Pulsating Torque and Geometry of Single-Phase Induction Motors

E. K. Strelbitskii

(Presented by the scientific seminar of the Electromechanical Faculty)

It is known that in single-phase electrical machines the electromagnetic torque, in a first approximation, varies with time according to the law [1, 2, 3]:

\[ M = \frac{E_m I_m}{2n}\cos\varphi - \frac{E_m I_m}{2n}\cos(2\omega_1 t - \varphi). \tag{1} \]

Here and below the following notation is used:

\(A\) — linear loading,
\(B_\delta\) — induction in the air gap,
\(D\) — diameter of the air gap,
\(G\) — weight,
\(E_m\) — amplitude of the e.m.f.,
\(I_m\) — amplitude of the current,
\(J_p\) — moment of inertia of the rotor,
\(J_{cm}\) — moment of inertia of the stator,
\(l\) — linear dimension of the machine,
\(M\) — torque,
\(M_{an}\) — amplitude of the pulsating torque,
\(n\) — rotational speed,
\(p\) — number of pole pairs,
\(P_2\) — rated shaft power,
\(P'\) — calculated power of the machine,
\(t\) — time,
\(\omega_1\) — angular frequency of the supply,
\(\varphi\) — phase-shift angle of current and voltage,
\(\eta\) — efficiency.

The first term of equation (1),

\[ \frac{E_m I_m}{2n}\cos\varphi, \]

represents the mean value of the torque, determined by the load on the shaft. The second term,

\[ \frac{E_m I_m}{2n}\cos(2\omega_1 t - \varphi), \]

represents the pulsating torque of the motor, whose frequency is equal to twice the supply frequency.

The pulsating torque causes torsional oscillations of the stator at this frequency, which are transmitted to the foundation and are the cause of the so-called structural noise. Since the pulsating torque

applied to the rotor to the same degree, then the rotor also performs forced torsional oscillations relative to coordinates rotating at a speed equal to the mean speed of the rotor. The presence of torsional oscillations of the rotating rotor is, in other words, an oscillation of the rotor speed with a frequency equal to twice the line frequency.

The nonuniformity of the rotor speed causes the appearance of higher harmonics of the pulsating torque and, consequently, of the stator vibrations caused by them. The torsional oscillations of the rotor also contain higher harmonics. The indicated oscillations of the rotor and stator cause noise and vibrations with frequencies \(2\omega_1\), \(4\omega_1\), \(6\omega_1\), \(8\omega_1\), etc. The amplitudes of the higher harmonics of the speed oscillations and of the stator vibrations depend on the tangent of the angle of inclination of the curve \(M_{an}=(fn)\) (Fig. 1). The weight of the higher harmonics, as is seen from Fig. 1, is proportional to the magnitude of the change in the amplitude of the pulsating torque \(\Delta M_{an}\) under a pulsating change of speed in the range \(2\Delta n\). It has been found that the weight of the higher harmonics of the pulsating torque and of the vibrations caused by them depends substantially on the dimensions of the segment \(ab\) of the motor characteristic within which the rotor speed oscillates [4].

In this case we are interested in the magnitude of the projection of this segment of the characteristic onto the torque axis, i.e., \(\Delta M_{an}\). The larger \(\Delta M_{an}\) is during one period of change of the supply voltage, the greater the weight of the higher harmonics [4]. For a constant \(\Delta n\), determined by the amplitude of the pulsating torque, the line frequency, and the moment of inertia of the rotor, \(\Delta M_{an}\) depends exclusively on the tangent of the angle of inclination of the curve \(M_{an}=f(n)\). Strictly speaking, \(\Delta n\) is also influenced somewhat by the moment of inertia of the stator. The point is that the angle of oscillation of the rotor must be measured not relative to fixed coordinates, but relative to coordinates located on the stator, which, as is known, deflects in the opposite direction by some angle whose magnitude is inversely proportional to the moment of inertia of the stator. But since the moment of inertia of the stator is 10–15 times greater than the moment of inertia of the rotor, we are justified in neglecting this angle.

Fig. 1.

On the other hand, for the very same curve \(M_{an}=f(n)\), the range of variation of the amplitude of the pulsating torque depends on the range of speed variation \(2\Delta n\) (Fig. 2).

The amplitude of the angular oscillations of the rotor is determined by the formula

\[ \Delta \alpha_p=\frac{M_{an}}{J\omega_1}. \tag{2} \]

It is known that, with a change in nominal power and nominal speed, \(M_{an}\) and \(J\) change according to different laws. It is therefore of interest to analyze how \(\Delta \alpha_p\), \(\Delta n\), and ultimately \(\Delta M_{an}\) will vary as functions of the nominal power of the motor—

… From formula (2) it follows that, for a definite number of poles of an asynchronous motor, the ratio \(\dfrac{M_{an}}{J_p}\) determines the amplitude of the angular oscillations of the rotor, the range of speed variation \(2\Delta n\), and, consequently, the weight of the higher harmonics.

The aim of the present work is to analyze the influence of the rated power of a single-phase induction motor on the weight of the higher harmonics of the pulsating torque and of the vibrations. In addition, the task is posed of determining the dependence of the amplitude of vibrations caused by the principal harmonic on the rated power of the machine.

Fig. 2.

The amplitude of the pulsating torque in synchronous watts is numerically equal to the so-called calculated power (2)

\[ M_{na} = IE = P' = \frac{P_{2n}}{\eta_n \cos \varphi_n}. \tag{3} \]

Since catalogues and reference books usually give not the calculated power \(P'\), but the rated shaft power \(P_{2n}\), let us consider separately the following functional dependences:

\[ M_{an} = P' = f_1\left(P_{2n}, \frac{\omega_1}{p}\right); \]

\[ J = f_2\left(P_{2n}, \frac{\omega_1}{p}\right). \tag{4} \]

For a series of geometrically similar machines, with constant synchronous speed (number of poles), the calculated power, numerically equal to the amplitude of the pulsating torque, depends on the linear dimension of the machine as follows (5):

\[ M_{an} = P' = l^4. \tag{5} \]

For machines of identical rating having different numbers of poles, the dependence

\[ M_{an}\, n = P' = l^4 n \tag{6} \]

is valid.

However, it should be borne in mind that formulas (5) and (6) are valid only under the condition that the electromagnetic loadings of the machine remain constant, i.e., when \(A = \mathrm{const}\) and \(B_\delta = \mathrm{const}\). In reality, the electromagnetic loadings of the machine are chosen as functions of the rated power. The electromagnetic loadings vary especially strongly in the region of small powers, i.e., for the machines that are the subject of the present consideration. For machines with power up to 600 W, N. S. Sergeev [5] gives the following dependence of the product of the electromagnetic loadings on the air-gap diameter and the speed of the machine:

\[ AB_\delta = D^x n^z, \tag{7} \]

where

\[ x \simeq 1, \]

\[ z \simeq 0.2 \ldots 0.3. \]

In other words, at \(n_c=\mathrm{const}\) the product of the electromagnetic loadings is directly proportional to the linear dimension of the machine:

\[ AB_{\delta}=l^1 . \tag{8} \]

In this case, the dependence of the amplitude of the pulsating torque on the linear dimension of the machine is expressed as follows:

\[ M_{pa}=P'=l^5 . \tag{9} \]

Let us return to formula (2) and consider the magnitude of the angular oscillations of the stator and rotor and the range of variation of the rotor speed, taking into account the refinement (9). As is known, the moment of inertia \(J\) is proportional to the flywheel moment

\[ J=\frac{GD^2}{4}. \tag{10} \]

Taking into account that

\[ G=l^3, \]

\[ D=l, \tag{11} \]

we obtain

\[ J=l^5. \tag{12} \]

Dividing (9) by (12), we obtain that the angular amplitude of the oscillations of the stator and rotor and the range of variation of the rotor speed due to the action of the pulsating torque remain constant over a wide power range:

\[ \Delta \alpha_{st}=l^0, \]

\[ \Delta \alpha_p=l^0, \]

\[ \Delta n=l^0. \tag{13} \]

The amplitude of the linear oscillations of the stator will obviously be expressed by the dependence

\[ a=\Delta \alpha_{st} R=l^1, \tag{14} \]

where \(R\) is the distance from the given point of the stator to the center of the machine.

Analyzing expressions (13, 14), the following may be noted.

1. The amplitudes of the angular oscillations of the stator and rotor caused by the pulsating torque, in the power range under consideration, do not depend on the rated power of the machine.

2. The range of variation of the rotor speed during its pulsating motion also does not depend on the rated power. Consequently, the weight of the higher harmonics of the pulsating torque and vibrations is the same for machines of different power and is determined exclusively by the magnitude of the operating slip, i.e., by the tangent of the angle of inclination of the characteristic.

3. The linear amplitude of the stator vibrations, when the rated power changes, is directly proportional to the linear dimension of the machine.

To confirm the conclusions obtained, the relationships occurring in existing motors manufactured in the Soviet Union (the unified AOLB series) and in the FRG (DA—AEG, MD—Siemens, and AD—Sachsenwerk) were analyzed. In Fig. 3,

Figs. 4, 5 present the dependences of weight, efficiency, and power factor on the rated power for four-pole micromotors of the indicated types. The data are taken from [6].

To represent the dependences shown in Figs. 3, 4, 5 in analytical form, we shall choose a power function, since it

Fig. 3.

Fig. 4.

is usually used in relationships of this kind and is most suitable for our analysis:

\[ \begin{aligned} G &= P_{2н}^{x},\\ \eta_{н} &= P_{2н}^{y},\\ \cos \varphi_{н} &= P_{2н}^{z}. \end{aligned} \tag{15} \]

Having plotted the envelopes of the dependencies in Figs. 3, 4, and 5 and drawn the mean curves (they are not shown in the figures in order to avoid cluttering the drawings), by selection we find such values of the exponents \(x\), \(y\), and \(z\) that curves (15) coincide, with a certain accuracy, with the aforementioned mean curves.

Fig. 5.

As a result of the selection we obtained the following values of the exponents:

\[ x=0.45,\qquad y=0.143,\qquad z=0.11. \]

Substituting the obtained values \(x\), \(y\), and \(z\) into formula (15), we obtain:

\[ \begin{gathered} G=P_{2n},\\ \eta_n=P_{2n},\\ \cos\varphi_n=P_{2n}. \end{gathered} \tag{16} \]

Substituting relations (16) into expressions (3), (12), and (2), we obtain:

\[ \Delta\alpha_{\mathrm{st}}=\Delta\alpha_r=P^0_{2n}, \tag{17} \]

that is, the amplitude of the angular oscillations of the stator and rotor remains constant over a wide range of powers. Expression (17) is equivalent to expression (13). Thus, it may be said that the propositions obtained by us, taking into account the dependence of the electromagnetic loads on power, are valid for actually existing machines.

Conclusions

1. In low-power single-phase induction motors, the amplitude of the angular oscillations of the stator and rotor arising under the action of the pulsating torque is constant over a wide range of rated powers.

2. The range of variation of the rotor speed under its pulsating motion also does not depend on the rated power.

1. The weight of the higher harmonics of the pulsating torque and of the vibrations caused by it does not depend on the motor power and is determined exclusively by the magnitude of the operating slip, i.e., by the tangent of the angle of inclination of the characteristic.

References

1. A. L. Kimball, P. L. Alger, Single-Phase Motor—Torque Pulsation. Tr. A.I.E.E., vol. 43, pp. 300–739, 1924.

2. K. P. Kovács, Pulsierendes Moment im asymmetrischen Betrieb von Wechselstrom-maschinen, A. f. E., Bd. 42, H. 2, 1955.

3. R. Beach, Physical Conceptions of Single-Phase Induction Motor Performance, Electrical Engineering, vol. 63, No. 7, pp. 254–263, 1943.

4. E. K. Strelbitskii. Pulsating torque of single-phase electric machines, Izvestiya vysshikh uchebnykh zavedenii, series Electromechanics, No. 9, 1959.

5. I. S. Sergeev. Design of Electrical Machines, GEI, 1956.

6. Review of foreign technology No. 131, Electric Motors of Low Power, TsBTI EI, 1958.