On the Calculation of Certain Mutators Using Fully Controlled Elements

A. I. Zaitsev, V. N. Mishin

(Presented by the scientific seminar of the Electromechanical Faculty)

Valve converters of alternating current into direct current have become widely used in various fields of technology, especially in recent times, which is connected with the development of semiconductor instrument engineering. However, when it is necessary to regulate the rectified voltage over wide limits, the power factor of the installation proves to be low. In order to improve the energy characteristics of such systems in mutators with a small number of phases, under an active-inductive load, it is proposed to shunt the latter with an uncontrolled valve [1], through which the energy of the electromagnetic field of the inductance is discharged after the corresponding phase voltage of the AC network passes through zero into the negative region. In addition, systems with asymmetric control of the valves were created, as well as numerous circuits with artificial commutation; in the latter, the ignition angle of the next valve is leading, which allows them to operate with the return of reactive power to the network. Of the systems with artificial commutation, only powerful installations with a relatively small range of regulation of the rectified voltage, developed at the Kiev Polytechnic Institute under the direction of I. M. Chizhenko, have found application.

The appearance of sufficiently powerful transistor switches, thyristor converters with artificial commutation, and also the development of fully controlled silicon valves make it possible to create converter devices also for relatively small powers, in which the principal drawback of valve circuits noted above is eliminated.

In ordinary valve systems with a shunting circuit, the duration of current pulses through the valves in the continuous load-current mode for a given converter circuit is determined uniquely by the switching-on (ignition) angle of the valves; in the discontinuous mode, by the switching-on angle and the load parameters. In mutators, where it is possible in one way or another [1], [3] to turn off the conducting power valve earlier than the conditions for natural commutation are created, the duration of valve operation will depend not only on the cir-

...the rectification scheme, but also on the signals of the control circuit. Therefore, here it is possible to operate with a power factor that is, to some extent, optimal for the given mode, by correspondingly varying in each particular case the firing angles and the duration of conduction of the power valves.

The use of fully controlled elements in valve systems also makes it possible, in the case of an active-inductive load, to operate with negative firing angles; here, for some time after the firing of the next power valve, an inverter mode will occur, with the energy stored in the load inductance being returned to the network. In such installations the need for a shunting valve disappears, and their operating mode corresponds to the maximum return of reactive energy.

Thus, the possibility of interrupting in one way or another the current in a conducting power valve (switching it off) makes it possible to create mutators for converting alternating current into direct current with higher energy characteristics.

The principal difference between such systems with semiconductor devices is that the power required for switching them off is small in comparison with ionic valves, and the turn-off time is insignificant, which makes it possible, without great error, to regard them as systems with instantaneously acting switches.

In the present work the operating modes of the direct-current and alternating-current circuits of an \(m\)-phase mutator with a shunting circuit using valves, which can be switched off in one way or another, are investigated. In this case the switching-off method itself is not considered, and the energy used for switching off the devices is not taken into account.

In the analysis we shall adopt the following assumptions:

1) the system is supplied from a network of infinite power, i.e., its active and inductive resistances are equal to zero;

2) the firing and switching-off of the valves occur instantaneously and without expenditure of additional energy;

Fig. 1

3) the voltage drops across the mutator valves are constant and equal to one another;

4) the load consists of an inductance, an ohmic resistance, and a d.c. emf, all unchanged in magnitude.

Under the assumptions made, the equivalent replacement circuit in the single-phase version will have the form shown in Fig. 1. In this circuit the controlled valve is represented by an automatic switch \(K\), which, at the corresponding instants of time, periodically and instantaneously performs the required switching in the main and shunting circuits. During these switchings, according to the adopted assumptions, the load current instantaneously passes, with unchanged magnitude, from the power circuit into the shunting circuit or vice versa, remaining unchanged in magnitude and direction in the load. The voltage drops across the valves are taken into account by an emf source \(E_{\mathrm{v}}\), without internal resistance, connected opposite to the current.

In accordance with the replacement circuit, the time during which switch \(K\) closes the main circuit will be called the conduction time, and the time during which the switch closes the shunting circuit will be called the pause time.

To simplify the derivations and for convenience in using the results obtained, in the investigation we shall employ the per-unit method developed, as applied to valve systems, by A. A. Bulgakov [2]. As base quantities we shall take the amplitude of the sinusoidal line voltage \(U_m\) and the short-circuit current determined by the amplitude of the line voltage and the active resistance of the d.c. circuit.

Let the relative voltage be denoted by

\[ \varepsilon = \frac{E + E_{\mathrm{v}}}{U_m} \tag{1} \]

and the relative current by

\[ j = \frac{i}{I_m} = \frac{iR}{U_m}, \tag{2} \]

where
\(E\) — emf of the d.c. circuit;
\(E_{\mathrm{v}}\) — voltage drop across the corresponding valve;
\(U_m\) — amplitude of the sinusoidal line voltage;
\(i\) — instantaneous value of the current;
\(I_m\) — base value of the current;
\(R\) — active resistance of the load circuit.

We shall take into account the electromagnetic inertia of the d.c. circuit by the load parameter

\[ \tan \Theta = \frac{\omega L}{R} = \omega T, \tag{3} \]

where \(\omega = 2\pi f\) — angular frequency of the a.c. network;
\(L\) — load inductance;
\(T\) — electromagnetic time constant of the load.

As in ordinary valve systems, the load current of the converters under consideration may be either continuous or discrete (intermittent). In the first case, the smallest possible firing angle, measured from the beginning of the positive half-waves of the line-voltage sinusoids, is determined by the value of the line voltage

\[ U_m \sin \vartheta_{\mathrm{v}} \geq 0 \tag{4} \]

or

\[ \vartheta_{\mathrm{v}} \geq 0. \tag{4a} \]

Analytically, the processes in the circuit in Fig. 1 are described by the following system of differential equations during conduction and pause, respectively:

\[ \left. \begin{aligned} U_m \sin(\vartheta_{\mathrm{on}}+\omega t) &= E+E_{\mathrm{v}}+iR+L\frac{di}{dt},\\ 0 &= E+E_{\mathrm{v}}-iR+L\frac{di}{dt} \end{aligned} \right\}, \tag{5} \]

where \(t\) is the time reckoned from the instant of closing of one or another circuit.

Dividing both equations of system (5) by \(U_m\) and replacing time by the angular coordinate \(\vartheta=\omega t\), we obtain, in relative units,

\[ \left. \begin{aligned} \sin(\vartheta_{\mathrm{on}}+\vartheta) &= \varepsilon+j+\operatorname{tg}\Theta\,\frac{dj}{d\vartheta},\\ 0 &= \varepsilon+j+\operatorname{tg}\Theta\,\frac{dj}{d\vartheta} \end{aligned} \right\}. \tag{6} \]

The solution of the differential equations (6) gives the following expressions for the load current at the instants of conduction and pause in relative units:

\[ \left. \begin{aligned} j_{\mathrm{on}} &= \left[\cos\Theta \sin(\vartheta_{\mathrm{on}}-\Theta+\vartheta)-\varepsilon\right] + \left[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{on}}-\Theta)+j_{0\mathrm{on}}\right] e^{-\vartheta \operatorname{ctg}\Theta},\\ j_{\mathrm{p}} &= (\varepsilon+j_{0\mathrm{p}})e^{-\vartheta \operatorname{ctg}\Theta}-\varepsilon \end{aligned} \right\}, \tag{7} \]

where \(j_{0\mathrm{on}}\) and \(j_{0\mathrm{p}}\) are the initial values of the currents at the instants of conduction and pause, respectively.

The initial values of the currents in the continuous mode are found as

\[ \left. \begin{aligned} j_{\mathrm{sp}} &= j_{\mathrm{on}} \quad \text{when} \quad \vartheta=\lambda,\\ j_{\mathrm{si}} &= j_{\mathrm{p}} \quad \text{when} \quad \vartheta=\frac{2\pi}{m}-\lambda \end{aligned} \right\}, \tag{8} \]

where \(\lambda\) is the duration of conduction of the power valves;
\(m\) is the number of phases of the mutator.

Substituting conditions (8) into the system of equations (7), we obtain

\[ \left. \begin{aligned} j_{0\mathrm{p}} &= \left[\cos\Theta\sin(\vartheta_{\mathrm{on}}-\Theta+\lambda)-\varepsilon\right] + \left[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{on}}-\Theta)+j_{\mathrm{si}}\right] e^{-\lambda \operatorname{ctg}\Theta},\\ j_{0\mathrm{on}} &= (\varepsilon+j_{0\mathrm{p}}) e^{-\left(\frac{2\pi}{m}-\lambda\right)\operatorname{ctg}\Theta} -\varepsilon \end{aligned} \right\}, \tag{9} \]

whence the initial values of the corresponding currents are determined as

\[ j_{0\mathrm{on}}= \frac{ \cos\Theta\left[ e^{\lambda\operatorname{ctg}\Theta}\sin(\vartheta_{\mathrm{on}}-\Theta+\lambda) -\sin(\vartheta_{\mathrm{on}}-\Theta) \right] }{ e^{\frac{2\pi}{m}\operatorname{ctg}\Theta}-1 } -\varepsilon, \tag{10} \]

\[ j_{\mathrm{on}}= \frac{ \cos \Theta \left[\sin(\vartheta_{\mathrm{v}}-\Theta+\lambda) - e^{-\lambda \operatorname{ctg}\Theta}\sin(\vartheta_{\mathrm{v}}-\Theta)\right] }{ 1-e^{-\frac{2\pi}{m}\operatorname{ctg}\Theta} } -\varepsilon \tag{11} \]

Fig. 2

The form of the current pulses in the various links of the mutator under consideration, with continuous load current, is shown in Fig. 2.

In the steady-state operating mode of the converter, the maximum angle of the duration of valve conduction is equal to \(2\pi/m\); in this case the pause mode is absent, and the current, under the adopted assumptions, instantaneously passes from one power valve to another. In this connection it should be noted that the presence of a shunting valve is necessary only over a relatively large range of regulation of the average rectified voltage. If, when one power ...

valve, another is simultaneously turned on; the need for a shunting circuit disappears.

Expressions (7) are also valid for ordinary converter systems with a shunting circuit operating with a lagging firing angle. In this case the duration of conduction of a valve of one phase or another will be determined, in the continuous-current regime, by the firing angles

\[ \lambda=\pi-\vartheta_{\mathrm{в}} \qquad \text{for} \qquad \vartheta_{\mathrm{в}}\geqslant \pi\,\frac{m-2}{m} \]

and

\[ \lambda=\frac{2\pi}{m} \qquad \text{for} \qquad \vartheta_{\mathrm{в}}\leqslant \pi\,\frac{m-2}{m}. \]

The expressions obtained are valid, as was indicated above, only in the region of existence of the continuous load-current regime. The conditions for the existence of this regime will be

\[ j_{\mathrm{on}}\geqslant 0,\ \text{if } \vartheta_{\mathrm{в}}\geqslant \arcsin \varepsilon, \tag{12} \]

and

\[ j_{\mathrm{i}}\geqslant 0 \quad \text{for } \vartheta=\arcsin \varepsilon-\vartheta_{\mathrm{в}},\ \text{if } \vartheta_{\mathrm{в}}\leqslant \arcsin \varepsilon. \tag{13} \]

Thus, the limiting values of the conduction-duration angles at which the continuous-current regime in the load still exists, taking into account conditions (12)—(13), are found from (7) and (10) as

\[ 0= \frac{ \cos\Theta\left[ e^{\hat{\lambda}_{\mathrm{rp}}\operatorname{ctg}\Theta} \sin(\vartheta_{\mathrm{в}}-\Theta+\hat{\lambda}_{\mathrm{rp}}) -\sin(\vartheta_{\mathrm{в}}-\Theta) \right] }{ e^{-\frac{2\pi}{m}\operatorname{ctg}\Theta}-1 } -\varepsilon, \tag{14} \]

if

\[ \vartheta_{\mathrm{в}}\geqslant \arcsin \varepsilon, \]

and

\[ 0=\cos\Theta\sin(\arcsin \varepsilon-\Theta)-\varepsilon+ \]

\[ -\left[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{в}}-\Theta)\right] +j_{\mathrm{on}}\,e^{-(\arcsin \varepsilon-\vartheta_{\mathrm{в}})\operatorname{ctg}\Theta}, \tag{15} \]

if

\[ \vartheta_{\mathrm{в}}\leqslant \arcsin \varepsilon. \]

Outside conditions (14)—(15), the load current is discontinuous. In this case, in the systems under consideration with a shunting circuit there may in turn occur three varieties of the discontinuous-current regime (Fig. 3).

1) The load current falls to zero during the pause (Fig. 3a);

2) The load current decreases to zero during operation of the next power valve in the range of angles (Fig. 3b)

\[ \vartheta<\arcsin \varepsilon \tag{16} \]

3) The current in the power valve falls to zero before the command for its forced or natural turn-off arrives (Fig. 3c).

The last variant of the discontinuous-current regime, which also occurs in valve systems with the ordinary (lagging) control angle, has been studied in detail in the work of A. A. Bulgakov [2] and is not considered here.

Let us examine the regions of existence of the first two cases of the discontinuous-current regime in greater detail.

1. The load current decreases to zero during the pause

Here the minimally possible firing angles of the power valves are determined by the values of the d.c. emf in the load circuit and by the voltage drop across the power valves

\[ \vartheta_{\mathrm{в}}\geqslant \arcsin \varepsilon \tag{17} \]

Fig. 3a.
Labels in the figure include: \(e=U_m\sin\omega t\), \(\vartheta_{\mathrm{on}}\), \(\lambda_i\), \(E\), \(\vartheta\), \(i_i\), \(i_p\), \(i_d\), \(E_{\mathrm{on}}\), \(e\), \(u_d\), \(\lambda_p\).

The load current in this mode is described by expressions (7), where the initial turn-on current is equal to zero, and the initial value of the pause current is found from condition (8), whence, after substituting the corresponding quantities, we obtain

\[ j_{0p}= \left[\cos\Theta\cdot\sin(\vartheta_{\mathrm{on}}-\Theta+\lambda)-\varepsilon\right] + \left[\varepsilon-\cos\Theta\cdot\sin(\vartheta_{\mathrm{on}}-\Theta)\right] e^{-\lambda\operatorname{ctg}\Theta}, \tag{18} \]

that is, the final expression for the load current in the pause mode will have the form

\[ j_p= \left\{ \cos\Theta\cdot\sin(\vartheta_{\mathrm{on}}-\Theta+\lambda) + \left[\varepsilon-\cos\Theta\cdot\sin(\vartheta_{\mathrm{on}}-\Theta)\right] e^{-\lambda\operatorname{ctg}\Theta} \right\} e^{-\vartheta\operatorname{ctg}\Theta} -\varepsilon . \tag{19} \]

The boundary conditions for the existence of this intermittent-current mode will be

\[ j_{p.l}=0 \qquad \text{when } \vartheta=\frac{2\pi}{m}-\lambda_{\mathrm{lim}1}, \tag{20} \]

if \(\vartheta'_{\mathrm{v}} > \arcsin \varepsilon\), and

\[ j_{\mathrm{p}} = 0 \quad \text{when} \quad \vartheta = \frac{2\pi}{m} - \arcsin \varepsilon - \lambda_{\mathrm{pr}1} + \vartheta'_{\mathrm{v}}, \tag{21} \]

if \(\vartheta'_{\mathrm{v}} < \arcsin \varepsilon\),

where
\(\lambda_{\mathrm{pr}1}\) is the duration of turn-on of the power valves at which the limiting case of existence of the considered intermittent-current mode occurs;
\(\vartheta'_{\mathrm{v}}\) is the angular coordinate corresponding to the application, to the next power valve, of the turn-on signal.

It follows from condition (21) that, in the given intermittent-current mode, when the control signal arrives the power valve will not begin operation, since the magnitude of the alternating voltage is less than the sum of the voltage drop across the valve and the back emf in the dc circuit, and the induc-

Fig. 3b. Diagram of waveforms. Visible labels: \(e = u_m \sin \omega t\), \(E\), \(\vartheta\), \(\vartheta_{\mathrm{v}1}\), \(\lambda_{\mathrm{i}1}\), \(\vartheta_{\mathrm{v}2}\), \(\lambda_{\mathrm{i}2}\), \(\frac{2\pi}{m}\), \(i_{\mathrm{k}2}\), \(i_{\mathrm{i}1}\), \(i_{\mathrm{p}}\), \(i_d\), \(E_{\mathrm{a}}\), \(e\), \(u_d\), \(\lambda_{\mathrm{a}}\), \(\varepsilon_{\mathrm{d}}\).

Fig. 3c

the load capacitance has already had time to discharge. If the control signal for switching on is maintained, the valve will begin to conduct current starting from the instant of time corresponding to

\[ \vartheta=\vartheta_{\mathrm{v}}=\arcsin \varepsilon . \]

Substituting conditions (20)—(21) into expression (19), we obtain equations for determining the limiting angles of the duration of switching on of the power valves

\[ 0=\left\{\cos\Theta\cdot\sin(\vartheta_{\mathrm{v}}-\Theta+\lambda_{\mathrm{rp}1})+ \left[\varepsilon-\cos\Theta\cdot\sin(\vartheta_{\mathrm{v}}-\Theta)\right]e^{-\lambda_{\mathrm{rp}1}\operatorname{ctg}\Theta}\right\} e^{-\left(\frac{2\pi}{m}-\lambda_{\mathrm{rp}1}\right)\operatorname{ctg}\Theta} -\varepsilon \tag{22} \]

for

\[ \vartheta'_{\mathrm{v}}=\vartheta_{\mathrm{v}}\geq \arcsin \varepsilon \quad \text{and} \]

\[ 0=\left\{\cos\Theta\cdot\sin(\vartheta_{\mathrm{v}}-\Theta+\lambda_{\mathrm{rp}1})+ \left[\varepsilon-\cos\Theta\cdot\sin(\vartheta_{\mathrm{v}}-\Theta)\right]\times e^{-\lambda_{\mathrm{rp}1}\operatorname{ctg}\Theta}\right\} e^{-\left(\frac{2\pi}{m}-\lambda_{\mathrm{rp}1}-\arcsin \varepsilon+\vartheta'_{\mathrm{v}}\right)\operatorname{ctg}\Theta} -\varepsilon \tag{23} \]

for

\[ \vartheta'_{\mathrm{v}}<\arcsin \varepsilon . \]

For analysis of the operation of the mutator, it is necessary to know not only the boundaries of the corresponding modes, but also the durations of the current pulses through the individual elements of the system when it operates outside the limiting conditions.

The angle of the duration of switching on the shunting valve in the discontinuous-current mode, when \(\lambda<\lambda_{\mathrm{gr}1}\), is found, for the given duration of switching on of the power valves \(\lambda\), from the condition

\[ j_{\mathrm{p}}=0 \quad \text{when } \vartheta=\lambda_{\mathrm{p}}, \tag{24} \]

whence, after substitution into (19), we obtain

\[ 0=\left\{\cos\Theta\sin(\vartheta_{\mathrm{v}}-\Theta+\lambda)+ \left[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{v}}-\Theta)\right]e^{-\lambda\operatorname{ctg}\Theta}\right\} e^{-\lambda_{\mathrm{p}}\operatorname{ctg}\Theta}-\varepsilon \tag{25} \]

or

\[ \lambda_{\mathrm{p}}=\operatorname{tg}\Theta\ln\frac{1}{\varepsilon} \left\{ \cos\Theta\sin(\vartheta_{\mathrm{v}}-\Theta+\lambda)+ \left[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{v}}-\Theta)\right]e^{-\lambda\operatorname{ctg}\Theta} \right\}. \tag{25a} \]

In those cases when \(\lambda>\lambda_{\mathrm{gr}1}\), we pass to case 2 of the discontinuous-current mode if \(\vartheta_{\mathrm{v}}<\arc\sin\varepsilon\); or to the continuous-current mode if \(\vartheta_{\mathrm{v}}>\arc\sin\varepsilon\).

2. The load current decreases to zero during operation of the power valves in the range of angles \(\vartheta<\arc\sin\varepsilon\).

This mode (Fig. 3b) occurs if the signal for switching on the next power valve arrives when the angular coordinate of the voltage satisfies inequality (16). In this case the valve is switched on and conducts current due to the emf of self-induction of the load inductance. If the amount of energy stored in the inductance is small, then the current flowing through the valve may fall to zero before the condition for its stable operation is fulfilled—\(\vartheta\geq\arc\sin\varepsilon\). After such a natural turn-off of the valve, provided that the turn-on signal is retained on its control elements, the device will resume operation when the angular coordinate becomes equal to \(\vartheta=\vartheta_{\mathrm{v}}=\arc\sin\varepsilon\). Upon repeated switching on of the valve, its current will begin to increase from zero.

If, however, the signal for switching on the valves arrives at instants of time \(\vartheta\geq\arc\sin\varepsilon\), the discontinuous-current mode under consideration cannot occur.

Thus, the current of the power valve in this mode, during the switching-on interval, consists of two pulses: the first of them, starting from some value, gradually decreases to zero, and the second increases from zero to some value at which the valve is artificially switched off.

This discontinuous-current mode exists for values of the angles of the duration of switching on of the power valves satisfying the following inequality:

\[ \lambda_{\mathrm{gr}}>\lambda>\lambda_{\mathrm{gr}1}. \tag{26} \]

It follows from inequality (26) that, on the one hand, the discontinuous-current mode under consideration bounds the region of continuous load currents, and on the other hand, the region of currents that fall to zero during the pause. In addition, the potential possibility of existence of the mode being analyzed must be checked by condition (16).

Mathematically, the load current in this mode, taking into account the system of equations (7), will be described by the following expressions:

\[ j_{\mathrm{n}1}= \left[\cos\Theta\sin(\vartheta_{\mathrm{v}1}-\Theta+\vartheta)-\varepsilon\right]+ \left[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{v}1}-\Theta)+j_{\mathrm{on}1}\right]e^{-\vartheta\operatorname{ctg}\Theta}, \tag{27} \]

\[ j_{\mathrm{i}2}=[\cos\Theta\sin(\vartheta_{\mathrm{v}2}-\Theta+\vartheta)-\varepsilon]+ \]
\[ +[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{v}2}-\Theta)]e^{-\vartheta\operatorname{ctg}\Theta}, \tag{28} \]

\[ j_{\mathrm{p}}=(\varepsilon+j_{\mathrm{op1}})e^{-\vartheta\operatorname{ctg}\Theta}-\varepsilon, \tag{29} \]

where \(\vartheta_{\mathrm{v}1}\) and \(\vartheta_{\mathrm{v}2}\) are the angular coordinates corresponding to the instant of the first and second turn-on of the power valves.

The initial values of the pause currents and of the first current pulse through the power valve will be found from the conditions

\[ j_{\mathrm{op}}=j_{\mathrm{i}2}\quad \text{for}\quad \vartheta=\lambda_{\mathrm{i}2}, \tag{30} \]

\[ j_{\mathrm{op1}}=j_{\mathrm{p}}\quad \text{for}\quad \vartheta=\lambda_{\mathrm{p}}=\frac{2\pi}{m}-\lambda_{\mathrm{i}2}-(\vartheta_{\mathrm{v}2}-\vartheta_{\mathrm{v}1}), \tag{31} \]

where \(\lambda_{\mathrm{i}2}\) is the duration of the second turn-on of the power valve.

Substituting conditions (30)—(31) into expressions (27)—(29), we obtain the following equations for determining the initial values of the load currents at various instants of time

\[ j_{\mathrm{op}}=[\cos\Theta\sin(\vartheta_{\mathrm{v}2}-\Theta+\lambda_{\mathrm{i}2})-\varepsilon]+ \]
\[ +[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{v}2}-\Theta)]e^{-\lambda_{\mathrm{i}2}\operatorname{ctg}\Theta}, \tag{32} \]

\[ j_{\mathrm{op1}}=j_{01}\{\cos\Theta\sin(\vartheta_{\mathrm{v}2}-\Theta+\lambda_{\mathrm{i}2})+ [\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{v}2}-\Theta)]\times \]
\[ \times e^{-\lambda_{\mathrm{i}2}\operatorname{ctg}\Theta}\} e^{-\left(\frac{2\pi}{m}-\lambda_{\mathrm{i}2}-\vartheta_{\mathrm{v}2}+\vartheta_{\mathrm{v}1}\right)\operatorname{ctg}\Theta} -\varepsilon \tag{33} \]

In calculations of the discontinuous-current mode under consideration, it is most convenient to use the time of the second turn-on of the power valves as the independent variable. In this case, the unknown angular coordinate of the boundary modes is the duration of the first turn-on of the power valve, which can be found from the condition

\[ j_{\mathrm{i}1}=0\quad \text{for}\quad \vartheta=\lambda_{\mathrm{i}1}. \tag{34} \]

After substituting condition (34) into equation (27), we obtain an expression from which the unknown angular quantity can be found,

\[ 0=[\cos\Theta\sin(\vartheta_{\mathrm{v}1}-\Theta+\lambda_{\mathrm{i}1})-\varepsilon]+ \]
\[ +[\varepsilon-\cos\Theta\sin(\vartheta_{\mathrm{v}1}-\Theta)+j_{\mathrm{op1}}]e^{-\lambda_{\mathrm{i}1}\operatorname{ctg}\Theta} \tag{35} \]

Expression (35) is transcendental and can be solved only by numerical and graphical methods.

Having determined the boundaries of existence of the load current modes and the duration of operation of the individual valves in each of them, we proceed to determine the rms and average values of the currents in the circuit elements.

The average and rms values of the load current in relative units, for any of the modes considered, are found as

\[ j_{\mathrm{avg}}=\frac{m}{2\pi}\left[ \int_{0}^{\lambda} j_{\mathrm{i}}(\vartheta)\,d\vartheta+ \int_{0}^{\lambda_{\mathrm{p}}} j_{\mathrm{p}}(\vartheta)\,d\vartheta \right], \tag{36} \]

\[ j_d=\sqrt{\frac{m}{2\pi}\left[\int_{0}^{\lambda} j_{u}^{2}(\vartheta)\,d\vartheta+\int_{0}^{\lambda_n} j_{u}^{2}(\vartheta)\,d\vartheta\right]} . \tag{37} \]

It must be borne in mind that, depending on the nature of the load current, the corresponding integrand functions and limits of integration must be substituted into expressions (36)--(37).

The integrals in square brackets in relations (36)--(37) are the mean-square areas of the current pulses through the corresponding valves of the system, from which the currents themselves can be found directly.

In the study of rectifier circuits it proves necessary to know the spectral composition of the current pulses passing through the power valves of the mutator. The expression for the current of the power valves (7) can be represented in the form of the following Fourier series [2]:

\[ j_u=j_{\mathrm{avg}}+\sum_{k=1}^{\infty} j_{ku}\sin(k\vartheta+\psi_k) =j_{\mathrm{avg}}+\sum_{k=1}^{\infty} a_k\sin k\vartheta+\sum_{k=1}^{\infty} b_k\cos k\vartheta, \tag{38} \]

where \(j_{\mathrm{avg}}\) is the average value of the current of the power valves of the system over the period of the supply voltage;

\[ j_{ku}=\sqrt{a_k^2+b_k^2} \tag{39} \]

is the amplitude of the \(k\)-th harmonic of the current pulse;

\[ \psi_k=\operatorname{arc\,tg}\frac{b_k}{a_k} \tag{40} \]

is the phase of the \(k\)-th harmonic.

The phase of the \(k\)-th harmonic relative to the AC line voltage is found as

\[ \varphi_k=-\operatorname{arc\,tg}\frac{b_k}{a_k}+\vartheta_v . \tag{41} \]

The coefficients of the Fourier series \(a_k\) and \(b_k\) are found by means of the integrals

\[ a_k=\frac{1}{\pi}\int_{0}^{\lambda} j_u(\vartheta)\sin k\vartheta\,d\vartheta , \tag{42} \]

\[ b_k=\frac{1}{\pi}\int_{0}^{\lambda} j_u(\vartheta)\cos k\vartheta\,d\vartheta . \tag{43} \]

If the current of the power valve over a period consists of two pulses (the second case of the discontinuous-current mode), then the coefficients of the Fourier series are determined by means of the following integrals:

\[ a_k=\frac{1}{\pi}\left[ \int_{0}^{\lambda_{u1}} j_{u1}(\vartheta)\sin k\vartheta\,d\vartheta + \int_{0}^{\lambda_{u2}} j_{u2}(\vartheta)\sin k(\vartheta+\vartheta_{v2}-\vartheta_{v1})\,d\vartheta \right], \tag{44} \]

\[ b_k=\frac{1}{\pi}\left[ \int_{0}^{\lambda_{u1}} j_{u1}(\vartheta)\cos k\vartheta\,d\vartheta + \int_{0}^{\lambda_{u2}} j_{u2}(\vartheta)\cos k(\vartheta+\vartheta_{v2}-\vartheta_{v1})\,d\vartheta \right], \tag{45} \]

where \(\lambda_{u1}\) and \(\lambda_{u2}\) are the durations of the first and second turn-ons of the power valves in the discontinuous-current mode;

\(\vartheta_{\mathrm{v}1}\) and \(\vartheta_{\mathrm{v}2}\) are the angular coordinates of the turn-on instants of the power valves;

\(\vartheta_{\mathrm{v}2}-\vartheta_{\mathrm{v}1}\) is the phase of the second valve-current pulse relative to the first pulse.

Using expressions (36)—(45), after integration and transformations, one can obtain, for any of the considered operating modes of the mutator, the values of the corresponding currents, in relative units, that flow in the various elements of the system, as well as their harmonic composition. The calculation formulas will be functions of the sought quantities with respect to the turn-on angles and the duration of conduction, in which the load characteristics \((\tg \Theta\) and \(\varepsilon)\) may be regarded as parameters. The result of such calculations will be the selection of converter elements and the calculation of its characteristics. It should be noted here that the possibility of full control of the mutator valves (independently of the control method) gives a certain range of turn-on angles and conduction durations that ensure, for given load parameters, constancy of the average current. That is, by adhering to a definite law of valve control, one can obtain the energy indices of the installation that are most desirable in each particular case.

Proceeding from the foregoing, the following procedure may be proposed for calculating the static modes of the mutators under consideration on fully controlled elements.

1. The load parameter \(\tg \Theta\) is determined and a definite law is specified for varying the turn-on angles, ensuring the required energy indices of the system.

2. The value of the e.m.f. in the direct-current circuit of the mutator is specified, i.e., the value \(\varepsilon\).

3. For the given \(\tg \Theta\) and \(\varepsilon\), taking into account the law of control of the turn-on angles, the boundaries of the various current modes of the converter are found by means of expressions (14)—(15), (22)—(23).

4. For a series of angles of conduction duration of the power valves, beginning with the zero value, the required currents and their spectral composition are found. In doing so, for each conduction duration one should check the operating mode of the mutator, so that subsequently the corresponding expressions may be used. As a result of such a calculation, the static characteristics of the given converter with a load having a definite \(\tg \Theta\) and direct-current e.m.f. are obtained.

5. The value of the direct-current e.m.f. \((\varepsilon)\) is varied, and the calculation is repeated, beginning with item 3.

Discrepancies between the calculated results obtained by the proposed method and practice may arise mainly as a consequence of neglecting the active and reactive resistances of the alternating-current network, and also the magnitude of the energy delivered to the load by the artificial-commutation circuit, if the latter is used for turning off the valves.

Thus, the method set forth will give a small error in the case where the mutator is supplied from a sufficiently powerful alternating-current network. Neglect of the energy of the artificial-commutation unit is justified when thyristors are used in the system under investigation that have a short recovery time of the control properties; an error arises here only under a load close to the no-load mode of the converter because of the commensurability of the energies entering the load from the network and from the artificial-commutation system. True, in some—

in which artificial-commutation circuits this effect may be absent.

Experimental studies carried out on an installation with silicon controlled valves showed that the experimental data are in good agreement with the theoretical results.

REFERENCES

1. I. I. Kaganov. Electronic and Ionic Converters, Gosenergoizdat, 1956.

2. A. A. Bulgakov. Fundamentals of the Dynamics of Controlled Valve Systems. Publishing House of the Academy of Sciences of the USSR, 1963.

3. A. I. Zaitsev, V. N. Mishin, A. A. Kuvshinov. “Pulse speed control in drives based on the UDP-D system with artificial commutation.” Reports to the Fourth All-Union Conference on Automated Electric Drive (collection General Problems of Electric Drive), VNIIEM, ONTI, Moscow, 1964.

4. Silicon Controlled Valves—“Thyristors”, translated from English, Energiya, 1964.