ARTIFICIAL SWITCHING UNIT ON THYRISTORS FOR A SIX-PHASE CONVERTER WITH AN INTERPHASE REACTOR

A. I. Zaitsev, Yu. M. Achkasov, M. A. Borovikov, V. N. Mishin,
A. A. Kuvshinnikov, V. P. Obrusnik, V. I. Chiryev

In most valve converters, the rectified voltage is regulated by changing the firing angle of the valves, $\alpha$. As the angle $\alpha$ is increased, the power factor of valve converters decreases sharply. At present, mercury rectifiers alone convert more than 15% of all electric power produced in the country. Therefore, increasing the power factor of converter units is a very important national-economic task.

The possibility of creating valve circuits with artificial commutation for generating reactive power was pointed out by G. I. Babat [1]; a number of circuits were developed by I. L. Kaganov [2], I. M. Chizhenko [3], and others.

Despite the large number of developed circuits and methods of artificial commutation, practical application has been found only by compensating units created at the Kiev Polytechnic Institute (KPI) under the direction of I. M. Chizhenko. However, the KPI circuit requires a relatively large increase in the installed capacity of the additional equipment, mainly on account of the three-phase interphase coil.

Below we consider a simple circuit for artificial current commutation in a six-phase rectifier with an interphase reactor, consisting of a commutating capacitor $C$ and commutating controlled valves $B_{\mathrm{I}}$ and $B_{\mathrm{II}}$ (Fig. 1).

Let us consider the operation of a converter with an artificial commutation unit, beginning from the instant when capacitor $C$ is charged with the polarity indicated in Fig. 1, and the voltage across it is

\[ U_c > e_{2a} - e_{2x}. \]

The timing diagrams explaining the operation and purpose of the individual elements of the circuit are presented in Fig. 2.

Assume that the power valves $B_1$ and $B_6$ are conducting current. At the instant when it is necessary to artificially turn off valve $B_1$, valve $B_1$ is opened. By this instant, a blocking potential must be present on the grid of valve $B_1$.

The capacitor \(C\) begins to discharge through the operating valve \(B_1\), passing through it a current in the nonconducting direction. Over a time interval equal to the commutation time \(\gamma_1\), the current from the main valve \(B_1\) transfers to the commutating valve \(B'\) and to the other main valve \(B_6\). Then

Fig. 1.

the commutating capacitor is discharged and recharged by a direct current equal to half the load current.

When the decreasing voltage \(U_c\) becomes equal to the phase voltage in the circuit of the valve \(B_1\) that is being switched out of operation, and then becomes smaller than it, conditions are created for reignition of this valve. To prevent this, the capacitance \(C\) must be chosen so that, for the given load current, the equality of the voltages \(U_c\) and \(e_{2a}\) occurs no earlier than after the time interval

\[ \Theta_{\min}=\gamma_1+\delta, \]

where \(\delta\) is the deionization time of the valve after extinction.

When the voltage on the capacitor reaches the value

\[ U_c=e_{2z}-e_{2v} \]

conditions are created for switching on valve \(B_3\). By this time there must be a positive potential on its grid.

Under the action of the increasing voltage \(e_{2v}\), over a time interval equal to the commutation time \(\gamma_2\), the current from the commutating valve \(B_1\) transfers to the main valve \(B_3\).

As a result, the capacitor \(C\) has become charged with the opposite polarity to the voltage

\[ U_c>e_{2z}-e_{2v}. \]

Then the commutating valve \(B_{\mathrm{II}}\) opens, the capacitor is recharged again, carrying out current commutation from valve \(B_6\) to valve \(B_4\).

Fig. 2.

Fig. 3.

This process is repeated at triple the frequency of the supply network, and the charge of both signs is usefully utilized.

In designing the artificial commutation unit, the problem is reduced to selecting the value of the commutating capacitance as a function of the load and the parameters of the converter. The equivalent circuit of the working loop of the artificial commutation unit during the transfer of current from the main valve being switched off to the commutating valve (the first commutation) is shown in Fig. 3.

We shall carry out the analysis under the following assumptions:

1. The parameters of the commutation circuit, the load current, and the phase voltages of the transformer remain unchanged during the turn-off interval.

2. The magnitudes of the voltage drops across the valves are equal and do not depend on the current flowing.

The process of current commutation from the valve being switched off into the commutating circuit is described by the following equations:

\[ \left. \begin{aligned} e_{2z}-\Delta U_a+\Delta U_a-e_{2a} &=L\frac{di_2}{dt}+i_2R-L\frac{di_1}{dt}-i_1R+U_c,\\[4pt] i_1&=\frac{I_d}{2}-i_c,\\[4pt] i_2&=\frac{I_d}{2}+i_c,\\[4pt] i_c&=C\frac{du_c}{dt}, \end{aligned} \right\} \tag{1} \]

where \(C\) is the value of the commutating capacitance;

$R$ and $L$ are the active resistance and inductance of the transformer phase;

$I_d, i_c, i_1, i_2$ are the currents of the load, the capacitor, and valves $B_1$ and $B_6$, respectively;

$\Delta U_a$ is the internal voltage drop in the valve.

Taking into account the initial conditions

\[ U_c = U_{co}, \quad i_c = 0 \]

we obtain the solution of the system of equations (1):

\[ U_c = e^{\sigma t}(U_{co}+\Delta E_1)\left(\frac{\sigma}{\beta}\sin \beta t-\cos \beta t\right)+\Delta E_1; \tag{2} \]

\[ i_c=\frac{1}{2L\beta}e^{\sigma t}(U_{co}+\Delta E_1)\sin \beta t, \tag{3} \]

where

\[ \Delta E_1=e_{2z}-e_{2a}-U_m\cos\left(\frac{2\pi}{3}-\alpha\right); \]

\[ \sigma=-\frac{R}{2L}=-\frac{1}{2T}; \]

\[ \beta=\sqrt{\frac{1}{2LC}-\frac{1}{4T^2}}. \]

The duration of the first commutation is found from the condition:

\[ (i_c)_{t=\gamma_1}=-\frac{I_d}{2}. \tag{4} \]

Substituting (4) into (3) and expanding the exponential and trigonometric functions in a Maclaurin series, we obtain approximately:

\[ \gamma_1=T\left(1-\sqrt{1-\frac{2I_dR}{U_{co}+\Delta E_1}}\right). \tag{5} \]

The condition for reliable turn-off of the power valve is

\[ (U_c)_{t=\gamma_1+\delta}=\Delta E_1. \tag{6} \]

Therefore, taking into account that after commutation the capacitor discharges with a current constant in magnitude,

\[ \Delta E_1=e^{\sigma \gamma_1}(U_{co}+\Delta E_1)\left(\frac{\sigma}{\beta}\sin \beta \gamma_1-\cos \beta \gamma_1\right)+\Delta E_1+\frac{I_d}{C}\delta. \tag{7} \]

Solving (5) and (7) jointly, we obtain the value of the initial voltage on the commutating capacitor:

\[ U_{co}=\frac{I_d}{3.14C}\left(\sqrt{2CL}+\delta\right)-\Delta E_1. \tag{8} \]

After the power valve has been extinguished, a constant current equal to $\dfrac{I_d}{2}$ flows through the capacitor. The time of flow of the constant current, under the condition

\[ (U_c)_{t=\gamma_1+t_n}=\Delta E_2 \tag{9} \]

is determined from the formula:

\[ t_n=\delta+\frac{2C(\Delta E_2-\Delta E_1)}{I_d}, \tag{10} \]

where

\[ \Delta E_2=e_{2z}-e_{2\mathrm{в}}=U_m\cos\left[\frac{\pi}{3}-\alpha+\omega(\gamma_1+t_n)\right]. \]

The equivalent circuit of the operating loop of the artificial commutation unit during transfer of current from the commutating valve to the main valve entering operation (the second commutation) is given in Fig. 4. It differs from the preceding one only in the values of the transformer phase voltages. Therefore the differential equation describing the process of the second commutation will have the same form as (1), except that \(\Delta E_2\) will enter it instead of \(\Delta E_1\).

Taking into account the initial conditions

\[ U_c=\Delta E_1,\quad i_c=-\frac{I_y}{2}, \]

we obtain the solution of equations (1) for the second commutation:

\[ U_c=\frac{I_d}{2C\beta}e^{\sigma t}\sin\beta t+\Delta E_2, \tag{11} \]

\[ i_c=\frac{I_d}{2\beta}e^{\sigma t}(\sigma\sin\beta t+\beta\cos\beta t). \tag{12} \]

The condition for completion of the recharging process of the commutating capacitor is:

\[ \left. \begin{aligned} (i_c)_{t=\gamma_2}&=0\\ (U_c)_{t=\gamma_2}&=U_{c0} \end{aligned} \right\} \tag{13} \]

Fig. 4.

Substituting (13) into (12), we obtain the duration of the second commutation:

\[ \gamma_2=\frac{2LRC}{2L-R^2C}\left(\sqrt{\frac{4T}{RC}-1}-1\right). \tag{14} \]

The total time of current transfer from one main anode to another must satisfy the condition:

\[ t_{k\Sigma}<\frac{1}{2f_k}, \tag{15} \]

where \(t_{k\Sigma}=\gamma_1+t_p+\gamma_2\);
\(f_k=150\ \text{Hz}\) is the commutation frequency.

After substituting (14) into (11) and expanding the exponential and trigonometric functions in Maclaurin series, we obtain:

\[ U_{c0}=\frac{I_d}{\sqrt{2}}\left(\sqrt{\frac{L}{C}}-R\right)+\Delta E_2. \tag{16} \]

Equating (8) and (16), we obtain a second-degree equation with respect to the capacitance \(C\). Its solution gives the minimum capacitance of the commutating capacitor sufficient for reliable commutation:

\[ C=\frac{I_d}{\Delta E^2}\left[0.44\Delta E+0.3I_dL-\sqrt{I_d(0.14L\Delta E+0.096I_dL^2)}\right], \tag{17} \]

where \(\Delta E=\Delta E_1+\Delta E_2\).

Knowing the capacitance and the voltage on the capacitor, it is possible to calculate the required installed power of the capacitors in the artificial commutation unit. Capacitors used in power systems to improve the power factor are usually rated at the industrial frequency of \(50\ \text{Hz}\). Therefore the installed capacitor power

tors, referred to a frequency of 50 Hz, under the assumption that the voltage waveform on the capacitors is close to sinusoidal, will be determined by the expression from [2]:

\[ Q_c=\frac{U_{c0}^{2}C}{2}\,\omega . \tag{18} \]

Substituting (16) into (18), we obtain:

\[ Q_c=\left[\frac{I_d}{\sqrt{2}}\sqrt{L}+\sqrt{C}\left(\Delta E_2-\frac{I_d}{\sqrt{2}}R\right)\right]^2\omega . \tag{19} \]

To estimate the power factor of the converter with artificial commutation, and also to determine the coefficient of effectiveness of utilization of the capacitors, let us assume that the load inductance is infinitely large, i.e., the load current is ideally smoothed. In this case the secondary current of the power transformer will have the form of trapezoids (Fig. 5).

Fig. 5.

After an approximate expansion of the alternating-current curve into a Fourier series, we obtain the rms value of the first harmonic of the current

\[ I_{(1)}=\frac{\sqrt{2}I_d}{\pi}\cos\frac{v_{k\Sigma}-v_{\mathrm{p}}}{4}\times \]

\[ \times \cos\left(\frac{\pi}{6}-\frac{v_{k\Sigma}+v_{\mathrm{p}}}{4}\right) \tag{20} \]

and the displacement angle of the first harmonic

\[ \varphi_{(1)}=\alpha-\frac{v_{k\Sigma}}{2}, \tag{21} \]

where \(v_{k\Sigma}=\omega t_{k\Sigma}\),

\[ v_{\mathrm{p}}=\omega t_{\mathrm{p}}. \]

The reactive power generated by the converter is:

\[ Q=3UI_{(1)}\sin\varphi_{(1)}= \]

\[ =3U\frac{\sqrt{2}I_d}{\pi}\cos\frac{v_{k\Sigma}-v_{\mathrm{p}}}{4}\cdot \cos\left(\frac{\pi}{6}-\frac{v_{k\Sigma}+v_{\mathrm{p}}}{4}\right)\cdot \sin\left(\alpha-\frac{v_{k\Sigma}}{2}\right), \tag{22} \]

where \(U\) is the rms value of the phase voltage of the transformer.

The ratio of expressions (22) and (19) gives the coefficient of effectiveness of utilization of the capacitors.

Experimental verification of the considered artificial-commutation circuit was carried out on an installation with TG1-5/3-type thyratrons. D-238g-type thyristors were used as the commutating valves. The experiments carried out showed good serviceability of the circuit. The coefficient of effectiveness of utilization of the capacitors reached a value of \(4.5 \div 5\).

Such circuits are recommended for application in those cases where a significant reduction of the rectified voltage, as in the PFC circuit, is not permissible.

References

1. G. I. Babat, Ya. A. Katsman. Thyristor converters with an improved power factor and thyristor compensators. Elektrotekhnika, 1937, No. 4.

2. I. L. Kaganov. Electronic and Ion Converters, Part III, GEI, 1956.

3. I. M. Chizhenko. “A circuit for converting electric current, ‘star–straight and reverse stars with equalizing coils and commutating capacitors’.” Izvestiya KPI, Vol. XXII, Kiev, 1957.