PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE named after S. M. KIROV

Vol. 117                                                                                                             1963

DETERMINATION OF THE MECHANICAL CHARACTERISTICS OF AN INDUCTION MOTOR BY MEASURING THE SPEED AND ACCELERATION OF THE MOTOR UNDER TRANSIENT CONDITIONS

A. I. ZAITSEV, M. P. TABINSKY, A. S. BOBROVSKY

In determining the mechanical characteristic of an induction motor, the greatest difficulties arise in measuring the motor torque. Installations for determining motor torque from the braking torque produced by shunt DC machines are bulky and have a large installed power, exceeding the power of the motor under test by a factor of 3–4. Therefore, a method of determining the torque from the acceleration developed by the motor during starting or braking without load, that is, at a static torque equal to zero \((M_{\text{st}} \simeq 0)\), is of considerable interest. In this case the equation of motion of the rotor and of the rotating masses connected with it has the form

\[ M_{\text{dv}} = J \frac{d\omega}{dt}, \tag{1} \]

where \(M_{\text{dv}}\) is the motor torque, kg·m;
\(J\) is the total moment of inertia of the rotating masses, kg·m·sec\(^2\);
\(\omega\) is the angular velocity of the motor in radians per second, \(1/\text{sec}\).

Thus, for a known moment of inertia, measurement of the motor torque \((M_{\text{dv}})\) is reduced to measurement of the acceleration \(\left(\frac{d\omega}{dt}\right)\). Having graphs of the speed \(\omega = f_1(t)\) and the acceleration \(\frac{d\omega}{dt} = f_2(t)\), it is easy to construct the function \(\omega = \varphi_1\left(\frac{d\omega}{dt}\right)\), or \(\omega = \varphi_2(M_{\text{dv}})\), that is, the mechanical characteristic of the motor.

If a loop oscillograph is available, recording the motor speed \(\omega = f_1(t)\) presents no difficulty. The acceleration of the motor can be determined by differentiating the voltage proportional to the speed by means of a capacitor. The circuit of the installation for determining \(\frac{d\omega}{dt} = f_2(t)\) is shown in Fig. 1.

A flywheel \((M)\) is mounted on the shaft of the motor under test \((AD)\). A tachogenerator \((TG)\), rigidly coupled to the motor shaft, produces a voltage proportional to the speed,

\[ U_{tg}=k\cdot \omega_{\mathrm{dv}}. \tag{2} \]

A capacitor \(C\) and the loop of an oscillograph \(SHO\), having resistance \(R\), are connected in series in the tachogenerator circuit. When the tachogenerator voltage \(U_{tg}\) changes, a charging current flows through the capacitor,

\[ i=C\frac{dU_c}{dt}. \tag{3} \]

Fig. 1.

A voltage is applied to the capacitor equal to

\[ U_c=U_{tg}-U_R=U_{tg}-iR, \tag{4} \]

therefore,

\[ i=C\frac{d}{dt}\left(U_{tg}-iR\right). \tag{5} \]

Separating the variables, we obtain

\[ i-RC\frac{di}{dt}=C\frac{dU_{tg}}{dt}. \tag{6} \]

If \(RC\) is so small that the term \(RC\,\dfrac{di}{dt}\) in equation (6) can be neglected, then we obtain, approximately,

\[ i\approx C\frac{dU_{tg}}{dt}=C\cdot k\frac{d\omega_{\mathrm{dv}}}{dt}. \tag{7} \]

The charging current of the capacitor, proportional to the acceleration of the motor, is recorded by the oscillograph loop.

The method of determining the acceleration of a motor by differentiating, by means of a capacitor, the voltage proportional to speed was proposed as early as 1912 by Itterberg [1, 2], but, despite its simplicity, it has not found wide application for the following reasons:

1) ordinary tachometric generators, as is known, produce a voltage containing commutator pulsations, which, upon differentiation, create such large disturbances that it is impossible to isolate the derivative. In order to reduce commutator pulsations, it is necessary to use machines with a large number of commutator segments (not fewer than 200) and to install smoothing filters [2, 3];

2) the shafts of the motor being tested and of the tachogenerator must be coupled rigidly (without elastic elements) and strictly coaxially. This requirement makes preparation of the experiment very laborious [2];

3) at low speeds the nonlinearity of the brush resistance introduces inaccuracies in the measurements [1].

The authors have succeeded in eliminating these shortcomings and in constructing an installation that makes it possible to record the mechanical characteristics of an asynchronous motor, which are observed on the screen of a cathode-ray oscilloscope. A block diagram of the installation is shown in Fig. 2.

A flywheel \(M\) is mounted on the shaft of the tested motor \(AD\). By means of a cardan shaft the asynchronous motor is connected to a differential \(D\), in which the speed of motor \(AD\) is added to the constant speed of an auxiliary asynchronous motor \(BD\). The need to use the differential will be justified below. The algebraic sum of the speeds of the asynchronous motor \(AD\) and the auxiliary motor \(BD\) is fed from the differential to the tachogenerator \(TG\). A two-phase asynchronous

Fig. 2.

motor with a hollow nonmagnetic rotor (type EM-4 or EM-8) is used as the tachogenerator. When one winding of the two-phase motor is supplied with an alternating current of a definite frequency, an alternating electromotive force of the same frequency, proportional to the rotational speed of the motor rotor, is induced in the other winding, which is in quadrature with the first.

A voltage of \(1 \div 5\) V at a frequency of \(1000\ \text{cps}\), obtained in the sinusoidal-voltage generator unit \(GCh\), is applied to the primary winding of the tachogenerator. Since the output voltage of the tachogenerator is small, it is amplified by the electronic amplifier \(U1\). After power amplification, the voltage from the output transformer \(Tr\) is fed into two channels: the speed channel \(KS\) and the acceleration channel \(KU\). In each channel the voltage is rectified by means of a full-wave semiconductor rectifier (\(VKS\) and \(VKU\)) and smoothed by a \(\Pi\)-type low-frequency filter (\(FS\) and \(FU\)).

The voltage from the speed channel passes through a divider to the cathode-ray oscilloscope (\(EO\)) for horizontal deflection of the beam. In the acceleration channel, the voltage proportional to the acceleration of the motor is picked off across the resistance \(R\) of the differentiating ce-

coil. After preliminary amplification in a DC push-pull amplifier, UTP, it is fed to the oscilloscope for vertical deflection of the beam.

When starting or reversing motor AD, the oscilloscope beam traces the mechanical characteristic on the screen (Fig. 3, a, b).

It was noted above that a diode bridge is used to rectify the alternating voltage proportional to speed. A drawback of the diode bridge is that the polarity of the voltage at the rectifier output does not change when the speed changes from \(-\omega\) to \(+\omega\). A phase discriminator cannot be used instead of the diode bridge, since the phase of the tachogenerator output voltage depends on the speed. This drawback is eliminated if the speeds of the motor under test, AD, and of an auxiliary motor, VD, rotating at constant speed, are added by means of a mechanical differential. In this case it is possible to ensure that, when the motor under test AD is reversed from \(-\omega\) to \(+\omega\), the speed at the differential output changes from 0 to \(j\omega\), where \(j\) is the transmission ratio of the reducer. With motor AD stationary, the speed at the differential output is equal to

\[ \frac{j\omega}{2}; \]

the voltage corresponding to this speed after rectification is taken as zero. When the speed of AD changes from 0 to \(+\omega\), the voltage at the rectifier-bridge output will increase, and when the speed changes from 0 to \(-\omega\), it will decrease.

Fig. 3a. Mechanical characteristic of motor IZO-31/4, \(P_n = 1.5\ \text{kW}\).

Fig. 3b. Mechanical characteristic of motor A-31/4, \(P_n = 0.6\ \text{kW}\).

In this way the mechanical characteristic can be obtained in all quadrants.

A ball bearing can successfully be used as the differential. The ball separator rotates at a speed proportional to the sum or difference of the angular velocities of the outer and inner rings (assuming that there is no slip of the balls).

From Fig. 4 it is seen that the velocity of point \(A\) is equal to

\[ V_A=\omega_1 R_1, \tag{8} \]

or

\[ V_A=\omega_4 R_1+\omega_3 R_3. \tag{9} \]

The velocity of point \(B\) is equal to

\[ V_B=\omega_2 R_2, \tag{10} \]

or

\[ V_B=\omega_4 R_2-\omega_3 R_3, \tag{11} \]

where \(\omega_1,\ \omega_2,\ \omega_3,\ \omega_4\) are the angular velocities of the inner ring, the outer ring, the balls, and the separator;
\(R_1,\ R_2,\ R_3\) are the radii of the inner ring, the outer ring, and the balls.

Fig. 4.

From equations (8) and (9) we have

\[ \omega_1 R_1=\omega_4 R_1+\omega_3 R_3, \tag{12} \]

and from equations (10) and (11)

\[ \omega_2 R_2=\omega_4 R_2-\omega_3 R_3. \tag{13} \]

Solving equations (12) and (13) with respect to \(\omega_4\), we obtain

\[ \omega_4=\frac{\omega_1 R_1+\omega_2 R_2}{R_1+R_2}. \tag{14} \]

In the same way we obtain the formula for the opposite direction of \(\omega_2\)

\[ \omega_4=\frac{\omega_1 R_1-\omega_2 R_2}{R_1+R_2}. \tag{15} \]

Structurally, a differential made from a ball bearing is easily feasible (Fig. 5). The rotation of the tested motor \(AD\) is transmitted by means of a cardan shaft (1) and a cage (2) to the outer ring—

with the ball bearing (3). The inner ring of the ball bearing is press-fitted onto the shaft of the auxiliary motor AM. To transmit rotation from the separator to the rotor of the two-phase tachogenerator, there is a coupling (4), whose cams enter the recesses of the separator, and a toothed (or friction) pair (5). The coupling rotates freely on the ball bearing (6), mounted on the shaft of the auxiliary motor. The experiments carried out showed that the mechanical characteristics obtained with the described setup with a ball-bearing differential are entirely analogous to the characteristics obtained with a setup using a gear differential. This shows that in this case there is no slipping of the balls relative to the rings.

The described setup has the following advantages:

1) the asynchronous tachogenerator provides a voltage with high accuracy (up to 0.1%) proportional to the motor speed;

Fig. 5.

2) owing to the use of a voltage with a frequency of 1000 Hz, after rectification it proved possible to use a smoothing filter with very low inertia;

3) high differentiation accuracy is obtained owing to the small time constant of the differentiating circuit. With capacitance \(C = 0.5\ \mu\text{F}\) and resistance \(R = 20\ \text{k}\Omega\), the time constant of the differentiating circuit is equal to

\[ T_c = R \cdot C = 20 \cdot 10^3 \cdot 0.5 \cdot 10^{-6} = 0.01\ \text{s}. \]

With this time constant, already after 0.03 sec the differentiation accuracy reaches 5%¹);

4) the setup has small overall dimensions and weight, and is assembled from readily available materials;

5) the setup is insensitive to motor vibrations. The torque scale when recording the mechanical characteristic can be determined by comparison with the starting torque of the motor, which is measured by means of a lever and a spring dynamometer (Fig. 6)

\[ M_{\text{p}}=P\cdot l,\ \text{kg m}, \]

where \(P\) is the dynamometer reading, kg;
\(l\) is the arm, m.

This starting torque is compared with the deflection of the oscillograph beam along the acceleration axis at a motor speed equal to zero. If the measured deflection is \(x_0\), mm, then the torque scale is determined from the formula

\[ \mu_M=\frac{M_{\text{p}}}{x_0},\ \frac{\text{kg m}}{\text{mm}}. \tag{16} \]

The motor torque at other sections of the oscillogram is determined from the formula

\[ M_i=x_i\mu_M=x_i\frac{P\cdot l}{x_0},\ \text{kg m}, \tag{17} \]

where \(x_i\) is the deflection of the oscillograph beam at speed \(\omega_i\). The relative error in determining the torque is calculated from the formula

\[ \frac{\Delta M}{M}=\frac{\Delta x_i}{x_i}+\frac{\Delta P}{P}+\frac{\Delta l}{l}+\frac{\Delta x_0}{x_0}. \tag{18} \]

If it is assumed that the quantities entering formula (18) are determined with a relative error of 2%, then the relative error in measuring the torque will not exceed 8–10%.

Fig. 6.

In the setup an EO-7 oscillograph with a 13LO36V cathode-ray tube (with afterglow) was used. As the auxiliary motor \(VD\), a three-phase induction motor DT-75, 75 W, 2800 rpm, was used.

In making the ball-bearing differential, ball bearing No. 202 was used.

¹) The solution of equation (6) has the form

\[ i=C\frac{dU_{m2}}{dt}\left(1-e^{-\frac{t}{T_c}}\right). \]

The accuracy of differentiation is determined by the binomial \(\left(1-e^{-\frac{t}{T_c}}\right)\). It is easy to calculate that \(\left(1-e^{-\frac{t}{T_c}}\right)=0.95\) already at time \(t=0.03\) sec.

An asynchronous two-phase motor with a hollow nonmagnetic rotor, EM-8, was used as the tachogenerator.

The installation can be used by machine-building plants for series testing of asynchronous motors, as well as by research laboratories.

REFERENCES

1. E. N. Sinelnikov, G. S. Somikhina. Measurement of the torques of asynchronous motors during run-up. Elektrichestvo, 1950, No. 7.

2. Fetter F., Basta J. Nove poznatky o měřeni zrychleni pomoci dynama. Sbornik II. Vědecké konference F. S. I., Prague, 1958 (see the journal of the Central Institute of Scientific and Technical Information of the Electrical-Engineering Industry and Instrument Engineering, series TS-1, P-5954, Moscow, 1961).

3. Führer H. und Waidmann L. Ein einfaches Verfahren zur Aufnahme der Drehmomenten—Drehzahlkennlinie von Asynchronmotoren, Elektrotechnik und Maschinenbau, No. 1, 1962.

4. V. A. Glebov. Experimental study of the starting processes of asynchronous motors. Elektrichestvo, No. 12, 1956.

Received by the editors
in May 1962.