ON THE STUDY OF CURRENTS, MAGNETOMOTIVE FORCES, AND MAGNETIC FIELDS IN ASYNCHRONOUS MACHINES WITH AN ASYMMETRIC ROTOR

E. V. KONONENKO

(Presented by the scientific-methodological seminar of the Electromechanical Faculty)

In machines operating in the asynchronous mode, three cases of rotor asymmetry are possible.

1. The rotor has both magnetic and electrical asymmetry. This is the general case of asymmetry. It includes synchronous salient-pole machines and synchronous-reluctance motors with a starting winding on the rotor.

2. The rotor has only electrical asymmetry. This case includes asynchronous machines with an asymmetric winding on the rotor.

3. The rotor has only magnetic asymmetry. This case includes synchronous salient-pole machines without a starting winding and with an open excitation winding.

The methodology that existed until recently for studying asynchronous motors with an asymmetric rotor is based on the method of symmetrical components, according to which the asymmetric system of rotor currents is decomposed into symmetrical components. Equivalent substitution circuits are drawn up, from which the components of the currents of positive and negative sequence are determined [1, 2, 5, 7, 9].

The methodology for studying asynchronous operating modes of synchronous salient-pole machines is based on the method of symmetrical components with the use of the theory of two reactions [2, 5].

The principal shortcoming of these methodologies is that they reflect the physical phenomena occurring in the machine insufficiently and without sufficient clarity.

In recent years Soviet scholars have developed new methods for studying the transient and steady-state operating modes of electrical machines, based on the application of operational calculus and vector analysis. The works of M. P. Kostenko, R. A. Lyuter, L. N. Gruzov, A. A. Gorev, A. D. Gorodsky, E. Ya. Kazovsky, M. N. Alyab’ev, and many others enjoy deserved authority in this field.

The most general of the methods developed is the method of rotating magnetic fields. As applied to the theory of alternating-current machines, the foundations of this method were first developed by the outstanding-

...by the Russian scholar M. O. Dolivo-Dobrovolsky. However, the method of rotating magnetic fields became widespread as a method of mathematical investigation of electrical machines only with the publication of works by L. N. Gruzov and E. Ya. Kazovsky [3, 6].

It should be noted that even in modern works the possibilities of the method of rotating magnetic fields have not been used fully.

The present work is devoted to the study of asynchronous operating conditions of three-phase machines with an asymmetrical rotor.

The investigation is based on the method of rotating magnetic fields. The paper attempts to make fuller use of the possibilities of this method for clarifying the physical processes occurring in machines with an asymmetrical rotor.

The following assumptions are adopted in the work.

1. On the stator side, the machine is supplied by a positive-sequence system of voltages from a source of infinitely large power.

2. The stator has one symmetrical system of windings.

3. Only the fundamental harmonic components of the magnetomotive forces (m.m.f.) and of the magnetic fields created by the windings are taken into account.

4. The effect of saturation of the steel is taken into account only by the choice of the corresponding parameters. Within the limits of the process under consideration, the parameters of the machine are assumed constant.

5. The influence of mechanical losses and losses in the steel is not taken into account in deriving the equations.

6. Zero-sequence currents are assumed equal to zero, since they do not affect the resultant magnetic field or the torque of the machine.

The investigation employs a system of relative units.

According to the method of rotating magnetic fields, voltages, currents, and flux linkages are represented by vectors that characterize the corresponding space waves.

Such a representation of all currents, voltages, and flux linkages by space vectors makes it possible to study simply and clearly the physical processes occurring in the machine by means of vector analysis.

The operational equations for the vectors of current \(i_s\) and of the resultant stator flux linkages \(\Psi_s\), in the presence of magnetic and electrical asymmetry of the machine along the rotor axes \((d, q)\), may be represented in the form [3, 6]

\[ i_s = \frac{\left[r + (p - j\omega)x_c(p)\right] U_s - (p + j\omega)x_u(p) U_s^{*}} {r^2 + rp\left[x_d(p) + x_q(p)\right] + (p^2 + \omega^2)x_d(p)x_q(p)} ; \tag{1} \]

\[ \Psi_s = \frac{\left[rx_c(p) + (p - j\omega)x_d(p)x_q(p)\right] U_s + rx_u(p) U_s^{*}} {r^2 + rp\left[x_d(p) + x_q(p)\right] + (p^2 + \omega^2)x_d(p)x_q(p)} , \tag{2} \]

where \(r\) is the active resistance of the stator winding,

\(\omega\) is the angular speed of rotation of the rotor,

\(x_d(p), x_q(p)\) are the operational reactive resistances along the longitudinal and transverse axes of the rotor,

\(U_s\) is the vector of voltages applied to the stator,

\[ x_c(p) = \frac{x_d(p) + x_q(p)}{2}; \qquad x_u(p) = \frac{x_d(p) - x_q(p)}{2}. \]

Here an asterisk denotes conjugate vectors.

Equations (1), (2) are the basic equations by means of which one can study both the transient and the steady-state operating regimes of asynchronous machines with an asymmetric rotor.

The transient regimes of asynchronous machines, taking rotor asymmetry into account, have been studied in considerable detail by E. Ya. Kazovskii [6].

In the present work only steady-state operating regimes are considered.

Taking into account that, owing to the mechanical inertia of the rotating masses, electromagnetic processes proceed much more rapidly than the change in rotational speed, it is assumed that the rotational speed remains constant at any instant of time.

If the voltage of phase \(a\) of the stator varies according to the law \(U_a=-U_m \cos(t+\delta_0)\), then the space vector of voltages in the rotor axes \((d, q)\) will have the form

\[ U_s = U_m e^{j[(1-\omega)t+\delta_0]}, \tag{3} \]

where \(\delta_0\) is the angle formed by the voltage vector \(U_s\) with the \(d\)-axis at the instant \(t=0\).

The angular velocity of the vector \(U_s\) relative to the rotor is equal to \((1-\omega)\). Since the vector \(U_s\) rotates in space at synchronous speed, we shall define the speed of the rotor relative to synchronous speed as the “synchronous slip,” equal to \(s_c=1-\omega\).

By slip, however (as is usually understood in the literature), we shall mean the rotational speed of the rotor relative to the rotating magnetic field of the stator.

Then equation (3) takes the form

\[ U_s = U_m e^{j(s_c t+\delta_0)}. \tag{4} \]

Taking this into account, the current vectors \(i_s\) and stator flux linkages \(\Psi_s\) in (1), (2) can be represented as consisting of two components:

\[ i_s = i_{s1}+i_{s2} = -\frac{[r+(p-j\omega)x_c(p)]U_m e^{j(s_c t+\delta_0)}}{A} + \]

\[ +\frac{-(p+j\omega)x_u(p)U_m e^{-j(s_c t+\delta_0)}}{A}; \tag{5} \]

\[ \Psi_s=\Psi_{s1}+\Psi_{s2} = \frac{[r x_c(p)+(p-j\omega)x_d(p)x_c(p)]U_m e^{j(s_c t+\delta_0)}}{A} + \]

\[ +\frac{r x_u(p)U_m e^{-j(s_c t+\delta_0)}}{A}, \tag{6} \]

where

\[ A=r^2+rp[x_d(p)+x_q(p)]+(p^2+\omega^2)x_d(p)x_q(p). \]

The equations of the steady-state operating regime, under the assumptions adopted, can be obtained from the operational equations (5), (6) by the simple replacement of \(p\) by \(\pm j s_c\) [3, 6].

Having performed the indicated substitution and inserting into equations (5), (6) the steady-state values of the operator reactances for a known number of short-circuited windings on the rotor, the equations of the steady-state operating condition after transformations can be represented in the form:

\[ i_s=i_{s1}+i_{s2}=I_1 e^{j(s_c t+\delta_0-\varphi_1)}+I_2 e^{-j(s_c t+\delta_0-\varphi_2)}, \tag{7} \]

\[ \Psi_s=\Psi_{s1}+\Psi_{s2}=\Psi_1 e^{j(s_c t+\delta_0-\psi_1)} +\Psi_2 e^{-j(s_c t+\delta_0-\psi_2)} . \tag{8} \]

where \(I_1,\ I_2,\ \Psi_1,\ \Psi_2\) are the moduli of the corresponding vectors, depending only on the parameters and the rotational speed of the rotor. At constant rotor speed they are constant quantities;

\(\varphi_1,\ \psi_1\) are the phase angles between the vectors \(i_{s1},\ \Psi_{s1}\) and \(U_s\);

\(\varphi_2,\ \psi_2\) are the phase angles between the vectors \(i_{s2},\ \Psi_{s2}\) and \(U_s^*\).

As is seen from equations (7) and (8), the components of the stator current and flux linkages, at constant rotor speed, have constant moduli and rotate relative to the rotor in opposite directions with angular velocity \(s_c\).

We shall agree to regard the vectors \(i_{s1},\ \Psi_{s1}\), rotating relative to the rotor in the positive direction, as vectors of the positive sequence, and the vectors \(i_{s2},\ \Psi_{s2}\), rotating in the negative direction, as vectors of the negative sequence.

The resultant vector of the stator current \(i_s\) in (7), after simple transformations, can be brought to the form:

\[ i_s=I_s(t)e^{j\beta}, \tag{9} \]

where

\[ I_s(t)= \sqrt{ I_1^2+I_2^2+2I_1I_2\cos 2\left( s_c t+\delta_0-\frac{\varphi_1+\varphi_2}{2} \right) } \]

is the amplitude of the resultant current vector;

\[ \beta=\operatorname{arctg}\left[ \frac{I_1-I_2}{I_1+I_2} \tg\left( s_c t+\delta_0-\frac{\varphi_1+\varphi_2}{2} \right) \right] -\frac{\varphi_1-\varphi_2}{2} \]

is the angle formed by the vector \(i_s\) with the rotor axis \(d\).

It follows from equation (9) that the amplitude of the current vector \(I_s(t)\) does not remain constant in time, but pulsates with twice the synchronous slip frequency \(s_c\), within limits from the maximum value, equal to

\[ I_{s\max}=I_1+I_2, \]

to the minimum, equal to

\[ I_{s\min}=I_1-I_2. \]

The angular velocity of rotation of the vector \(i_s\) relative to the rotor can be determined by the equation

\[ \frac{d\beta}{dt} = \frac{(I_1^2-I_2^2)s_c} {I_1^2+I_2^2+2I_1I_2\cos 2\left( s_c t+\delta_0-\frac{\varphi_1+\varphi_2}{2} \right)} . \tag{10} \]

Analysis of this equation shows that the resulting current vector rotates relative to the rotor with variable angular velocity.

Since, in the system of relative units, currents and magnetomotive forces are expressed by the same vectors, equations (9), (10) also represent the law of variation of the m.m.f. produced by the stator winding.

Consequently, the m.m.f. of the stator winding with an asymmetric rotor will rotate relative to the rotor with variable angular velocity, while at the same time varying in magnitude.

The vector of the resulting flux linkages \(\Psi_s\) (8), after simple transformations, can be reduced to the form

\[ \Psi_s=\Psi_s(t)e^{j\psi}, \tag{11} \]

where

\[ \Psi_s(t)= \sqrt{ \Psi_1^2+\Psi_2^2+2\Psi_1\Psi_2\cos 2\left(s_c t+\delta_0-\frac{\psi_1+\psi_2}{2}\right) } \]

is the amplitude of the resulting flux-linkage wave;

\[ \psi=\operatorname{arc\,tg}\left[ \frac{\Psi_1-\Psi_2}{\Psi_1+\Psi_2} \operatorname{tg}\left(s_c t+\delta_0-\frac{\psi_1+\psi_2}{2}\right) \right] -\frac{\psi_1-\psi_2}{2} \]

is the angle formed by the vector \(\Psi_s\) with the rotor axis \(d\).

Since, in the system of relative units, flux linkages and magnetic fields are determined by the same vectors, equation (11) also represents the law of variation of the stator magnetic field. The angular velocity of rotation of the stator magnetic field relative to the rotor can be determined as the derivative \(\dfrac{d\psi}{dt}\). This velocity, expressed in relative units, is called the slip in the literature.

Consequently, the slip is equal to

\[ s=\frac{d\psi}{dt} = \frac{(\Psi_1^2-\Psi_2^2)s_c} {\Psi_1^2+\Psi_2^2+2\Psi_1\Psi_2\cos 2\left(s_c t+\delta_0-\frac{\psi_1+\psi_2}{2}\right)} . \tag{12} \]

It follows from equations (11), (12) that the stator magnetic field has a variable amplitude and rotates relative to the rotor with variable angular velocity. Figure 1 presents curves of the variation of the magnetic-field amplitude \(\Psi_s(t)\) and of the slip \(s\) as functions of time. The amplitude of the magnetic field varies from the maximum value, equal to \(\Psi_{s\max}=\Psi_1+\Psi_2\), to the minimum value, equal to \(\Psi_{s\min}=\Psi_1-\Psi_2\), with a frequency equal to twice the synchronous slip.

At the same time the slip \(s\) varies from the minimum value, equal to

\[ s_{\min}=\frac{\Psi_1-\Psi_2}{\Psi_1+\Psi_2}s_c, \]

to the maximum, equal to

\[ s_{\max}=\frac{\Psi_1+\Psi_2}{\Psi_1-\Psi_2}s_c, \]

with the maximum of the magnetic field coinciding in time with the minimum of the slip, and vice versa.

Slip is a periodic function. It is not difficult to prove that its mean value is equal to the synchronous slip \(s_{cp}=s_c\).

Fig. 1.

Studies show that the limits of variation of the amplitude of the magnetic field and of the slip depend on the degree and type of asymmetry, as well as on the magnitude of the active resistance of the stator winding. The greatest limits of variation occur in the limiting case of electrical asymmetry, when the rotor has only a single-axis winding.

The character of the variation of the angular velocity of the magnetic field relative to the stator is determined by the equation

\[ \omega_\psi = \omega + s, \tag{13} \]

where \(s\) is the slip determined by equation (12).

It follows from equations (12), (13) that the rotational speed of the magnetic field in space does not remain constant in time, but varies with a frequency of doubled synchronous slip within the limits from

\[ \omega_{\psi \max}=\omega+s_{\max} \]

to

\[ \omega_{\psi \min}=\omega+s_{\min}. \]

Fig. 2.

Figure 2 presents the limits of variation of the rotational speed of the magnetic field as a function of the rotational speed of the rotor, calculated from equation (13) for an induction motor with a single-phase rotor having the following parameters: \(r=0.042\); \(x_d=2.6\); \(T_d=82\); \(T_d^1=4.22\). As can be seen from the graph shown, the limits of variation of the rotational speed of the magnetic field increase as the rotor speed \(\omega\) changes from 0 to 0.5 and decrease as \(\omega\) changes from 0.5 to 1.0.

At \(\omega = 0.5\) we have the greatest limits of speed variation. This is explained as follows. The limits of speed variation depend on the amplitudes of the magnetic fields of the positive \(\Psi_1\) and negative \(\Psi_2\) sequences. At \(\omega = 0.5\) the negative-sequence field is stationary relative to the stator; therefore the stator winding has no damping effect on \(\Psi_2\), which in this case reaches its greatest value.

Thus, during the operation of an alternating-current machine with an asymmetrical rotor, the mmf and the magnetic field of the stator rotate in space with variable angular velocity, while at the same time changing in magnitude.

To determine the phase quantities of the current and flux linkages, the resultant vectors \(\dot{i}_s\), \(\Psi_s\) are projected onto the phase axes [3, 6].

Consequently, the instantaneous values of the current in the stator phases can be determined as

\[ i_a = R_e\left[i_s e^{j(\omega t+\gamma_o)}\right]; \qquad i_b = R_e\left[i_s e^{j(\omega t+\gamma_o-120^\circ)}\right]; \]

\[ i_c = R_e\left[i_s e^{j(\omega t+\gamma_o-240^\circ)}\right], \tag{14} \]

where \(\gamma_o\) is the angle formed by the rotor axis \(d\) with the axis of phase \(a\) at the instant \(t=0\).

The current in phase \(a\), taking equation (9) into account, can be represented in the form

\[ i_a = I_s(t)\cos\left[t+\alpha_o-\varphi(t)\right], \tag{15} \]

where \(\varphi(t)\) is the variable phase angle,
\(I_s(t)\) is the current amplitude.

It follows from equation (15) that the current in the stator phases is, in the general case, nonsinusoidal.

Taking equation (7) into account, the current \(i_a\) can be represented as the sum of two sinusoidal components of different frequencies,

\[ i_a = I_1\cos(t+\alpha_o-\varphi_1) + I_2\cos\left[(1-2s_c)t+\alpha_o-2\gamma_o-\varphi_2\right]. \tag{16} \]

In studying the currents in the stator phases, it is convenient to use both equation (15) and equation (16).

Fig. 3.

At a rotor speed close to synchronous, an oscillogram of the current in a stator phase is shown in Fig. 3. The current envelope represents the variation of the amplitude \(I_s(t)\), which pulsates with a frequency equal to twice the synchronous slip, from its maximum value, equal to \(I_{s\max}=I_1+I_2\), to its minimum value, equal to \(I_{s\min}=I_1-I_2\). Taking this into account, from the oscillogram one can determine the amplitudes of the currents of the positive \(I_1\) and negative \(I_2\) sequences.

\[ I_1 = \frac{I_{s\max} + I_{s\min}}{2}; \qquad I_2 = \frac{I_{s\max} - I_{s\min}}{2}. \]

When the rotor speed is reduced, another characteristic form of the current curve can be obtained (Fig. 4). The envelope of the current \(i_a\) is a negative-sequence current; the curve inscribed between the envelopes is a positive-sequence current.

Fig. 4.

Figures 3 and 4 show two characteristic forms of the current curves \(i_a\). An intermediate one is a more complex curve, from which it is not possible to determine the components \(I_1\) and \(I_2\). One such curve is shown in Fig. 5.

Fig. 5.

When the rotor speed changes, the shape of the current curve \(i_a\) changes and will have one of the types of curves described above.

Conclusions

1. When machines with any rotor asymmetry operate in an asynchronous mode, with a symmetrical system of applied voltages and a symmetrical stator, the stator-winding m.m.f. and the resultant magnetic field rotate in space with variable angular velocity, while simultaneously varying in magnitude.

2. If slip is defined as the rotational speed of the resultant magnetic field of the stator relative to the rotor, then, at constant rotor speed, the slip does not remain constant, but changes according to the law determined by equation (12).

In studying asynchronous operating modes, it is convenient to express the rotor speed as a speed relative to synchronous speed. For this purpose the concept of synchronous, or mean, slip is introduced.

1. The limits of variation of slip, as well as of the rotational speed of the magnetic field, depend on the degree of rotor asymmetry, and also on the magnitude of the active resistance of the stator winding. The greatest limits of variation of slip occur in the limiting

case of electrical asymmetry, when there is a single-phase winding on the rotor at the subsynchronous rotational speed of the rotor.

1. The magnetic field, rotating relative to the stator winding with a variable angular velocity, generally induces in it a nonsinusoidal e.m.f., which leads to a distortion of the shape of the current curve in the phases.

REFERENCES

1. Berger A. Ya. et al. The Asynchronous Motor under Abnormal Conditions, Moscow, 1939.

2. Veshenevsky S. N. Calculation of Characteristics and Resistances for Electric Motors, GEI, 1953.

3. Gruzov L. N. Methods of Mathematical Investigation of Electrical Machines. GEI, 1953.

4. Kostenko M. P., Lyuter R. A., Kazovsky E. A. Problems in the development of the modern theory of the synchronous machine. VEP, No. 2, 1952.

5. Kostenko M. P. Electrical Machines. Special Part. GEI, 1949.

6. Kazovsky E. Ya. Transient processes in asynchronous machines with allowance for rotor asymmetry. El-vo, No. 4, 1950.

7. Merkin G. B., Nezgovorova E. D. Operation of an asynchronous motor with asymmetry in the rotor. Proceedings of LPI, No. 2, issue 1, 1938.

8. Syromyatnikov I. A. Operating Regimes of Asynchronous Motors. GEI, 1950.

9. Shturman G. I. Asymmetry of secondary circuits in asynchronous machines. Moscow, 1935.