Several studies have been published on this subject concerning especially the calculation of the transformation integral and the plotting of the field spectrum. The method presented in this paper covers larger topics including field strength, circulation, and flux and offers solutions either in closed form or, if not possible, resorting to numerical integration and avoiding, for the considered cases, the utilization of any numerical method like the finite element method. In the present work, the usage of a symbolic language is presented and certain new results have been obtained. The method permits applying the conformal transformation avoiding the hand calculation of integrals for cases in which this calculation is very difficult by traditional procedures because it involves finding some changes of variable suitable for the considered case. For this purpose, the Maple 12 software has been used. An application for electrical engineering which has been considered led the author to find a new solution superior to the known ones.

Conformal transformation is a very useful tool for the study of various potential fields. The procedure offers closed form solutions for many interesting cases, when it is possible, avoiding the utilization of any numerical method like the finite element method, finite difference method, and so forth. In the present work, the usage of a symbolic language, for this purpose, is presented. The procedure permits applying the conformal transformation avoiding the traditional hand calculation which involves some changes of variable difficult to be found and carried out. If the expression of the indefinite integral may not be obtained, the numerical integration is used. Therefore, the program permits controling the procedure, which may be difficult or even impossible by traditional calculation. For this purpose, we have considered and used the Maple 12 software.

An application for electrical engineering for the calculation of the magnetic field in a rotating electrical machine has been considered. Simultaneously, some new results which show important differences relatively to the traditional treatments have been found.

Because in the literature the subject is treated in various manners, we will recall the principle and properties of conformal transformation in the manner we have used, for being able to pass consistently to the usage of symbolic programs in this field. Consider two complex planes,

Equation (

Analogously, assuring the correspondence of the points on the two planes

As known, a transformation like that above is called conformal transformation because two curves, which cross each other at a certain angle on plane

As also known, at the base of the conformal transformation lies the Riemann theorem: the interior of a simply connected surface, of any domain

In many applications it is necessary to make successively two conformal transformations, one from the

For any complex function as in the following expression:

The field strength, having on a plane

which with the above relations, and for simplicity, denoting

At each point of the domain

The flux through an arc of curve which ties two points denoted by 1 and 2 on domain

For the sake of clearness, we can consider the tube of flux of a plane-parallel configuration crossing the mentioned arc of curve and having the thickness equal to unity. The described flux in domain

With the potential and flux being the same for the case of each of the two planes it is convenient to carry out the calculations in the case in which they are simpler. For instance, in many cases the most convenient case is that of a rectangular domain, say on a

The passage from the upper half-plane to a rectangle is related to a strip, and is accomplished by the known conformal transformation

Then, the calculation of

In several applications, in Maple 12, it may be necessary to replace symbol

For practically solving a problem by using a conformal transformation, the following stages have to be browsed:

To find the adequate analytic function for transforming the given domain configuration into another convenient for solving the problem, in most cases it is the case of the rectangle explained above. Sometimes, several transformations are necessary for this purpose.

If inside the domain

The computation of potential differences and fluxes should be performed using the complex potential function choosing that of the domains

In many cases it suffices to obtain the function for conformal transformation of the domain

A very interesting case is that of the interior domain of a polygonal contour into the

The transformation of this domain into that of the upper half-plane is given by the relation

There follows

A transformation of the form (

For the previously mentioned cases, the known calculations are difficult and their following is also difficult. In these cases, the symbolic languages for programming allow the possibility of avoiding both difficulties. We will explain the method on an example known in literature, in order to permit an easy comparison. We have used the Maple 12 language.

The advantages are the following

There is no need to look for the obtained expression, in closed form, of the integral, if it exists, the indefinite integral is automatically returned by the software, even if the expression is so long that the traditional solving could not be expected. Moreover, the display of the integral in closed form is not always necessary because the evaluation of the stored result for the given data is also automatically performed by the program. If the expression in closed form of the indefinite integral cannot be obtained, the program continues to work correctly as explained in Section

For the preparation of the program, it is convenient to keep unchanged the values of certain variables or names during the calculation, according to the required conditions. For this purpose, as we have experienced, the use of

An important remark, as principle, is that for conformal transformations we must work not with physical lengths that should be expressed in metres but with the corresponding relative quantities. Therefore, the lengths which occur should be divided by another length, usually a segment which exists in the considered configuration.

As we have realized, the usage of a program prepared in Maple language permits avoiding the calculation of fluxes, which can be replaced by using the complex potential function that remains unchanged by conformal transformations. Therefore, it suffices to calculate the difference of two coordinates of two points for obtaining the flux passing through the respective tube of flux, in any transformed configuration.

At the same time, for obtaining the point of the minimum value of the component of the field strength, it has been possible to calculate by Maple the derivative of the considered function, avoiding the classical treatments [

As known, in classical treatments, the representation of the normal component of the field strength versus abscissa of the given configuration is relatively difficult. We succeeded in avoiding this complication by using a parametric plot by a procedure found in Maple 12 language.

It is useful to warn concerning calculations for the

The problem chosen for presenting the procedure is the plane-parallel configuration of Figure

Configuration of a magnetic system.

The top continuous line is considered to be at any magnetic potential

The gap between the two armatures, upper and lower, called air-gap, has its minimum thickness denoted by

If the slot did not exist, the flux between the two armatures had any value

By using, as already mentioned, a symbolic language, we obtained much easier the solution and did a deeper analysis. As a result, it may be mentioned that the values of the field strength do not differ from those of [

At the beginning, we consider that some mentions concerning the calculation of integrals would be very useful, as follows. Maple 12 can perform both literal and numerical integration, respectively. If Maple cannot find the expression, in closed form of the indefinite integral, or for definite integrals with one or both limits floating-point numbers (i.e., not fixed number of digits), for example, 3.04,

Before presenting the accurate solution we have obtained, it may be added that Maple allows also for a simple calculation of any flux, without implying the calculation in closed form of the occurring integral. It suffices to express the formula of the conformal transformation and the relation for determining the fluxes by using the expression of the complex potential function. For the limit zero, in the considered case, instead of the value zero, a negative value close to zero will be introduced for

The steps of the program will now be presented as follows.

The Schwarz-Christoffel transformation (

Normal component of the field strength for the case

In applying the above formula, when other requirements are not necessary, we will take the constant

In the program of Algorithm

It is useful to note that in the program list, the comments are marked by the presence before them of the sign

We will begin by expressing the complex potential function for the case of a rectangle with particular boundary conditions described in Section

The scalar potential, according to the described conditions, should be

Formula (

There follows

The minimum of

Although the computation returns the values of the components in terms of

by putting

general

The results may be seen in Figures

Normal component of the field strength for the case

For the calculation of the flux reduction due to the existence of the slot that will be denoted by

Due to the performed transformations, the calculation could refer to the

The last mention is of outstanding interest, not found in traditional treatments, because it means that to obtain the total flux it suffices to know only half of it, either for positive abscissa or for the negative abscissa.

In the case of infinite length as in [

For easier solving of this and other cases, we have no more used the transformation formula based on relation (

The computing expressions we obtained may be followed in Algorithms

represent expressions in the calculation and have been put in a form to facilitate their usage.

verifications.

the used abscissa.

In this stage, the obtained formula can be used for one of the two purposes: to calculate the difference for a finite length of the armature but keeping the symmetry of the configuration, which cannot be performed by the results of the known literature, and for the case of the infinite length, given in the known literature, but surely, due to the mentioned circumstances, we obtained different results. Namely, for the relative value (i.e., length divided by

In practice, one of the most important results is the Carter factor, which refers to an alternative succession of slots and teeth, the pitch, that is, the distance between the axes of two successive slots having any value

With the formulae we established previously, with much better approximation, we can calculate it. It is the ratio between the flux in the case in which the thickness of the air-gap was constant and equal to its minimum value and the actual flux in the real case. Both results from the complex potential function and the relations are given in the program of Algorithm

According to the above definition, the Carter factor will be given by the formula:

In the literature there are several Maple programs concerning the Schwarz-Christoffel transformation, mainly devoted to the calculation of the transformation and to plotting the field spectrum.

An interesting analysis on the Schwarz-Christoffel transformation has been presented in [

Moreover, we have established that, in several cases we have experimented, numerical integration leads to a precision very near to that by using the solution in closed form. As explained, the fluxes and circulations have been calculated avoiding the classical procedures which involve supplementary integrations.

A new method, efficient in many cases, for using the conformal transformation for the calculation of potential fields has been presented. The method is based on the utilization of symbolic languages and permits to directly obtain the integral of the Schwarz-Christoffel formula, regardless of its complication, provided the solution in closed form exists, which is not possible by the traditional methods. The method correctly works even if the expression in closed form of the indefinite integral cannot be obtained. Moreover, the proposed procedure has allowed the obtaining of new results of higher accuracy than previously. Also, the analysis of the singular point could be carried out and taken into consideration.