FEATURES OF USING THE ROTATING-FIELD METHOD TO ANALYZE THE OPERATION OF ASYNCHRONOUS MOTORS WITH A NONUNIFORM AIR GAP

V. A. Sagaidak

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

General remarks. When analyzing the operation of asynchronous micromotors by the rotating-field method, it is possible to take into account, relatively simply, the spatial higher harmonics of the magnetic field in the air gap and the losses in the steel core. By the rotating-field method one can obtain equations that make it possible, with sufficient accuracy, to calculate the starting and operating characteristics, including those of motors of very low power—on the order of several watts and below—which have concentrated asymmetrical stator windings shifted in space by an arbitrary angle \[1, 2\]. Especially accurate equations can be obtained as applied to motors with a solid rotor, having distributed parameters of the rotor winding and a saturated magnetic system.

Certain difficulties arise when analyzing, by the indicated method, the operation of motors with a nonuniform air gap—for example, motors with split poles. The difficulty in using the rotating-field method in this case is due to the fact that one or another system of currents of constant amplitude, forming a rotating wave of magnetomotive forces, creates magnetic fluxes of different magnitude at different instants of time. This phenomenon arises because the magnetic reluctance in the path of the main flux of a machine with a nonuniform gap is different for different directions of action of the magnetomotive forces.

Below, one possible way of overcoming the noted difficulty in using the rotating-field method is set forth.

Scheme for Deriving the Basic Equations in Machines with a Uniform Gap

The derivation of the basic equations by the rotating-field method for a machine with a uniform gap is usually carried out in the following sequence:

1. Notations are introduced for the as yet unknown currents of all windings of the machine. In doing so, it is assumed that in the rotor, in the general case, there are two basic current systems (direct and reverse), as well as a number of systems of currents of higher orders.

1. The magnetomotive forces created by the stator windings and by the current systems of the short-circuited rotor winding are determined.

2. The pulsating magnetomotive forces of the stator windings are expanded into a series of pulsating spatial harmonics of the magnetomotive forces in the air gap.

3. Each pulsating harmonic is resolved into two waves rotating in opposite directions.

4. The total (from all the machine windings) “forward” and “backward” waves of the magnetomotive forces are calculated, and then the corresponding rotating magnetic fields.

5. Voltage-balance equations are written for each stator winding and for each rotor current system separately.

The resulting system of equations is the starting point for further analysis of the operation of the machine or for calculating its characteristics.

Distinctive features of a possible scheme for deriving the fundamental equations in machines with a nonuniform gap

In order to overcome the difficulty set out above, it is necessary to sum not the magnetomotive forces, but the like-named rotating magnetic fields created by individual stator windings or by individual rotor current systems.

The calculation of the magnetic fields created by one of the stator windings may be carried out according to the following scheme:

1. Determine the form of the pulsating magnetomotive force of the winding.

2. Determine the form of the pulsating magnetic field in the air gap, taking into account the nature of the variation of the air gap along the armature circumference.

3. Expand the pulsating magnetic field of complex form into a series of pulsating spatial harmonics.

4. Represent each elementary pulsating field in the form of two circular fields rotating in opposite directions. It is useful to emphasize that in the latter case, just as when expanding a pulsating field of complex form into a series of pulsating spatial harmonics, we are no longer concerned with either the magnitude or the form of the air gap. In both cases one simply makes an equivalent replacement of some magnetic fields by others.

At first glance, the procedure described above seems rather cumbersome; however, in the most frequently encountered cases the calculations are greatly simplified. Consider, for example, a motor with split poles, having concentrated windings on the stator and an air gap of constant magnitude \((\delta_1)\) under the salient poles. The magnetomotive force, for example, of the auxiliary winding has a rectangular form. The form of the magnetic field repeats the form of the magnetomotive force and, consequently, the calculation of the amplitude of the fundamental harmonic of the magnetic field (similarly, of the higher harmonics) can be carried out under the assumption that the air gap over the entire armature circumference is identical and equal to \(\delta_1\). Indeed, regardless of the nature of the gap outside the span of the auxiliary winding \((y_3)\), the form and magnitude of the field within \(y_3\) remain unchanged. When calculating the amplitude of the harmonic component it is, of course, necessary to take into account the pitch-shortening factor of the corresponding harmonic, bearing in mind that the winding pitch is less than the pole pitch (see the Appendix).

Calculation of the magnetic fields produced by a certain system of rotor currents is performed in an analogous manner:

1. First, the rotating wave of magnetomotive force, for example, of the principal positive-sequence rotor-current system ($\overline{F}_{2f}$), is replaced by two equivalent magnetomotive forces $F_{2d}$ and $F_{2q}$, pulsating respectively along and across the axis of the magnetic system.

2. Next, the magnetic fields produced by the stationary pulsating magnetomotive forces described above are calculated.

In the latter case it is necessary to take into account that a sinusoidal magnetomotive force in space (for example $F_2$ or $F_{2q}$, in a machine with a nonuniform air gap) produces a nonsinusoidal field. The principal harmonic of the pulsating fields (in our example $\Phi_{2d}$ and $\Phi_{2q}$) may be calculated by a method analogous to that known in the theory of salient-pole synchronous machines. For this purpose, as is known, it is sufficient, when calculating the pulsating magnetomotive forces, to introduce the corresponding correction factor $\kappa_d$ or $\kappa_q$, and then to calculate the principal harmonic of the field assuming the air gap to be constant and equal to $\delta_1$ over the entire circumference of the armature.

The method set forth above is illustrated below by an example of deriving the basic equations of a motor with split poles. For simplicity of exposition, the spatial higher harmonics of the magnetic field in the gap are not taken into account below. A more detailed analysis of the operation of asynchronous motors with split poles, taking into account, among other things, the spatial higher harmonics, is given in [2].

Conclusion

In conclusion, let us note several conclusions that follow from the foregoing, and also from the example considered in the Appendix.

1. The rotating-field method can be applied without analysis of the operation and calculation of asymmetrical asynchronous motors with a nonuniform air gap.

2. The rotating-field method makes it possible to take into account simply the magnetic losses and the spatial higher harmonics of the field in the air gap.

3. In analysis by the rotating-field method, the mathematical developments are naturally accompanied by clear physical interpretations.

4. The rotating-field method makes it possible to reduce to a minimum the number of coefficients in the basic system of equations that are due to mutual-induction fields. Thus, for example, when analyzing the operation of a motor with split poles, it proves possible to manage with only two coefficients of this kind: $x_p$—due to the field of “common” mutual induction, linked with all windings (stator and rotor); $x_m$—due to the field of “partial” mutual induction, linked only with the stator windings.

When analyzing the operation of the same motors by the cross-field method, it is necessary to introduce six or more analogous coefficients.

1. The reactance $x_p$ is calculated on the assumption that the air gap is equal to $\delta_1$ over the entire circumference of the armature, i.e., by a method well known from the theory of symmetrical asynchronous motors.

1. The distinctive features of a motor with a nonuniform air gap are taken into account by introducing into the calculation the winding coefficients and the coefficients $\kappa_d$ and $\kappa_g$, whose calculation presents no difficulty.

Appendix

An example of deriving the basic system of equations for the operating process of a motor with split poles by the rotating-field method (without taking into account the spatial higher harmonics of the field in the air gap).

1. Flux of the main winding

\[ \dot{\Phi}_1 = \frac{\sqrt{2}\, I_{1r}\, w'_1}{2p} \cdot \frac{4}{\pi}\gamma_\mu = 2\kappa_1 w'_1 \kappa \dot{I}_1 . \tag{1} \]

Here

\[ 2\kappa_1 = \frac{\sqrt{2}}{2p} \cdot \frac{4}{\pi} \cdot \gamma_\mu, \tag{2} \]

$\gamma_\mu$ is the magnetic conductance of the magnetic circuit with a uniform air gap;
$\kappa$ is the complex coefficient accounting for losses in the steel,

\[ w' = w \cdot k_w . \tag{3} \]

1. Direct and reverse circular components of the field of the main winding

\[ \overline{\Phi}_{1f} = \kappa_1 \cdot w'_1 \cdot \kappa \cdot \dot{I}_1, \]

\[ \overline{\Phi}_{1b} = \kappa_1 \cdot w'_1 \cdot \kappa \cdot \dot{I}_1 . \tag{4} \]

1. Similar fluxes of the auxiliary winding, oriented with respect to the axis of the main winding,

\[ \overline{\Phi}_{3f} = \kappa_1 \cdot w'_3 \cdot \kappa \cdot \dot{I}_3 \cdot e^{j\beta}, \]

\[ \overline{\Phi}_{3b} = \kappa_1 \cdot w'_3 \cdot \kappa \cdot \dot{I}_3 \cdot e^{-j\beta}. \tag{5} \]

1. Magnetizing forces of the direct and reverse systems of rotor currents

\[ \overline{F}_{2f} = \frac{\kappa_1}{\gamma_\mu} \cdot w'_2 \cdot m_2 \cdot \dot{I}_f, \]

\[ \overline{F}_{2b} = \frac{\kappa_1}{\gamma_\mu} \cdot w'_2 \cdot m_2 \cdot \dot{I}_b . \tag{6} \]

1. Total components of the two magnetizing forces pulsating along and across the axis of the magnetic system

\[ F_{2d} = \frac{\kappa_1}{\gamma_\mu} \cdot w'_2 \cdot m_2 \cdot \left( \dot{I}_f + \dot{I}_b \right), \]

\[ F_{2g} = \frac{\kappa_1}{\gamma_\mu} \cdot w'_2 \cdot m_2 \left( j\dot{I}_f - j\dot{I}_b \right). \tag{7} \]

1. Pulsating longitudinal and transverse fields due to the rotor currents. Taking into account the nonuniformity of the gap $(\kappa_d,\ \kappa_g)$,

\[ \dot{\Phi}_{2d} = \kappa_d \cdot \kappa_1 \cdot w'_2 \cdot m_2 \cdot \kappa \left( \dot{I}_f + \dot{I}_b \right), \]

\[ \dot{\Phi}_{2g} = j \cdot \kappa_d \cdot \kappa_1 \cdot w'_2 \cdot m_2 \cdot \kappa \left( \dot{I}_f - \dot{I}_b \right). \tag{8} \]

1. Let us take into account that in space the axes of the fields \(\Phi_{2d}\) and \(\Phi_{2g}\) are displaced by \(\pi/2\) electrical radians. Then the resultant forward and backward rotating fields from the rotor currents are

\[ \overline{\Phi}_{2f} = \kappa_1 \cdot w'_2 \cdot m_2 \cdot \kappa \left( \dot I_f \frac{\kappa_d+\kappa_g}{2} + \dot I_b \frac{\kappa_d-\kappa_g}{2} \right), \]

\[ \overline{\Phi}_{2b} = \kappa_1 \cdot w'_2 \cdot m_2 \cdot \kappa \left( \dot I_f \frac{\kappa_d-\kappa_g}{2} + \dot I_b \frac{\kappa_d+\kappa_g}{2} \right). \tag{9} \]

1. The resultant forward and backward fields created by all three rotor windings are

\[ \overline{\Phi}_f = \kappa_1 \cdot w'_1 \kappa \left( \dot I_1 + e^{j\beta}\dot I'_3 + \frac{\kappa_d+\kappa_g}{2}\cdot \dot I'_f + \frac{\kappa_d-\kappa_g}{2}\dot I'_b \right), \]

\[ \overline{\Phi}_b = \kappa_1 \cdot w'_1 \cdot \kappa \left( \dot I_1 + e^{-j\beta}\dot I'_3 + \frac{\kappa_d-\kappa_g}{2}\cdot \dot I'_f + \frac{\kappa_d+\kappa_g}{2}\cdot \dot I'_b \right). \tag{10} \]

1. The e.m.f.s induced in the windings of the machine with the equivalent stationary rotor are

\[ \dot E_{2f}=-j\kappa_0 \cdot w'_2 \cdot \overline{\Phi}_f, \]

\[ \dot E_{2b}=-j\kappa_0 \cdot w'_2 \cdot \overline{\Phi}_b, \]

\[ \dot E_{1\mu}=-j\kappa_0 \cdot w'_1 \cdot \left(\overline{\Phi}_f+\overline{\Phi}_b\right), \]

\[ \dot E_{3\mu}=-j\kappa_0 \cdot w'_3 \left(e^{-j\beta}\overline{\Phi}_f+e^{j\beta}\overline{\Phi}_b\right). \tag{11} \]

Or, introducing the impedance of the magnetizing circuit

\[ z_\mu=j\kappa_0 \cdot \kappa_1 \left(w'_1\right)^2 \cdot \kappa = jx_\mu \cdot \kappa \tag{12} \]

and reducing the rotor and auxiliary windings to the number of turns and winding factor of the main winding,

\[ \dot E'_{2f} = -z_\mu \left( \dot I_1 + e^{j\beta}\dot I'_3 + \frac{\kappa_d+\kappa_g}{2}\dot I'_f + \frac{\kappa_d-\kappa_g}{2}\dot I'_b \right), \]

\[ \dot E'_{2b} = -z_\mu \left( \dot I_1 + e^{-j\beta}\dot I'_3 + \frac{\kappa_d-\kappa_g}{2}\cdot \dot I'_f + \frac{\kappa_d+\kappa_g}{2}\dot I'_b \right), \tag{13} \]

\[ \dot E_{1\mu} = -2z_\mu \left( \dot I_1 + \cos\beta\, \dot I'_3 + \frac{\dot I'_f+\dot I'_b}{2} \right), \]

\[ \dot E_{3\mu} = -2z_\mu \left( \dot I_1\cos\beta + \dot I'_3 + \frac{\dot I'_f+\dot I'_b}{2}\cos\beta - j\frac{\dot I'_f-\dot I'_b}{2}\cdot \sin\beta \right). \]

1. Taking into account further the e.m.f.s due to the field of partial mutual induction and leakage fluxes, as well as the active resistances of the windings, we compose the voltage-balance equations for the circuit of the main and auxiliary windings and the circuits of the forward and backward rotor currents:

\[ \dot U_1 = \dot I_1\left(z_1+2z_\mu+z_m\right) + \dot I'_3\left(2z_\mu\cos\beta+k_m z_m\right) + \dot I'_f z_\mu + \dot I'_b z_\mu; \]

\[ 0=\dot I_1(2z_\mu\cos\beta+k_m z_m)+\dot I'_3(z'_3+2z_\mu+k_m^3 z_m)+ \]
\[ +\dot I'_f z_\mu e^{-j\beta}+\dot I'_b z_\mu e^{j\beta}; \]

\[ 0=\dot I_1 z_\mu+\dot I'_3 z_\mu e^{j\beta}+\dot I'_f\left(z'_f+z_\mu\frac{K_d+K_g}{2}\right)+\dot I'_b z_\mu\frac{K_d-K_g}{2}; \]

\[ 0=\dot I_1 z_\mu+\dot I'_3 z_\mu e^{-j\beta}+\dot I'_f z_\mu\frac{K_d-K_g}{2}+\dot I'_b\left(z'_b+z_\mu\frac{K_d+K_g}{2}\right). \]

REFERENCES

1. V. A. Sagaydak. Study of low-power single-phase asynchronous electric motors by the method of rotating fields. Proceedings of NETI, vol. 1, 1953.

2. V. A. Sagaydak. Analysis of the operation of single-phase asynchronous electric motors with split poles by the method of rotating fields. Proceedings of NETI, vol. 3, 1961.