CALCULATION OF A CONVERTER UNIT WITH SINGLE-STAGE FORCED COMMUTATION

V. A. Shubenko

(Presented by Professor, Doctor V. K. Shcherbakov)

Circuits with controlled valve devices have now become widely used in various fields of technology for rectifying and inverting current, as well as for frequency conversion.

It is known that all circuits with valve devices possess the property of consuming, to a greater or lesser extent, a current lagging in phase behind the voltage of the system to which the converter is connected. Owing to the leakage reactance of the system, even rectifiers operating without ignition delay of the next valve possess this property, but it is manifested especially sharply in controlled rectifiers and inverters. In converting considerable electric powers, this property must be taken into account, since operation of the converter under such conditions is associated with considerable reactive power consumed from the alternating-current system.

It is known, for example, that the shortage of reactive power arising at the inverter end of direct-current power-transmission lines leads to the necessity of installing reactive-power generators, which substantially affect the economics of direct-current lines.

Under conditions of converting considerable electric powers, because of a number of positive qualities, the six-phase circuit proposed by Professor Larionov, or cascade-bridge circuits based on the Larionov circuit, proves advantageous.

The equation for the variation of the current decreasing during the commutation time, as applied to this circuit [1], is expressed as

\[ i_{з}=I_d\, \frac{\cos\alpha+\cos\left(\vartheta-\frac{\pi}{3}\right)} {\cos\alpha-\cos(\alpha+\gamma)} . \tag{1} \]

To determine the overlap angle, the following equation may be used:

\[ \cos \alpha - \cos(\alpha + \gamma) = \frac{2 I_d X_a}{E_{\mathrm{lm}}}. \tag{2} \]

Relations 1 and 2 are valid when the inductance in the converter dc circuit is sufficiently large.

In these equations:

\(I_d\) — the magnitude of the current in the dc circuit,

\(X_a\) — the leakage reactance of the transformer and of the alternating-current transmission system,

\(E_{\mathrm{lm}}\) — the amplitude of the secondary line voltage,

\(\alpha\) — the firing angle of the valves, measured from the instant of firing in rectification,

\(\gamma\) — the overlap angle,

\(\vartheta = \omega t\).

When operating in the rectifier mode with \(\alpha \ne 0\), and especially in the inverter mode, this circuit is also characterized by consumption of reactive power from the ac system.

In addition to the usual methods of generating reactive power for converters, numerous so-called forced-current-commutation circuits have been proposed, which are commonly divided into two-stage and single-stage circuits. Among the latter, several circuits are known [2] based on the introduction of a reactor with a tapped midpoint and parallel-connected capacitors, or of an auxiliary transformer whose secondary windings are closed through capacitors (Fig. 1).

Fig. 1.

The presence in the circuit of such an auxiliary transformer with capacitors in the secondary windings leads to the appearance in the ac circuit of additional emfs superposed on the fundamental sinusoidal emfs. The distortion of the latter makes it possible to increase the angle \(\alpha\) and to change the phase-shift angle between current and voltage, so that the inverter, for example, can be freed from reactive-power consumption or even operate in the mode of generating it. The curves of currents and voltages characterizing the operation of a converter with forced commutation are shown in Fig. 2.

To evaluate the stability of inverter operation, as well as the magnitude of the inverter reactive power, it is necessary to be able to determine the angle \(\gamma\).

Let us obtain an equation characterizing the change of current during overlap, as well as an equation for determining the angle \(\gamma\) under forced-

forced commutation. In order to simplify the solution of the problem, we shall use the following simplifying assumptions:

Fig. 2.

1. We do not take into account the active and equivalent active resistances of the electric circuit.

2. We neglect the magnetizing current of the auxiliary transformer.

3. We disregard the effect of the capacitance of the transformer windings, its terminals, and the valves on the nature of the commutation process.

4. We assume that in the direct-current circuit of the converter there is a reactor with sufficiently large inductance.

In the calculation we use the following notation for the quantities:

\(i_{ca}, i_{cb}, i_{cc}\) — currents through the capacitors of phases \(a, b\), and \(c\) of the auxiliary transformer,

\(i_3, i_5\) — the valve currents 3 and 5 varying during the overlap,

\(e_{ca}, e_{cb}, e_{cc}\) — voltages across the capacitors of phases \(a, b\), and \(c\),

\(E_{ca0}, E_{cb0}, E_{cc0}\) — voltage across the capacitors of phases \(a, b\), and \(c\) at the instant of the beginning of commutation,

\(X_a\) — the sum of the leakage reactances of the main and auxiliary transformers and the reduced reactance of the alternating-current system,

\[ X_c = \frac{1}{\omega C} \]

— the capacitive reactance,

\[ \frac{E_{\ell m}}{X_c - X_a} = A; \qquad \sqrt{\frac{-X_c}{X_a}} = B; \]

\[ E_{cb0} - E_{cc0} = D. \]

Using the condition of magnetic equilibrium of the auxiliary transformer, it is possible to relate the primary and secondary currents. This relation is obtained more simply for a transformation ratio \(k_{\mathrm{tv}} = 1\).

In the interval

\[ \alpha - \frac{\pi}{3} + \gamma < \vartheta < \alpha \]

\[ \begin{gathered} I_d-i_{ca}+i_{cb}-I_d=0,\\ I_d-i_{cb}+i_{cc}=0,\\ i_{ca}+i_{cb}+i_{cc}=0,\\ i_{ca}=i_{cb}=\frac{1}{3}I_d,\\ i_{cc}=-\frac{2}{3}I_d . \end{gathered} \tag{3} \]

In the interval \(\alpha<\vartheta<\alpha+\gamma\),

\[ \left. \begin{aligned} &I_d-i_{ca}+i_{cb}-i_3=0,\\ &i_3-i_{cb}+i_{cc}-i_5=0,\\ &i_3+i_5=I_d,\\ &i_{ca}+i_{cb}+i_{cc}=0,\\ &i_{ca}=\frac{1}{3}I_d, \end{aligned} \right\} \tag{4a} \]

\[ i_{cb}=i_3-\frac{2}{3}I_d, \tag{4b} \]

\[ i_{cc}=i_5-\frac{2}{3}I_d, \tag{4c} \]

Fig. 3.

For the interval \(\alpha+\gamma<\vartheta<\alpha+\dfrac{\pi}{3}\), in an analogous manner we obtain

\[ i_{ca}=i_{cb}=\frac{1}{3}I_d,\qquad i_{cb}=-\frac{2}{3}I_d . \tag{5} \]

Thus, outside the commutation periods, the currents through the capacitors are expressed in terms of the current \(I_d\).

The voltage balance for the commutation circuit according to Fig. 3 in the interval \(\alpha<\vartheta<\alpha+\gamma\)

\[ \left. \begin{aligned} &e_{53}-e_{cb}-e_b+e_c-i_{55}+e_{cc}=0,\\ &e_s=-X_a\frac{di}{d\vartheta},\\ &e_c=E_{c0}+X_c\int i_c\,d\vartheta . \end{aligned} \right\} \tag{6} \]

From 4(b) and 4(v) it follows that

\[ i_{cb}-i_{cc}=i_3-i_5=i. \tag{7} \]

When the angles are measured from the instant at which commutation begins,

\[ e_b-e_c=E_{\ell m}\sin(\vartheta+\alpha). \]

Thus, equation (6) can be replaced by the differential equation characterizing the change in the current difference \(i\) during the overlap period,

\[ \frac{d^2 i}{d\vartheta^2}+\frac{X_c}{X_a}\,i = -\frac{E_{\ell m}}{X_a}\cos(\vartheta+\alpha). \tag{8} \]

To solve equation (8) by the operational method, the initial conditions are determined.

At the instant commutation begins, for \(\vartheta=0\),

\[ i_0=I_d \]

and

\[ \left(\frac{di}{d\vartheta}\right)_0 = -\frac{E_{\ell m}\sin\alpha+D}{X_a}. \]

The solution of equation (8) as applied to the inverter mode, when \(\beta=\pi-\alpha\), gives:

\[ i=A\cos(\vartheta-\beta)+(I_d-A\cos\beta)\cos B\vartheta+ \]

\[ +\left(\frac{D}{BX_a}+AB\sin\beta\right)\sin B\vartheta. \tag{9} \]

Solving jointly (4a) and (7), we obtain

\[ i_3=\frac{i+I_d}{2}, \tag{10a} \]

\[ i_5=\frac{I_d-i}{2}. \tag{10b} \]

Substitution of (9), for example into (10a), does not give a definite relation \(i_3=f(\vartheta)\), since the constant \(D\), among the other determined quantities, is unknown.

The constant \(D\) can be determined by using the conditions of periodicity and equilibrium of the three-phase nonsinusoidal system of currents and voltages.

From Fig. 2 it follows that

\[ E_{ca0}=E_{cb0}-X_c I_d\frac{\pi}{9}, \]

\[ E_{cc0}=E_{cb0}-X_c I_d\frac{2\pi}{9} + X_c\int_0^\gamma i_3\,d\vartheta, \tag{11} \]

\[ D=E_{cb0}-E_{cc0} = X_c I_d\frac{2\pi}{9} - X_c\int_0^\gamma i_3\,d\vartheta. \]

Substituting equation (10a), with (9) taken into account, into (11) and integrating, we obtain

\[ D=\frac{1}{1+\cos B\gamma}\left[ 2X_c I_d\left(\frac{2\pi}{9}-\frac{\gamma}{9}\right) -AX_c\sin(\gamma-\beta) \right. \]
\[ \left. {}-BX_a\left(I_d-A\cos\beta\right)\sin B\gamma -AX_c\sin\beta\cdot\cos B\gamma \right]. \tag{12} \]

In equation (12) the constant \(D\) is expressed in terms of the unknown overlap angle \(\gamma\) under forced commutation of the currents.

If we use the condition that, for \(\theta=\gamma\), \(i_3=0\) for (10a), we obtain:

\[ \frac{A}{2}\cos(\gamma-\beta) +\frac{1}{2}\left(I_d-A\cos\beta\right)\cos B\gamma \]
\[ {}-\frac{1}{2}\left(\frac{D}{BX_a}+AB\sin\beta\right)\cdot\sin B\gamma +\frac{1}{2}I_d=0. \tag{13} \]

Equations (12) and (13), for given \(I_d\), \(E_{\text{sh}}\), \(X_c\), \(X_a\), and \(\beta\), constitute a system of two transcendental equations with two unknowns, \(D\) and \(\gamma\), from which these quantities can be determined.

Substitution of the constant \(D\) into equation (10a), with (9) taken into account, gives a definite law of current variation during the overlap interval under forced commutation.

Thus, instead of a fairly simple and readily solved equation for determining \(\gamma\) under natural commutation of currents, we have a system of two transcendental equations for determining the analogous quantity under forced commutation of currents.

Equations (11, 12, 13) also make it possible to determine other quantities characteristic of a converter with single-stage forced commutation.

REFERENCES

1. K. A. Krug, Electromagnetic Processes in Installations with Controlled Mercury Rectifiers. ONTI, 1935.
2. Gen. El. Rev., Willis, 1932, vol. 35, p. 32.