STUDY OF ELECTROMECHANICAL TRANSIENT PROCESSES IN ROLLER-TABLE MOTORS ON AN ANALOG COMPUTER TAKING INTO ACCOUNT STEEL SATURATION BY LEAKAGE FLUXES

E. V. Kononenko, B. A. Danchinov

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

The basic equations describing transient processes in asynchronous machines, when the combined action of all three phases is expressed by generalized vectors in a coordinate system rotating with an arbitrary angular velocity \(\omega_k\), have the form:

a) voltage-balance equations

\[ \overline{u}_s = r_s \overline{i}_s + \frac{d\overline{\Psi}_s}{dt} + j\omega_k \overline{\Psi}_s, \tag{1} \]

\[ \overline{u}_r = r_r \overline{i}_r + \frac{d\overline{\Psi}_r}{dt} + j(\omega_k-\omega)\overline{\Psi}_r; \]

b) equations of drive motion

\[ \frac{J}{p}\cdot \frac{d\omega}{dt}=M-M_c. \tag{2} \]

where

\[ \overline{\Psi}_s=L_s\overline{i}_s+L_m\overline{i}_r, \]

\[ \overline{\Psi}_r=L_r\overline{i}_r+L_m\overline{i}_s, \]

\[ M=\frac{3}{2}p\left|\overline{\Psi}_s\times \overline{i}_s\right| = \frac{3}{2}p\frac{L_m}{L_r' L_s}\left|\overline{\Psi}_r\times \overline{\Psi}_s\right| = \frac{3}{2}p\frac{L_m}{L_s' L_r}\left|\overline{\Psi}_r\times \overline{\Psi}_s\right|, \]

\[ L_s'=L_s-\frac{L_m^2}{L_r}, \]

\[ L_r'=L_r-\frac{L_m^2}{L_s}. \]

In these expressions

\[ \overline{U}_s(\overline{U}_r),\ \overline{i}_s(\overline{i}_r),\ \overline{\Psi}_s(\overline{\Psi}_r) \]
— resultant vectors of voltage, current, and flux linkage of the stator (rotor),

\[ r_s(r_r) \]
— active resistance of the stator (rotor),

\[ \omega \]
— angular velocity of the rotor,

\[ \omega_k \]
— angular velocity of the coordinate axes,

\[ L_s(L_r) \]
— total inductance of the stator (rotor),

\[ L_m \]
— mutual inductance of the windings,

\[ L_s'(L_r') \]
— transient inductance of the stator (rotor),

$M$ — driving torque of the motor,
$M_c$ — load resisting torque,
$J$ — moment of inertia referred to the motor shaft,
$p$ — number of pole pairs,
$t$ — time.

The system of equations (1) and (2) is a system of nonlinear differential equations that has no rigorous analytical solution. If one takes into account that the actual parameters of the machine $r_s$, $r_r$, $L_s$, $L_r$, $L_m$, which enter these equations as coefficients, are in the general case variables depending on the rotor speed and on the currents, then the analytical study of transients becomes still more complicated.

Therefore, the most rational method for comprehensive investigations of transients should be recognized as the method of mathematical modeling on continuous-action analog computers.

It should be noted that the nature of the course of transients in asynchronous machines is substantially influenced by their parameters. In most cases the parameters of machines are considered constant; their values are chosen depending on the problem under study, and it is recommended that these values be chosen to correspond to the initial condition of the transient process [1].

The assumption that the parameters are constant considerably simplifies the study of transients; however, in some cases it may lead to erroneous results [2, etc.].

Recently a number of works have been published devoted to taking account of the varying parameters of asynchronous machines with the use of analog and digital computers [2–5, etc.].

Taking variable parameters into account is associated with the difficulty of obtaining their real dependences as functions of rotational speed or current. Modeling transients in the presence of such dependences, however, presents no difficulty. The change in motor parameters is due to the phenomenon of skin effect (current displacement) and saturation of the machine steel, and above all of the tips of its teeth.

An analysis of the parameters of AR-series roller-table motors showed that their active resistances and leakage reactances do not depend on the phenomenon of current displacement. They are practically independent of saturation by the main flux, as is the mutual-induction reactance $X_m$. Processing of short-circuit test data established that, for most types of motors, the leakage reactances depend to a significant degree on the saturation of the tooth tips by leakage fluxes; moreover, within the range of stator-current variation from the no-load current to the starting value, these reactances vary according to a linear law of the form

\[ X_k = X_1 + X_2' = a - b I_k, \tag{3} \]

where
$X_k$ — short-circuit reactance,
$X_1$ — stator leakage reactance,
$X_2'$ — rotor leakage reactance referred to the stator.

At currents smaller than $I_0$ and greater than $I_{\text{start}}$, $X_1$ and $X_2'$ may be considered constant: in the first case unsaturated, in the second saturated.

Thus, when calculating transients in roller-table motors, it is necessary to take into account the variation of the stator and rotor leakage reactances due to saturation of the tooth layer by leakage fluxes; the remaining parameters may be considered constant.

It is known that mathematical modeling by equations (1) and (2) in complex notation is very difficult; therefore it is more convenient to express them in a rectangular coordinate system with axes \(\alpha, \beta\) of the complex plane, i.e., by the equations of an idealized two-phase machine.

The differential and algebraic equations of a short-circuited asynchronous motor in a form convenient for modeling, in relative units, in the coordinate system \(\alpha, \beta\), rotating at synchronous speed, can be represented as

\[ \frac{d\Psi_{s\alpha}}{dt}=u_{s\alpha}-r_s i_{s\alpha}+\Psi_{s\beta}, \]

\[ \frac{d\Psi_{s\beta}}{dt}=u_{s\beta}-r_s i_{s\beta}-\Psi_{s\alpha}, \]

\[ \frac{d\Psi_{r\alpha}}{dt}=-r_r y_s\Psi_{r\alpha}+r_r y_m\Psi_{s\alpha}+\Psi_{r\beta}-\omega\Psi_{r\beta}, \]

\[ \frac{d\Psi_{r\beta}}{dt}=-r_r y_s\Psi_{r\beta}+r_r y_m\Psi_{s\beta}-\Psi_{r\alpha}+\omega\Psi_{r\alpha}, \]

\[ i_{s\alpha}=y_r\Psi_{s\alpha}-y_m\Psi_{r\alpha}, \tag{4} \]

\[ i_{s\beta}=y_r\Psi_{s\beta}-y_m\Psi_{r\beta}, \]

\[ i_s=\sqrt{i_{s\alpha}^{2}+i_{s\beta}^{2}}, \]

\[ M=\frac{3}{2}I_m(\Psi_{r\alpha}\Psi_{s\beta}-\Psi_{s\alpha}\Psi_{r\beta}), \]

\[ \frac{d\omega}{dt}=\frac{1}{J}(M-M_c). \]

Here

\[ u_{s\alpha},\ u_{s\beta} \]
— components of the stator voltage vector along the \(\alpha\) and \(\beta\) axes;

\[ \Psi_{s\alpha},\ \Psi_{s\beta}\ (\Psi_{r\alpha},\ \Psi_{r\beta}) \]
— components of the stator (rotor) flux-linkage vector along the \(\alpha\) and \(\beta\) axes;

\[ i_{s\alpha},\ i_{s\beta} \]
— components of the stator current vector along the \(\alpha\) and \(\beta\) axes;

\[ i_s \]
— resultant stator-current vector.

Allowance for the variation of leakage resistances was made by introducing nonlinear functions of the resultant stator current:

\[ y_s=\frac{x_s}{x_sx_r-x_m^2}=\frac{1}{x_r\sigma}=\frac{1}{x'_r}=f_1(i_s), \]

\[ y_r=\frac{x_r}{x_sx_r-x_m^2}=\frac{1}{x_s\sigma}=\frac{1}{x'_s}=f_2(i_s), \tag{5} \]

\[ y_m=\frac{x_m}{x_sx_r-x_m^2}=\frac{k_r}{x_s\sigma}=\frac{k_s}{x_r\sigma}=f_3(i_s), \]

where

\[ x_s=x_1+x_m\ (x=x'_2+x_m) \]
— total inductive reactance of the stator (rotor),

\[ x'_s=x_s-\frac{x_m^2}{x_r}\quad \left(x'_r=x_r-\frac{x_m^2}{x_s}\right) \]
— transient inductive reactance of the stator (rotor),

\[ \sigma=\frac{x_sx_r-x_m^2}{x_sx_r} \]
— Blondel leakage coefficient,

\[ k_s=\frac{x_m}{x_s}\quad\left(k_r=\frac{x_m}{x_r}\right) \]

— coupling coefficient of the stator (rotor).

The actual values \(y_s\), \(y_r\), and \(y_m\) were obtained as functions of the rms values of the stator phase current. Modeling them as functions of the modulus of the vector of the resultant current is justified by the fact that, in relative units, the amplitude values (equal to the resultant current) and the rms values of the phase currents are equal to one another.

Figure 1 presents the structural diagram of the model compiled for solving equations (4) and (5), whose implementation was carried out on an MN-14 analog computer.

Fig. 1. Structural diagram of the model for the system of equations (4) and (5)

The variable parameters \(y_s\), \(y_r\), and \(y_m\) were reproduced on universal nonlinear blocks H2, H3, and H4 with linear approximation, as shown in Fig. 2, where the variables \(i_s\), \(y_s\), \(y_r\), and \(y_m\) are represented in machine variables—voltages.

Fig. 2. Functional dependence accounting for saturation, for the AR 53-6 motor

The proposed model circuit makes it possible, when necessary, to perform calculations also with constant parameters. For this purpose, by setting the transfer coefficients \(K_8\) equal to zero, the nonlinear blocks H2—H4 are disconnected, and through the voltage dividers \(K_9\), \(K_{10}\), \(K_{11}\) voltages are supplied that correspond to the values of the constant parameters \(y_s\), \(y_r\), \(y_m\). In calculations taking account of parameter variation, the transfer coefficients \(K_8\) are set equal to unity, connecting the nonlinear blocks, while the coefficients \(K_9\), \(K_{10}\), and \(K_{11}\) are set equal to zero.

Figure 3 gives calculated oscillograms, recorded by a two-coordinate recording instrument DRP-2, of the starting and reversal of the roller-table motor AR 53-6 with the following parameters in relative units: \(r_s=0.0545\), \(r_r=0.206\), \(x_m=2.05\), \(J=61.5\), \(M_c=0\).

It is evident from the oscillograms that the resultant current \(i_s\) in the initial stage of the transient process pulsates strongly and, at the end of the process, smoothly decreases to the steady-state value. The current pulsations at the beginning of the transient process are explained by the appearance of its free (aperiodic) component, which, being superposed on the forced (periodic) component, causes pulsations of the transient current. After the aperiodic component has decayed, the pulsations disappear.

In accordance with the change in the current \(i_s\), the parameters \(y_s\), \(y_r\), \(y_m\) also change, assuming at the end of the transient process constant unsaturated values.

Figure 4 presents oscillograms of braking by short-circuiting the stator winding after disconnecting it from the supply network, and of braking by counterconnection of the same motor.

The proposed model circuit and the method for taking account of the change in parameters due to saturation by leakage fluxes make it possible to investigate various

Fig. 3. Calculated oscillograms of starting (a) and reversing (b) of an AR 53-6 motor.

modes: starting, reversal with a decayed and a nondecayed rotor field, reclosing at any initial speed, plugging braking, and braking by short-circuiting the stator winding after disconnecting it from the supply. In this case, the parameters of the motor under investigation may be either constant or varying according to any nonlinear law.

a)  b)

Fig. 4. Calculated oscillograms of braking by short-circuiting the stator winding after disconnecting it from the network (a) and of plugging braking (b) of an AR 53-6 motor

REFERENCES

1. M. M. Sokolov, L. P. Petrov, L. B. Masandilov, V. L. Ladenzon. Electromagnetic transient processes in an asynchronous electric drive. “Energiya,” 1967.

2. M. G. Zhuk, N. I. Zenkin, V. M. Kirpichnikov, N. I. Tomashevsky. Approximate allowance for variable parameters in calculating transient processes in an asynchronous motor. Izv. vuzov, “Gornyi zhurnal,” 1968, No. 6.

3. Wu Zha Han. Study of the influence of current displacement on the dynamics of an asynchronous motor. Izv. vuzov, “Elektromekhanika,” No. 7, 1966.

4. Wu Zha Han. Study of the influence of saturation on the starting dynamics of an asynchronous motor. “Elektrotekhnika,” No. 8, 1967.

5. V. F. Sivokobylenko, Ya. A. Oirekh. Study of the dynamic modes of a high-power asynchronous electric drive on an electronic analog computer. Izv. vuzov, “Elektromekhanika,” No. 4, 1968.