Proceedings of the Tomsk Polytechnic Institute named after S. M. Kirov, holder of the Order of the October Revolution and the Order of the Red Banner of Labor

Vol. 200　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　1974

Replacement of the Squirrel-Cage Rotor Winding of an Induction Motor with an Equivalent Two-Phase Winding

E. V. Kononenko

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

In the study of various transient operating conditions of induction motors, in cases where the task is not to determine the currents in the rotor bars, the squirrel-cage winding is usually replaced, when setting up the equations, by two equivalent windings. It is assumed here that the equivalent windings are concentrated and arranged along mutually perpendicular axes (longitudinal and transverse), and that the width of the winding coils is equal to the pole pitch.

However, the replacement of the squirrel-cage rotor winding of an induction motor by an equivalent two-phase winding has not received sufficiently complete treatment in the literature. Often, when considering transient processes of squirrel-cage induction motors, it is assumed that there is a symmetrical three-phase winding on the rotor, the parameters of which are referred to the stator winding. Such an approach makes it difficult to determine the relationship between the calculated parameters used in the theory of the steady-state regime and the parameters used in the theory of transient processes. In this connection, the present article considers the question of replacing the squirrel-cage rotor winding by an equivalent two-phase winding.

Since, in setting up the voltage equilibrium equations of induction machines, only the fundamental harmonic components are taken into account, then, when replacing real windings by equivalent ones, one should proceed from the equality of the fundamental harmonic magnetomotive forces (m.m.f.) of the real and equivalent windings. To determine the m.m.f. produced by the real rotor windings, it is necessary to know the spatial distribution of the currents in the bars of the winding. As a first approximation, the distribution of the current components along the \(d\) and \(q\) axes in the bars of the rotor winding is taken to be sinusoidal. Equating the amplitudes of the fundamental harmonic m.m.f.’s produced by the longitudinal components of the currents in the bars and by the equivalent winding along the longitudinal axis, obtained in the same way as in synchronous machines [1], we find that the number of turns of the equivalent rotor winding along the longitudinal axis per pole will be equal to

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

where \(\alpha_c\) is the skew angle of the rotor slots in electrical radians;
\(n_c\) is the number of rotor slots per pole.

Since the slots of the squirrel-cage winding are distributed uniformly around the circumference of the rotor, we have

\[ \alpha_c=\frac{\pi}{n_c};\qquad n_c=\frac{Z_2}{2p}. \]

It is not difficult to see that, owing to the symmetry of the rotor, the numbers of turns of the equivalent windings along the longitudinal and transverse axes are identical and will be equal to

\[ W_{\mathrm{e}}=W_{\mathrm{e}d}=W_{\mathrm{e}q}=\frac{Z_2}{8p}. \tag{2} \]

In order to be able, in analytical investigations, to use the equations of the equivalent rotor windings, it is necessary to determine the parameters of these windings. In doing so, it is sufficient to find their active resistances and leakage inductive reactances, since the mutual-induction reactances between the equivalent rotor windings and the stator windings are equal to the mutual-induction reactances between the real windings. In replacing the real windings it was assumed that the energy relations in the machine must remain unchanged.

In induction motors with a squirrel-cage rotor, the active resistances and leakage inductive reactances of the equivalent windings can be calculated by the same formulas as in synchronous machines [1]. Since in induction motors the equivalent windings along the longitudinal and transverse axes are identical, one may restrict the calculation of parameters to the winding along the longitudinal axis alone.

However, the calculation formulas for induction motors can be simplified. In the transformations, let us first assume that the active resistance of the elements of the short-circuiting ring may be neglected. Since the resistance of all bars is the same, the active resistance of the equivalent winding can be represented in the form

\[ R_r=4p\cdot R_c\left(\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_1}{2}+\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_2}{2}+\ldots+ \right. \]

\[ \left. +\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}+\ldots\right) =pR_c\left(n_c-\frac{\sin n_c\alpha_c}{\sin\alpha_c}\right) \tag{3} \]

or

\[ R_r=R_c\cdot\frac{Z_2}{2}, \tag{4} \]

where \(R_c\) is the active resistance of a bar.

The active resistance of the equivalent rotor windings, referred to the stator winding, is equal to

\[ r_2=R_r\cdot m_i m_u, \tag{5} \]

where the current transformation coefficient \(m_i\) and voltage transformation coefficient \(m_u\) of the induction motor, in accordance with the general laws for referring windings of electrical machines, can be represented in the form [2]

\[ m_i=\frac{m_1}{2}\cdot\frac{W_1\cdot K_{\mathrm{ob}1}}{2p\cdot W_{\mathrm{e}}} =\frac{2m_1\cdot W_1\cdot K_{\mathrm{ob}1}}{Z_2}, \tag{6} \]

\[ m_u=\frac{2}{m_1}\cdot m_i =\frac{4W_1\cdot K_{\mathrm{ob}1}}{Z_2}. \tag{7} \]

Similarly, one can find the calculation formulas for the leakage inductance of the equivalent winding. If, in the transformations, one neglects

by the leakage of the elements of the short-circuiting ring, then the leakage inductance of the equivalent winding can be represented in the form

\[ L_{\sigma r}=4p\cdot L_c\left(\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_1}{2} +\sin^2\frac{\pi}{\tau}\cdot\frac{\tau_2}{2}+\ldots+ \sin^2\frac{\pi}{\tau}\cdot\frac{\tau_k}{2}+\ldots\right) =pL_c\left(n_c-\frac{\sin n_c\alpha_c}{\sin \alpha_c}\right), \tag{8} \]

or

\[ L_{\sigma r}=L_c\cdot\frac{Z_2}{2}, \tag{9} \]

where \(L_c\) is the inductance of a bar of the rotor winding.

The leakage reactance of the equivalent windings of the rotor of an induction motor, referred to the stator winding, is equal to

\[ x_{\sigma r}=2\pi f\cdot L_{\sigma r}\cdot m_i\cdot m_u. \tag{10} \]

Substituting into (5) and (10) the values of the reduction coefficients \(m_i\) and \(m_u\), we obtain

\[ r_2=R_c\cdot\frac{Z_2}{2}\cdot \frac{8m_1\cdot (W_1\cdot K_{\mathrm{w}1})^2}{Z_2^2} = R_c\cdot\frac{4m_1(W_1\cdot K_{\mathrm{w}1})^2}{Z_2}, \tag{11} \]

\[ x_{\sigma r}=x_c\cdot\frac{Z_2}{2}\cdot \frac{8m_1\cdot (W_1\cdot K_{\mathrm{w}1})^2}{Z_2^2} = x_c\frac{4m_1(W_1\cdot K_{\mathrm{w}1})^2}{Z_2}, \tag{12} \]

where \(x_c=2\pi f_1L_{\sigma r}\) is the leakage reactance of a bar of the rotor winding.

Comparing expressions (11) and (12) with the well-known formulas for calculating the active resistance and the leakage reactance of a phase of the secondary winding of an induction motor with a squirrel-cage rotor, referred to the stator winding, under the same assumptions, we see that they are identical. Consequently, when using formulas (11) and (12), in order to improve the accuracy of calculating the resistances, \(R_c\) and \(x_c\) must be determined with allowance for the influence of the short-circuiting rings, as is customary in induction machines.

It follows from the results obtained that, when calculating the active resistance and leakage reactance of the equivalent rotor windings of an induction motor referred to the stator winding, one may use formulas known from the general theory of induction machines. However, it must be remembered that the resistance reduction coefficients \(m_z=m_i m_u\) of the two-phase equivalent windings (6) and (7) differ from the case considered in the general course, when in an induction motor with a squirrel-cage rotor the number of phases of the rotor winding is taken to be \(Z_2\), and the number of turns per phase is taken to be \(1/2\).

References

1. E. V. Kononenko. Synchronous Reactive Machines. Energiya Publishing House, 1970.

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