STUDY OF THE MAGNETIC FIELD IN THE AIR GAP OF AN IMPROVED SYNCHRONOUS RELUCTANCE MACHINE USING AN ELECTROLYTIC TANK

E. V. Kononenko, A. N. Aifert, I. P. Guk

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

In view of the presence in the rotor body of an improved synchronous reluctance machine (SRM) of nonmagnetic interlayers along the path of the transverse magnetic flux, the magnetic field in the air gap of such a machine has a complex form. An analytical calculation of this field is extremely difficult. Therefore, to determine the coefficients characterizing the magnetic field in the air gap of an improved SRM [1], it is expedient to use the method of modeling the magnetic field in an electrolytic tank.

In the investigation the following basic assumptions were adopted:

1. The magnetic permeability of the steel is infinitely large in comparison with the permeability of air. The magnetic permeability of the material of the internal nonmagnetic interlayers was taken to be equal to the magnetic permeability of air.

2. The magnetic field was regarded as plane-parallel.

3. Only the first harmonic of the magnetomotive force created by the stator winding was taken into account.

4. The stator surface was considered smooth.

As is known, the magnetic field in the gap between stator and rotor may be regarded as potential if there are no current sources in it. In this case the magnetic field is described by Laplace’s equation

\[ \frac{\partial^2 \varphi}{\partial x^2} + \frac{\partial^2 \varphi}{\partial y^2}=0, \]

where \(\varphi=f(x,y)\) is the scalar magnetic potential.

In an electrical analogy such a field can be represented by the field of current in a conducting medium, if the electrode model coincides in shape with the cross section of the machine.

The cross section of the improved SRM under investigation is shown in Fig. 1 (hereinafter variant I). In addition, the magnetic field was investigated for the following modifications:

a) the central nonmagnetic interlayer—1—is absent (variant II),

b) nonmagnetic interlayer 2 is absent (variant III).

Furthermore, all investigations were carried out for different values of the air gap \((\delta)\) and pole overlap \((\alpha)\).

Taking into account that in investigations and calculations of SRMs the two-reaction theory is used, the magnetic field was investigated separately along the transverse and longitudinal axes. Since the pattern of the magnetic field in the electri-

Fig. 1. Cross section of the rotor of the improved SRM.

Since in an electrical machine it is repeated periodically, it is sufficient to investigate the region \(q—o—q'\) in the case of the longitudinal field and the region \(d—o—d'\) in the case of the transverse field (Fig. 1). The shape of the electrode models for both cases is shown in Fig. 2.

Fig. 2. Shape of the electrode models in modeling the magnetic field:
a) for the longitudinal field,
b) for the transverse field.

If the \(d\)-axis is taken as the origin of reference, then the potential of the stator-bore surface in the case of the longitudinal field (Fig. 2, a) will vary according to a cosine law, while the potential of all steel surfaces of the rotor and of the lines \(oq\) and \(oq'\) is equal to zero. In the case of the transverse field (with the same origin of reference), the potential of the stator-bore surface will vary according to a sinusoidal law. The potential of the line passing along the axis of symmetry of the central nonmagnetic interlayer is equal to zero. The surfaces of regions A and B are equipotential. In the absence of the central nonmagnetic interlayer, the potential of the surface of region A becomes equal to zero.

To satisfy the boundary conditions on the surface of the stator bore, it is necessary to approximate the given continuous function by a stepwise one. In [2] a method is indicated for determining the minimum number and dimensions of equipotential electrodes which approximate, with a specified accuracy, the continuous boundary function; this method was used in the present work. The boundaries of the region along the \(d\) and \(q\) axes were made of metal plates with zero potential, in order to take into account the change in sign of the boundary function in adjacent quadrants. It is known that installing a conducting wall changes the field of the electrodes in the same way as if, in addition to the system under consideration, there existed its mirror image, but with the opposite sign.

The model constructed was ten times larger than the original. The electrodes were made of brass. As the electrolyte, a weak solution of copper sulfate in distilled water was used, in a layer 3 cm thick, with specific conductivity

\[ \gamma = 10^{-2}\ \frac{\text{cm}}{\Omega\cdot\text{cm}^3}. \]

As the detector, an amplifier was used whose sensitivity voltage was \(0.01\ \text{V}\), responding well to the slightest deviation of the probe from the equipotential. The electrolytic tank was supplied from an alternating-current source with a voltage of \(40\ \text{V}\) and a frequency of \(200\ \text{Hz}\).

As a result of the investigations carried out, maps of the magnetic-field distribution were constructed for each particular case. Then, after constructing the curves of induction distribution along the surf—

Fig. 3. Magnetic-field coefficients for the first design variant.

of the stator bore and expanding them in a harmonic series, coefficients characterizing the magnetic field in the air gap of the improved SRM were calculated. These coefficients are presented in Figs. 3 and 4 as functions of the pole-arc ratio.

Fig. 4. Magnetic-field coefficients for design variants II and III.
— variant II,
--- variant III.

The studies showed that the additional nonmagnetic layers have practically no effect on the longitudinal magnetic field. Therefore, the coefficients characterizing the longitudinal magnetic field of the improved SRM do not differ from the corresponding coefficients of simple SRMs. At the same time, such layers substantially reduce the transverse magnetic field, thereby increasing the ratio

\[ \frac{x_d}{x_q}, \]

which leads to an increase in the maximum power of the SRM.

As can be seen from Figs. 3 and 4, among the three designs considered, the SRM of design variant I has the greater power, since the \(K_q\) curves for this variant lie below the corresponding curves for variants II and III. The curves for the coefficients \(K_d\), as well as \(\alpha_d\) and \(K_{\varphi d}\), coincide for all three variants; therefore they are not given in Fig. 4.

The coefficients obtained make it possible to calculate the magnetic circuit of the SRM completely, using the method set forth in [3].

To assess the correctness of the results obtained, an experimental determination was made of the unsaturated values of the induk-

…of the reactive resistances \(x_q\) of experimental specimens having the rotor designs considered above. The values of these resistances were then calculated by means of the coefficients given in Figs. 3 and 4. The values of the experimental and calculated parameters are given in Table 1.

Table 1

All values of the resistances \(x_q\) are given for an air-gap value \(\delta = 0.6\) mm. It is evident from the table that the calculated values agree quite well with the experimental ones. Thus, the coefficients obtained may be recommended for use in the design of improved SRMs of the constructions considered.

REFERENCES

1. E. V. Kononenko, A. N. Aifert. On the calculation of magnetic fields of synchronous reluctance machines. Izv. TPI, vol. 145, 1966.

2. G. M. Gershtein, I. N. Salii. On the discrete approximation of a discontinuous function at the boundary of a region in electrical modeling. Elektrichestvo, No. 11, 1963.

3. E. V. Kononenko, A. N. Aifert, B. P. Garganeev. Accounting for the influence of saturation on the parameters of synchronous reluctance motors. In the present collection.