CALCULATION OF MOTOR CHARACTERISTICS IN AUTOMATED MSA–D SYSTEMS BY THE METHOD OF MAGNETIC-AMPLIFIER LINEARIZATION

M. A. BOROVIKOV

(Presented by the scientific seminar of the Electromechanical Faculty)

Existing graphical methods for calculating motor characteristics in magnetic-amplifier systems with self-saturation—direct-current motor (MSA–D) systems [1–3]—have the shortcomings inherent in all graphical methods: considerable bulkiness, laboriousness, and limited accuracy, so that in a number of cases the possibility of taking nonlinearities into account remains only potential and does not lead to the desired accuracy.

In this connection it is expedient to use calculation methods based on the linearization of the characteristics of drive elements over specified sections. In this case the principal computational work can be carried out analytically, while graphical constructions may be used only to determine parameters.

If the region of small load currents is excluded from consideration, in which the inductance of saturated reactors and magnetizing currents manifest themselves strongly, then, for calculation purposes, one may introduce the concept of the “no-load characteristic” of the MA and apply methods for calculating characteristics analogous to those used in analyzing the classical G–D system. The “no-load” characteristic of an MA operating in continuous-current mode can be constructed from the demagnetization characteristics of the cores (Figs. 1 and 2) on the basis of the equations of the mechanical characteristics of the motor [4, 5].

For a single-phase amplifier we have [4]:

\[ U_0 = \frac{2U_m}{\pi}\left(1 - \frac{\beta_y}{2\xi}\right); \tag{1} \]

for a three-phase amplifier [5]:

\[ U_0 = \frac{U_m}{\pi}\left(1 - \frac{\beta_y}{\xi}\right). \tag{2} \]

Assigning values of the m.m.f. or of the control current, from the demagnetization curves of Fig. 1 or 2 we find \(\beta_y/\xi\) and calculate \(U_0\).

Fig. 1. Demagnetization characteristic of the cores of a single-phase MVS.

Figure 3 shows the “no-load” characteristic for a three-phase amplifier with an output power of 5.5 kW, used in the serial PMU-9M drive. Three-phase amplifiers are manufactured for higher powers. Therefore it is more advantageous to make them with two control windings: the actual control winding and the bias winding. In this case there is no need to operate with the m.m.f.; all calculations are simpler to carry out with respect to the control current, since control is effected by only one winding. For this reason Fig. 3 gives the dependence of the voltage \(U_o\) on the control current \(I_y\); the curve of the amplification factor is also plotted there:

\[ k = \frac{\Delta U_o}{\Delta I_y}. \tag{3} \]

Below we shall describe the calculation procedure as applied to systems with a three-phase amplifier. Incidentally, this method is more suitable precisely for three-phase systems, since the region of linearity of external ...

characteristics is considerably greater for them. In Fig. 4, as an example, two drive circuits with a three-phase MU are shown: in circuit a) a PMU drive is used; in circuit b) an intermediate magnetic amplifier \(MU_2\) is employed. In both cases, positive current feedback is introduced into the common loop, which makes it possible to bring these systems closer to a circuit with speed feedback. We shall derive the expression for the speed characteristic for the circuit of Fig. 4a.

Fig. 2. Demagnetization characteristic of the cores of a three-phase MUS.

Labels in the figure: \(\Delta B\), \(\mathrm{kGs}\); \(B_y/\xi\); \(\Delta B=f(I\omega)_y\); \(B_y/\xi=f(I\omega)_y\); \((I\omega)_{\mathrm{st}}\); \((I\omega)_y\); \(0, 30, 60, 90, \alpha\).

The control current of the MU is determined by the expression:

\[ I_y=\frac{\sigma U_p}{R_y}-\frac{U_y}{R_y}+\frac{U_{\mathrm{ot}}}{R_y}, \tag{4} \]

where \(\sigma\) is the relative displacement of the slider of the setting potentiometer;

\(R_p\) is its resistance;

\[ R_y=r_y+r_y+\sigma R_p(1-\sigma) \]

is the total resistance of the comparison circuit.

The resistance \(R_t\), across which the current-feedback voltage \(U_{\mathrm{ct}}\) is developed, is not included in the total resistance, since the internal resistance of the source of positive feedback in the direction of flow of the control current \(I_y\) is close to zero (valves in the forward direction).

The voltage at the armature may be represented as follows:

\[ U_{ya}=U_{0}-I_{ya}R_{mu}, \tag{5} \]

Fig. 3.

Labels in the figure: \(U_0\), V; \(K\), V/A; \(U_0=f(I_{ya})\); \(K=f(I_{ya})\); \(A_4\); \(A_3\); \(I_{ya}\), mA.

where \(R_{mu}\) is the equivalent resistance of the magnetic amplifier, which determines the slope of its external characteristic.

The no-load voltage of the magnetic amplifier in the general case is determined by the expression

\[ U_0=A+kI_{ya}, \tag{6} \]

where \(A\) is a certain constant voltage when a piecewise-linear approximation is used (Fig. 3).

The expression for the voltage of the current-feedback signal in the general case has the form

\[ U_{ot}=C+\beta I_{ya}, \tag{7} \]

where \(C\) is a certain constant voltage (Fig. 5) of the source of the current-feedback signal;

Fig. 4.

Fig. 5.

\(\beta\) is the coefficient of current feedback (current compensation). Substituting (5)–(7) into (4), after minor transformations, we obtain the final expression for the control current:

\[ I_y=\frac{\sigma U_{\mathrm{p}}}{R_y+k}+\frac{C-A}{R_y+k}+\frac{\beta+R_{\mathrm{mu}}}{R_y+k}I_a . \tag{8} \]

The motor speed in the general form is determined by the expression

\[ n=\frac{U_0}{c_E}-\frac{I_a}{c_E}R_{\mathrm{e}}, \tag{9} \]

where \(R_{\mathrm{e}}\) is the resistance of the armature circuit of the motor when it is supplied from the MU (in the general case

\[ R_{\mathrm{e}}>r_a+2r_{\mathrm{v}}+2r_{\mathrm{r}}=R \]

because of the influence of commutation).

Substituting (8) into (6), and the expression obtained into (9), we derive the equation of the speed characteristic of the motor in the circuit of Fig. 4a:

\[ n=\frac{A}{c_E}+\frac{k\sigma U_{\mathrm{p}}}{(R_y+k)c_E} +\frac{(C-A)k}{(R_y+k)c_E} -\frac{I_a}{c_E}\left[ R_{\mathrm{e}}-\frac{k(\beta+R_{\mathrm{mu}})}{R_y+k} \right]. \tag{10} \]

Expressions for the circuit of Fig. 4b (and, in general, for any circuit) can be obtained analogously:

\[ I_{y2}=\frac{\sigma U_{\mathrm{p}}}{R_{y2}+k_1k_2} +\frac{C-(A+Bk_1)}{R_{y2}+k_1k_2} +\frac{\beta+R_{\mathrm{mu}}}{R_{y2}+k_1k_2}I_a, \tag{11} \]

\[ n=\frac{A+Bk_1}{c_E} +\frac{k_1k_2\sigma U_{\mathrm{p}}}{c_E(R_{y2}+k_1k_2)} +\frac{[C-(A+Bk_1)]k_1k_2}{c_E(R_{y2}+k_1k_2)} -\frac{I_a}{c_E} \left[ R_{\mathrm{e}}-\frac{k_1k_2(\beta+R_{\mathrm{mu}})}{R_{y2}+k_1k_2} \right], \tag{12} \]

where \(I_{y2}\) is the control current of the intermediate magnetic amplifier;

\(k_2=\Delta I_{y1}/\Delta I_{y2}\) is the gain coefficient of the intermediate amplifier;

\(I_{y1}=B+k_2 I_{y2}\) is the control current of the power MU;

\(B\) is the initial control current of the intermediate MU according to the control characteristic (Fig. 6) under its linear approximation.

The total resistance of the comparison circuit in this case is also determined by the expression

\[ R_{y2}=r_{y2}+r_a+R_{\mathrm{p}}\sigma(1-\varepsilon). \]

Expressions (10) and (12) make it possible to take into account the nonlinearity of the characteristics of the elements of the systems in Fig. 4. In this case the calculation is carried out by the method of successive approximations: for a given \(\sigma\), the motor speed is determined approximately, \(n\cong\sigma n_{\mathrm{nom}}\), and the output voltage of the amplifier \(U_0\); from the value of \(U_0\), using the characteristics of Figs. 3, 5, and 6, \(k_1\), \(k_2\), \(A\), \(B\), and \(C\) are found; then, with the aid of expressions (8) or (11), the selection of these quantities is refined. After satisfactory refinement results have been obtained, the characteristic is calculated for the specified \(\sigma\).

The method described gives results quite acceptable for practice; in addition, expressions (10) and (12) can be used for selecting the parameters of the system in its design. This is the main advantage of the linearization method.

Fig. 6.

In conclusion it is necessary to emphasize that expressions (11), (12) are valid when there is a source of smoothed reference voltage. Supplying the control circuits from a source of pulsating voltage, as is the case in the standard PMU circuits [6], significantly lowers the sensitivity of the measuring circuit; moreover, the pulsating voltage is transformed from one winding into another, creating unidirectional currents in the control circuits by feeding them from rectifier sources (for example, when an intermediate magnetic amplifier is present). All this disrupts the normal operation of the power magnetic amplifier and makes adjustment difficult. In this connection, where it is necessary to have high rigidity of the motor characteristics, it is recom—

It is necessary to use smoothing filters for the control voltages, and in the bias circuit of the magnetic amplifier, when it is supplied from pulsating-voltage sources, it is advisable to install smoothing chokes in order to reduce interference in the control circuits, which ultimately leads to a significant reduction in the gain factor of the magnetic amplifier.

REFERENCES

1. D. A. Alenchikov and V. S. Kulebakin. A method for calculating the characteristics of direct-current motors with choke control. Avtomatika i telemekhanika, vol. XX, No. 7, 1959.

2. D. A. Alenchikov. A method for calculating the characteristics of a direct-current electric drive with choke control, based on the use of families of external characteristics of rectifier-choke converters. Transactions of the Scientific Research Institute of the Electrical Industry, vol. 5, 1959.

3. M. A. Borovikov. A graphical method for calculating the characteristics of direct-current motors in the magnetic-amplifier—motor system. Izvestiya TPI, vol. 117, 1963.

4. M. A. Borovikov. Calculation of the characteristics of a direct-current motor controlled by means of a single-phase self-saturating magnetic amplifier. Interuniversity collection on problems of production automation. Novosibirsk, 1964.

5. M. A. Borovikov. Operation of a three-phase self-saturating magnetic amplifier on the armature of a separately excited direct-current motor. Proceedings of the Fourth All-Union Conference on Automated Electric Drive. Publishing House “Energiya,” 1965.

6. A. Ya. Petrovsky, Ya. B. Rozman. Controlled Electric Drive with Magnetic Amplifiers. Publishing House “Energiya,” 1964.