Graphical Method for Calculating the Characteristics of DC Motors in the Magnetic Amplifier–Motor System

M. A. Borovikov

(Presented by the scientific seminar of the Electromechanical Faculty)

In modern industrial electric drives, magnetic amplifiers (MAs) have found wide application for controlling DC motors in systems with a wide range of smooth speed regulation and as the final stage of amplifying tracking systems with DC motors [1 and 2]. The widespread use of MAs is explained by the fact that they have a number of advantages over other types of amplifiers: they possess high structural strength, have no rotating parts, require no maintenance, and are immediately ready for operation after being switched on. In this connection, the development of simple engineering methods for calculating the speed characteristics \(n = f(I_a)\) and mechanical characteristics \(n = f(M)\) of motors controlled by magnetic amplifiers is an important task.

Knowledge of the characteristics of the MA–motor system is necessary when calculating tracking systems in order to determine the maximum possible torque, speed, and acceleration at the output shaft, and to determine the minimum possible response time of the system to various disturbing influences. Knowledge of the mechanical and speed characteristics is absolutely necessary for analyzing the technical capabilities of drives regulated by means of magnetic amplifiers.

The magnetic amplifiers most widely used in automated electric drives are those connected according to circuits with self-excitation (with internal feedback), since such circuits give the same effect as circuits with external positive current feedback, but provide smaller dimensions and a higher efficiency of the amplifier for the same output power [1]. The purpose of the present work is to calculate the speed \(n = f(I_a)\) or mechanical \(n = f(M)\) characteristics of a separately excited DC motor controlled on the armature side by a magnetic amplifier connected according to a self-excitation circuit.

Existing methods for calculating the characteristics of a DC motor controlled by a magnetic amplifier are based on representing the operation of the amplifier on the motor in the form of a vector diagram

[3 and 4]. Thus, Fig. 1 shows the schematic circuit of an MA operating into an active load, the equivalent circuit of the amplifier, and the vector diagram for the equivalent sinusoidal voltages of the equivalent circuit, referred, for example, to the rectified-current side. All resistances of the equivalent circuit are variable when considering a motor load [4].

The initial data for calculations in all cases [3–5] are the volt-ampere characteristics of the working circuit of the MA, that is, the dependences of the rms voltage in the working circuit (the modulus of the vector \(cb\)) on the rms current \(U_{\text{MA}} = f(I_-)\) for different premagnetizing ampere-turns. For amplifier circuits without internal feedback [3 and 5], the volt-ampere characteristics can be constructed from the simultaneous magnetization curves of the steel, which are given in the extensive literature on the calculation of magnetic amplifiers. In the case of amplifier circuits with internal feedback,

Fig. 1. \(w_{\sim}\)—working winding of the magnetic amplifier (MA); \(w_y\)—control winding of the MA; \(w_{\mathrm{cm}}\)—bias winding; \(X_{\mathrm{MA}}\)—equivalent inductive resistance of the working winding; \(r_{\mathrm{MA}}\)—active resistance of the working winding; \(r_{\mathrm{v}}\)—valve resistance; \(U_{\mathrm{MA}}\)—voltage in the working circuit of the MA; \(U_{\mathrm{n}}\)—voltage across the load; \(I_{\mathrm{n}}\)—load current in average values.

to obtain satisfactory accuracy in calculating motor characteristics, it is necessary to use experimentally obtained volt-ampere characteristics [4].

With such a choice of initial data, the methods for calculating motor characteristics prove to be rather cumbersome, especially when the MA–motor system contains both internal and external feedbacks.

In the present work it is proposed, as the initial data for calculating the speed characteristics \(n=f(I_{\mathrm{a}})\) or mechanical characteristics \(n=f(M)\) of the motor, to take not the volt–ampere characteristics of the working circuit of the magnetic amplifier, but its load characteristics \(I_{\mathrm{n}}=f(Iw)_{\mathrm{y}}\), for various values of the active load resistance \(R_{\mathrm{n}}\) (Fig. 2, a), obtained, for example, according to the circuit of Fig. 1 by means of magnetoelectric instruments. Since the voltage across the load at constant \(R_{\mathrm{n}}\) is proportional to the current \(U_{\mathrm{n}}=I_{\mathrm{n}}\cdot R_{\mathrm{n}}\), the moduli of the vector \(\overline{ac}\) of the vector diagram in Fig. 1 are taken as known in this case. This, as will be shown below, leads to a very simple and clear graphical method, equally convenient for calculating motor characteristics both in open-loop and in closed-loop systems.

The use of the family of characteristics \(I_{\mathrm{n}}=f(Iw)_{\mathrm{y}}\) for various \(R_{\mathrm{n}}=\mathrm{const}\) (Fig. 2, a) is based on representing a direct-current motor in steady-state operation by an active resistance [4 and 6], whose magnitude depends on the speed, flux, and armature current and is determined by the expression

\[ R_{\mathrm{e}}=\frac{U_{\mathrm{a}}}{I_{\mathrm{a}}} =\frac{k_{\mathrm{e}}\Phi n}{I_{\mathrm{a}}}+R_{\mathrm{a}} . \tag{1} \]

For separately excited motors, neglecting armature reaction in the first approximation, this expression can be rewritten in the form

\[ R_{\mathrm{e}}=\frac{c_{\mathrm{e}}n}{I_{\mathrm{a}}}+R_{\mathrm{a}}, \tag{2} \]

Having the load characteristics of the magnetic amplifier \(I_{\mathrm{n}}=f(Iw)_{\mathrm{y}}\) for various values of \(R_{\mathrm{n}}\) (Fig. 2, a) and using expression (2), one can calculate the speed characteristic \(n=f(I_{\mathrm{a}})\) of the motor for fixed control ampere-turns, i.e., in an open-loop system (for example, the motor is connected according to the circuit of Fig. 1 as a load). To do this, by determining, for given \((Iw)_{\mathrm{y}}=\mathrm{const}\), the load currents \(I_{\mathrm{n}}\) from the load characteristics of Fig. 2, a for each \(R_{\mathrm{n}}\), and equating these resistance values to the equivalent resistance of the motor, from (2) one can each time find the speed \(n\) that, at the given current \(I_{\mathrm{n}}=I_{\mathrm{a}}\), ensures the selected equivalent input resistance of the motor \(R_{\mathrm{e}}=R_{\mathrm{n}}\). The expression for the speed is then obtained in the form

\[ n=\frac{(R_{\mathrm{e}}-R_{\mathrm{a}})\cdot I_{\mathrm{a}}}{c_{\mathrm{e}}}. \tag{3} \]

As a result of the calculations we obtain the dependence \(n=f(I_{\mathrm{a}})\) at \((Iw)_{\mathrm{y}}=\mathrm{const}\), that is, the speed characteristic of the motor. The mechanical characteristic at \(\Phi=\mathrm{const}\) differs from the speed characteristic only in scale.

In order to accelerate the process of calculating the speed characteristics \(n=f(I_{\mathrm{a}})\), one can construct auxiliary dependences of the armature current on speed when the motor operates with a constant equivalent input resistance, that is, the dependences \(I_{\mathrm{a}}=f(n)\) at

\[ \frac{U_{\mathrm{a}}}{I_{\mathrm{a}}}=R_{\mathrm{e}}=\mathrm{const}, \]

constructed according to the expression

\[ I_{\mathrm{a}}=\frac{c_{\mathrm{e}}n}{R_{\mathrm{e}}-R_{\mathrm{a}}}. \tag{4} \]

Neglecting the nonlinearity of the armature resistance, that is, taking \(R_{\mathrm{a}}=\mathrm{const}\), we obtain the dependences \(I_{\mathrm{a}}=f(n)\) for various \(R_{\mathrm{e}}=\mathrm{const}\) according to (4)

Fig. 2.
a) Load characteristics of the magnetic amplifier, \(I_n = f(I_{oy})\);
b) auxiliary characteristics \(I_a = f(n)\) at \(R_e = \mathrm{const}\) and speed characteristics \(n = f(I_a)\) of the motor for \(I_{oy} = 150\ \mathrm{A}\) and \(I_{oy} = 25\ \mathrm{A}\); dashed lines are calculated characteristics; solid lines are experimental characteristics.

Visible labels in the figure:

• Panel a:
• \(I_n\)
• \(I_n = f(I_{oy})\)
• \((I_{oy}),\ \mathrm{A}\)
• \(R_n = 10\ \Omega\)
• \(20\ \Omega\)
• \(30\ \Omega\)
• \(40\ \Omega\)
• \(50\ \Omega\)
• \(70\ \Omega\)
• \(100\ \Omega\)
• \(R_n = 200\ \Omega\)
• points \(1, 2, 3, 4, 5, 6, 8\)

• scale marks \(50,\ 100,\ 150\)

• Panel b:

• \(I_a\)
• \(I_a = f(n)\)
• \(n = f(I_a)\)
• \(n,\ \mathrm{rev/min}\)
• \(n_0\)
• \(R_e = 10\ \Omega\)
• \(20\ \Omega\)
• \(30\ \Omega\)
• \(40\ \Omega\)
• \(50\ \Omega\)
• \(70\ \Omega\)
• \(100\ \Omega\)
• \(R_e = 200\ \Omega\)
• \((I_{oy}) = 25\)
• \((I_{oy}) = 150\)
• points \(1, 2, 3, 4, 5, 6, 8\)
• scale marks \(500,\ 1000,\ 1500\)

in the form of straight lines issuing from the origin of the coordinate axes \(n, I_{\text{a}}\). Choosing the values \(R_{\text{e}}\) from the family of load characteristics of the MA, we plot the obtained dependences in the axes \(n, I_{\text{a}}\), as shown in Fig. 2,b, and obtain a reflection of the amplifier load characteristics in the axes \(n, I_{\text{a}}\). With the relative arrangement of the coordinate axes of the amplifier and the motor chosen in Fig. 2, the construction of the speed characteristic of the motor in a system without feedback, i.e., at \((Iw)_{\text{y}} = \text{const}\), is performed very simply: from the points of intersection of the vertical \((Iw)_{\text{y}} = \text{const}\) (points 1, 2, 3, 4, etc.) with the MA characteristics \(I_{\text{n}} = f(Iw)_{\text{y}}\), it is necessary to draw horizontals until they intersect the corresponding auxiliary characteristics of the motor (points \(1'\), \(2'\), \(3'\), \(4'\), etc.). Connecting the points found, we obtain the speed characteristic \(n = f(I_{\text{a}})\) of the motor (in Fig. 2 the characteristics are plotted for two values of the control ampere-turns: \(25a\) and \(150a\)).

This method gives fairly good agreement with experimentally obtained characteristics. The error in calculating the characteristic \(n = f(I_{\text{a}})\) of the motor in the load range from no-load to \((1.5 \div 2) I_{\text{a nom}}\) (that is, at control ampere-turns close to the nominal ones), for the MA circuit according to Fig. 1, does not exceed \(\pm 5\%\). If necessary, when constructing the speed characteristics \(n = f(I_{\text{a}})\), the nonlinearity of the armature resistance and the armature reaction can be taken into account. For this, the auxiliary characteristics of the motor \(I_{\text{a}} = f(n)\) at \(R_{\text{e}} = \text{const}\) must be plotted with allowance for the dependences \(R_{\text{a}} = f(I_{\text{a}})\) and \(\Phi = f(I_{\text{a}})\), and then, by the method described above, the speed characteristics obtained. From the speed characteristics \(n = f(I_{\text{a}})\), the mechanical characteristics of the motor are plotted with the aid of the expression [7]

\[ M = \frac{1}{1.03}\, k_{\text{e}} \Phi I_{\text{a}} = \frac{1}{1.03}\, \frac{E}{n}\, I_{\text{a}} . \tag{5} \]

This method is also applicable to series- or compound-excitation motors, for which the family of auxiliary characteristics \(I_{\text{a}} = f(n)\) at \(R_{\text{e}} = \text{const}\) can be plotted if universal operating characteristics \(\nu = f(i)\) and \(\mu = f(i)\) [7] are available, or if experimentally obtained natural speed characteristics \(n = f(I_{\text{a}})\) are available.

The use of the MA characteristics \(I_{\text{n}} = f(Iw)_{\text{y}}\) and of the motor \(I_{\text{a}} = f(n)\) for the same \(R_{\text{n}} = R_{\text{e}} = \text{const}\) makes it possible to plot the speed characteristics \(n = f(I_{\text{a}})\) of the motor also in systems with feedback. Let us consider the procedure for constructing motor characteristics in systems with various feedbacks only with respect to one parameter; moreover, for simplicity, we shall consider cases in which the feedback is introduced by means of a separate control winding of the MA (Fig. 3).

The speed characteristics of the motor are plotted very simply in the presence of external positive current feedback (Fig. 3,a). In this case the control ampere-turns of the MA are determined as the sum of the reference ampere-turns and the ampere-turns of the current-feedback winding:

\[ (Iw)_{\text{y}} = (Iw)_{\text{z}} + (Iw)_{\text{t}}, \tag{6} \]

where \((Iw)_{\text{z}}\) are the ampere-turns of the reference winding,

\((Iw)_{\text{t}}\) are the ampere-turns of the current-feedback winding.

Since the reference ampere-turns are a constant quantity, \((Iw)_{\text{z}} = \text{const}\), and the ampere-turns of the current winding are proportional to the load current, \((Iw)_{\text{t}} = c_1 \cdot I_{\text{n}}\), the total control ampere-turns are also a function of the load current, \((Iw)_{\text{y}} = f(I_{\text{n}})\). This dependence must be plotted

Fig. 3. Variants of the MA—motor system with separate feedbacks.
\(w_z\)—set-point winding; \(w_t\)—winding of external positive current feedback; \(w_u\)—winding of negative voltage feedback; \(w_n\)—winding of negative speed feedback.

in the MA coordinate system and, from it, find the speed characteristic of the motor \(n=f(I_a)\). For this purpose, in the left-hand coordinate system of Fig. 4 the load characteristics of the MA \(I_n=f(Iw)_y\) are plotted, and in the right-hand coordinate system the auxiliary characteristics \(I_a=f(n)\) of the motor for the same resistances \(R_n=R_e\). Next, in the left-hand coordinate system we draw the vertical \(AB\), representing the value of the setting ampere-turns \((Iw)_z\), the ray \(O_1C\), representing the dependence of the ampere-turns of the current winding on the load current \((Iw)_t=f(I_n)\), and the ray \(AD\), determining the total ampere-turns \((Iw)_y=f(I_n)\)—the trajectory of the operating point of the amplifier when the load changes in a circuit with external current feedback. From the intersection points 1, 2 ... 7 of the straight line \(AD\) with the MA load characteristics, in the right-hand coordinate system of Fig. 4 we find the speed characteristic of the motor. In doing so, the influence of the feedback on the stiffness of the motor characteristic is clearly visible.

When the MA–motor system has only one negative voltage feedback (Fig. 3, b), the resultant ampere-turns of the amplifier will be determined by the expression

\[ (Iw)_y=(Iw)_z-(Iw)_u, \tag{7} \]

where \((Iw)_u\) are the ampere-turns of the voltage-feedback winding.

The ampere-turns of the voltage winding \((Iw)_u\) are proportional to the voltage across the MA load, \((Iw)_u=c_y\cdot U_n\); consequently, the resultant ampere-turns of the amplifier are a function of the load voltage, \((Iw)_y=f(U_n)\). From this point of view it is more convenient to characterize the operation of the amplifier, in axes \(U_n,(Iw)_y\), by load voltage curves at the load \(U_n=f(Iw)_y\), while it is more convenient to construct the motor characteristics \(n=f(I_a)\) from the current load curves \(I_n=f(Iw)_y\). Therefore, in the present case, in addition to the current load curves it is necessary to plot voltage load curves at the load for the same load resistances \(R_n\). The curves \(U_n=f(Iw)_y\) will differ from the corresponding curves \(I_n=f(Iw)_y\) only by scale, since \(U_n=I_n\cdot R_n\), and \(R_n=\mathrm{const}\).

In Fig. 5, in the left-hand coordinate system, the characteristics \(U_n=f(Iw)_y\) and \(I_n=f(Iw)_y\) are plotted for three values of the load resistance, so as not to obscure the drawing excessively. In the construction, for simplicity, the shunting influence of the feedback circuit has not been taken into account, since its resistance is usually sufficiently large (in the case considered, 12 times greater than the largest load resistance \(R_t\)) and distorts the MA load characteristics only very slightly. If necessary, the shunting influence of the feedback circuit can be taken into account. Next, in the same coordinate system we draw the vertical \(AB\), representing the setting ampere-turns \((Iw)_z=\mathrm{const}\), and the feedback characteristic \((Iw)_u=f(U_n)\), laying it off from point \(A\) to the left, since condition (7) holds. The resulting ray \(AC\) will represent the dependence of the resultant ampere-turns of the MA on the load voltage \((Iw)_y=f(U_n)\), or the trajectory of the operating point of the amplifier in the axes \(U_n,(Iw)_y\). From the points of intersection of the ray \(AC\) with the characteristics \(U_n=f(Iw)_y\) (points \(1',2',7'\)), transferring them vertically to the corresponding curves \(I_n=f(Iw)_y\), we obtain the trajectory of the operating point of the amplifier \((Iw)_y=f(I_n)\) in the axes \(I_n,(Iw)_y\) (curve 127). From this curve, in the right-hand coordinate system of Fig. 5, the speed characteristic \(n=f(I_a)\) of the motor is found in the usual manner. The course of the construction is shown in Fig. 5 by arrows.

The calculation of the speed characteristics of the motor in a system with negative speed feedback, introduced into its own winding,

Fig. 4. Construction of the characteristics \(n=f(I_a)\) of the motor in the presence of external positive current feedback.

Fig. 5. Construction of motor characteristics \(n=f(U_a)\) in a circuit with feedback by voltage.

(Fig. 3, a) must be carried out by the method of successive approximation, since the operation of the MA is characterized by trajectories of the operating point in the axes \(I_{\mathrm{n}}\) (or \(U_{\mathrm{n}}\)) and \((I w)_y\), which are not functionally related to the coordinate along which the feedback is introduced. Therefore, each time it is necessary to find such an operating point of the amplifier, and from it the speed of the motor, so that the condition is satisfied:

\[ (I w)_y = (I w)_з - (I w)_n, \tag{8} \]

where \((I w)_n\) is the ampere-turns of the speed-feedback winding.

As usual, we construct the load characteristics \(I_{\mathrm{n}} = f((I w)_y)\) of the amplifier and the auxiliary characteristics \(I_{\mathrm{a}} = f(n)\) for the same resistances (Fig. 6). In the axes \(I_{\mathrm{n}}, (I w)_y\) we lay off the vertical \(AB\) of the specified ampere-turns; in the axes \(n, I_{\mathrm{a}}\) we draw an auxiliary ray at an angle of \(45^\circ\) and, using the widely used method of transferring coordinate axes [8], construct the dependence of the ampere-turns of the speed-feedback winding on the speed, \((I w)_n = f(n)\), in the coordinate axes \(I_{\mathrm{n}}, (I w)_y\). We shall lay off this dependence from point \(A\) to the left of the vertical \(AB\), since the feedback is negative. The ray \(AD\) represents the dependence \((I w)_n = f(n)\) constructed in the axes \(I_{\mathrm{n}}, (I w)_y\). Now suppose, for example, that the motor operates in such a way that its equivalent resistance is equal to \(R_2\); choose arbitrarily point 1 on the MA operating curve corresponding to this resistance to the left of the vertical \(AB\), since condition (8) holds. From point 1 we find point 2 on the line \(R_2 = \mathrm{const}\) in the axes \(n, I_{\mathrm{a}}\), that is, the motor speed. Transferring this speed value into the left-hand coordinate system (point 3), we find the value of the feedback ampere-turns (point 4). If points 1 and 4 lie on the same vertical, then condition (8) is satisfied and point 2 of the motor operation has been found correctly; if, however, 1 and 4 do not lie on the same vertical, then the construction must be repeated, choosing a new position of point 1, until the points coincide vertically. After two or three constructions, any point of the characteristic \(n = f(I_{\mathrm{a}})\) of the motor when operating with speed feedback is found correctly. The course of the construction is analogous in all cases to that described and is shown in Fig. 6 by arrows.

Using these methods, it is possible to calculate the characteristics \(n = f(I_{\mathrm{a}})\) of the motor also in systems with several feedbacks introduced into different windings, and in cases where several feedback signals are combined on one winding.

In addition, the use of load characteristics \(I_{\mathrm{n}} = f((I w)_y)\) makes it possible to select the parameters of the elements of the MA–motor system in order to obtain the desired form of the motor characteristics \(n = f(I_{\mathrm{a}})\) or \(n = f(M)\). Thus, by specifying the required motor characteristic \(n = f(I_{\mathrm{a}})\) (Fig. 7), we obtain the trajectory of the amplifier operating point \((I w)_y = f(I_{\mathrm{n}})\) (curve \(2 \ldots 7\)), which in particular cases can be functionally related to the coordinates of the system, or, with the aid of a number of inverse constructions, the types and characteristics can be selected and the parameters of the feedback circuits that realize such a trajectory can be determined; that is, the problem of synthesizing an MA–motor system with the desired properties in the static regime can be solved.

Thus, the use of MA load characteristics leads to a simple graphical method for calculating motor characteristics in the MA–motor system. The best calculation results, naturally, can be achieved when experimental curves are used, but in the absence of the latter calculated load characteristics may also be used. Recently, [[unclear: continuation on next page]]

Fig. 6. Construction of the characteristic \(n=f(I_a)\) of a motor in a circuit with speed feedback.

Visible diagram labels include:

• \((I_y)_n=f(I_n)\)
• \((I_y)_v=f(I_a)\)
• \(n=f(I_a)\)
• \(n\), rpm
• \(1000\)
• \(I_a\)
• \(I_m\)
• \(O_1\), \(O_2\)
• \(R_1, R_2, R_3, R_4, R_5, R_6, R_7\)
• \(\beta\)
• Points \(1, 2, 3, 4, 5, 6, 7, 8\)

Fig. 7. Determination of the trajectory of the operating point of the magnetic amplifier for obtaining a motor characteristic \(n=f(I_{\mathrm{a}})\) of the desired form.

Visible labels in the diagram: \(aI_y\), \((I_y)_y/a\), \(aI_{\mathrm{a}}\), \(n\), rpm; \(O_1\), \(O_2\); \(60\), \(100\); \(500\), \(1000\), \(1500\); \(R_1\), \(R_2\), \(R_3\), \(R_4\), \(R_5\), \(R_6\), \(R_7\), \(R_8\); \((I_y)_y=f(I_{\mathrm{a}})\); \(n=f(I_{\mathrm{a}})\); points \(1\)–\(7\).

The calculation of magnetic amplifiers from standard characteristics [1], that is, load characteristics in relative units for amplifiers connected according to the same circuit and having cores with identical magnetic properties, has become widespread. Having standard characteristics and using simple relationships [1], one can quickly calculate, from the rated data, a family of load characteristics of an actual amplifier. Therefore, for series of magnetic amplifiers produced by industry, it is necessary to provide standard characteristics in the catalogs.

All the characteristics and constructions presented in this paper pertain to a real amplifier made of steel grade E42; lamination Sh40U; stack thickness 36 mm; number of turns of one load winding \(w_{\sim} = 600\); numbers of turns of the control windings \(w_1 = w_2 = w_3 = 1000\); \(w_4 = 40\); rectifiers of type D304. Motor of type MI-12F; \(U_{\mathrm{nom}} = 110\) V; \(I_{a\,\mathrm{nom}} = 1.46\) A; \(n_{\mathrm{nom}} = 2000\) rpm; \(R_a = 2.2\) ohm.

Conclusions

1. The choice of the load characteristics of the magnetic amplifier as the initial data makes it possible to use a simple and clear graphical method for calculating the characteristics of DC motors in a magnetic-amplifier--motor system.

2. The proposed method can also be used for the synthesis of elements that ensure the desired static properties of the system.

3. In order to be able to use this method in designing electric drives according to the MA--motor system, it is necessary, for series of amplifiers intended for DC motors, to provide standard characteristics in the catalogs.

References

1. T. Kh. Stefanovich. Magnetic Amplifiers for the Automation of Industrial Installations. Gosenergoizdat, 1961.

2. H. Winkler. A New Range of Drives Adjustable by Means of Magnetic Amplifiers, with Power from 2.8 to 187 kW. AEG Mitteilungen, vol. 49, no. 10/11, 1959.

3. V. S. Kulebakin. On the use of semiconductor rectifiers in an automated electric-drive system. Proceedings of the Academy of Sciences of the USSR, Technical Sciences Section, no. 2, 1958.

4. D. A. Alenchikov, V. S. Kulebakin. A method for calculating the characteristics of DC motors with choke control. Automation and Remote Control, vol. XX, no. 7, 1959.

5. O. B. Rossenbaul and I. N. Selivokhin. Construction of the characteristics of a DC drive with choke control. Electricity, no. 10, 1958.

6. A. A. Sirotin. Automatic Control of Electric Drives. Gosenergoizdat, 1959.

7. V. P. Andreev and Yu. A. Sabinin. Fundamentals of Electric Drive. Gosenergoizdat, 1956.

8. A. V. Basharin. Calculation of Dynamics and Synthesis of Nonlinear Control Systems. Gosenergoizdat, 1960.

Received by the editors
in June 1962.