ON THE CALCULATION OF ELECTROMAGNETIC PROCESSES IN A SIX-PHASE COMPENSATING CONVERTER WITH AN INTERPHASE REACTOR

A. I. ZAITSEV, A. A. KUVSHINOV

The operation of a compensating converter is accompanied by complex electromagnetic processes, the course of which depends on many parameters of the circuit, which contains nonlinear elements (valves) and magnetic couplings.

Fig. 1

The currents flowing through the transformer windings, the valves, and the commutating capacitors contain a broad spectrum of higher harmonics. The voltages applied to the individual elements of the circuit are also nonsinusoidal.

Determining the converter currents because of valve overlap is a nonlinear problem, since the duration of overlap, owing to the nonlinearity of the valves, depends on the load condition. In the six-phase compensating converter with an interphase reactor considered here (Fig. 1), the investigation of electromagnetic processes is complicated by the fact that commutation is two-stage and that, in the artificial-commutation node, there are additional nonlinear elements—the valves. As a result,

the compensating converter operates in different modes with a different number and combination of switched-on power and commutating valves. Each of these modes has its own specific features in the course of electromagnetic processes. In essence, the steady-state operating mode of a compensating valve converter is a continuous series of transient processes that arise when valves are unlocked or locked.

There is a fairly large number of methods for analyzing and calculating circuits with nonlinear elements. However, an analytical study of electromagnetic processes in such circuits is associated with considerable difficulties. The equations obtained as a result of such a study prove to be very complex and often altogether unsolvable.

The most expedient and very illustrative method for analyzing electromagnetic processes in converters with two-stage forced commutation, making it possible to obtain instantaneous and integral characteristics fairly simply, is the method of piecewise-linear approximation. The volt-ampere characteristics of the valves are replaced by segments of straight lines, as a result of which they may be regarded as linear within definite time intervals. If the entire operating period of a compensating converter is divided into a number of intervals within which all elements of the circuit are linear, then consideration of electromagnetic processes in a nonlinear circuit is reduced to the solution of a linear problem. However, the process as a whole remains nonlinear, since the instants at which the process in the nonlinear element passes from one section of its characteristic to another depend on the currents and voltages in the circuit [1].

The application of this method is further justified by the fact that the volt-ampere characteristics of semiconductor valves currently being produced are very well approximated by a small number of rectilinear segments, which substantially simplifies the solution of the problem.

In studying electromagnetic processes in a compensating converter we adopt the following assumptions:

1. The magnetizing currents of the transformer and of the equalizing reactor are not taken into account when considering the processes in the circuit.

2. The active resistances of the windings of the transformers and of the equalizing reactor, and the self-capacitances of the circuit elements, are small and are not taken into account when considering the processes in the circuit.

3. The voltage drops across the valves in the conducting state are neglected; the resistances of the valves in the nonconducting state are infinitely large.

4. The parameters of the power transformer and the load current are constant.

The assumptions adopted are customary and widely used in the study of electromagnetic processes in three-phase-current valve converters [2]. The magnitude of the resulting error depends on the power of the converter under study and on the form of the valve characteristics. As the converter power increases and the valve characteristics approach the ideal, the magnitude of the resulting error decreases.

The operating features of a six-phase compensating converter with an equalizing reactor were considered in [3]. Under normal loads in this converter either one or two power valves are switched on in succession. When the load current increases above a certain limit, which depends on the parameters of the forced-commutation unit, forced commutation breaks down. This is an emergency mode, in which it is necessary either to disconnect the converter or to transfer it to operation with natural commutation. These functions must be performed by the control circuit.

Therefore, for the study of electromagnetic processes in a compensating converter with two-stage forced commutation

it is sufficient to consider in detail two modes: the mode with two power valves switched on simultaneously and the mode with one power valve switched on.

Mode with two power valves switched on simultaneously

Let us find the equations for the currents and voltages in the interval \(0 < \vartheta < \dfrac{\pi}{3} - \vartheta_{\mathrm{p}}\), when power valves 1 and 2 are switched on. This interval (1) can be divided into three sections (Fig. 2):

Fig. 2.

1) switching on of valve 1, \(0 < \vartheta < \vartheta_{\mathrm{k}2}\);

2) operation with the artificial-commutation unit switched on, \(\vartheta_{\mathrm{k}2} < \vartheta < \dfrac{\pi}{3} - \vartheta_{\mathrm{p}} - \vartheta_{\mathrm{k}1}\);

3) switching off of valve 2, \(\dfrac{\pi}{3} - \vartheta_{\mathrm{p}} - \vartheta_{\mathrm{k}1} < \vartheta < \dfrac{\pi}{3} - \vartheta_{\mathrm{p}}\).

Here \(\vartheta_{\mathrm{k}1}\) is the duration of the first commutation;

\(\vartheta_{\mathrm{p}}\) is the duration of direct-current flow through the capacitor;
\(\vartheta_{\mathrm{k}2}\) is the duration of the second commutation.

During operation of the converter with the commutation unit switched off, the current of the commutating capacitor \(i_c\) is equal to zero; the voltage across it is equal to the constant value \(-I_{c0}\); the anode currents are equal to one another and constitute one half of the load current \(I_d\).

When valve 1 is switched on and valve 2 is switched off, the commutation circuit is the same (the secondary windings of transformer \(a\) and \(y\) and the commutating capacitor \(C\)); therefore, the electromagnetic processes occurring in the circuit when valve 1 is switched on and valve 2 is switched off are described by identical equations:

\[ i_a=\frac{I_d}{2}+i_c,\qquad i_y=\frac{I_d}{2}-i_c, \]

\[ i_c=C\frac{du_c}{dt},\qquad I_d=i_a+i_y. \tag{1} \]

Writing the equation of e.m.f. in the commutation circuit and taking (1) into account, we obtain

\[ 2LC\frac{d^2u_c}{dt^2}+u_c=-I_m\cos\left(\frac{\pi}{6}+\psi_0+\vartheta\right), \tag{2} \]

where
\(i_a,\ i_y\) are the anode currents;
\(C\) is the capacitance of the commutating capacitor;
\(L\) is the inductance of one transformer phase;
\(I_m\) is the amplitude of the transformer secondary voltage;
\(I_c\) is the voltage of the commutating capacitor;

\[ \psi_0=\frac{\pi}{6}-\vartheta_{\mathrm{g}}+\vartheta_{\mathrm{k}1}+\vartheta_{\mathrm{p}}, \]

\(\vartheta_{\mathrm{g}}\) is the extinction angle, counted to the left from the point of natural firing of the valves.

The boundary conditions for equation (2) will be different for the periods of the first and second commutations.

For the second commutation:

\[ (u_c)_{\vartheta=0}=-U_m\cos\left(\frac{\pi}{6}+\psi_0\right),\qquad (i_c)_{\vartheta=0}=-\frac{I_d}{2}. \]

For the first commutation:

\[ (u_c)_{\vartheta=\frac{\pi}{3}-\vartheta_{\mathrm{p}}-\vartheta_{\mathrm{k}1}} =-U_{c0},\qquad (i_c)_{\vartheta=\frac{\pi}{3}-\vartheta_{\mathrm{p}}-\vartheta_{\mathrm{k}1}} =0. \]

Then the solution of equation (2) for the period of the second commutation is written as follows:

\[ u_c=\frac{U_m}{\varepsilon^2-1} \left[ \cos\left(\frac{\pi}{6}+\psi_0\right)\cos\varepsilon\vartheta -\varepsilon\sin\left(\frac{\pi}{6}+\psi_0\right)\sin\varepsilon\vartheta -\varepsilon^2\cos\left(\frac{\pi}{6}+\psi_0+\vartheta\right) \right] -I_d\omega L\varepsilon\sin\varepsilon\vartheta, \tag{3} \]

\[ i_c= \frac{\omega C\varepsilon U_m}{\varepsilon^2-1} \left[ \varepsilon\sin\left(\frac{\pi}{6}+\psi_0+\vartheta\right) -\cos\left(\frac{\pi}{6}+\psi_0\right)\sin\varepsilon\vartheta -\varepsilon\sin\left(\frac{\pi}{6}+\psi_0\right)\cos\varepsilon\vartheta \right] -\frac{I_d}{2}\cos\varepsilon\vartheta, \tag{4} \]

where

\[ \varepsilon=\frac{1}{\omega\sqrt{2LC}}. \]

The solution of equation (2) for the period of the first commutation has the following form:

\[ u_c=- \left[ U_{c0} -\frac{\varepsilon^2 U_m}{\varepsilon^2-1} \cos\left(\frac{2\pi}{3}-\vartheta_{\mathrm{g}}\right) \right] \cos\varepsilon\left(\vartheta-\frac{\pi}{3}+\vartheta_{\mathrm{k}1}+\vartheta_{\mathrm{p}}\right) - \]

\[ -\frac{\varepsilon U_m}{\varepsilon^2-1} \sin\left(\frac{2\pi}{3}-\vartheta_{\mathrm{g}}\right) \sin\varepsilon\left(\vartheta-\frac{\pi}{3}+\vartheta_{\mathrm{k}1}+\vartheta_{\mathrm{p}}\right) +\varepsilon\cos\left(\frac{\pi}{6}+\psi_0+\vartheta\right), \tag{5} \]

\[ i_c=\omega C\varepsilon\left\{\left[U_{co}-\frac{\varepsilon^2U_m}{\varepsilon^2-1}\cos\left(\frac{2\pi}{3}-\vartheta_\Gamma\right)\right]\sin\varepsilon\left(\vartheta-\frac{\pi}{3}+\vartheta_{\mathrm{k}1}+\vartheta_{\mathrm{p}}\right) -\right. \]
\[ \left.{}-\frac{\varepsilon U_m}{\varepsilon^2-1}\left[\sin\left(\frac{2\pi}{3}-\vartheta_\Gamma\right)\cos\varepsilon\left(\vartheta-\frac{\pi}{3}+\vartheta_{\mathrm{k}1}+\vartheta_{\mathrm{p}}\right)-\sin\left(\frac{\pi}{6}+\psi_0+\vartheta\right)\right]\right\}. \tag{6} \]

The values of the anode currents are obtained by substituting (4) and (6) into (1).

Mode with one power valve switched on (recharging of the commutating capacitor by direct current)

Since the load current is a constant quantity, a constant current equal to one half of the load current flows through the commutating capacitor in the interval \(\frac{\pi}{3}-\vartheta_{\mathrm{p}}<\vartheta<\frac{\pi}{3}\) (section II, Fig. 2). The voltage on the commutating capacitor varies during this time according to a linear law:

\[ u_c=\frac{I_d}{2\omega C}\left(\vartheta-\frac{\pi}{3}\right)+(u_c)_{\vartheta=\frac{\pi}{3}} . \tag{7} \]

Substituting into (7) the value \((u_c)_{\vartheta=\frac{\pi}{3}}=(-u_c)_{\vartheta=0}\), we obtain

\[ u_c=\frac{I_d}{2\omega C}\left(\vartheta-\frac{\pi}{3}\right)+U_m\cos\left(\frac{\pi}{6}+\psi_0\right). \tag{8} \]

The anode current in this interval is equal to the load current \(I_d\).

Determination of the commutation duration and the initial voltage on the commutating capacitor

The durations of the first and second commutations \(\vartheta_{\mathrm{k}1}\) and \(\vartheta_{\mathrm{k}2}\), the duration of operation of one power valve \(\vartheta_{\mathrm{p}}\), and the initial voltage on the commutating capacitor \(I_{co}\) are to be determined. To determine them, we have the following four conditions:

1) at \(\vartheta=\vartheta_{\mathrm{k}2}\), the current \(i_c\), determined by expression (4), is equal to zero;

2) at \(\vartheta=\vartheta_{\mathrm{k}2}\), the voltage \(u_c\), determined by expression (3), is equal to \(-I_{co}\);

3) at \(\vartheta=\frac{\pi}{3}-\vartheta_{\mathrm{p}}\), the current \(i_c\), determined from formula (6), is equal to \(\frac{I_d}{2}\);

4) at \(\vartheta=\frac{\pi}{3}-\vartheta_{\mathrm{p}}\), the voltage \(u_c\), determined from expression (5), is equal to the voltage \(u\), determined from expression (8).

Conditions 1 and 2 follow from the very principle of operation of the forced-commutation unit. Conditions 3 and 4 follow from the requirements of correct matching of the modes at the boundaries of successive time intervals in which different combinations of power and commutating valves are switched on.

As a result we obtain four transcendental equations which, in the general form, are unsolvable.

Simplification of the solution. Neglecting the variation of phase EMFs in the operating intervals of the forced-commutation unit

The above equations are considerably simplified if, in studying the electromagnetic processes occurring in the commutation circuit, one assumes that the phase EMFs of the converter transformer are constant in the operating intervals of the forced-commutation unit (the first commutation, the period of operation of one power valve, the second commutation).

Taking the above into account, we obtain the following simplified equations:

Second commutation:

\[ u_c=-I_d\omega L\varepsilon \sin \varepsilon\vartheta-U_m\cos\left(\frac{\pi}{6}+\psi_0\right), \tag{9} \]

\[ i_c=-\frac{I_d}{2}\cos \varepsilon\vartheta. \tag{10} \]

Taking into account conditions 1 and 2, from (9) and (10) we obtain:

\[ \vartheta_{k2}=\frac{\pi}{2\varepsilon}; \tag{11} \]

\[ U_{c0}=I_d\omega L\varepsilon+U_m\cos\left(\frac{\pi}{6}+\psi_0\right). \tag{12} \]

First commutation:

\[ u_c=-\left[ 2U_m\cos\left(\vartheta_\Gamma-\frac{\vartheta_{k1}+\vartheta_n}{2}\right) \sin\left(\frac{\pi}{6}-\frac{\vartheta_{k1}+\vartheta_n}{2}\right)+ \right. \]

\[ \left. +I_d\omega L\varepsilon \right] \cos\varepsilon\left(\vartheta-\frac{\pi}{3}+\vartheta_{k1}+\vartheta_n\right) -U_m\cos\left(\frac{2\pi}{3}-\vartheta_\Gamma\right), \tag{13} \]

\[ i_c=\omega C\varepsilon\left[ 2U_m\cos\left(\vartheta_\Gamma-\frac{\vartheta_{k1}+\vartheta_n}{2}\right) \sin\left(\frac{\pi}{6}-\frac{\vartheta_{k1}+\vartheta_n}{2}\right)+ \right. \]

\[ \left. +I_d\omega L\varepsilon \right] \sin\varepsilon\left(\vartheta-\frac{\pi}{3}+\vartheta_{k1}+\vartheta_n\right). \tag{14} \]

Recharging of the Commutating Capacitor by Direct Current

The equation for the voltage of the commutating capacitor \(u_c\) has the same form (8) as before.

From conditions 3 and 4 we obtain two transcendental equations:

\[ I_d(1\sin\varepsilon\vartheta_{k1}) = 4\varepsilon\omega C U_m \cos\left(\vartheta_\Gamma-\frac{\vartheta_{k1}+\vartheta_n}{2}\right) \sin\left(\frac{\pi}{6}-\frac{\vartheta_{k1}+\vartheta_n}{2}\right) \sin\varepsilon\vartheta_{k1}, \tag{15} \]

\[ I_d(\varepsilon\vartheta_n\tg\varepsilon\vartheta_{k1}-1) = 4\varepsilon\omega C U_m \sin\left(\vartheta_\Gamma-\frac{\vartheta_{k1}+\vartheta_n}{2}\right) \cos\left(\frac{\pi}{6}-\frac{\vartheta_{k1}+\vartheta_n}{2}\right) \tg\varepsilon\vartheta_{k1}. \tag{16} \]

Equations (15) and (16) make it possible, for given \(\vartheta_\Gamma\), \(L\), \(C\), and \(I_d\), to determine, by assigning a series of values to the angle \(\vartheta_n\), two curves

\[ \vartheta_{k1}=f_1(\vartheta_n) \qquad \text{and} \qquad \vartheta_{n1}=f_2(\vartheta_n). \]

The point of intersection of these curves determines the values \(\vartheta_{k1}\) and \(\vartheta_n\) corresponding to the given converter parameters.

In order to estimate the error introduced into the calculations by assuming constancy of the phase emf during the commutation periods, the principal quantities characterizing the electromagnetic processes of the compensating converter were calculated on a digital computer from formulas (3)—(8) for various \(L\), \(C\), and \(\vartheta_2\). For these same parameters, a calculation was made using the simplified formulas (9)—(16). The calculation results for all extinction angles \(\vartheta_2\) differed by no more than 15%.

In Fig. 2, calculated curves are plotted for the anode currents \(i_a\), \(i_y\), \(i_z\), \(i_b\), the current \(i_c\), and the voltage \(u_c\) of the commutating capacitor for the case when

\[ L=0.157\cdot 10^{-3}\ \text{H},\quad C=350\cdot 10^{-6}\ \text{F},\quad \vartheta_2=\frac{\pi}{3},\quad I_d=500\ \text{A},\quad U_m=278\ \text{V}. \]

The curve of the current \(i_a\) was constructed in the following manner. In the intervals \(0<\vartheta<\vartheta_{k2}\) and \(\frac{\pi}{3}-\vartheta_n-\vartheta_{k1}<\vartheta<\frac{\pi}{3}-\vartheta_n\), according to equations

\[ (1),\ (10)\ \text{and}\ (14) \]

the sections of the curves \(i_a\) and \(i_y\) are plotted. In the interval \(\vartheta_{k2}<\vartheta<\frac{\pi}{3}-\vartheta_n-\vartheta_{k1}\), when the artificial-commutation unit is not operating, the anode currents \(i_a\) and \(i_y\) are equal to one half of the load current \(I_d\). In the interval of operation of one

of the power valve, \(\frac{\pi}{3} - \vartheta_{p} < \vartheta < \frac{\pi}{3}\), the anode current \(i_a\) is equal to the load current \(I_d\). Then, by shifting the section of the curve \(i_y\), constructed in the interval \(0 < \vartheta < \frac{\pi}{3} - \vartheta_{p}\), by an angle \(\frac{\pi}{3}\) to the right, the entire curve of the anode current \(i_a\) was obtained.

From expressions (9), (13), and (8), one half-period of the voltage curve on the commutating capacitor \(u_c\) is constructed. The second half-period is constructed as the mirror image of the first, shifted by an angle \(\frac{\pi}{3}\) to the right.

Knowing the magnitudes and shapes of the anode-current curves and of the voltage on the commutating capacitor, one can determine all the other quantities of interest to us (the primary current and the voltage on the secondary windings of the transformer, the voltage on the interphase reactor, the voltage on the power valves, etc.).

REFERENCES

1. S. R. Glinternik. Electromagnetic Processes and Operating Conditions of High-Power Static Converters. Nauka Publishing House, Leningrad, 1968.

2. M. P. Kostenko, L. R. Neiman, G. N. Blavadzevich. Electromagnetic Processes in Systems with High-Power Rectifier Installations. Publishing House of the Academy of Sciences of the USSR, Moscow--Leningrad, 1946.

3. A. I. Zaitsev, Yu. M. Achkasov, M. A. Borovikov, V. N. Mishin, A. A. Kuvshinov, V. P. Obrusnik, V. N. Chiryev. A node of artificial commutation on thyristors for a six-phase converter with an interphase reactor. Izvestiya TPI, vol. 161, 1967.