STUDY OF SELF-OSCILLATIONS OF A SYNCHRONOUS RELUCTANCE MOTOR OPERATING FROM A CONTROLLED GENERATOR

E. V. Kononenko, A. L. Kislitsyn, A. F. Fink

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

Owing to the simplicity and reliability of their design, and to the absence of an excitation winding on the rotor, synchronous reluctance motors have found wide application in low-power drives wherever synchronous rotational speed is required. They are also used in the controlled-generator—synchronous-reluctance-motor system, which is employed to drive the pendulum of a hydraulic-turbine speed regulator.

It is known from practice that, for certain ratios of the parameters of the synchronous reluctance motor and the supplying generator, self-excited oscillations (self-oscillations) arise in this system. The essence of this phenomenon is that, after receiving a random external impulse, the motor subsequently rotates with an oscillating speed under periodic variations of current and voltage. Moreover, these oscillations do not decay, but have a constant or even increasing amplitude. Investigation of the stability of synchronous reluctance motors is important for practice; however, this question has not been considered in the literature.

The purpose of the present study is to choose a method for determining the influence of the parameters of the supplying generator and the motor on self-excited oscillations and to establish the limits of stable operation of a synchronous reluctance motor.

The shaft of the controlled generator is rigidly connected to the shaft of the hydrogenerator; therefore, owing to the inertia of the rotating masses, it may be assumed that the rotational speed during the transient process remains unchanged, and consequently the frequency of the supplying generator is constant. Taking into account that the controlled generator is excited by permanent magnets, it may be assumed that the no-load e.m.f. of the generator remains unchanged during the investigation of transients. Under load in the controlled generator, voltage drops occur in the active resistance \(r\) and inductive reactance \(x\). This circumstance can, with considerable accuracy, be imitated by connecting active resistances and inductive resistances in the form of chokes with linear volt-ampere characteristics in series with the stator circuit [1].

On the basis of the foregoing, for convenience in studying transient processes, the system under consideration is reduced to an equivalent circuit (Fig. 1). The voltage applied to this circuit,

equal to the e.m.f. of the no-load condition of the controlled generator and is a constant quantity. The voltage drops in the active and inductive resistances of the stator winding of the controlled generator are taken into account

Fig. 1. Equivalent replacement circuit of the synchronous-reluctance motor—controlled generator system.

by connecting in series with the stator winding of the synchronous-reluctance motor the resistances

\[ r=R_{\mathrm{G}}, \quad x=\frac{x_{d\mathrm{G}}+x_{q\mathrm{G}}}{2}, \]

where \(R_{\mathrm{G}}\) is the active resistance of the stator winding of the controlled generator,

\(x_{d\mathrm{G}}\) is the inductive resistance of the controlled generator along the direct axis,

\(x_{q\mathrm{G}}\) is the inductive resistance of the controlled generator along the quadrature axis.

These assumptions considerably simplify the theoretical analysis, since they make it possible to pass from the study of two machines to the study of a synchronous-reluctance motor connected to a network of infinitely large power \((U=\mathrm{const})\).

To study transient electromechanical processes of a synchronous-reluctance motor, which include self-excited oscillations, it is necessary to consider jointly the voltage-balance equations and the torque equation (the equation of rotor motion).

The phenomena occurring in a synchronous-reluctance motor are very diverse; therefore, in order to simplify the equations, in the investigation we adopt the following generally accepted assumptions:

1. The windings of the synchronous-reluctance motor are symmetrical.

2. The rotor winding is reduced to a two-phase winding arranged along mutually perpendicular axes \(d\) and \(q\).

3. The applied voltages are sinusoidal.

4. Only the fundamental harmonic components of the spatial curves of magnetomotive forces and magnetic fluxes are taken into account.

5. The influence of saturation is taken into account by choosing the corresponding parameters of the machine; within the limits of the process under consideration the parameters remain constant.

6. The influence of mechanical losses and iron losses is not taken into account in determining the currents.

1. A system of relative units is used.

Under the adopted assumptions, the voltage equilibrium equations of a synchronous reluctance motor in the system of \(d\)- and \(q\)-axes rotating at synchronous speed can be represented in operator form as [2, 3]:

\[ \begin{aligned} U_d &= p\psi_d + r i_d - \psi_q p\gamma;\\ U_q &= p\psi_q + r i_q + \psi_d p\gamma;\\ 0 &= p\psi_{rd} + i_{rd} r_{rd};\\ 0 &= p\psi_{rq} + i_{rq} r_{rq}, \end{aligned} \tag{1} \]

where \(\psi_d, \psi_q\) are the flux linkages of the stator windings along the direct and quadrature axes, respectively,

\(\psi_{rd}, \psi_{rq}\) are the flux linkages of the rotor windings along the direct and quadrature axes, respectively,

\(i_d, i_q\) are the currents of the stator windings along the direct and quadrature axes,

\(i_{rd}, i_{rq}\) are the currents of the rotor windings along the direct and quadrature axes.

The flux linkages in equations (1), in relative units, are equal to:

\[ \begin{aligned} \Psi_d &= x_d i_d + x_{ard}\cdot i_{rd}; \qquad \Psi_{rd} = x_{ard} i_d + x_{rd} i_{rd};\\ \Psi_q &= x_q i_q + x_{arq}\cdot i_{rq}; \qquad \Psi_{rq} = x_{arq} i_q + x_{rq} i_{rq}, \end{aligned} \tag{2} \]

where

\[ x_d = \omega L_d \]

is the inductive reactance of the stator winding along the direct axis of the synchronous reluctance motor;

\[ x_q = \omega L_q \]

is the inductive reactance of the stator winding along the quadrature axis;

\[ x_{ard} = \omega L_{ard} \]

is the inductive reactance due to the mutual induction of the rotor winding and the stator winding along the direct axis;

\[ x_{arq} = \omega L_{arq} \]

is the inductive reactance due to the mutual induction of the rotor winding and the stator winding along the quadrature axis;

\[ x_{rd} = \omega L_{rd} \quad \text{and} \quad x_{rq} = \omega L_{rq} \]

are the inductive reactances of the rotor winding along the direct and quadrature axes, respectively.

Without presenting the intermediate derivations and transformations, which are set forth in detail by A. A. Gorev [3] and A. I. Vazhnov [2], the equation of rotor motion can be written in the form

\[ M_{\text{st}} + H p\omega = \frac{3}{2}\left(-\Psi_q i_d + \Psi_d i_q\right), \tag{3} \]

where

\[ H = 2{,}74\,\frac{GD^2 n_s^2}{3U_n I_n}\cdot 10^{-6}\;(\text{sec}) \]

is the inertia constant of the synchronous-reluctance motor,

\[ GD^2 \]

is the flywheel moment of the rotating masses,

\[ M_{\text{st}} \]

is the static torque.

In equations (1, 2, 3), the values of the variables in the nominal operating mode are taken as the base quantities of the relative system of units.

The system of equations (1, 3), when the rotational speed varies, is nonlinear; therefore a direct solution of such a system in general form is impossible. Self-excited oscillations, as a type of static-

... stability can be investigated with the aid of the theory of small oscillations [2, 3]. In this case it is assumed that the rotor of the synchronous-reluctance motor performs small stable harmonic oscillations relative to a certain mean position, characterized by the angle \(\Theta_0\). A deviation of the rotor by an angle \(\Delta \Theta\) from its initial position will lead to a corresponding change in all quantities—currents, voltages, flux linkages—which enter into the system of equations (1, 3). Therefore, from the general equations, which are nonlinear, linear equations of the small relative motion are formed by expanding in powers of the small coordinates of this perturbed motion and discarding their higher powers. To find the linear equations that must be satisfied by the small coordinates \(\Delta i_d\), \(\Delta i_q\), \(\Delta U_d\), \(\Delta U_q\), etc., it is necessary to do the following:

1. Substitute the new values of the variables into equations (1, 3):

\[ i_d = i_{d_0} + \Delta i_d,\quad i_q = i_{q_0} + \Delta i_q,\quad \Theta_1 = \Theta + \Delta \Theta, \]

\[ i_{rd} = i_{rd_0} + \Delta i_{rd},\quad U_d = U_{d_0} + \Delta U_d \text{, etc.} \]

1. Eliminate the terms of the equations corresponding to the mean angle \(\Theta_0\), and neglect products of increments of the variables.

A system of linear equations in the new coordinates \(\Delta U_d\), \(\Delta U_q\), \(\Delta \Theta\), etc., is obtained; moreover, these equations will have constant coefficients.

To simplify the analysis, small periodic swings may be reduced to the consideration of small harmonic oscillations. Therefore we shall assume that the angular deviation \(\Delta \Theta\) obeys the law

\[ \Delta \Theta = \Delta \Theta_m \cdot \cos h\tau, \tag{4} \]

where \(\Delta \Theta_m\) is the amplitude of the change in the angle \(\Delta \Theta\),

\[ h = \frac{f_k}{f_1}, \]

\(f_k\) is the frequency of the rotor oscillations,
\(f_1\) is the frequency of the supply network.

In the case of sinusoidal oscillations, in the presence of sinusoidal voltages, the increments of the variables \(\Delta U_d\), \(\Delta U_q\), \(\Delta i_d\), etc., will be harmonic functions of the same frequency.

After the transformations listed above, the system of equations (1, 3) can be represented in the form

\[ \begin{aligned} \Delta U_d &= p\Delta \Psi_d + r\Delta i_d - \Delta \Psi_q + \Psi_{q_0}p \cdot \Delta \Theta;\\ \Delta U_q &= p\Delta \Psi_q + r\Delta i_q + \Delta \Psi_d - \Psi_{d_0}p\Delta \Theta;\\ 0 &= p\Delta \Psi_{rd} + r_{rd}\Delta i_{rd};\\ 0 &= p\Delta \Psi_{rq} + r_{rq}\Delta i_{rq};\\ \Delta M &= \Delta M_{\mathrm{st}} - HpS, \end{aligned} \tag{5} \]

where

\[ \Delta M = -\left(\Delta \Psi_q i_{d_0} + \Delta i_d \Psi_{q_0} - i_{q_0}\Delta \Psi_d - \Psi_{d_0}\Delta i_q\right). \]

The static torque on the shaft of the synchronous-reluctance motor \(M_{\mathrm{st}}\) remains unchanged; therefore the last equation of system (5) can be written as

\[ HpS + \Delta M = 0, \tag{6} \]

where

\[ S = \frac{\omega_s - \omega}{\omega_s} \]

is the slip.

In the obtained system of equations (5), the variables with the symbol \(\Delta\) are regarded as coordinates of motion, and the variables without the symbol \(\Delta\) as constant quantities.

The most complete and profound analysis of the operation of a synchronous-reluctance motor can be obtained by solving the linearized equations of the system jointly. There exist well-known criteria of Hurwitz, Mikhailov, Routh, and others, which make it possible to determine the stability of a system without resorting to the computation of the roots of the characteristic equation [2, 3].

However, in all cases difficulties arise if it is necessary to establish the influence of some parameter on stability, since the stability conditions are expressed by a complicated combination of the coefficients of the equation, while the latter, in turn, are complicated functions of the parameters of the synchronous-reluctance motor and of the controlled generator. Therefore it is very difficult to determine within what limits the parameters of the system must be varied in order to find the boundary of stable operation.

In the case under consideration it is most expedient to use a method of investigation based on the frequency criterion [4, 5]. The essence of the method reduces to the consideration of small harmonic oscillations of the rotor of the synchronous-reluctance motor and to the analysis of the quantities determining the change in electromagnetic torque, the so-called coefficients of synchronizing and asynchronous torques. This method, which uses the torque characteristics of the machine, makes it possible to determine directly which parameter of the system causes instability.

By varying the parameters of the system, it is possible, on the basis of the equations of the frequency method, to construct the boundaries of stable operation of the synchronous-reluctance motor.

Let us first consider how to determine the electromagnetic torque \((\Delta M)\) under the assumption that the active resistance of the stator-winding circuit is equal to zero. In the case of sinusoidal oscillations of the angle \(\Delta \Theta\), there is no need, in order to determine \(\Delta M\), to solve the equations in operator form. The determination of the currents \(\Delta i_d, \Delta i_q\) may be carried out by the complex method [3]. According to this method, instead of a harmonic function of time, a complex quantity is introduced into the consideration; for example, instead of the variable \(\Delta i_d = \Delta i_{dm}\cos h\tau\), one operates with the complex quantity \(\Delta \dot{i}_d = \Delta i_{dm} e^{jh\tau}\). The investigation of oscillations requires only the determination of the currents in the stator winding; therefore, in what follows we shall operate only with the voltage equations of the stator windings. In this case the currents of the rotor circuits, which are contained in the flux linkages \(\Psi_d\) and \(\Psi_q\), are eliminated by means of the equations of the rotor circuits.

After substituting the expressions for the currents and flux linkages into the equation of the electromagnetic torque, we obtain

\[ \Delta \dot{M} = \left\{ U^2 \left( \frac{1}{x_q} - \frac{1}{x_d} \right) \cos 2\Theta + U^2 \sin^2 \Theta \left[ \frac{1}{x_d(jh)} - \frac{1}{x_d} \right] + \right. \]
\[ \left. + U^2 \cos^2 \Theta \left[ \frac{1}{x_q(jh)} - \frac{1}{x_q} \right] \right\}\Delta \dot{\Theta}, \tag{7} \]

where

\[ x_d(jh) = x_d - \frac{jh x_{ard}^2}{jh x_{rd} + r_{rd}} \quad \text{is the operator reactive resistance} \]

along the direct axis,

\[ x_q(jh) = x_q - \frac{jh x_{arq}^2}{jh x_{rq} + r_{rq}} \quad \text{is the operator reactive resistance} \]

along the quadrature axis.

The expression for the electromagnetic torque was obtained under the assumption that the active resistance in the stator-winding circuit is zero \((R=0)\). The influence of this resistance, as studies have shown, proves to be substantial, especially for the asynchronous torque. In the general case the additional asynchronous torque depends significantly not only on \(R\), but also on \(R^2\), which must be taken into account if

\[ \frac{R}{h}>0.01 \quad [3,4,6]. \]

The remaining terms, containing \(R^3\) and higher powers, may be neglected because of their smallness.

As applied to the synchronous-reluctance motor, in accordance with [6], the expressions for the additional asynchronous torques have the form

\[ \begin{aligned} M_{dr_1}={}&-R\left\{\left[\left(\frac{x_{dh}}{z_{dh}^{2}}-\frac{1}{x_q}\right)U\sin\Theta\right]^2 +\left[\left(\frac{x_{qh}}{z_{qh}^{2}}-\frac{1}{x_d}\right)U\cos\Theta\right]^2 \right.\\ &\left.{}-U^2\frac{r_{dh}}{z_{dh}^{2}}\sin^2\Theta -U^2\frac{r_{qh}}{z_{qh}^{2}}\cos^2\Theta\right\}\\ &+\frac{2R}{h}\left[-\frac{U^2\cos\Theta\sin\Theta}{x_d}\cdot\frac{r_{dh}}{z_{dh}^{2}} +\frac{U^2\sin\Theta\cos\Theta}{x_q}\cdot\frac{r_{qh}}{z_{qh}}\right] \end{aligned} \tag{8} \]

—is the component of the additional asynchronous torque that depends on \(R\).

\[ M_{dr_2}=-\frac{R^2}{h}U\left\{ -\cos^2\Theta\,\frac{A'x_{dh}-B'r_{dh}}{x_dx_q} +\cos^2\Theta\,\frac{A'x_{qh}+B'r_{qh}}{z_{qh}^{2}} +\sin^2\Theta\,\frac{A'x_{dh}+B'r_{dh}}{z_{dh}^{2}} +\frac{B'r_{qh}-A'x_{qh}}{x_dx_q} \right\} \tag{9} \]

—is the component of the additional asynchronous torque that depends on \(R^2\), where

\[ A'=\frac{r_{dh}x_{qh}}{z_{dh}^{2}z_{qh}^{2}}+\frac{r_{qh}x_{dh}}{z_{dh}^{2}z_{qh}^{2}}, \]

\[ B'=\frac{x_{dh}x_{qh}}{z_{dh}^{2}z_{qh}^{2}}-\frac{r_{dh}r_{qh}}{z_{dh}^{2}z_{qh}^{2}}, \]

\(r_{dh}, x_{dh}, z_{dh}\) are the resistances calculated respectively as the resultant active, reactive, and total resistance from the equivalent circuit (Fig. 2) for the longitudinal axis of the synchronous-reluctance motor.

Fig. 2. Equivalent substitution circuit of the rotor winding of a synchronous-reluctance motor along the longitudinal axis.

\(r_{qh}, x_{qh}, z_{qh}\) are the resistances calculated from the equivalent circuit (Fig. 3) respectively as the resultant active, reactive, and total resistance for the transverse axis of the synchronous-reluctance motor.

Then, taking into account the active resistance of the stator-winding circuit, after transformations we obtain the following equation of rotor motion in complex form:

\[ \left[-h^2T_j\omega_s^2+M_{s_0}+M_{s_g}+jh\left(M_{d_0}+M_{dr_1}+M_{dr_2}\right)\right]\Delta\dot{\Theta}=0, \tag{10} \]

where

\[ M_{s_0}=U^2\left(\frac{1}{x_q}-\frac{1}{x_d}\right)\cos 2\Theta \]

is the principal synchronizing torque due to the difference of the resistances along the \(d\) and \(q\) axes;

\[ M_{sg}=U^2\cos^2\Theta\left(\frac{1}{x''_q}-\frac{1}{x_q}\right) \frac{h^2T_q^{\prime\prime 2}}{1+h^2T_q^{\prime\prime 2}} + U^2\sin^2\Theta\left(\frac{1}{x''_d}-\frac{1}{x_d}\right) \frac{h^2T_d^{\prime\prime 2}}{1+h^2T_d^{\prime\prime 2}} \]

Fig. 3. Equivalent circuit of the rotor winding of a synchronous-reluctance motor along the transverse axis.

—is the additional synchronizing torque due to the presence of a winding on the rotor along the direct and quadrature axes;

\[ M_{d_0}=U^2\cos^2\Theta\left(\frac{1}{x''_q}-\frac{1}{x_q}\right) \frac{T''_q}{1+h^2T_q^{\prime\prime 2}} + U^2\sin^2\Theta\left(\frac{1}{x''_d}-\frac{1}{x_d}\right) \frac{T''_d}{1+h^2T_d^{\prime\prime 2}} \]

—is the principal asynchronous torque due to the presence of a winding on the rotor.

Denoting the real part of equation (10)—the total synchronizing torque—by \(X(h)\), and the imaginary part—the total asynchronous torque—by \(Y(h)=h\sum M_a\), we obtain

\[ M(jh)=X(h)+jY(h). \]

Plotting \(Y(h)\) along the ordinate axis and \(X(h)\) along the abscissa axis, it is easy to construct the hodograph of the torque-frequency characteristic if \(h\) is varied within the limits from \(0\) to \(\infty\).

Proceeding from the formulation of the stability criterion in terms of the torque-frequency characteristic [4], the system is stable if the total asynchronous torque at the resonance of the forced oscillations is positive, i.e., for \(X(h_0)=0\)

\[ Y(h_0)>0. \]

Therefore, to determine the boundary of stable operation, we set the total synchronizing torque equal to zero and determine the resonant frequency \(h_0\).

In the stability analysis it is necessary to determine the boundary of stable operation as a function of the parameters of interest. The most convenient for this purpose are plane diagrams as functions of two parameters of the regulated generator, \(r\) and \(x\) (Fig. 1).

Substituting the values \(h_0\), \(\omega_s\), \(x\) into the equation \(jY(h_0)\) and setting \(Y(h_0)\) equal to zero, we determine the critical resistance at which

Fig. 4. Comparison of the theoretical and experimentally determined regions of stable operation of a synchronous reluctance motor with a starting winding only along the direct axis: — theoretical curve; — — — experimental curve.

Fig. 5. Comparison of the theoretical and experimentally determined regions of stable operation of a synchronous reluctance motor with a starting winding along the direct and quadrature axes: — theoretical curve; — — — experimental curve.

the system is at the stability boundary. By assigning other values of $x$, we similarly obtain other points that determine the region of stable operation.

Experimental verification of the method was carried out on two RO42-2 synchronous reluctance motors having different rotors:

1. A rotor with a starting winding along the direct and quadrature axes.
2. A rotor with a starting winding only along the direct axis.

Figures 4 and 5 present curves determining the boundaries of stable operation of the RO42-2 synchronous reluctance motor, obtained experimentally and by calculation. All parameters necessary for the calculations were determined experimentally.

The theoretical and experimental analysis carried out makes it possible to draw the following conclusions:

1. The study of self-excited oscillations in the system synchronous reluctance motor—controlled generator can be reduced to the study of a synchronous reluctance motor connected to a network of infinitely large power, according to the circuit of Fig. 1.

2. Application of the frequency method makes it possible most simply to determine the boundary of stable operation of a synchronous reluctance motor.

3. As the active resistance of the stator-winding circuit increases, the degree of stable operation of the motor decreases.

4. To eliminate self-excited oscillations of a synchronous reluctance motor, the ratio of the parameters $x/r$ (Fig. 1) must lie within certain limits. For the RO42-2 motor, $x/r$ must be no less than $0.5 \div 0.6$. Self-excited oscillations that arise can be eliminated by connecting inductive resistances in series in the stator-winding circuit.

5. At small operating angles $\Theta$, the starting winding of the rotor along the quadrature axis has a large influence on the stability of operation of a synchronous reluctance motor. The investigations showed that, in motors having such a winding, the boundary of stable operation is shifted toward larger values.

REFERENCES

1. V. A. Venikov, A. V. Ivanov-Smolenskii, Physical Modeling of Electrical Machines. GEI, 1956.

2. A. A. Gorev. Transient Processes of a Synchronous Machine. GEI, 1950.

3. A. I. Vazhnov. Fundamentals of the Theory of Transient Processes of a Synchronous Machine. GEI, 1960.

4. I. D. Urusov. Linear Theory of Synchronous Machines. Publishing House of the Academy of Sciences of the USSR, 1960.

5. E. P. Popov. Dynamics of Automatic-Control Systems. GITTL, Moscow, 1954.

6. R. A. Lyuter. Calculation of the damping-torque coefficient with allowance for the influence of the stator-winding resistance. VEI, No. 5, 1953.