STATIC STABILITY OF SYNCHRONOUS RELUCTANCE GEARED MOTORS WITH A SHORT-CIRCUITED WINDING ON THE ROTOR

A. A. Buimov, B. P. Garganeev, E. V. Kononenko

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

Low-speed motors with electromagnetic reduction of rotational speed most successfully solve the problem of low-speed operation. Owing to a number of advantages, they are used in various automation systems as actuating motors. There are several types of low-speed motors, which differ in design, number of windings, arrangement of windings, type of supply voltage, etc. In this article the static stability of synchronous reluctance geared motors (SRGM) is considered, since these motors have a comparatively simple manufacturable design, which ensures high operating reliability.

The use of SRGM in automation systems imposes a requirement that consists in the fact that not only the average, but also the instantaneous rotational speed of the rotor must be equal to the synchronous speed, i.e., speed oscillations relative to the average must be absent. The causes of oscillations in the rotor rotational speed may be both external factors—changes in voltage, frequency, load—and internal ones—the relation between the motor parameters. Therefore, when studying the operating modes of an SRGM it is necessary to investigate the influence of the parameters on static stability.

Loss of static stability is an electromechanical transient process, and the behavior of a synchronous reluctance geared motor in this mode is described by a complete system of equations, which, for variable rotor rotational speed, is nonlinear:

\[ \begin{aligned} U_d &= p\Psi_d - \Psi_q k_p \omega_p + r i_d;\\ U_q &= p\Psi_q + \Psi_d k_p \omega_p + r i_q;\\ 0 &= p\Psi_{rd} - \Psi_{rq}(k_p - 1)\omega_p + r_r i_{rd};\\ 0 &= p\Psi_{rq} + \Psi_{rd}(k_p - 1)\omega_p + r_r i_{rq}; \end{aligned} \tag{1} \]

\[ H \frac{d\omega_p}{dt} = (\Psi_d i_q - \Psi_q i_d) - \frac{k_p - 1}{k_p} (i_{rd}\Psi_{rq} - i_{rq}\Psi_{rd}) - k_M \omega_p - M_C \]

\[ p\Theta = \omega_1 - k_p \omega_p, \]

where \(k_p\) is the reduction coefficient;
\(H\) is the mechanical constant;
\(\omega_p\) is the rotor rotational speed;
\(k_M\) is the coefficient of mechanical damping.

In deriving the equations it was assumed that there are two equivalent short-circuited windings on the rotor, located along the longitudinal and transverse axes.

Existing methods make it possible to analyze the stability of a system without requiring integration of the differential equations. The most accessible of them is the so-called direct Lyapunov method. The direct Lyapunov method makes it possible to investigate the stability of dynamic systems for arbitrary large deviations of the variables, provided only that the original differential equations of the system remain valid.

However, in many problems the determination of the Lyapunov functions needed for stability analysis by the direct method involves great difficulties. Therefore one may confine oneself to considering stability for small deviations of the variables, examining the linearized differential equations (equations of the first approximation).

To linearize the equations, the disturbances are taken to be infinitesimal and harmonic. In this case the position of the rotor at any instant of time can be determined by the angle [1]:

\[ \Theta=\Theta_0+\Delta\Theta . \]

A deviation of the rotor through the angle \(\Delta\Theta\) from its initial position leads to a corresponding change in all quantities: currents, voltages, flux linkages, and speed, which enter the system of equations. Therefore, in system (1) it is necessary to make the substitution:

\[ U_d=U_{d0}+\Delta U_d,\quad i_d=i_{d0}=\Delta i_d \ \text{and so on.} \]

After substitution into the equations, the quantities with the subscript zero characterize the initial steady-state regime, while the quantities with the sign \(\Delta\) are due to small oscillations of the rotor.

Excluding from the equations the terms corresponding to the steady-state regime and neglecting products of increments of quantities having the second order of smallness, they can be represented in the form of equations with constant coefficients:

\[ \begin{aligned} \left(x_d-\frac{x_{ad}^2}{x_{rd}}\right)p\Delta i_d ={}&-r\Delta i_d +\omega_{p0}\left[x_q k_p-\frac{x_{ad}}{x_{rd}}x_{aq}(k_p-1)\right]\Delta i_q \\ &+\frac{x_{ad}}{x_{rd}}r_r\Delta i_{rd} +\omega_{p0}\left[x_{aq}k_p-\frac{x_{ad}}{x_{rd}}x_{rq}(k_p-1)\right]\Delta i_{rq} \\ &-\left[\frac{x_{ad}}{x_{rd}}(k_p-1)(x_{aq}i_{q0}+x_{rq}i_{rq0}) -k_p(x_q i_{q0}+x_{aq}i_{rq0})\right]\Delta\omega_p \\ &-U\cos\Theta\,\Delta\Theta; \end{aligned} \tag{2} \]

\[ \begin{aligned} \left(x_q-\frac{x_{aq}^2}{x_{rq}}\right)p\Delta i_q ={}&-\omega_{p0}\left[x_d k_p-\frac{x_{aq}}{x_{rq}}x_{ad}(k_p-1)\right]\Delta i_d -r\Delta i_q \\ &-\omega_{p0}\left[x_{ad}k_p-\frac{x_{aq}}{x_{rq}}x_{rd}(k_p-1)\right]\Delta i_{rd} +\frac{x_{aq}}{x_{rq}}r_r\Delta i_{rq} \\ &+\left[\frac{x_{aq}}{x_{rq}}(k_p-1)(x_{ad}i_{d0}+x_{rd}i_{rd0}) -k_p(x_d i_{d0}+x_{ad}i_{rd0})\right]\Delta\omega_p \\ &-U\sin\Theta\,\Delta\Theta; \end{aligned} \]

\[ \begin{gathered} \left(x_{rd}-\frac{x_{ad}^{2}}{x_d}\right)p\Delta i_{rd} = \frac{x_{ad}}{x_d}r\Delta i_d -\omega_{p0}\left[k_p x_q\frac{x_{ad}}{x_d}-x_{aq}(k_p-1)\right]\Delta i_q-\\ -r_r\Delta i_{rd} -\omega_{p0}\left[x_{aq}k_p\frac{x_{ad}}{x_d}-x_{rq}(k_p-1)\right]\Delta i_{rq}-\\ -\left[\frac{x_{ad}}{x_d}k_p(x_q i_{q0}+x_{aq}i_{rq0})-(k_p-1)(x_{aq}i_{q0}+\right.\\ \left.+x_{rq}i_{rq0})\right]\Delta\omega_p +\frac{x_{ad}}{x_d}U\cos\Theta\,\Delta\Theta; \\[1em] \left(x_{rq}-\frac{x_{aq}^{2}}{x_q}\right)p\Delta i_{rq} = -\left[x_{ad}(k_p-1)\omega_{p0}-x_d k_p\omega_{p0}\frac{x_{aq}}{x_q}\right]\Delta i_d+\\ +\frac{x_{aq}}{x_q}r\Delta i_q +\omega_{p0}\left[x_{ad}k_p\frac{x_{aq}}{x_q}-x_{rd}(k_p-1)\right]\Delta i_{rd} -r_r\Delta i_{rq}+\\ +\left[\frac{x_{aq}}{x_q}k_p(x_d i_{d0}+x_{ad}i_{rd0})-(k_p-1)(x_{ad}i_{d0}+x_{rd}i_{rd0})\right]\Delta\omega_p+\\ +\frac{x_{aq}}{x_q}-U\sin\Theta\,\Delta\Theta; \\[1em] Hp\Delta\omega +\left[(x_d-x_q)i_{q0}+(n x_{ad}-x_{aq})i_{rq0}\right]\Delta i_d +\left[(x_d-x_q)i_{d0}+\right.\\ \left.+(x_{ad}-n x_{aq})i_{rd0}\right]\Delta i_q +\left[(x_{ad}-n x_{aq})i_{q0}+n(x_{rd}-\right.\\ \left.-x_{rq})i_{rq0}\right]\Delta i_{rd} +\left[(n x_{ad}-x_{aq})i_{d0}+n(x_{rd}-x_{rq})i_{rd0}\right]\Delta i_{rq} -k_M\Delta\omega_p; \\ P\Delta\Theta=-k_p\Delta\omega_p. \end{gathered} \]

The characteristic equation of system (2) for an SRGM with a short-circuited winding on the rotor has the sixth degree

\[ a_0p^6+a_1p^5+a_2p^4+a_3p^3+a_4p^2+a_5p+a_6=0. \]

The coefficients \(a_0, a_1,\ldots,a_6\) can be obtained by expanding the characteristic determinant composed of the coefficients of the system of equations.

A. M. Lyapunov was the first to rigorously define the conditions under which investigation of stable unperturbed motion by the equations of the first approximation is admissible. According to Lyapunov, if the linearized differential equations (the equations of the first approximation) of a system have a characteristic equation whose roots have negative real parts, then the unperturbed motion is stable. There exist criteria that make it possible to determine the negativity of the real parts of all roots of the characteristic equation without solving it: the algebraic criteria of Hurwitz and Routh, the frequency criteria of Nyquist, Mikhailov, and others.

Taking into account the large volume of computations, the calculation of static stability must be performed with the aid of digital computers. On a digital computer it is more convenient to use algebraic criteria, since in this case operations are carried out only with real numbers, and judgments about stability are made depending on the sign of the numbers obtained in the calculation.

To obtain the coefficients of characteristic equation (3), it is necessary to expand the characteristic determinant composed of the coefficients of system (2). Expansion of a determinant of the sixth order is associated with a large amount of computational work; therefore it is advisable to use standard subroutines available for digital computers. In the present work a subroutine based on the Danilevsky method was used.

For a characteristic equation of degree higher than the fourth, among the algebraic criteria it is more convenient to use the Routh criterion. The Routh criterion for equation (3) is written in the form of Table 1.

Table 1

In order for the system corresponding to characteristic equation (3) to be stable, it is necessary and sufficient that the coefficients of the first column of the Routh table be positive quantities. If at least one of these coefficients becomes negative, stable operation of the SRRM is disrupted. Consequently, the boundary of stable operation of the SRRM will correspond to those values of the parameters entering equations (2) for which one of the coefficients of the first column of Table 1 becomes zero. A change in the sign of the coefficient $a_6$ means crossing the pull-out boundary, while a change in the sign of $c_{16}$ means crossing the self-oscillation boundary. Thus the regions of stable and unstable operation are found in the plane of any two parameters, with all the remaining parameters unchanged.

When the parameters of the SRRM change, the boundaries of stable operation change. Calculations show that the influence of the parameters on the regions of stable operation is manifested as follows.

The overload capacity (slip) of the SRRM depends on the ratio of the synchronous inductive reactances along the $d$, $q$ axes and on the magnitudes of the active resistances of the rotor and stator windings. As the ratio $x_d/x_q$ increases and the active resistances of the rotor and stator windings decrease, the overload capacity increases.

The region of self-oscillation is affected mainly by the mechanical constant, the coefficient of friction, and the active resistances of the windings.

Figure 1a presents the pull-out and self-oscillation boundaries as functions of the magnitude of the active resistance of the stator winding. As follows from the figure, as the resistance of the stator winding increases from 0.01 to 0.5, the overload capacity decreases by 25–30%.

Stable operation is possible at resistance values less than 0.03 relative units and at values exceeding 0.3. As the load increases, the self-oscillation region shifts toward higher resistances, expanding at small $r$ and decreasing at large ones.

The influence of the active resistance of the rotor winding on stability

Fig. 1. Regions of stable operation of the SRGM: a) \(r_r=4;\ k_M=0.3;\ H=5\) el. sec.; b) \(H=5\) el. sec.; \(r=0.08;\ \dfrac{x_d}{x_q}=1.4;\ k_M=0.3;\) I — region of stable operation; II — region of self-oscillation; III — pull-out region

operation of the SRGM is shown in Fig. 1b. Calculations show that the motor, at an active resistance of the stator winding of 0.1, operates stably up to \(r_r=1.2\). In the range \(r_r=1.2\div1.5\) there is a sharp narrowing of the stable-operation region. This narrowing occurs due to unstable operation at small loads. A further increase in \(r_r\) has little effect on the stable-operation region. The overload capacity of the motor, as follows from Fig. 2, decreases as \(r_r\) increases. The most significant decrease is observed up to \(r_r=1.5\), while with a further increase in the active resistance of the rotor winding it is manifested more weakly. The decrease in the overload capacity of the motor is explained by the fact that, as it increases, the asynchronous component of the torque caused by the rotor winding decreases.

Fig. 2. Region of stable operation of the SRGM: \(k_M=0.2;\ \dfrac{x_d}{x_q}=1.5;\ r_r=10;\) I — region of stable operation; II — region of self-oscillation; III — pull-out region

The influence of the moment of inertia of the rotating masses of an SRGM with a winding on the rotor appears qualitatively in the same way as in motors without a winding on the rotor [3]. In Fig. 2 the boundaries of stable operation are given for variation of the mechanical constant \(H\). From the curves presented it is evident that, in synchronous reluctance geared motors, as the moment of inertia increases, the stable-operation region contracts. At small values of \(H\) the motor operates stably practically over the entire range of variation of \(r\).

The magnitude of the coefficient of mechanical damping has a significant influence on the stability of SRGM operation; however, the character of the stable-operation regions when \(k_M\) is varied is analogous to that of a motor without a winding on the rotor [3].

Generalizing the results of the study of the influence of the rotor winding on the static st—

stability of a synchronous reluctance geared motor, it may be concluded that a short-circuited rotor winding expands the region of stable operation of the SRGM in the synchronous mode. However, a substantial reduction in the region of self-starting occurs in the range of small values of the active resistance of the rotor winding, at which no-load starting of the motors becomes impossible.

REFERENCES

1. E. V. Kononenko. Synchronous Reluctance Machines. Moscow, Energiya, 1970.

2. B. M. Kagan, T. M. Ter-Mikaelyan. Solving Engineering Problems on Digital Computers. Moscow–Leningrad, Energiya, 1964.

3. A. A. Buimov, E. V. Kononenko, R. P. Laas, B. P. Garganeev. Increasing the static stability of low-speed synchronous motors with electromagnetic speed reduction. In: Design of Electrical Equipment and Electric Drive, vol. 2, “Electromechanical Devices and Elements of Technology.” Moscow, Energiya, 1973.