ON DETERMINING THE PARAMETERS OF THE EQUIVALENT CIRCUIT OF AN INDUCTION MOTOR WITH INDUCTIVE REACTANCE IN THE ROTOR CIRCUIT

V. G. Vlasov

(Recommended by the scientific seminar of the departments of electrical machines and general electrical engineering)

Induction motors with inductive reactance (IR) in the rotor circuit, owing to the inclusion in the rotor circuit of a nonlinear frequency-dependent resistance that changes its magnitude with the change in the rotor-current frequency, and to the removal of the greater part of the slip losses from the volume of the machine, possess, in comparison with normal squirrel-cage motors, an increased continuously permissible torque at low rotational speeds [1].

The inductive reactance is a transformer whose primary winding is a coil placed inside a massive steel disk. The induction disks are mounted on the motor shaft, and the coils are connected to the terminals of the rotor phase winding without sliding contact.

One of the fundamental questions in the theory of an induction machine with IR in the rotor circuit is the determination of the equivalent electrical parameters of the massive disks, i.e., the active and reactive resistances referred to the stator winding and making it possible, by using an equivalent circuit, to carry out a complete calculation of the machine with determination of its principal characteristics.

During operation of an induction motor with IR, complex electromagnetic processes take place in the massive disks, caused by changes in the magnitude and frequency of the rotor current.

The electromagnetic wave created by the magnetomotive force of the IR winding penetrates into the thickness of the massive disk and produces eddy currents in it, which are concentrated on the inner surface of the disk facing the coil.

In considering the processes associated with the penetration of an electromagnetic wave into a ferromagnetic body, the starting equations are Maxwell’s equations without allowance for displacement currents:

\[ \operatorname{rot}\mathbf{H}=\gamma \mathbf{E} \tag{1} \]

\[ \operatorname{rot}\mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}, \tag{2} \]

where

\[ \begin{aligned} \mathbf{H} &\text{ — magnetic-field intensity;}\\ \gamma &\text{ — specific electrical conductivity;}\\ \mathbf{E} &\text{ — electric-field intensity;}\\ \mathbf{B}=\mu\mu_0\mathbf{H} &\text{ — magnetic induction.} \end{aligned} \]

\(\mu\) is the relative magnetic permeability;
\(\mu_0\) is the magnetic constant.

Solving equations (1) and (2) is possible only under the assumption that the electrical \((\gamma)\) and magnetic \((\mu)\) properties of the medium do not depend on the intensities of the electric and magnetic fields. In reality, however, the magnetic permeability of the steel magnetic cores of an IS depends strongly on the magnetic-field intensity, whose magnitude decreases with distance from the surface of the magnetic core into its interior.

Therefore, if the amplitude of the field intensity on the disk surface has a sufficiently large value, then \(\mu\) at first increases with distance from the surface, reaches a maximum value, and then decreases.

However, the problem of determining the equivalent parameters of an IS can be solved in a direct way, using the concept of the complex magnetic resistance of a solid magnetic core [2].

The concept and definition of the complex magnetic resistance of a solid magnetic core subjected to the action of a periodically varying magnetizing force is based in the work of L. R. Neiman on an approximate analytical solution of the process of propagation of an electromagnetic wave in ferromagnetic bodies.

Considering the complex magnetic permeability as a definite function depending only on the geometric coordinate \(x\), which determines the distance from the surface of the solid core, for the case of a well-pronounced surface effect, L. R. Neiman succeeded in finding a function \(\mu=\varphi(x)\) which made it possible to solve equations (1) and (2) and at the same time determined a relation \(\mu=f(H)\), very close to the actual magnetization curve, expressed analytically by him in the form

\[ B=K_1\cdot H^{\frac{1}{m}}, \tag{3} \]

where \(K_1\) is a coefficient depending on the material;
\(m\) is the order of the parabola.

As a result, values were obtained for the amplitudes of the first harmonics of the electric- and magnetic-field intensities, as well as the period-average value of the Poynting vector as functions of the coordinate \(x\).

A second important conclusion from the work of L. R. Neiman is the assertion that the modulus of the complex magnetic permeability differs very little from the magnetic permeability determined from the fundamental magnetization curve.

This circumstance makes it possible, from the known rms value of the current in the inductor (in our case, in the IS coil) and, consequently, from the known rms value of the magnetic-field intensity on the disk surface, to determine the modulus of the complex magnetic permeability at the surface, using the fundamental magnetization curve.

As shown in the work of L. R. Neiman, the solution of equations (1) and (2), with the adopted law of variation \(\mu=\varphi(x)\), describes quite accurately the phenomenon of propagation of an electromagnetic wave in a solid ferromagnetic body throughout the entire region where the magnetic-field intensity has a significant magnitude.

According to the theory [3], between the complex magnetic resistance of the core \((Z_m)\) and its total equivalent electrical resistance \((\dot Z)\), referred to a winding wound on the solid core, there is the relation

\[ \dot Z=j\frac{1}{Z_m}\cdot \omega\cdot W_{\mathrm{IS}}^2, \tag{4} \]

where \(\omega\) is the angular frequency,
\(W_{\mathrm{ic}}\) is the number of turns in the ID winding.

According to the data of [2], the complex magnetic reluctance of a massive magnetic core acted upon by an alternating m.m.f. is determined by the expression

\[ Z_m=(0.45+j0.74)\cdot \frac{l_\mu}{U_\mu}\sqrt{\frac{\omega\cdot \gamma}{\mu\cdot \mu_0}}, \tag{5} \]

where \(l_\mu\) is the length of the magnetic core, equal for the ID to the perimeter of the cross section of the inner cavity of the core;

\(U_\mu\) is the perimeter of the cross section of the magnetic core. For the ID

\[ U_\mu=\pi D_{\mathrm{av}}, \]

where \(D_{\mathrm{av}}\) is the mean diameter of the inner cavity of the core.

Equation (5) takes into account the nonconstancy of the magnetic permeability of the material as the wave advances into the depth of the core, and the eddy-current and hysteresis losses. It follows from (5) that the ratio of the components of the complex magnetic reluctance is constant and equal to:

\[ \frac{R_m}{X_m}=0.6 . \]

On the basis of (4) and (5), one can write expressions for the total equivalent electrical impedance of the magnetic core of the induction disks:

\[ Z=(2.14+j1.3)\frac{U_\mu}{l_\mu}\sqrt{\frac{\mu\cdot \mu_0\cdot f_2}{\gamma}}\,W_{\mathrm{ic}}^2, \tag{6} \]

where \(f_2\) is the rotor-current frequency.

The total equivalent electrical impedance of the magnetic core of the induction disks, referred to the stator winding, in accordance with the equivalent circuit of an asynchronous machine with ID, taking into account that \(f_2=f_1\cdot s\), will be determined as

\[ Z'_{\mathrm{ic}}=\frac{Z_{\mathrm{ic}}\cdot o}{\sqrt{s}}, \tag{7} \]

where

\[ Z_{\mathrm{ic}}\cdot o=K\cdot K_{\mathrm{r}}\cdot W_{\mathrm{ic}}^2 . \tag{8} \]

Here

\[ K=(2.14+j1.3)\frac{U_\mu}{l_\mu}\cdot \sqrt{\frac{\mu\cdot \mu_0\cdot f_1}{\gamma}} . \]

\[ K_{\mathrm{r}}=\frac{m_1}{m_2}\left(\frac{W_1\cdot K_{\mathrm{w}1}}{W_2\cdot K_{\mathrm{w}2}}\right)^2 \]

is the reduction factor;

\(f_1\) is the stator-current frequency.

For convenience of calculation and to make it possible to determine the equivalent parameters of induction disks differing in geometric dimensions and in the number of turns in the coil, we introduce the concept of the unit electrical impedance of a massive core.

If we assume that

\[ \frac{U_\mu}{l_\mu}=1 \]

and \(f_1=50\) Hz, then the unit impedance of the massive magnetic core, referred to one turn, will be equal to:

\[ Z_{\mathrm{ic}.(1)}=17.7\sqrt{\frac{\mu\cdot \mu_0}{\gamma}} . \tag{9} \]

Substituting into (9) the value of \(\mu\) from (3), as

\[ \mu = K_1 \cdot H^{\frac{1-m}{m}} \]

and, taking into account that

\[ \mu_0 = 4\pi \cdot 10^{-7}\ \frac{\mathrm{H}}{\mathrm{m}} \quad \text{and} \quad \gamma = 6.5 \cdot 10^6\ \frac{1}{\Omega \cdot \mathrm{m}} \]

(for a temperature of \(100^\circ\mathrm{C}\)), we obtain

\[ Z_{\mathrm{ir}(1)} \approx 8.5 \cdot 10^{-2} \cdot H^{-\frac{3}{7}}, \tag{10} \]

where \(H\) is the magnetic-field strength in oersteds, for \(K_1 = 1100\) and \(m = 7\).

The analytical curve \(Z_{\mathrm{ir}(1)} = f(H)\) coincides very closely with the experimental curve constructed on the basis of processing a large number of experimental data (Fig. 1).

Fig. 1. Dependence of \(Z_{\mathrm{ir}}(1)\) on the specific ampere-turns of the inductive resistance.

For convenience of application in practical calculations, it is advisable to reconstruct the curve \(Z_{\mathrm{ir}(1)}\) as a function of the specific ampere-turns \(aW\), taking into account that

\[ aW = 0.4\pi \cdot H, \]

where

\[ aW = \frac{l_{\mathrm{ir}} W_{\mathrm{ir}}}{l_\mu}, \tag{11} \]

\(I_{\mathrm{ir}}\) is the current of the IR coil, equal to the rotor current when the IR coils are connected in a “star.”

Expressions (6)—(11) make it possible, from the known rms value of the current in the IR coil and, consequently, the rms value of the magnetic-field strength at the surface of the core, to determine the total equivalent electrical resistance of the solid disks referred to the stator winding.

Since the rotor current is not known before determining the parameters of the solid disks, it is necessary initially to assign the rotor current in the operating mode of interest to us, determine the specific magnetizing ampere-turns, and, from (10) or from the curve \(Z_{\mathrm{ir}(1)} = f(aW)\), find the unit electrical resistance of the solid core, and then determine:

\[ Z'_{is.a}=Z_{is.(1)}\cdot \frac{U_{\mu}}{i_{\mu}}\cdot W_{is}^{2}\cdot K_{p}. \tag{12} \]

Having determined \(Z_{is}\) from (7), one can ultimately determine the rotor current of the asynchronous motor with IR and compare it with the value specified initially. If the rotor current found differs from the specified one, it is necessary to recalculate for another value of the rotor current.

It should be noted that, as is evident from (5) and (6), the power factor of the IR over wide limits of variation of the slip (practically from \(\infty\) to 0.2) remains constant and equal to:

\[ \cos\varphi_{is}=0.83. \]

This is also confirmed by experimental data.

Using the developed method, an electromagnetic calculation was carried out for explosion-proof asynchronous electric motors with IR in the rotor circuit of the VAOI series from frame size 4 to frame size 7 (modifications of the unified VAO series).

Experimental investigations of prototype samples confirmed the correctness of the method for calculating the IR parameters.

References

1. B. G. Vlasov, L. L. Antonovskii. Throttle-type explosion-proof adjustable electric drive with an asynchronous electric motor with induction resistance. Proceedings of the Gipronisel'elektrokhim Institute, issue 1, Kemerovo Book Publishing House, Kemerovo, 1967.

2. L. R. Neiman. The surface effect in ferromagnetic bodies. Gosenergoizdat, 1949.

3. L. R. Neiman, P. L. Kalantarov. Theoretical Foundations of Electrical Engineering. Part III, 1959.