Investigation of the Influence of Synchronous-Reluctance Motor Parameters on the Stability of Its Operation

S. V. Kononenko, A. L. Kislitsyn

(Presented by the scientific seminar of the Electromechanical Faculty)

In order to design a synchronous-reluctance motor (SRM) operating stably under specified conditions, it is necessary to determine the influence of its various parameters on self-oscillations (self-swinging).

The method for investigating the stability of an SRM is considered in detail in [1]. However, the question of the influence of SRM parameters—apart from the active resistance of the stator winding and the inductive reactance of the external circuit—on the region of stable operation has not previously been considered.

In the present work, assuming that the motor operates at constant line voltage, the influence on the region of stable operation of an SRM is considered for the following parameters: the reactance of armature reaction along the direct axis \(x_{ad}\), the ratio \(\dfrac{x_{ad}}{x_{aq}}\), the frequency of the supply network, the moment of inertia of the rotating masses, and the magnitude of the load.

In analyzing electromechanical transient processes, to which the phenomenon of self-swinging belongs, it is necessary to consider jointly the voltage-balance equations and the torque equation. These equations, written in the \(d\) and \(q\) axes, are nonlinear. On the basis of the theory of small perturbations, by linearizing the equations, linear equations of perturbed motion are obtained.

Then, solving the obtained system of equations, the rotor equation of motion can be represented in the following form [1]:

\[ \left[T_j p^2 + M_{so} + M_{sq} + p\left(M_{do} + M_{dr_1} + M_{dr_2}\right)\right]\Delta \Theta = 0, \tag{1} \]

where \(T_j\) is the inertia constant of the rotating masses;
\(M_{so}\) is the principal synchronizing torque;
\(M_{sq}\) is the additional synchronizing torque caused by the presence of a winding on the rotor along the direct and quadrature axes;
\(M_{do}\) is the principal asynchronous torque;
\(M_{dr_1}\) is the additional asynchronous torque depending on the active resistance of the stator winding in the first degree;

$M_{dr_2}$ is an additional asynchronous torque, depending on the active resistance of the stator winding to the second power.

In the case under consideration it is most expedient to use a method of investigation based on the frequency criterion. The essence of the method reduces to considering small harmonic oscillations of the rotor of the SRM and analyzing the quantities that determine the change in electromagnetic torque. This method, which uses the torque characteristics of the machine, makes it possible to determine directly the influence of the SRM parameters on the position of the boundary of stable operation.

The boundary of the region of stable operation of the SRM when the frequency criterion is used is determined by the passage of the curve of the hodograph of the torque characteristic through the origin of coordinates of the complex plane [1].

Substituting into equation (1) $p = jh$ and separating the real and imaginary parts, we obtain

\[ X(h) + jY(h) = 0. \tag{2} \]

Here $X(h)$ and $Y(h)$ are polynomials in powers of $h$, whose coefficients contain the unknown parameters $\lambda$ and $r_{s\mathrm{kr}}$, whose influence on stability is to be investigated.

From equation (2) we determine two equations defining the desired solution

\[ X(\lambda, r_{s\mathrm{kr}}) = 0, \]

\[ Y(\lambda, r_{s\mathrm{kr}}) = 0 \tag{3} \]

where $r_{s\mathrm{kr}}$ is the critical active resistance of the stator-winding circuit, determining the boundary of the region of stable operation;

$\lambda$ is the parameter whose influence on stability is to be established.

Solving equations (3), one can determine the boundary of the region of stable operation in the plane of the parameters $\lambda$ and $r_{s\mathrm{kr}}$.

The existence of a periodic solution of equation (1), which is determined by a numerical method, is possible not for all values of the sought parameter. This makes it necessary, when analyzing the influence of parameters on the position of the boundary of the region of stable operation, to calculate a large number of variants.

The use of electronic digital computers [2] considerably reduces the time expenditure in these investigations.

This paper presents the results of investigations performed for the RO-42-4 motor, which has a rotor with filled interpole spaces.

According to the technical specifications, RO-42 motors must operate stably when the frequency varies within the limits $(30 \div 200)\%$ of the nominal value. In accordance with this, the influence of the parameters was determined over the entire operating frequency range.

In investigating the influence of a certain parameter $\lambda$ on self-oscillation, all the remaining motor parameters are taken as unchanged, with the exception of the active resistance of the stator winding.

In designing an SRM, the correct choice of the value of the reactance $x_{ad}$ is of great importance [3]. In the investigation it was assumed that the inductive reactance of the armature reaction $x_{ad}$ varies within the limits from 1 to 2. Here and below the parameters are expressed in relative units.

Figure 1 presents the boundaries of the region of stable operation, obtained by calculation, for various values of the frequency $\omega$. The region,

enclosed between each curve and the abscissa axis is the region of unstable operation; the region enclosed between each curve and the ordinate axis is stable. As can be seen from Fig. 1, with an increase in the inductive reactance \(x_{ad}\), the stability of the motor increases proportionally, and vice versa. Since the reactance \(x_{ad}\) is inversely proportional to the magnitude of the air gap, to increase stability it is necessary to choose the air gap as small as possible.

Fig. 1. Effect of the reactance \(x_{ad}\) on the location of the boundary of the region of stable operation for \(\dfrac{x_{ad}}{x_{aq}}=\mathrm{const}\).

One of the important factors determining the energy performance of an SRM is the ratio of reactances \(\dfrac{x_{ad}}{x_{aq}}\), which depends on the rotor design. The results of calculating the stability boundary as a function of \(\dfrac{x_{ad}}{x_{aq}}\), carried out for an SRM having all the other parameters the same as those of the RO-42-4 motor, are presented in Fig. 2. From the graphs shown it is seen that, as the ratio \(\dfrac{x_{ad}}{x_{aq}}\) increases, the stability of motor operation decreases and, consequently, the probability of self-oscillation increases.

In analyzing the theoretical and calculated data it was established that a change in the frequency of the supply network, with a corresponding proportional change in voltage, has a great influence on the location of the boundary of stable operation. As can be seen from the curves in Figs. 1 and 2, a decrease in the frequency of the supply network shifts the boundary of the region of stable operation toward smaller values of active resistance, and vice versa.

Synchronous reluctance motors RO-42 under operating conditions usually operate at load angles \(\Theta\) approximately equal to \(10^\circ\), and with a flywheel mass on the rotor shaft equal to \(1\ \mathrm{kg\,m^2}\). Such a mode

is rather close to no-load operation. An increase in the load on the shaft has a greater influence on the position of the boundary of the region of stable

Fig. 2. Influence of the ratio \(\dfrac{x_{ad}}{x_{aq}}\) on the position of the boundary of the region of stable operation at \(x_{ad}=\mathrm{const}\).

Fig. 3. Influence of the load of a synchronous reluctance motor on the position of the boundary of the region of stable operation.

operation. Figure 3 shows the stability boundaries at various load angles \(\Theta\), obtained for several values of the supply-network frequency. It should be noted that, as the load angle increases, the stability of the motor increases. When the load angle changes from \(0^\circ\) to \(20^\circ\), the stability increases only slightly.

The results of investigations carried out for different values of the flywheel mass on the motor rotor shaft showed that increasing the flywheel mass by a factor of 3–4 compared with the existing one somewhat increases the stability of the SRM.

The investigations performed showed that, in addition to the active resistance of the stator winding, other motor parameters also have a large influence on the boundaries of stable operation of the SRM, especially the magnitude of the air gap and the ratio

\[ \frac{x_{ad}}{x_{aq}} . \]

To determine the influence of the parameters on the position of the boundary of the stable operating region of the SRM, it is advisable to use the frequency stability criterion.

REFERENCES

1. E. V. Kononenko, A. L. Kislitsyn, A. F. Fink. Investigation of self-oscillations of a synchronous reluctance motor when operating from a regulating generator. Izvestiya TPI, vol. 132, 1964.

2. E. V. Kononenko, A. L. Kislitsyn, A. S. Gitman, A. F. Fink. Application of an ACVM for investigating self-oscillations of a synchronous reluctance motor by the frequency method. Izvestiya TPI, vol. 138, 201, 1965.

3. E. V. Kononenko, B. P. Garganeev, A. L. Kislitsyn. Some questions of the theory and design of three-phase synchronous reluctance motors. Interuniversity collection of papers, issue 3, “Questions of the Theory and Design of Electrical Machines,” Novosibirsk, 1963.