables number.
The final set of optimization variables contains only two variables: the lamination width, and the insulation width. The dependence between the rotor dimensions, the number of rotor lamination, the rotor insulation sheets and their width, determinates that this quantities are implicit functions, so one obtains the number independent variables, for a given values of rotor radius.
3.1.The basic mechanism of the genetic algorithm
The genetic algorithm method [5] codes parameters of the problem’s search area as a finite length string. It works with the code of the parameter set, not with the parameter themselves. The objective function evaluates the quality of the solution coded as a string. This result is used to search for a better solution.
The genetic algorithm mechanism works as follows:

One initialises the population of strings
One computes the objective function for each member of the considered population
One selects and one reproduces individs to form a new population according to each member’s objective function.
One performs the genetic operator on the population
If the stop condition is not satisfied, one return to the objective function computation.
3.2 Encoding
The encoding is one of the most important stages of the genetic algorithm. By encoding the proper quantities and user the proper encoding one influences significantly the algorithm efficiency.
For the axially-laminated rotor optimisation, as variable geometric rotor data one considers the width and the number of insulation sheets.
3.3 The genetic operators
The genetic algorithm uses the genetic operator of selection, crossover and mutation to manipulate the strings in a population. One combines the strings in different arrangements to find the string that optimise the objective function. For the proposed problem the objective function of is the saliency ratio of the reluctance machine, L d
/ L q
.
The search space is n

[2,200], and
 iz


0 .
05 ; 10
 mm. The adopted dimensions include the appropriated values of rotor dimensions, which allows to obtain the required machine performances.
The adopted value of the selection operator used to obtain a new generation, is an elitist type one. This operator preserves the best individual of each generation.
The size of the chromosome is 20 bits string, 8 bits for the variable n (the insulation sheets number) and 12 bits for
 iz
(the width of an insulation sheet). Because the insulation sheets number belongs to the natural number set, the encoding process needs 8 bits.
The low variable number allows the use of a larger bits number for each variable. In this case, the searching accuracy becomes:
 iz

10

0 .
05
2
12

2 .
429 .
10

3 (8)
The encoding accuracy of
 iz is a very good one, due to the very low number of the design variable.
The initially population must be bigger than the individual dimension. In this case it was adopted a population of 30 individuals.

The crossover operator is two-point crossover type. The crossover probability is 0.8.
The mutation operator is an adaptive type one. This operator produces a high probability at the algorithm beginning and a low one at the algorithm ending. The analytical formula of the probability mutation in this case is:
0.2 and the number of generation is 200. p m
( t )
 p m e
( t

1 )

(9) where: p m
(t) is the mutation probability at first generation , p m

0 .
25 and

4. Conclusion
The experimental model has the rated power 0.55kW, two poles, the phase voltage
110V, the supply frequency 50 Hz.
The stator data are: the inner stator diameter D=63.2mm, the outer stator diameter
D ext
=114 mm, the air gap length

=0.35mm, the stack length is l=55mm and the number of turn per phase is N
1
=214.
The validation of the machine inductances was made using the d.c. decay test [2],
[3], [4].
The optimum value of saliency ratio was obtained and for
 iz
=0.35 mm.

A large combination of rotor data can be faster investigated with a computing program including an adequate optimization subroutine, because the running time is significantly shorter in comparison with the finite elements method based program.
So, the machine performances can be estimated with a good accuracy before the prototype manufacture.
References
[1] I. Boldea, “ Reluctance synchronous machines and drives” , Clarendon Press, Oxford, 1996.
[2] I. Torac, “ An analytical model of the synchrono us reluctance machine with axially laminated rotor”,
Ph.D. thesis, Universitatea "Politehnica" Timisoara, 1999.
[3] I. Boldea, “ Electric machine parameters - identification, estimation and validation”, Ed. Academiei
Romane, Bucuresti, 1991.
[4] P. Vas, “ Pa rameter estimation, condition monitoring and diagnosis of electrical machines”, Clarendon
Press, Oxford, 1993.
[5] A. Argeseanu, M. Pop, “Algoritmi genetici – Teorie si aplicatii”, Ed. Politehnica, Timisoara, 2006