FUNDAMENTAL EQUATIONS OF SYNCHRONOUS RELUCTANCE GEARED MOTORS

A. A. BUIMOV, E. V. KONONENKO

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

Low-speed synchronous motors with electromagnetic speed reduction are finding ever broader application in various branches of industry, owing to their advantages over high-speed motors with mechanical reducers. Often such motors do not have a DC excitation winding. In this case they are called synchronous reluctance geared motors. However, their application is restrained by such shortcomings as a limited starting torque and a tendency toward speed oscillations in the synchronous mode. To eliminate these shortcomings, various additional devices are used that improve starting and the uniformity of rotation of the motor. One such device is a squirrel-cage winding located in the rotor slots [6]. The use of a squirrel-cage winding is possible for low-speed motors with a ratio of teeth on the rotor and stator equal to \(z_2 = z_1 + 2p\), since only in this case does the rotor rotate in the same direction as the stator field.

When a squirrel-cage winding is present on the rotor in a synchronous reluctance geared motor, in addition to the synchronous electromagnetic torque, an asynchronous torque also arises, which affects operation both during starting and in the established synchronous mode. The magnitude of the asynchronous torque depends on the parameters of the squirrel-cage winding. In designing motors with a squirrel-cage winding on the rotor, it is necessary to investigate for which combinations of the stator- and rotor-winding parameters the best variant of a synchronous reluctance geared motor is possible, one possessing reliable starting and sufficient damping of oscillations of the rotor speed. The most complete investigations of various operating regimes can be carried out using a system of differential equations that includes the voltage-balance equations of the windings and the equation of motion of the rotor. The questions of formulating equations for synchronous reluctance geared motors have been considered most thoroughly in [1, 2, 4]. However, for a more complete account of the phenomena occurring in these motors, it becomes necessary once again to return to the formulation and transformation of the fundamental equations of synchronous reluctance geared motors with a squirrel-cage winding on the rotor.

In deriving the equations it is assumed that the stator has a three-phase two-pole winding. Since allowing for the influence of the squirrel-cage winding presents certain difficulties in the study, in the case where the task of studying the currents in the bars is not posed, it may be

replace it with two equivalent windings with mutually perpendicular axes. When replacing the real winding by equivalent windings, it is assumed that the magnetomotive forces of the real and equivalent windings are equal. Since the bars of the short-circuited winding are located in each slot of the rotor, i.e., uniformly around the circumference of the rotor, the number of turns of the equivalent windings along both axes will be the same. Under the condition of a sinusoidal current distribution along the rotor surface, the amplitude of the fundamental harmonic of the m.m.f. created by the currents \(i_r\) in the bars of the real winding, per pole, is equal to [7]:

\[ F_{r1}=F_1+F_2+\ldots+F_k =\frac{4}{\pi} I_r\left(\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_1}{2} +\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_2}{2} +\ldots+\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}\right) =\frac{4}{\pi} I_r\left(\frac{n_c}{4}-\frac{\sin n_c\alpha_c}{4\sin\alpha_c}\right), \]

where \(\alpha_c\) is the angle between two adjacent bars; \(n_c\) is the number of bars per pole pitch; \(\tau_k\) is the angle between the bars forming the short-circuited circuit.

If it is assumed that a current flows through the equivalent rotor winding whose maximum value is \(I_r\), then the amplitude of the fundamental harmonic of the m.m.f. created by this winding is

\[ F_{\mathrm{e}1}=\frac{4}{\pi} I_r w_{\mathrm{e}}. \]

Equating \(F_{r1}\) and \(F_{\mathrm{e}1}\), we obtain:

\[ w_{\mathrm{e}}=\frac{1}{4}\left(n_c-\frac{\sin n_c\alpha_c}{\sin\alpha_c}\right). \]

After substituting the values \(\alpha_c=\frac{\pi}{n_c}\), \(n_c=\frac{z_2}{2p}\), we obtain the number of turns of the equivalent winding:

\[ w_{\mathrm{e}}=\frac{z_2}{8p}. \]

The axis of one winding is aligned with the axis of one of the rotor teeth and is denoted \(pd\), while the axis of the second winding \(pq\) leads the axis \(pd\) by 90 geometrical degrees. Both windings are stationary relative to the rotor.

Thus, in a synchronous reluctance geared motor there are three phase windings on the stator and two equivalent windings on the rotor. In the general case the rotor rotates at speed \(\omega_p\). The fundamental harmonic of the air-gap permeance, when the rotor rotates with angular speed \(\omega_p\), rotates at speed \(k_p\omega_p\), where \(k_p=\frac{z_2}{2p}\) is the reduction coefficient of the motor. This latter circumstance complicates the derivation and transformation of the equations.

In order to simplify the analytical expressions in deriving the fundamental equations, generally known assumptions are adopted concerning the symmetry of all windings and the linearity of the magnetic system. Higher harmonics of the magnetic field are not taken into account.

For the stator windings and the equivalent rotor windings, the equations of voltage equilibrium in the phase axes have the form:

\[ U_a=p\Psi_a+i_a r; \]

\[ \begin{aligned} U_b &= p\Psi_b+i_b r;\\ U_c &= p\Psi_c+i_c r;\\ 0 &= p\Psi_{pd}+i_{pq} r_r;\\ 0 &= p\Psi_{pq}+i_{pq} r_r; \end{aligned} \tag{1} \]

where \(U_a, U_b, U_c, i_a, i_b, i_c, i_{pd}, i_{pq}\) are the instantaneous values of the voltages and currents;

\(r, r_r\) are the active resistances of the windings;

\(\Psi_a, \Psi_b, \Psi_c, \Psi_{pd}, \Psi_{pq}\) are the total flux linkages of the windings;

\[ \frac{d}{dt}=p \]

is the differentiation operator.

The total flux linkages are determined by the expressions:

\[ \begin{aligned} \Psi_a &= L_a i_a+M_{ab} i_b+M_{ac} i_c+M_{apd} i_{pd}+M_{apq} i_{pq};\\ \Psi_b &= M_{ab} i_a+L_b i_b+M_{bc} i_c+M_{bpd} i_{pd}+M_{bpq} i_{pq};\\ \Psi_c &= M_{ac} i_a+M_{bc} i_b+L_c i_c+M_{cpd} i_{pd}+M_{cpq} i_{pq};\\ \Psi_{pd} &= M_{apd} i_a+M_{bpd} i_b+M_{cpd} i_c+L_{pd} i_{pd}+M_{pdq} i_{pq};\\ \Psi_{pq} &= M_{apq} i_a+M_{bpq} i_b+M_{cpq} i_c+M_{pqd} i_{pd}+L_{pq} i_{pq}, \end{aligned} \tag{2} \]

where \(L, M\) are the inductances and mutual inductances of the windings.

An investigation of the permeance of the air gap, caused by the toothed structure of the stator and rotor, shows that it varies along the stator bore according to a law close to sinusoidal and has a constant component [4]. Neglecting the higher harmonics of the magnetic permeance, it can be represented in the form:

\[ \lambda=\lambda_0+\lambda_2\cos\gamma \]

where \(\gamma=k_p\omega_p t\) is the angle determining the position of the axis of maximum magnetic permeance \(d\) relative to the axis of phase “a.”

With a stationary rotor (\(\omega_p=0,\ \gamma=\mathrm{const}\)), the wave of magnetic permeance is stationary relative to the stator, and at every point of the air gap it is a constant quantity. If the rotor rotates at the speed \(\omega_p\), then the wave of magnetic permeance of the air gap will also move along the stator bore at the speed \(k_p\omega_p\). In this case the inductances and mutual inductances of the stationary stator windings are periodic functions of the angle \(\gamma\):

\[ \begin{aligned} L_a &= L_0+L_2\cos 2\gamma;\\ L_b &= L_0+L_2\cos(2\gamma+120^\circ);\\ L_c &= L_0+L_2\cos(2\gamma-120^\circ);\\ M_{ab} &= M_0+M_2\cos(2\gamma-120^\circ);\\ M_{ac} &= M_0+M_2\cos(2\gamma+120^\circ);\\ M_{bc} &= M_0+M_2\cos 2\gamma. \end{aligned} \tag{3} \]

The equivalent rotor windings rotate together with the rotor at the speed \(\omega_p\), and the angle determining the position of the rotor relative to the stator is equal to \(\gamma_p=\omega_p t\). Therefore the self-inductances of the equivalent rotor windings will be a periodic function of \(\gamma-\gamma_p\):

\[ \begin{aligned} L_{pd} &= L_{p0}+L_{p2}\cos 2(\gamma-\gamma_p);\\ L_{pq} &= L_{p0}-L_{p2}\cos 2(\gamma-\gamma_p). \end{aligned} \tag{4} \]

The mutual inductance between the rotor windings is absent for a uniform air gap. However, with a varying magnetic permeance of the air gap, two mutually perpendicular windings have a mutual-inductance coefficient [5], which is also a periodic function of the angle \(\gamma-\gamma_p\):

\[ M_{pdq}=M_{pqd}=M_{p2}\sin 2(\gamma-\gamma_p). \tag{5} \]

The mutual inductances between the stator and rotor windings consist of two components. One component is due to the rotation of the rotor windings relative to the stator at the speed \(\omega_p\), and the second is due to the rotation of the magnetic permeance relative to the stator at the speed \(\kappa_p\omega_p\) and relative to the rotor at \(\kappa_p\omega_p-\omega_p\). Therefore the mutual inductances have the form:

\[ \begin{aligned} M_{apd}&=M_{02}\cos\gamma_p+M_{22}\cos(2\gamma-\gamma_p);\\ M_{bpd}&=M_{02}\cos(\gamma_p-120^\circ)+M_{22}\cos(2\gamma-\gamma_p-120^\circ);\\ M_{cpd}&=M_{02}\cos(\gamma_p+120^\circ)+M_{22}\cos(2\gamma-\gamma_p+120^\circ); \end{aligned} \tag{6} \]

\[ \begin{aligned} M_{apq}&=-M_{02}\sin\gamma_p+M_{22}\sin(2\gamma-\gamma_p);\\ M_{bpq}&=-M_{02}\sin(\gamma_p-120^\circ)+M_{22}\sin(2\gamma-\gamma_p-120^\circ);\\ M_{cpq}&=-M_{02}\sin(\gamma_p+120^\circ)+M_{22}\sin(2\gamma-\gamma_p+120^\circ). \end{aligned} \]

Equations (1), taking into account (3), (4), (5), and (6), are differential equations with periodic coefficients, whose solution in general form is impossible. In order to simplify the equations under consideration and eliminate the periodic coefficients, coordinate transformations [3, 5] are used. The elimination of periodic coefficients in the equations of synchronous reluctance geared motors is made possible by the choice of a new coordinate system rigidly connected with the magnetic permeance of the air gap. The axis of maximum magnetic permeance of the air gap is taken as the longitudinal axis \(d\), while the transverse axis \(q\) leads the longitudinal axis by 90 electrical degrees and coincides with the axis of minimum magnetic permeance of the air gap.

In transforming the equations, first a transition is made from the actual variables in the stator axes \(a,b,c\) to new variables in the axes \(\alpha,\beta\). The relation between the variables is defined by expressions of the form:

\[ i_\alpha=\frac{2}{3}\left[i_a-\frac{1}{2}(i_b+i_c)\right]; \]

\[ i_\beta=\frac{i_b-i_c}{\sqrt{3}}. \]

Then a transition is made from the variables in the axes \(\alpha,\beta\) to variables in the axes \(d,q\), rotating at the speed \(\kappa_p\omega_p\):

\[ i_d=i_\alpha\cos\gamma+i_\beta\sin\gamma; \]

\[ i_q=-i_\alpha\sin\gamma+i_\beta\cos\gamma. \]

The transition from the axes \(pd,pq\) of the equivalent rotor windings to the axes \(d,q\) is carried out directly with the aid of the expressions:

\[ i_{rd}=i_{pd}\cos(\gamma-\gamma_p)+i_{pq}\sin(\gamma-\gamma_p); \]

\[ i_{rq}=-i_{pd}\sin(\gamma-\gamma_p)+i_{pq}\cos(\gamma-\gamma_p). \]

Currents, voltages, and flux linkages of the windings are subjected to transformations of this kind. As a result of these transformations, a system of differential equations with constant coefficients is obtained:

\[ \begin{aligned} U_d &= p\Psi_d - \Psi_q \cdot p\gamma + r i_d;\\ U_q &= p\Psi_q + \Psi_d p\gamma + r i_q;\\ 0 &= p\Psi_{rd} - \Psi_{rq} p(\gamma-\gamma_p) + r_r i_{rd};\\ 0 &= p\Psi_{rq} + \Psi_{rd} p(\gamma-\gamma_p) + r_r i_{rq}, \end{aligned} \tag{7} \]

where

\[ \begin{aligned} \Psi_d &= L_d i_d + M'_{rd} i_{rd};\\ \Psi_q &= L_q i_q + M'_{rq} i_{rq};\\ \Psi_{rd} &= M_{rd} i_d + L_{rd} i_{rd};\\ \Psi_{rq} &= M_{rq} i_q + L_{rq} i_{rq}; \end{aligned} \]

\[ L_d = L_0 - M_0 + \frac{3}{2}L_2; \qquad L_q = L_0 - M_0 - \frac{3}{2}L_2; \]

\[ M'_{rd} = M_{02} + M_{22}; \qquad M'_{rq} = M_{02} - M_{22}; \]

\[ M_{rd} = \frac{3}{2}(M_{02}+M_{22}); \qquad M_{rq} = \frac{3}{2}(M_{02}-M_{22}); \]

\[ L_{rd} = L_{p0}+L_{p2}; \qquad L_{rq} = L_{p0}-L_{p2}. \]

To investigate electromechanical transient processes, the system of equations (7) must be supplemented by the equation of motion of the rotor:

\[ I p\omega_p = M - M_c. \tag{8} \]

To simplify the investigation of the equations, all quantities must be expressed in relative units. As base quantities it is expedient to choose the generally accepted ones:

1) The amplitude value of the rated phase voltage of the stator winding

\[ U_b = U_{mn}. \]

2) The amplitude value of the rated phase current of the stator

\[ I_b = I_{mn}. \]

3) The synchronous angular frequency

\[ \omega_b = 2\pi f_n. \]

4) The rated power of the motor

\[ P_b = 3U_n I_n = \frac{3}{2}U_b \cdot I_b. \]

5) The torque

\[ M_b = \frac{P_b \cdot k_p}{\omega_b}. \]

6) The time corresponding to the rotation of the conductance, at the base angular frequency, through 1 electrical radian,

\[ t_b = \frac{1}{\omega_b} = \frac{1}{2\pi f_n}\ \text{el. sec.} \]

In order not to choose, in calculations, different systems of base quantities for the stator and rotor windings, it is necessary to refer the rotor windings to the stator windings. It is expedient to determine the reduction coefficients on the basis of equality of the fundamental harmonics of the m.m.f. of the reduced and equivalent windings. Equating the amplitudes of the first harmonics

the m.m.f. of the stator winding and of the equivalent rotor winding, one can determine the current reduction coefficients:

\[ m_{Jd}=m_{Jq}=\frac{m}{2}\frac{w_1 k_{\mathrm{w}1}}{2p w_{\mathrm{e}}}. \]

In the transformed rotor equations, as a result of the linear transformations the mutual inductances are increased by a factor of \(3/2\) and therefore become nonreciprocal. In order for the mutual inductances to become reciprocal, it is necessary to choose the voltage reduction coefficients equal to

\[ m_{Ud}=m_{Uq}=\frac{2}{m}m_{Jd}. \]

After substituting the reduced values of the currents into the voltage equations of the rotor windings and multiplying the equations by the voltage reduction coefficient, the active resistances of the equivalent windings and the inductances take the form:

\[ R'_r=m_U\cdot m_J\cdot r_r;\qquad L'_r=m_U\cdot m_J\cdot L_r. \]

It should be noted that writing the system of equations in relative units does not differ from the system of equations (7), if it is borne in mind that all quantities in these equations are expressed in relative units.

The expression for the electromagnetic torque of the motor can be obtained by considering the power balance. The power supplied to the stator winding, in relative units, is equal to:

\[ P=\frac{2}{3}(U_a i_a+U_b i_b+U_c i_c). \tag{9} \]

Transforming the phase quantities of currents and voltages through their components, one can obtain the power in the \(d, q\) axes:

\[ P=U_d i_q+U_q i_d. \]

After substituting the values of \(U_d\) and \(U_q\) from (7), this formula takes the form:

\[ P=(i_d p\Psi_d+i_q p\Psi_q)+p\gamma(i_q\Psi_d-i_d\Psi_q)+r(i_d^2+i_q^2). \tag{10} \]

The power consumed by the motor from the network is equal to the sum of the power going to change the magnetic energy of the machine, the electromagnetic power transmitted through the air gap, and the power expended on heating the stator windings.

The electromagnetic power \(P_{\mathrm{m}}\), transmitted through the air gap, is in turn expended on creating the electromagnetic torque of rotation and on losses in the rotor winding. If the losses in the rotor winding are subtracted from \(P_{\mathrm{m}}\) and the remaining mechanical power is divided by the angular velocity of the rotor, then, after transformations, we obtain the formula for the electromagnetic torque of rotation in relative units:

\[ M=(\Psi_d i_q-i_d\Psi_q)-\frac{k_p-1}{k_p}(\Psi_{rq} i_{rd}-\Psi_{rd} i_{rq}). \tag{11} \]

In the investigation of electromechanical transient processes, various forms of writing the complete system of equations are possible. However, for the convenience of solving it on analog computing machines, it is necessary to express certain constant and variable quantities in the form in which they can be measured experimentally on real machines, and in the form in which they can be most simply calculated

taken into account. Therefore, in the equations the derivatives of the angles must be replaced by the rotor speed:

\[ p\gamma = \kappa_p \omega_p; \]

\[ p(\gamma-\gamma_p)=(\kappa_p-1)\omega_p. \]

It is expedient to express the stator and rotor currents through the flux linkages and the parameters of the motor windings:

\[ x_d=x_{ad}+x_{ss}; \quad x_{rd}=x_{ad}+x_{sr}; \]

\[ x_q=x_{aq}+x_{ss}; \quad x_{rq}=x_{aq}+x_{sr}, \]

where \(x_{ad}, x_{aq}\) are the mutual-induction reactances of the stator windings;
\(x_{ss}, x_{sr}\) are the leakage reactances of the stator and rotor windings.

To determine the effect of the load, as well as the nature of the speed oscillations in the synchronous mode, it is necessary to introduce, as in ordinary synchronous machines, the angle \(\Theta\). Since the selected coordinate system \(d, q\) rotates with speed \(\kappa_p \omega_p\), and the vector of the voltage applied to the stator winding rotates with a constant synchronous speed, the position of the magnetic conductance of the air gap is determined by the angle \(\Theta\) between the \(q\) axis and the resultant voltage vector:

\[ \Theta=(\omega_1-\kappa_p \omega_p)t. \tag{12} \]

Then the components of the voltage along the \(d, q\) axes, expressed as functions of the angle \(\Theta\), take the form

\[ U_d=-U\sin\Theta; \quad U_q=U\cos\Theta. \tag{13} \]

Taking these transformations into account, we obtain the system of equations most convenient for solution on analog computers:

\[ p\Psi_d=-U\sin\Theta+\kappa_p\omega_p\Psi_q-r\cdot i_d; \]

\[ p\Psi_q=U\cos\Theta-\kappa_p\omega_p\Psi_d-r\cdot i_q; \]

\[ p\Psi_{rd}=(\kappa_p-1)\omega_p\Psi_{rq}-r_r i_{rd}; \]

\[ p\Psi_{rq}=-(\kappa_p-1)\omega_p\Psi_{rd}-r_r i_{rq}; \]

\[ i_d=\frac{x_{rd}}{x_d x_{rd}-x_{ad}^{2}}\Psi_d-\frac{x_{ad}}{x_d x_{rd}-x_{ad}^{2}}\Psi_{rd}; \tag{14} \]

\[ i_q=\frac{x_q}{x_q x_{rq}-x_{aq}^{2}}\Psi_q-\frac{x_{aq}}{x_q x_{rq}-x_{aq}^{2}}\Psi_{rq}; \]

\[ i_{rd}=-\frac{x_{ad}}{x_d x_{rd}-x_{ad}^{2}}\Psi_d+\frac{x_d}{x_d x_{rd}-x_{ad}^{2}}\Psi_{rd}; \]

\[ i_{rq}=-\frac{x_{aq}}{x_q x_{rq}-x_{aq}^{2}}\Psi_q+\frac{x_q}{x_q x_{rq}-x_{aq}^{2}}\Psi_{rq}; \]

\[ H_p\omega_p=(\Psi_d i_q-\Psi_q i_d)-\frac{\kappa_p-1}{\kappa_p}(\Psi_{rq}i_{rd}-\Psi_{rd}i_{rq})-M_c; \]

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

where

\[ H=\frac{I\omega^3}{P_b\kappa_p^2} \]

is the mechanical constant of the rotating masses.

This system of equations makes it possible to study the influence of parameters and load on various operating modes of a synchronous reluctance geared motor. For example, by varying over a wide-

within which the value of the active resistance of the short-circuited rotor winding can be varied, it is possible to analyze the influence of \(r_r\) on the possibility and duration of starting, on the magnitude of the starting currents and torque, and on the nature of oscillations of the rotor speed both during starting and in the synchronous mode.

References

1. P. Yu. Kaasik. Equations for investigating the operating modes of low-speed gearless electric machines. In: Low-Power Electric Motors. Publishing House of the Academy of Sciences of the USSR, Leningrad, 1971.

2. P. Yu. Kaasik, Yu. F. Kokunov, A. A. Pukhov. On the theory of subsynchronous reluctance motors. In: Low-Power Electric Machines. Leningrad, 1970.

3. A. I. Vazhhnov. Fundamentals of the Theory of Transient Processes of a Synchronous Machine. Gosenergoizdat, 1960.

4. P. Yu. Kaasik, B. V. Sidelnikov. A method of mathematical analysis of nonstationary modes of subsynchronous reluctance motors. Proceedings of Higher Educational Institutions of the Lithuanian SSR, Electronics and Automation, VI, 1970.

5. L. N. Gruzov. Methods of Mathematical Investigation of Electric Machines. Gosenergoizdat, 1953.

6. A. S. Kurakin. A short-circuited cage as a starting means for synchronous reduction motors. Electromechanics, 1969, No. 1.

7. E. V. Kononenko. Synchronous Reluctance Machines. Energia Publishing House, 1970.