APPLICATION OF ANALOG COMPUTERS TO THE ANALYSIS OF ELECTROMAGNETIC PROCESSES IN THREE-PHASE RECTIFIERS

V. A. KOCHEGUROV, A. A. TERESHCHENKO

(Presented by the scientific seminar of the Research Institute of Nuclear Physics)

The widespread introduction of rectifier installations into various branches of industry poses the problem of developing a sufficiently simple and accurate method for calculating them. The objectives of the calculation should be the selection of the parameters of the main equipment and the construction of the operating characteristics of the rectifier for various modes of its operation.

In the general case, an analytical calculation of currents and voltages in the circuits of rectifier systems is difficult and makes it possible to obtain a final result only for comparatively simple variants of rectifier circuits. In addition, the design and adjustment of complex control and automatic-regulation systems containing converters cannot be carried out without knowledge of the processes occurring in them. Of particular interest is the question of determining the actual dynamic parameters of an ionic converter together with grid control and a load operating in a closed automatic-control system.

The use of continuous-action computing machines considerably facilitates the conduct of such investigations and makes it possible to solve problems that were practically inaccessible by the calculation methods usually employed. An electronic computer makes it possible quickly to obtain clearly observable oscillograms of currents and voltages in the circuits of a rectifier system, from which one can visually judge the influence of the circuit parameters on the character of the electromagnetic processes occurring in it. Increasing the accuracy of calculation with the aid of an electronic computer substantially facilitates the selection of optimal parameters and the investigation of re—

Fig. 1.

operating modes of rectifiers. Electronic modeling of single-phase and two-phase rectifier circuits is given in the literature [1, 2].

Below is a method for calculating, on analog computers, single-way and double-way three-phase uncontrolled rectifiers.

a) Model of a single-way three-phase rectifier

The circuit diagram of a single-way three-phase rectifier is shown in Fig. 1. The elements \(r\) and \(L_S\) take into account the active resistances and leakage inductances of the primary and secondary windings of the transformer. In accordance with the notation given in Fig. 1, the operation of the circuit is described by the following equations, compiled on the basis of Kirchhoff’s laws:

\[ \left. \begin{aligned} e_1 &= L_S \frac{di_1}{dt} + r i_1 + r_{B_1}\, i_1 + U_{\mathrm{н}}; \\[0.8em] e_2 &= L_S \frac{di_2}{dt} + r i_2 + r_{B_2}\, i_2 + U_{\mathrm{н}}; \\[0.8em] e_3 &= L_S \frac{di_3}{dt} + r i_3 + r_{B_3} i_3 + U_{\mathrm{н}}; \\[0.8em] i_1 + i_2 + i_3 &= i_{\mathrm{н}}; \\[0.3em] U_{\mathrm{н}} &= i_{\mathrm{н}} Z_{\mathrm{н}}; \\[0.3em] Z_{\mathrm{н}} &= r_{\mathrm{н}} + pL_{\mathrm{н}}; \end{aligned} \right\} \tag{1} \]

where \(r_{B_1}, r_{B_2}, r_{B_3}\) are the valve resistances.

Solving the system of equations (1) with respect to the currents \(i_1, i_2, i_3\), we obtain the initial equations for constructing and calculating the model circuit. The block diagram of the mathematical model of the circuit of a single-way three-phase rectifier is shown in Fig. 2.

Fig. 2.

The volt-ampere characteristics of the rectifier valves \(i_v = f(U_v)\) are reproduced in the model by means of

tube diodes 6Kh2P connected in the feedback circuit of the corresponding operational amplifiers.

The equations of the individual blocks of the model, taking into account the variable-transformation equations, are written as:

\[ \begin{gathered} \frac{i_1}{M_{i_{\mathrm{v}}}} = -\frac{1}{p}\left( \kappa_{11}\frac{e_1}{M_e} -\kappa_1\frac{i_1}{M_{i_{\mathrm{v}}}} -\kappa_{\mathrm{B}1}\frac{i_1}{M_{i_{\mathrm{v}}}} -\kappa_{12}\frac{U_{\mathrm{n}}}{M_u} \right); \\[6pt] \frac{i_2}{M_{i_{\mathrm{v}}}} = -\frac{1}{p}\left( \kappa_{21}\frac{e_2}{M_e} -\kappa_2\frac{i_2}{M_{i_{\mathrm{v}}}} -\kappa_{\mathrm{B}2}\frac{i_2}{M_{i_{\mathrm{v}}}} -\kappa_{22}\frac{U_{\mathrm{n}}}{M_u} \right); \\[6pt] \frac{i_3}{M_{i_{\mathrm{v}}}} = -\frac{1}{p}\left( \kappa_{31}\frac{e_3}{M_e} -\kappa_3\frac{i_3}{M_{i_{\mathrm{v}}}} -\kappa_{\mathrm{B}3}\frac{i_3}{M_{i_{\mathrm{v}}}} -\kappa_{32}\frac{U_{\mathrm{n}}}{M_u} \right); \\[6pt] \frac{i_{\mathrm{n}}}{M_i} = -\left( \kappa_{41}\frac{i_1}{M_{i_{\mathrm{v}}}} +\kappa_{42}\frac{i_2}{M_{i_{\mathrm{v}}}} +\kappa_{43}\frac{i_3}{M_{i_{\mathrm{v}}}} \right); \\[6pt] \frac{U_{\mathrm{n}}}{M_u} = -\left( \kappa_{51}\frac{i_{\mathrm{n}}}{M_i} +\kappa_5 p \frac{i_{\mathrm{n}}}{M_i} \right). \end{gathered} \tag{2} \]

The variable-transformation equations have the form:

\[ \left. \begin{array}{lll} i_1=M_{i_{\mathrm{v}}}U_1; & e_1=M_e e'_1; & i_{\mathrm{n}}=M_iU_4;\\ i_2=M_{i_{\mathrm{v}}}U_2; & e_2=M_e e'_2; & U_{\mathrm{n}}=M_uU_5;\\ i_3=M_{i_{\mathrm{v}}}U_3; & e_3=M_e e'_3; & t=M_t t_{\mathrm{m}}. \end{array} \right\} \tag{3} \]

Hence the correspondence of the model elements to the parameters of the real circuit is determined by the following relations:

\[ \left. \begin{gathered} L_{\mathrm{n}}=\frac{\kappa_{12}\kappa_5 M_e M_i}{\kappa_{11}M_i}; \qquad L_s=\frac{M_e M_t}{\kappa_{41}\kappa_{11}M_{i_{\mathrm{v}}}}; \\[6pt] r_{\mathrm{n}}=\frac{\kappa_{12}\kappa_{51}M_e}{\kappa_{11}M_i}; \qquad r=\frac{\kappa_1 M_e}{\kappa_{11}M_{i_{\mathrm{v}}}}; \qquad r_{\mathrm{B}}=\frac{\kappa_{\mathrm{B}}M_e}{\kappa_{11}M_{i_{\mathrm{v}}}}. \end{gathered} \right\} \tag{4} \]

where \(M_i, M_{i_{\mathrm{v}}}\) are current scales;

\(M_e, M_u\) are voltage scales;

\(M_t\) is the time scale;

\(\kappa_{11}, \kappa_{12}\ldots\) are the transfer coefficients of the operational amplifiers.

Oscillograms of the rectified current and voltage, obtained on the electronic computer and corresponding to the steady-state operating mode of the circuit with an active-inductive load, are given in Fig. 3.

The phase electromotive forces of the rectifier, \(e_1, e_2, e_3\), in the model (Fig. 2) are represented in the form of electromotive forces \(e'_1, e'_2, e'_3\), which are generated in a specially designed source of low-frequency three-phase voltage. The block diagram of this source is given in Fig. 4. The synchronous motor SM rotates, by means of worm pair 1 and friction mechanism 3, a disk with probes 6. The probes are immersed in bath 5 with electrolyte, where a uniform electric field has been created by means of parallel electrodes 7. When the disk rotates, the amplitudes of the voltages picked off by the probes vary according to a harmonic law. The voltages from the probes are fed through sliding contacts 4 to the inputs of cathode followers. The frequency of the electromotive force is determined by the rotational speed of the disk and can be smoothly regulated

Fig. 3.

Fig. 4.

by means of variator 2 from 0.5 to 10 units. The amplitude is adjusted by means of potentiometers \(P_1\) and \(P_2\).

b) Model of a double-way three-phase rectifier

Figure 5 shows the schematic diagram of a double-way three-phase rectifier and gives the designations of its elements and parameters, as well as of currents and voltages. In this circuit, two three-phase single-way rectifier sections are connected in series; one of them is assembled on valves 1, 2, 3 and develops the rectified voltage \(U_{v1}\). The second is on valves 4, 5, 6 with rectified voltage \(U_{v2}\). Both valve groups use one and the same three-phase secondary winding of the transformer. Valves 1, 2, 3, connected to the transformer phases by their anodes and having a common cathode point, form the cathode group of valves. Valves 4, 5, 6, connected to the same transformer phases by their cathodes, have a common anode point and form the anode group of valves. Consequently, the voltages \(U_{v1}\) and \(U_{v2}\), which are respectively the upper and lower envelopes of the phase e.m.f.’s, are added together, producing the rectified voltage of the circuit \(U_n\). For the converter circuit shown in Fig. 5, we have the following system of equations for voltages and currents:

Fig. 5.

\[ \left. \begin{aligned} e_1 &= Lsp\,(i_1+i_6)+r(i_1+i_6)+r_{v1}\,i_1+U_{v1};\\ e_2 &= Lsp\,(i_2+i_5)+r(i_2+i_5)+r_{v2}\,i_2+U_{v1};\\ e_3 &= Lsp\,(i_3+i_4)+r(i_3+i_4)+r_{v3}\,i_3+U_{v1};\\ e_1 &= Lsp\,(i_1+i_6)+r(i_1+i_6)+r_{v6}\,i_6+U_{v2};\\ e_2 &= Lsp\,(i_2+i_5)+r(i_2+i_5)+r_{v5}\,i_5+U_{v2};\\ e_3 &= Lsp\,(i_3+i_4)+r(i_3+i_4)+r_{v4}\,i_4+U_{v2};\\ i_1+i_2+i_3 &= i_n;\\ i_4+i_5+i_6 &= i_n;\\ U_{v1}+U_{v2} &= U_n;\\ U_n &= i_n Z_n;\\ Z_n &= r_n+pL_n. \end{aligned} \right\} \tag{5} \]

Solving the system of equations (5) with respect to the currents \(i_1, i_2, i_3, i_4, i_5, i_6\), as in the case of the single-way three-phase rectifier, we obtain the original equations for composing and calculating the model circuit. The structural diagram of the model for solving the equations of voltages and currents in the circuits of the double-way three-phase rectifier is shown in Fig. 6. Correspond-

Fig. 6.

The correspondence between the elements of the model and the parameters of the actual circuit is determined in the same way as for the case of the model of a single-cycle three-phase rectifier.

Oscillograms of the machine calculation of currents and voltages in the circuits of the rectifier-bridge circuit are given in Fig. 7.

Comparison of the results obtained from calculation on an analog computer and in actual circuits showed good agreement of the characteristics for all the rectifier circuits and operating modes investigated. Electronic models of more complex rectifier circuits, as well as of controlled rectifiers, can be constructed on the basis of the circuits considered in the present article.

Fig. 7.

REFERENCES

1. V. G. Vasil'ev, V. A. Zverev. Electronic model of a rectifier-bridge circuit. Elektromekhanika, No. 1, 1961.

2. V. Ya. Yakubovskii. Application of a continuous-action electronic computer for calculating the steady-state operating mode of a two-phase rectification circuit. Elektromekhanika, No. 5, 1962.