Bulletin of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov

Vol. 117　　　　　　　　　　　　　　　　　　　　　　　　　　　1963

APPARATUS FOR RECORDING THE DYNAMIC MECHANICAL CHARACTERISTICS OF AN ASYNCHRONOUS MOTOR

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

In the works of E. A. Kazovsky [1], V. A. Glebov [2], L. P. Petrov [3, 4], and other authors, it has been shown that for an accurate calculation of the electromechanical transients of asynchronous motors it is necessary to use dynamic mechanical characteristics. These characteristics are absent from technical reference literature; therefore, for each individual case they must be determined experimentally. Below is a description of a simple apparatus for recording the dynamic mechanical characteristics of a motor.

The block diagram of the apparatus is shown in Fig. 1. The motor under investigation, \(AD\), is connected to an asynchronous tachogenerator \(TG\). As the tachogenerator, a two-phase motor with a hollow nonmagnetic rotor (a Ferraris motor) of type \(EM\)-8 is used. To the primary

Fig. 1.

winding of the tachogenerator \(O1\), a sinusoidal voltage at a frequency of \(1000 \div 1500\) Hz and a voltage of about 5 V is supplied. In the secondary winding \(O2\), an alternating emf is induced whose magnitude is proportional to the rotational speed of the motor, whose frequency is equal to the frequency of the voltage applied to the primary winding, and whose phase depends on the speed and direction of rotation of the tachogenerator rotor. With the aid of the electronic amplifier \(U\), this emf is amplified in voltage and in

power. From the output transformer Tr, the amplified voltage enters the speed channel \(KS\) and the acceleration channel \(KU\), where it is rectified by diode bridges \(VS\) and \(VU\) and smoothed by low-frequency filters \(FS\) and \(FU\). The rectified voltage in the speed channel is taken from a potentiometer and fed to the horizontally deflecting plates of the electron-beam oscilloscope EO. The voltage rectified in the acceleration channel is fed to a differentiating \(R-C\) circuit. Across the resistance \(R\) a voltage is developed that is proportional to the acceleration of the motor; it is amplified by the direct-current amplifier UP and fed to the vertically deflecting plates of the electron-beam oscilloscope.

During starting or braking of the motor under test \(AD\), the electron beam on the oscilloscope screen traces a curve representing the dependence of speed on motor acceleration,

\[ \omega = f\!\left(\frac{d\omega}{dt}\right). \]

On the basis of D’Alembert’s theorem, the angular acceleration in the rotation of a body is equal to the sum of the moments of the driving forces about the axis of rotation divided by the moment of inertia about the same axis,

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

where \(J_{\text{dv}}\) is the moment of inertia of the motor, kg·m·sec\(^2\);

\(M_{\text{dv}}\) is the motor torque, kg·m;

\(M_{\text{st}}\) is the moment of static resistance, kg·m;

\(\dfrac{d\omega}{dt}\) is the angular acceleration, \(\dfrac{1}{\text{sec}^{2}}\).

During starting or braking of the motor without load, the moment of static resistance may be neglected. In this case, proceeding from formula (1), the motor torque is equal to

\[ M_{\text{dv}} \simeq J_{\text{dv}}\,\frac{d\omega}{dt}. \tag{2} \]

Thus, the motor torque is proportional to the angular acceleration, with \(J_{\text{dv}}\) being the coefficient of proportionality. Consequently, from the graphical representation of the dependence

\[ \omega = f\!\left(\frac{d\omega}{dt}\right) \]

it is easy to pass to the dependence \(\omega=\varphi(M_{\text{dv}})\), that is, to the mechanical characteristic of the motor; for this purpose it is necessary to multiply the acceleration scale \(\mu_{d\omega/dt}\) by the constant value of the moment of inertia \(J_{\text{dv}}\).

Fig. 2.

Let us determine the accuracy with which the angular acceleration of the motor is determined when differentiating a voltage proportional to speed. For this purpose let us consider an electric circuit consisting of a capacitance \(C\) and a resistance \(R\) (Fig. 2). The voltage \(U_{\text{in}}\) applied to the circuit is equal to the sum of the voltage drops across \(R\) and \(C\):

\[ U_{\text{in}}=U_c+U_{\text{out}}. \tag{3} \]

Differentiating equality (3), we obtain

\[ \frac{dU_{\text{in}}}{dt}=\frac{dU_c}{dt}+\frac{dU_{\text{out}}}{dt}. \tag{4} \]

The voltage drop across the resistance \(R\) is equal to

\[ U_{\text{out}}=iR=RC\frac{dU_c}{dt}. \tag{5} \]

Hence

\[ \frac{dU_c}{dt}=\frac{U_{\text{out}}}{RC}. \tag{6} \]

Substituting (6) into formula (4), we obtain

\[ \frac{dU_{\text{in}}}{dt}=\frac{U_{\text{out}}}{RC}+\frac{dU_{\text{out}}}{dt}, \]

or

\[ RC\frac{dU_{\text{out}}}{dt}+U_{\text{out}}=RC\frac{dU_{\text{in}}}{dt}. \tag{7} \]

When the input voltage \(U_{\text{in}}\) varies linearly with time, that is,

\[ U_{\text{in}}=\frac{dU_{\text{in}}}{dt}\cdot t, \]

then the exact solution of equation (7) has the form

\[ U_{\text{out}}=RC\frac{dU_{\text{in}}}{dt}\left(1-e^{-\frac{t}{RC}}\right). \tag{8} \]

From this expression it is evident that theoretically the output voltage reaches the steady-state value

\[ U_{\text{out}}=RC\frac{dU_{\text{in}}}{dt} \]

only after an infinitely long time. But if a small error is allowed in determining \(U_{\text{out}}\), for example \(2\%\), then in practice the output voltage reaches the steady-state value already after a time equal to

\[ t=4RC. \]

From analysis of oscillograms of electromechanical transient processes it has been established that during the time \(t=0.002\ \text{sec}\) the motor speed may be considered to vary according to a linear law [2], [3]. Therefore the time constant of the differentiating circuit must be no greater than

\[ RC\leq \frac{t}{4}=\frac{0.002}{4}=0.0005\ \text{sec}. \]

With a capacitance \(C=0.5\ \mu\text{F}\), the output resistance must be equal to

\[ R=\frac{0.0005}{0.5}\cdot 10^6=1000\ \Omega. \]

To determine the absolute value of the acceleration from the deviation

of the beam on the oscilloscope screen, it is necessary to know the acceleration scale. It can be determined in several ways.

First method. A sinusoidal voltage with known amplitude and frequency is applied to the input of the differentiating circuit,

\[ U_{\mathrm{in}}=U_m \sin(\omega t+\varphi). \]

At the output of the differentiating circuit, the voltage is proportional to the derivative of the input signal (with some error inherent in the differentiating circuit),

\[ U_{\mathrm{out}}=RC\,\frac{dU_{\mathrm{in}}}{dt} =RC\cdot U_m\cdot \omega \cdot \cos(\omega t+\varphi). \]

The maximum deflection of the oscilloscope beam is proportional to the maximum value of the derivative,

\[ \left(\frac{dU_{\mathrm{in}}}{dt}\right)_{\max}=U_m\cdot \omega . \]

Let us denote this deflection by the letter \(x\).

The scale is found from the relation

\[ \mu_{\frac{dU}{dt}}=\frac{U_m\cdot \omega}{x},\ \frac{v}{\mathrm{sec}\cdot \mathrm{mm}} . \tag{9} \]

When taking the mechanical characteristic, the input voltage varies in proportion to the motor speed,

\[ U_{\mathrm{in}}=k\cdot n_{\mathrm{dv}}, \]

where

\[ k=\frac{U_0}{n_0} \]

is the velocity coefficient of the tachogenerator and amplifier, equal to the ratio of the voltage at the filter output at no-load speed to this speed.

Consequently, the acceleration scale is equal to

\[ \mu_{\frac{dn}{dt}} =\mu_{\frac{dU}{dt}}\cdot k =\frac{U_m\cdot \omega \cdot n_0}{x\cdot U_0}, \ \frac{\mathrm{rev}}{\mathrm{min}\cdot \mathrm{sec}\cdot \mathrm{mm}} . \tag{10} \]

Second method. A small pulley of radius \(R\) is mounted on the shaft of the tachogenerator TG (Fig. 3), around which several turns of cord are wound. A weight \(G\) is attached to the cord and allowed to descend freely. Because of its small magnitude, friction in the tachogenerator bearings may be neglected.

According to d’Alembert’s theorem, the forces acting on the material system and the inertia forces are equal:

\[ R\left(G-\frac{G}{g}\cdot a\right)=J\frac{d\omega}{dt}, \tag{11} \]

where \(J\) is the moment of inertia of the rotor of the tachogenerator and the pulley, \(\mathrm{kg\,m\,sec^2}\);

\(a\) is the acceleration of the translationally moving weight \(G\), \(\frac{\mathrm{m}}{\mathrm{sec}^2}\);

\[ g=9.81\,\frac{\mathrm{m}}{\mathrm{sec}^2} \]

is the acceleration due to gravity.

We use the relation between angular and linear acceleration

\[ a = R \frac{d\omega}{dt}. \tag{12} \]

Substituting (12) into (11), after a number of transformations we obtain

\[ \frac{d\omega}{dt} = \frac{G}{\dfrac{J}{R}+\dfrac{RG}{g}} \tag{13} \]

or

\[ \frac{dn}{dt} = \frac{30G}{\pi\left(\dfrac{J}{R}+\dfrac{RG}{g}\right)}. \tag{14} \]

Having measured on the oscilloscope screen the deflection of the beam \(x\) in mm corresponding to this acceleration, we find the acceleration scale

Fig. 3.

\[ \mu_{\frac{dn}{dt}} = \frac{30G}{\pi\left(\dfrac{J}{R}+\dfrac{RG}{g}\right)\cdot x}, \quad \frac{\mathrm{rev}}{\mathrm{min}\cdot \mathrm{sec}\cdot \mathrm{mm}}. \tag{15} \]

The motor torque corresponding to the deflection \(x_1\) of the electron beam along the acceleration axis when taking the mechanical characteristic is found from the formula

\[ M_{\mathrm{dv}} = \frac{J_{\mathrm{dv}}\pi}{30}\, x_1 \mu_{\frac{dn}{dt}}. \tag{16} \]

Conclusions

The described setup makes it possible to obtain, with sufficient accuracy, the dynamic mechanical characteristics of an asynchronous motor. Unlike other setups [2], [3], [4], it is simple to manufacture and is assembled from readily available materials.

The mechanical characteristics are observed directly on the oscilloscope screen and can easily be photographed with an ordinary camera. To facilitate visual observation, it is recommended to use in the oscilloscope a cathode-ray tube with afterglow.

Fig. 4.

Labels in the figure:

• \(n\)
• \(\dfrac{\omega}{\min}\)
• Static characteristic
• Dynamic characteristic
• \(M\), kg·m

Fig. 5.

Labels in the figure:

• \(M\), kg·m
• Static characteristic
• Dynamic characteristic
• \(n\)
• \(\dfrac{\omega}{\min}\)

In Figs. 4, 5, and 6 the dynamic mechanical characteristics of motor A 31-4 are given: for the motoring mode (Fig. 4), the plugging mode (Fig. 5), and dynamic braking (Fig. 6). In the figures, for comparison, the static mechanical characteristics of the same motor are also given, obtained on an analog setup, but with the use of a mechanical differential, which is necessary for distributing the mechanical characteristics of the various modes among the corresponding quadrants. The additional flywheel increases the duration of the mechanical transient processes of the motor, owing to which the mechanical characteristic approaches the static one.

Fig. 6.

LITERATURE

1. E. Ya. Kazovskii. Some Questions of Transient Processes in Alternating-Current Machines. Gosenergoizdat, 1953.

2. V. A. Glebov. Experimental investigation of starting processes of asynchronous motors. Elektrichestvo, No. 12, 1956.

3. L. P. Petrov. Dynamic characteristics of asynchronous squirrel-cage motors. VEI, No. 5, 1959.

4. L. P. Petrov. New methods for the experimental investigation of the mechanical characteristics of asynchronous motors. Scientific Notes of the Odessa Polytechnic Institute, 1958, Vol. XVII.

Received by the editors
in June 1962.