PARAMETERS OF EQUIVALENT ROTOR WINDINGS OF SYNCHRONOUS RELUCTANCE MACHINES

E. V. Kononenko

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

To simplify analytical investigations of starting and transient processes, the actual rotor windings along the direct and quadrature axes of synchronous reluctance machines are replaced by two equivalent windings [1]. In order for such a replacement to be possible, it is necessary to find a method for determining the parameters of the equivalent windings from the parameters of the actual rotor windings.

In the design and calculation of various operating modes of synchronous machines, semi-empirical formulas are usually used to determine the parameters of equivalent damper windings [2, etc.]. Recently, more rigorously substantiated expressions have been obtained for these resistances [3, 4, 5]. Since, structurally, the rotor of synchronous reluctance machines differs substantially from that of ordinary synchronous machines, the question of the parameters of the equivalent rotor windings must be considered separately.

Taking into account the equality of the mutual-inductance resistances between the equivalent rotor windings and the stator windings [1], when determining the parameters of the equivalent rotor windings it is sufficient to find formulas that make it possible to calculate the active resistances and leakage reactances of these windings. When replacing the actual rotor windings by equivalent ones, it is necessary to proceed from the fact that the energy processes in the machine remain unchanged. As a first approximation, we take the distribution of the current components in the bars of the rotor winding along the \(d\) and \(q\) axes to be sinusoidal [1, 3, 4].

A. Active Resistances

In determining the active resistances, we proceed from the equality of the losses in the equivalent and actual rotor windings. If we denote the active resistances of the equivalent rotor windings along the direct and quadrature axes of the rotor respectively by \(R_{rd}\) and \(R_{rq}\), then the losses in these windings are determined by the equations

\[ p_{ed} = \frac{1}{2} I_{rd}^{2} R_{rd}, \tag{1} \]

\[ p_{eq} = \frac{1}{2} I_{rq}^{2} R_{rq}, \tag{2} \]

where \(I_{rd}\), \(I_{rq}\) are the maximum values of the currents in the equivalent windings.

Losses in the actual rotor winding due to the current components along the direct axis are determined as the sum of the losses in the individual bars. Taking into account the sinusoidal distribution of currents along the rotor surface (Fig. 1), the losses in any bar can be determined as

\[ p_{kd}=\frac{1}{2} I_{kd}^{2} R_{k\mathrm{sc}}, \tag{3} \]

where \(I_{rq}\sin \dfrac{\pi}{\tau}\dfrac{\tau_k}{2}=I_{kd}\) is the maximum value of the direct-axis component of the current in the \(k\)-th bar;

\(R_{k\mathrm{sc}}\) is the active resistance of the \(k\)-th bar, taking into account the influence of the short-circuiting rings.

Fig. 1. Diagram of the rotor winding of a synchronous reluctance machine with an odd number of bars per pole.

The total losses in the rotor winding are equal to:

\[ p_{rd}=2p\left[ R_{1\mathrm{sc}}\sin^{2}\frac{\pi}{\tau}\frac{\tau_1}{2} + R_{2\mathrm{sc}}\sin^{2}\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+ R_{k\mathrm{sc}}\sin^{2}\frac{\pi}{\tau}\frac{\tau_k}{2} +\ldots+ \frac{1}{2}R_{Q\mathrm{sc}} \right] I_{rd}^{2}. \tag{4} \]

Equating the left-hand sides of equations (1) and (4), we find the expression for the active resistance of the equivalent winding along the direct axis:

\[ R_{rd}=4p\left( R_{1\mathrm{sc}}\sin^{2}\frac{\pi}{\tau}\frac{\tau_1}{2} + R_{2\mathrm{sc}}\sin^{2}\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+ R_{k\mathrm{sc}}\sin^{2}\frac{\pi}{\tau}\frac{\tau_k}{2} +\ldots+ \frac{1}{2}R_{Q\mathrm{sc}} \right). \tag{5} \]

In an analogous manner one can determine the active resistance of the equivalent rotor winding along the quadrature axis. The losses in any bar due to the current component along the quadrature axis are determined by the expression

\[ p_{kq}=\frac{1}{2} I_{kq}^{2} R_{k\mathrm{sc}}, \tag{6} \]

where \(I_{kq}=I_{rq}\cos \dfrac{\pi}{\tau}\dfrac{\tau_k}{2}\) is the maximum value of the transverse component of the current in the \(k\)-th bar.

The losses in the real rotor winding due to the transverse components of the currents in the bars are determined as the sum of the losses in the individual bars (Fig. 1):

\[ p_{rq}=2p\left(R_{1ck}\cos^2\frac{\pi}{\tau}\frac{\tau_1}{2} +R_{2ck}\cos^2\frac{\pi}{\tau}\frac{\tau_2}{2}+\ldots+\right. \]

\[ \left. +R_{kck}\cos^2\frac{\pi}{\tau}\frac{\tau_k}{2} +\ldots+\frac{1}{2}R_{ock}\right)I_{rq}^{2}. \tag{7} \]

Equating the left-hand sides of equations (2) and (7), we find the expression for the active resistance of the equivalent winding along the transverse axis

\[ R_{rq}=4p\left(R_{1ck}\cos^2\frac{\pi}{\tau}\frac{\tau_1}{2} +R_{2ck}\cos^2\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+R_{kck}\cos^2\frac{\pi}{\tau}\frac{\tau_k}{2} +\ldots+\right. \]

\[ \left. +\frac{1}{2}R_{ock}\right). \tag{8} \]

When there is an even number of bars per pole, the term \(\dfrac{1}{2}R_{ock}\) in equation (8) is absent. Equations (5) and (8) make it possible to calculate the active resistances of the equivalent rotor windings in the case when bars of different configuration are present on the poles (as, for example, in an improved rotor). In ordinary synchronous reluctance machines, identical bars are usually placed on the poles. In this case it may be assumed that

\[ R_{ock}=R_{1ck}=R_{2ck}=\ldots=R_{kck}=R_{ck}. \tag{9} \]

Then equations (5) and (8) can be considerably simplified. Equation (5) may be written in the form

\[ R_{rd}=4p\left(R_{ck}A_d+\frac{1}{2}R_{Qck}\right), \tag{10} \]

where

\[ A_d=\sin^2\frac{\pi}{\tau}\frac{\tau_1}{2} +\sin^2\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+\sin^2\frac{\pi}{\tau}\frac{\tau_k}{2}+\ldots \tag{11} \]

If the numbers of short-circuited circuits and bars on the pole shoe are respectively \(n\) and \(n_c\), and the angle between two adjacent bars is \(\alpha_c\), then, for both even and odd \(n_c\), equation (11) may be represented in the form

\[ A_d=\frac{n_c}{4}-\frac{\sin n_c\alpha_c}{4\sin\alpha_c}. \tag{12} \]

Then equation (10) will have the form

\[ R_{rq}=pR_{ck}\left(n_c-\frac{\sin n_c\alpha_c}{\sin\alpha_c}\right)+2pR_{Qck}. \tag{13} \]

Equation (8) may be written in the form

\[ R_{rd}=4pR_{ok}A_q, \tag{14} \]

\[ A_q=\frac{1}{2} +\cos^2\frac{\pi}{\tau}\frac{\tau_1}{2} +\cos^2\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+\cos^2\frac{\pi}{\tau}\frac{\tau_n}{2}. \tag{15} \]

For an odd number of bars on the pole shoe, \(n_c=2n+1\);

\[ \frac{\pi}{\tau}\frac{\tau_n}{2}=2n\alpha_c \]

equation (15) will be equal to:

\[ A_q=\frac{n_c}{4}+\frac{\sin n_c\alpha_c}{4\sin\alpha_c}. \tag{16} \]

Then equation (14) will take the form:

\[ R_{rq}=pR_{ck}\left(n_c+\frac{\sin n_c\alpha_c}{\sin\alpha_c}\right). \tag{17} \]

It is not difficult to prove that, for an even number of bars per pole, equation (17) does not change.

Equations (13), (17) make it possible to calculate the active resistances of the equivalent windings of a conventional rotor with a uniform distribution of identical bars on the pole shoe. For an improved rotor, or with a nonuniform distribution of bars on the pole shoe, it is necessary to use equations (5) and (8).

In order to find the values of the active resistances of the equivalent rotor windings referred to the stator winding, it is necessary to multiply \(R_{rd}\) and \(R_{rq}\) by the resistance reduction coefficients, equal to \(m_z=m_I m_u\) [1].

\[ \begin{aligned} r_{rd}&=R_{rd}\cdot m_{Id}\cdot m_{ud},\\ r_{rq}&=R_{rq}\cdot m_{Iq}\cdot m_{uq}. \end{aligned} \tag{18} \]

B. Leakage inductive reactances

When determining the leakage inductive reactances, it is necessary to proceed from the condition of equality of the magnetic energy of the leakage fields in the equivalent and real rotor windings. If the leakage inductance of the equivalent rotor windings along the \(d\)- and \(q\)-axes is denoted respectively by \(L_{srd}\) and \(L_{srq}\), then the magnetic energies of the leakage fluxes in these windings are determined by the equalities

\[ A_{\mathrm{e}d}=\frac{1}{2}I_{rd}^{2}L_{srd}, \]

\[ A_{\mathrm{e}q}=\frac{1}{2}I_{rq}^{2}L_{srq}. \tag{19} \]

The magnetic energies from the leakage fields in the real rotor winding due to the direct- and quadrature-axis components of the currents are determined as sums of the energies in the individual bars. With a sinusoidal distribution of currents along the rotor surface, by analogy with equations (4) and (7), one may write

\[ A_{rd}=2p\left(L_{1ck}\sin^2\frac{\pi}{\tau}\frac{\tau_1}{2} +L_{2ck}\sin^2\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+ L_{kck}\sin^2\frac{\pi}{\tau}\frac{\tau_k}{2} +\ldots+\frac{1}{2}L_{Qck}\right)I_{rd}^{2}, \tag{20} \]

\[ A_{rq}=2p\left(L_{1ck}\cos^2\frac{\pi}{\tau}\frac{\tau_1}{2} +L_{2ck}\cos^2\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+ L_{kck}\sin^2\frac{\pi}{\tau}\frac{\tau_k}{2} +\ldots+\frac{1}{2}L_{Qck}\right)I_{rq}^{2}, \]

where \(L_{k\mathrm{sk}}\) is the inductance of the \(k\)-th bar, taking into account the influence of short-circuited bars.

Equating the left-hand sides of equations (19) and (20), we find expressions for the leakage inductances of the equivalent windings

\[ L_{srd}=4p\left(L_{1\mathrm{sk}}\sin^2\frac{\pi}{\tau}\frac{\tau_1}{2} +L_{2\mathrm{sk}}\sin^2\frac{\pi}{\tau}\frac{\tau_2}{2}+\ldots+\right. \]

\[ \left. +L_{k\mathrm{sk}}\sin^2\frac{\pi}{\tau}\frac{\tau_k}{2} +\ldots+\frac{1}{2}L_{Q\mathrm{sk}}\right), \tag{21} \]

\[ L_{srq}=4p\left(L_{1\mathrm{sk}}\cos^2\frac{\pi}{\tau}\frac{\tau_1}{2} +L_{2\mathrm{sk}}\cos^2\frac{\pi}{\tau}\frac{\tau_k}{2}+\ldots\right. \]

\[ \left. +L_{k\mathrm{sk}}\sin^2\frac{\pi}{\tau}\frac{\tau_2}{2} +\ldots+\frac{1}{2}L_{Q\mathrm{sk}}\right). \]

In the case where identical bars are located on the pole shoe, it may be assumed that

\[ L_{0\mathrm{sk}}=L_{1\mathrm{sk}}=L_{2\mathrm{sk}}=\ldots=L_{k\mathrm{sk}}=L_{\mathrm{sk}}. \tag{22} \]

Then equations (21), taking into account transformations (10)—(17), can be written in the form

\[ L_{srd}=pL_{\mathrm{sk}}\left(n_c-\frac{\sin n_c\alpha_c}{\sin\alpha_c}\right)+2pL_{Q\mathrm{sk}}, \tag{23} \]

\[ L_{srd}=pL_{\mathrm{sk}}\left(n_c+\frac{\sin n_c\alpha_c}{\sin\alpha_c}\right). \]

In order to find the values of the leakage inductive reactances of the equivalent rotor windings, referred to the stator winding, it is necessary to multiply \(L_{srd}\) and \(L_{srq}\) by the angular frequency of the stator current and by the resistance conversion factors:

\[ x_{srd}=2\pi f_1\cdot L_{srd}\cdot m_{Id}\cdot m_{ud}, \]

\[ x_{srq}=2\pi f_1\cdot L_{srq}\cdot m_{Iq}\cdot m_{uq}. \tag{24} \]

The formulas obtained in the present work also make it possible to take into account changes in the parameters of the equivalent rotor windings caused by the influence of the phenomenon of current displacement in the rotor bars. For this purpose, the values \(R_{k\mathrm{sk}}\) and \(L_{k\mathrm{sk}}\), determined with allowance for current displacement, must be substituted into the calculation formulas, as is usually done in the calculation of induction motors [2].

REFERENCES

1. E. V. Kononenko. Reduction of rotor windings of synchronous reluctance machines. — In the present collection.

2. I. M. Postnikov. Design of Electrical Machines, Gostekhizdat, Ukrainian SSR, 1960.

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

4. Ya. B. Danilevich, V. V. Dombrovskii, E. Ya. Kazovskii. Parameters of AC Electrical Machines, Nauka, 1965.

5. Ya. B. Danilevich, Yu. A. Kulik. Theory and Calculation of Damper Windings of Synchronous Machines, Publishing House of the Academy of Sciences of the USSR, 1962.