REDUCTION OF ROTOR WINDINGS OF SYNCHRONOUS RELUCTANCE MACHINES

E. V. Kononenko

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

In synchronous reluctance machines, for the purposes of asynchronous starting and for increasing the stability of operation, a special short-circuited winding is placed on the rotor. Usually the rotor winding is a cast aluminum squirrel cage. As a rule, not only the slots present in the rotor core are filled with aluminum, but also the interpole spaces. Two possible variants of the rotor design are shown in Fig. 1. When the theory of two reactions is applied, the rotor winding may be represented by two systems of short-circuited circuits whose magnetic axes coincide with the rotor axes \(d\) and \(q\).

Fig. 1. Rotor designs of synchronous reluctance machines: a) conventional rotor; b) improved rotor.

In studying starting and nonstationary transient processes, it is expedient, in order to simplify the investigations, to replace the real multicircuit rotor windings along the direct and quadrature axes of synchronous reluctance machines by two equivalent windings whose width is equal to the pole pitch. When replacing real rotor windings by equivalent ones, it must be borne in mind that such transformations should not affect the processes occurring in the machine.

Rotor windings are distributed. In analytical

in studies, usually only the fundamental harmonic components of the magnetomotive forces (m.m.f.) produced by distributed windings are taken into account. Accordingly, when replacing the actual rotor windings along the \(d\) and \(q\) axes by equivalent ones, we shall proceed from the equality of the fundamental harmonic m.m.f.’s produced by the windings. To determine the m.m.f.’s produced by the actual rotor windings, it is necessary to find the spatial distribution of the currents in the bars of the rotor winding. As a first approximation, the distribution of the current components along the \(d\) and \(q\) axes in the bars of the rotor winding may be assumed sinusoidal [1, 2]. Then the m.m.f.’s produced by the rotor windings can be determined on the basis of the diagram shown in Fig. 2.

Fig. 2. Diagram of the rotor winding of a synchronous reluctance machine with an even number of bars per pole.

The components of the currents in the bars of the rotor winding along the longitudinal axis are induced by the pulsating magnetic field produced by the m.m.f. of the stator winding \(F_d\). With a sinusoidal distribution of currents in the bars along the rotor surface, the maximum value of the currents \(I_{rd}\) will be in the bars displaced from the pole axis by a distance \(\frac{\tau}{2}\), i.e., in the bars located in the interpolar space. The current in any pair of bars displaced from the pole axis by a distance \(\frac{\tau_k}{2}\) and forming a short-circuited loop will be equal to

\[ I_{kd}=I_{rd}\sin\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}. \tag{1} \]

The amplitude of the fundamental harmonic m.m.f. produced by the currents flowing in the \(k\)-th pair of bars is determined by the expression

\[ F_{kd}=\frac{4}{\pi} I_{rd}\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}. \tag{2} \]

The amplitude of the fundamental harmonic m.m.f. produced by the direct-axis components of the currents in the bars, per pole, is determined as the sum

\[ F_{rd1}=F_{1d}+F_{2d}+\ldots+F_{kd}+\ldots+F_{Qd} =\frac{4}{\pi} I_{rd}\left(\sin^{2}\frac{\pi}{\tau}\frac{\tau_{1}}{2}+\right. \]

\[ \left. +\sin^{2}\frac{\pi}{\tau}\frac{\tau_{2}}{2}+\ldots+\sin^{2}\frac{\pi}{\tau}\frac{\tau_{k}}{2}\right) +\frac{2}{\pi}I_{rd}. \tag{3} \]

Let the number of short-circuited circuits on the pole shoe be equal to \(n\). In synchronous reluctance machines the bars on the pole shoe are distributed uniformly. Denote the angle between two adjacent bars by \(\alpha_c\), and the number of bars on the pole shoe by \(n_c\). In this case, irrespective of the number of bars on the pole (even or odd),

\[ \sin^{2}\frac{\pi}{\tau}\frac{\tau_{1}}{2} +\sin^{2}\frac{\pi}{\tau}\frac{\tau_{2}}{2} +\ldots+ \sin^{2}\frac{\pi}{\tau}\frac{\tau_{n}}{2} = \frac{n_c}{4}-\frac{\sin n_c\alpha_c}{4\sin\alpha_c}. \tag{4} \]

Then equation (3) can be represented in the form:

\[ F_{rd1}=\frac{4}{\pi}I_{rd} \left( \frac{n_c}{4} -\frac{\sin n_c\alpha_c}{4\sin\alpha_c} +\frac{1}{2} \right). \tag{5} \]

If it is assumed that, in the equivalent rotor winding along the \(d\)-axis, a current flows whose maximum value is equal to \(I_{rd}\), then the amplitude of the fundamental harmonic m.m.f. produced by this winding is determined by the expression

\[ F_{\mathrm{ed}1}=\frac{4}{\pi}I_{rd}\cdot W_{\mathrm{ed}}, \tag{6} \]

where \(W_{\mathrm{ed}}\) is the number of turns of the equivalent rotor winding.

Equating, in accordance with the adopted reduction conditions, the left-hand sides of equations (5) and (6), we find that

\[ W_{\mathrm{ed}}=\frac{1}{4} \left( n_c+2-\frac{\sin n_c\alpha_c}{\sin\alpha_c} \right). \tag{7} \]

Analogous transformations may also be performed for the rotor winding along the quadrature axis. The components of the currents in the bars of the rotor winding along the quadrature axis are induced by the pulsating magnetic field produced by the m.m.f. \(F_q\) (Fig. 2). With a sinusoidal distribution of currents in the bars of the rotor winding, the current in any pair of bars forming a short-circuited circuit along the \(q\)-axis is determined by the expression:

\[ I_{kq}=I_{rq}\cos\frac{\pi}{\tau}\frac{\tau_k}{2}. \tag{8} \]

For an even number of bars on the pole shoe, the amplitude of the fundamental harmonic m.m.f. produced by the quadrature-axis components of the currents in the bars, per pole, is determined as the sum of the first harmonics of the m.m.f. of the short-circuited circuits along the \(q\)-axis.

\[ F_{rq1}=F_{1q}+F_{2q}+\ldots+F_{kq}+\ldots+F_{nq}= \]

\[ =\frac{4}{\pi}I_{rq} \left( \cos^{2}\frac{\pi}{\tau}\frac{\tau_1}{2} +\cos^{2}\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+ \cos^{2}\frac{\pi}{\tau}\frac{\tau_n}{2} \right) = \]

\[ =\frac{4}{\pi}I_{rq} \left( \frac{n_c}{4} +\frac{\sin n_c\alpha_c}{4\sin\alpha_c} \right). \tag{9} \]

It can be shown that, with an odd number of bars per pole, the equation for the m.m.f. remains unchanged.

The amplitude of the fundamental harmonic produced by the equivalent rotor winding along the \(q\)-axis when the current \(I_{rq}\) flows through it is determined by the expression

\[ F_{\mathrm{e}q1}=\frac{4}{\pi} I_{rq} W_{\mathrm{e}q}, \tag{10} \]

where \(W_{\mathrm{e}q}\) is the number of turns of the equivalent rotor winding along the transverse axis.

Comparing equations (9) and (10), we find

\[ W_{\mathrm{e}q}=\frac{1}{4}\left(n_c+\frac{\sin n_c\alpha_c}{\sin \alpha_c}\right). \tag{11} \]

In synchronous reluctance machines the stator and rotor windings are electromagnetically coupled. In order not to choose different systems of base quantities for the stator and rotor windings in calculations, and also when drawing up equivalent circuits, it is necessary to refer the rotor windings to the stator windings. In referring the windings, one must be guided by the condition that the energy relations in the machine must not depend on the reference.

To refer the equivalent rotor windings to the stator windings, it is necessary to determine the reference coefficients for currents \(m_I\), voltages \(m_u\), and resistances \(m_r=m_I\cdot m_u\), which make it possible to recalculate the equivalent rotor windings to the number of turns and phases of the stator winding.

Taking into account that the distribution of the m.m.f. produced by the rotor windings is assumed to be sinusoidal, the coefficient \(m_I\) is determined from the equality of the fundamental harmonics of the m.m.f. of the referred and equivalent windings.

The maximum values of the currents in the equivalent rotor windings are equal to \(I_{rd}\) and \(I_{rq}\), and we shall denote their referred values by \(I'_{rd}\) and \(I'_{rq}\). Then the amplitudes of the first harmonics of the m.m.f. produced by the stator winding when the currents \(I'_{rd}\) and \(I'_{rq}\) flow through the winding will be equal to:

\[ F_{ad1}=\frac{m}{2}\frac{4}{\pi}\frac{W_1 k_{ob1}}{2p} I'_{rd}, \tag{12} \]

\[ F_{aq1}=\frac{m}{2}\frac{4}{\pi}\frac{W_1 k_{ob1}}{2p} I'_{rq}, \tag{13} \]

where \(W_1\), \(k_{ob1}\), \(m\) are the number of turns, the winding factor, and the number of phases of the stator winding;

\(p\) is the number of pole pairs.

Equating the left-hand sides of equations (6) and (12), (10) and (13), we obtain:

\[ m_{Id}=\frac{I_{rd}}{I'_{rd}}=\frac{m}{2}\frac{W_1 k_{ob1}}{2p W_{\mathrm{e}d}}, \tag{14} \]

\[ m_{Iq}=\frac{I_{rq}}{I'_{rq}}=\frac{m}{2}\frac{W_1 k_{ob1}}{2p W_{\mathrm{e}q}}. \tag{15} \]

The current reference coefficients (14), (15) are obtained from the condition of equality of the fundamental harmonics of the m.m.f. of the equivalent and referred rotor windings. It is not difficult to show that, if the m.m.f. produced by the rotor winding varies in space according to a sinusoidal law, the expressions for the reference coefficients (14) and (15), for the adopted

assumptions does not change if they are determined on the basis of equality of the fundamental harmonic inductions in the air gap, produced by the actual and the reduced rotor windings.

When determining the voltage reduction factors \(m_{ud}\), \(m_{uq}\), it is necessary to follow the general rules for transforming the windings of electrical machines [3].

In analytical studies of synchronous reluctance machines, the theory of \(2T = x\) reactions is used. When transforming the equations to the \(d\) and \(q\) axes, the power of the stator winding is reduced by a factor of \(\dfrac{m}{2}\), while the power of the rotor winding remains unchanged. In this connection, the mutual inductances between the phase windings of the stator and the rotor windings in the transformed equations become nonreciprocal. To restore reciprocity of the mutual inductances, the power of the reduced rotor windings must be reduced by a factor of \(\dfrac{m}{2}\).

The current reduction factors were obtained independently of the conditions considered above. In order that the mutual inductances in the equations of synchronous reluctance machines be reciprocal, the voltage reduction factors must be chosen equal to:

\[ m_{ud} = \frac{2}{m} m_{Id} = \frac{W_1 k_{\mathrm{ob}1}}{2p W_{\mathrm{Э}d}}, \tag{16} \]

\[ m_{uq} = \frac{2}{m} m_{Iq} = \frac{W_1 k_{\mathrm{ob}1}}{2p W_{\mathrm{Э}q}} \tag{17} \]

The mutual-induction reactance of the stator winding of synchronous reluctance machines along the \(d\)-axis can be determined from the equality

\[ x_{ad} = 2\pi f_1 \frac{\psi_{ad1}}{i_d}, \tag{18} \]

where \(\psi_{ad1}\) is the flux linkage produced by the fundamental harmonic of induction in the air gap that arises when the current \(i_d\) flows through the stator winding.

\[ \psi_{ad1} = W_1 k_{\mathrm{ob}1}\Phi_{ad1} = \frac{6}{\pi^2}\frac{(W_1 k_{\mathrm{ob}1})^2}{p}\,\tau l\, \frac{\mu_0 k_d}{k_\delta k_{\mu d}\delta}. \tag{19} \]

Then

\[ x_{ad} = \frac{4}{\pi} m f_1 \left(\frac{W_1 k_{\mathrm{ob}1}}{p}\right)^2 \tau l \frac{\mu_0 k_d}{k_\delta k_{\mu d}\delta}\,[\Omega]. \tag{20} \]

The mutual-induction reactance of the stator winding on the \(q\)-axis can, by analogy, be written in the form:

\[ x_{ad} = \frac{4}{\pi} m f_1 \frac{(W_1 k_{\mathrm{ob}1})^2}{p}\, \tau l \frac{\mu_0 k_d}{k_\delta k_{\mu d}\delta}\,[\Omega], \tag{21} \]

where \(k_d\), \(k_q\) are the field-form factors along the \(d\) and \(q\) axes;
\(k_{\mu d}\), \(k_{\mu q}\) are the saturation coefficients of the magnetic circuit along the \(d\) and \(q\) axes;
\(\delta\), \(\tau\), \(l\) are the air gap, pole pitch, and calculated length of the machine.

It is not difficult to prove that, with the reduction factors adopted above, the mutual-induction reactances between the equivalent wind-

rotor windings and stator windings along the \(d\) and \(q\) axes can be calculated by formulas (20) and (21), since

\[ x_{ard}=x_{ad}, \qquad x_{arq}=x_{aq}. \tag{22} \]

The formulas presented in this paper make it possible to reduce the actual rotor windings along the direct and quadrature axes to equivalent stator windings.

REFERENCES

1. M. E. Telat. A new approach to determining the inductive reactances of a synchronous machine, GEI, 1959.

2. Ya. B. Danilevich, V. V. Dobrovolsky, E. Ya. Kazovsky. Parameters of AC Electrical Machines, Nauka, 1965.

3. I. I. Talalov. Transformation of circuits in the analysis of electrical machines, Elektrichestvo, No. 4, 1961.