ANALYSIS AND CALCULATION METHODS FOR AN AUTONOMOUS SWITCHING DEVICE

A. I. ZAITSEV, B. A. LAPIN

(Presented by the scientific seminar of the Electromechanical Faculty)

In modern converter technology, pulse devices using thyristors operating in the forced-commutation mode are widely employed. Their operational reliability, control characteristics, and range of operating frequencies depend to a significant extent on the switching device that performs the forced turn-off of the power valve.

The circuits of such devices published in the domestic and foreign literature differ from one another mainly in the different number and combinations of valve and reactive elements \[1, 2, 3\]. Their common fundamental basis is a method of charging the commutating capacitance by means of the current flowing through the load, and changing the polarity of the voltage on this capacitance by means of resonant recharging through the power valve. The use of a power thyristor in the process of preparing the switching device for operation leads to a substantial deterioration of the converter’s control characteristics; in particular, the range of regulation of the duty factor of the output pulses is reduced and the load range is limited from below. In addition, the flow of the recharging current through the power valve causes its additional heating.

Below, two variants of circuits of an autonomous switching device developed by the authors are considered, as applied to systems with single-ended pulse-width modulation, in which the disadvantages listed above are absent.

In the circuits shown in Figs. 1 and 2, commutation of the power thyristor $T_1$ is accomplished by connecting capacitor $C_1$ through thyristor $T_2$ in parallel with the load. This capacitor must first be charged to a voltage exceeding the voltage of the primary source $E$. In the devices under consideration, for this purpose an oscillatory charging of the capacitor through the inductance $L_1$ is used. The shock excitation of the oscillatory circuit $L_1 C_1$, required to obtain an excess voltage across the capacitance, is accomplished by its rapid discharge through the load and a special discharge circuit $L_2—C_2, R$. This circuit is connected to the capacitance by means of an addi-

of the auxiliary thyristor \(T_3\) (Fig. 1) or of the commutating valve \(T_2\) (Fig. 2), and forces its discharge at low load currents, including the no-load mode \((I_{\mathrm{н}}=0)\). In the circuit of Fig. 1, in order to reduce the number of control signals, valve \(T_2\) is switched on through a differentiating circuit from thyristor \(T_3\).

Fig. 1.

Fig. 2.

Diode \(D_2\) (Fig. 2) provides decoupling of the elements \(L_2, C_2, R\) from the power circuit of the converter. To investigate the commutating device we shall use the equivalent circuit shown in Fig. 3. The switches \(K_1, K_2, K_3\) in this figure correspond to the valves \(D_1, T_2, T_3\) in the schematic diagram of Fig. 1.

In calculating the parameters of the circuit we shall proceed from the fact that the duration of the discharge of capacitance \(C_1\) is much less than the period of natural oscillations of the \(L_1 C_1\) circuit. In this case the increment of current in \(L_1\) during the capacitance-discharge time may be neglected, and, in analyzing the oscillatory process in the \(L_1 C_1\) circuit, the initial value of \(i_1\) may be taken equal to zero. Under this condition it may be assumed that the process of resonant charging of the commutating capacitance begins at the instant of simultaneous closing of switch \(K_1\) and opening of \(K_2, K_3\). The expressions for the current and voltage over the capacitance-charging interval have the form:

Fig. 3.

\[ I_1(t)=(E-U_{3.0})\sqrt{\frac{C_1}{L_1}}\sin \omega_1 t. \tag{1} \]

\[ U_1(t)=U_{3.0}+(E-U_{3.0})(1-\cos \omega_1 t), \tag{2} \]

where

\[ \omega_1=\sqrt{\frac{1}{C_1L_1}}; \]

\(U_{3.0}\) is the initial voltage of the capacitance in the charging interval, equal to the final voltage in the preceding discharge interval.

The duration of the charge \(T_3\), determined by the instant of opening of switch \(K_1\) (i.e., by the turn-off of diode \(D_1\)), is found from (1) for \(I_1(t)=0\):

\[ T_3=\pi\sqrt{L_1C_1}. \tag{3} \]

Since this time must not exceed the commutation period \(T_k\), the natural frequency of the charging circuit \(f_1\) and the inductance \(L_1\) must satisfy the inequalities:

\[ f_1 > 0.5 f_k, \tag{4} \]

\[ B_1 < \frac{1}{(f_k)^2}\cdot \frac{1}{C_1}, \tag{5} \]

where \(f_k\) is the commutation frequency.

The voltage across the capacitor at which commutation of the power valve occurs is equal to the final voltage in the charging interval:

\[ U_{\text{z.k.}} = 2E - U_{\text{z.o.}}. \tag{6} \]

The average and rms currents consumed by the commutating device are determined by the following expressions:

\[ I_{1\text{ avg}}=\frac{1}{T_k}\int_{0}^{T_z} I_1(t)\,dt = (2E-U_{\text{z.o.}})\,C_1 f_k; \tag{7} \]

\[ I_{1\text{ rms}}= \sqrt{\frac{1}{T_k}\int_{0}^{T_z} I_1^2(t)\,dt} = (E-U_{\text{z.o.}})\,C_1\pi\sqrt{f_1\cdot f_k}. \tag{8} \]

Since the energy losses in the circuit are proportional to the rms current, it is advisable to make it minimal by choosing, in accordance with (4), the smallest frequency \(f_1 \cong 0.5 f_k\). Then

\[ I_{\text{rms.min}}=\frac{E}{\sqrt{2}}\,\pi f_k C_1. \tag{9} \]

The low natural frequency of the charging circuit makes it possible without difficulty to construct it with a high quality factor and to reduce the energy losses during charging of the commutating capacitor to a negligible value. For comparison, we note that in known commutation circuits using resonant recharging of a capacitor through a power thyristor, the natural frequency of the circuit usually exceeds the commutation frequency by a factor of 5–10. Energy losses in such circuits during one recharging cycle reach 20–30%.

The choice of the parameters of the circuit \(L_2C_2\) can be made on the basis of analysis of the discharge process of the capacitor \(C_1\), which begins when the valves \(T_2\) and \(T_3\) are turned on. During the discharge of the capacitor, the state of the switches \(K_1, K_2, K_3\) (closed or open) depends on the value of the voltage across the capacitor \(C_1\) (Fig. 3). Therefore, depending on the value of \(U_1\), the entire discharge interval can be divided into three time sections:

1. \(K_1\) is open;
   \(K_2, K_3\) are closed;
2. \(K_1, K_2, K_3\) are closed;
3. \(K_1, K_3\) are closed,
   \(K_2\) is open.

The end of section 3 coincides with the instant at which switch \(K_3\) opens.

An exact analysis of this circuit in general form is difficult because of the transcendental character of the expressions determining the boundaries of the time sections. To simplify the derivations, it is expedient to neglect the currents through the inductance \(L_1\) and the resistance \(R\), which do not play a principal role in the capacitor-discharge process. This neglect of the conductances \(L_1\) and \(R\) does not lead to substantial errors if the duration

of the discharge process is negligible. The expressions for the current and voltage in this case have the form:

\[ I_2(t)=U_{\mathrm{p.o}}\sqrt{\frac{C_{\mathrm{e}}}{L_2}}\sin \omega_{\mathrm{e}}t -I_{\mathrm{n}}\frac{C_2}{C_1+C_2}(1-\cos \omega_{\mathrm{e}}t), \tag{10} \]

\[ \begin{aligned} U_1(t)&=U_{10}-\frac{I_{\mathrm{n}}}{C_1}t -U_{10}\frac{C_2}{C_1+C_2}(1-\cos \omega_{\mathrm{e}}t)+{}\\ &\quad + I_{\mathrm{n}}\left(\frac{C_1}{C_1+C_2}\right)^2 \sqrt{\frac{L_2}{C_{\mathrm{e}}}}\left(\omega_{\mathrm{e}}t-\sin \omega_{\mathrm{e}}t\right), \end{aligned} \tag{11} \]

where

\[ C_{\mathrm{e}}=\frac{C_1\cdot C_2}{C_1+C_2}, \]

\[ \omega_{\mathrm{e}}=\sqrt{\frac{1}{L_2C_{\mathrm{e}}}}. \]

In deriving (10), (11), the initial voltage across capacitance \(C_2\) was taken to be zero. This condition is mandatory for normal operation of the circuit. Therefore the resistance \(R\) must be sufficiently small to ensure complete discharge of \(C_2\) during the commutation period \(T_{\mathrm{k}}\).

Since the purpose of the \(L_2C_2\) circuit is to force the discharge process of capacitance \(C_1\) under light-load and no-load operation of the converter \((I_{\mathrm{n}}=0)\), the choice of its parameters must be based on the following conditions:

1. The capacitance \(C_1\) at no load of the converter must discharge through the \(L_2C_2\) circuit completely and in a time that ensures excitation of the oscillatory circuit \(L_1C_1\);

2. leakage of charge from capacitance \(C_1\) through the \(L_2—C_2\) circuit at large load currents must be negligible.

The maximum value of the discharge time \(T_{\mathrm{p}}\) is found from (5), for \(I_{\mathrm{n}}=0\), from the condition \(I_2(t)=0\):

\[ T_{\mathrm{p}}=\pi\sqrt{L_2C_{\mathrm{e}}}. \tag{12} \]

The final voltage after the discharge of the capacitance, which is the initial voltage in the next charging interval, is found from (11) for \(I_{\mathrm{n}}=0\) and \(t=T_{\mathrm{p}}\):

\[ U_{\mathrm{p.k}}=U_{\mathrm{p.o}}\frac{C_1-C_2}{C_1+C_2}. \tag{13} \]

To ensure turn-off of thyristor \(T_2\), this voltage must be negative or equal to zero. Therefore

\[ C_2\geq C_1. \tag{14} \]

If \(C_1\ne C_2\), when the commutating device is switched on the currents and voltages reach their steady-state values over several repetition periods. In this case the initial voltage in each subsequent charging cycle of \(C_1\) is related to the final voltage in the preceding charging cycle by the relation:

\[ U_{\mathrm{z.o}}(n)=U_{\mathrm{z.k}}(n-1)\cdot q, \tag{15} \]

where \(n\) is the period number;

\[ q=\frac{C_1-C_2}{C_1+C_2}. \]

Using (15) and (6), it is easy to obtain a recurrence expression for the amplitude value \(U_1\) in any repetition period:

\[ \begin{aligned} U_{\text{z.k.}(1)} &= 2E,\\ U_{\text{z.k.}(2)} &= 2E - 2E \cdot q,\\ U_{\text{z.k.}(3)} &= 2E - 2E \cdot q + 2E \cdot q^2,\\ &\ldots\ldots\ldots\ldots\ldots\ldots\\ U_{\text{z.k.}(n)} &= \sum_{r=1}^{n} 2E[-q]^{(r-1)} . \end{aligned} \tag{16} \]

The condition for convergence of the series (16) is the inequality \(|q|<1\), which is satisfied if \(C_1 \ne C_2\); the established amplitude of the voltage on the capacitance \(C_1\) is found from (16) as a result of a limiting transition.

\[ U_{\text{z.k.}}=\lim_{n}\sum_{r=1}^{n}2E[-q]^{(r-1)} =E\left(1+\frac{C_2}{C_1}\right). \tag{17} \]

The initial voltage in the steady-state regime will be equal to:

\[ U_{\text{z.o}}=U_{\text{z.k.}}\cdot q =E\left(1-\frac{C_2}{C_1}\right). \tag{18} \]

Substituting (18) into (7), we obtain an expression for the current consumed by the switching device at no load of the converter:

\[ I_{1\ \text{n.l.}}=E(C_1+C_2)\cdot f . \tag{19} \]

In order to reduce this current, it is advisable to choose the minimum capacitance \(C_2\) according to inequality (14). In this case the initial voltages \(U_{\text{z.o}}\) and \(U_{1\text{.o}}\) at \(I_{\text{n}}=0\) will be equal, respectively, to \(0\) and \(2E\). At \(I_{\text{n}}\ne 0\), the voltages \(U_{\text{z.o}}\) and \(U_{1\text{.o}}\) are practically independent of the value of the capacitance \(C_2\), and in calculations they should likewise be taken as equal to \(0\) and \(2E\).

Experimental investigation of the circuit showed that the maximum duration of discharge of the capacitance \(C_1\), at which a sufficiently intense excitation of the \(LC\) circuit still occurs, is \(\frac{1}{5}\div\frac{1}{6}\) of its natural period. Taking this into account, from formula (12) one can find the upper limit for \(L_2\):

\[ L_2 \le \left(\frac{1}{6}\div\frac{1}{9}\right)\frac{C_1}{C_2}L_1 . \tag{20} \]

To determine the minimum permissible values of \(L_2\), let us examine the influence of the parameters of the \(L_2C_2\) circuit on the commutation process at \(I_{\text{n}}\ne 0\). Denote by \(t_\Delta\) the time that would be available to the thyristor \(T_1\) for restoring its control properties in the absence of the \(LC\) circuit (Fig. 4). This time is found from (11), if \(U_1(t)=E\) and \(i_{\text{n}}=0\) are assumed:

\[ t_\Delta=\frac{(U_{10}-E)}{i_{\text{n}}}\,C_1 . \tag{21} \]

Fig. 4

Owing to the additional discharge of the capacitance through the \(L_2C_2\) circuit, the real time \(t_\delta\) available for recovery is less than \(t_\Delta\) by a certain amount (Fig. 4). Since finding an exact expression for \(\tau\) is not possible, we estimate its value approximately. With this

For this purpose, let us rewrite (21), for \(U_1(t)=E\), in the following form:

\[ E=U_{10}-\frac{I_{\mathrm{n}}}{C_1}\,t-R(t), \tag{22} \]

where

\[ R(t)=U_{\mathrm{p}.0}\frac{C_2}{C_1+C_2}\left(1-\cos\omega_{\mathrm{e}}t\right) -I_{\mathrm{n}}\left(\frac{C_2}{C_1+C_2}\right)^2 \times \]

\[ {}\times \sqrt{\frac{L_2}{C_3}}\left(\omega_{\mathrm{e}}t-\sin\omega_{\mathrm{e}}t\right). \tag{23} \]

Comparing (21) and (22), taking into account the fact that connecting the circuit \(L_2C_2\) only accelerates the discharge of the capacitance \(C_1\), one may conclude that \(R(t)\) is a positive quantity. Taking this into account, on the basis of (21), (22), we form the obvious inequalities:

\[ t_{\Delta}>t_0>t_{\Delta}-R(t')\frac{C_1}{I_{\mathrm{n}}}. \tag{24} \]

\[ \tau<R(t')\frac{C_1}{I_{\mathrm{n}}}. \tag{25} \]

Here \(t'\) denotes the time from the interval \(0-t_{\Delta}\) at which the function \(R(t)\) assumes its greatest value.

To find \(R(t')\), we expand the functions \(\sin\omega_{\mathrm{e}}t\) and \(\cos\omega_{\mathrm{e}}t\) entering into (23) in a Taylor series, restricting ourselves to the linear part and taking the remainder term into account.

Then

\[ R(t)=U_{10}\frac{C_2}{C_1+C_2}\sin\omega_{\mathrm{e}}\lambda\cdot(\omega_{\mathrm{e}}t) -I_{\mathrm{n}}\left(\frac{C_2}{C_1+C_2}\right)\times \]

\[ {}\times \sqrt{\frac{L_2}{C_3}}\, \frac{\sin\omega_{\mathrm{e}}\nu\cdot(\omega_{\mathrm{e}}t)^2}{2}, \tag{26} \]

where \(0<\lambda,\nu<t\).

It is obvious that the greatest value \(R(t')\) in the time interval \(0\le t<t_{\Delta}\) does not exceed the following quantity:

\[ R(t')<R=U_{10}\frac{C_2}{C_1+C_2}\left(\omega_{\mathrm{e}}t_{\Delta}\right)^2. \tag{27} \]

Expression (27) was obtained from (26) for \(\lambda=t=t_{\Delta}\) and \(\nu=0\). Now, with the aid of (25) and (27), it is possible to estimate the decrease in \(t_{\Delta}\) caused by leakage of charge through the circuit \(L_2C_2\):

\[ \tau<\frac{U_{10}}{I_{\mathrm{n}}}\frac{t_{\Delta}^2}{L_2}. \tag{28} \]

It is more convenient to express \(\tau\) as a percentage of \(t_{\Delta}\):

\[ \tau_0=\frac{\tau}{t_{\Delta}}\cdot100\% < \frac{U_{10}}{I_{\mathrm{n}}}\frac{t_{\Delta}}{L_2}\cdot100\%= \]

\[ = \frac{U_{10}(U_{10}-E)}{I_{\mathrm{n}}^2}\cdot\frac{C_1}{L_2}\cdot100\%. \tag{29} \]

As the current \(I_{\mathrm{n}}\) increases, \(\tau_0\) decreases rapidly and, at maximum load, usually does not exceed one percent.

Thus, having selected \(L_2\) according to (20), by (29) one can estimate the decrease in the time allowed to the power valve for restoring its control properties, caused by the additional discharge of the switching capacitance through the circuit \(L_2C_2\).

The advantage of the described switching circuits is especially pronounced when they are used in converter devices with

deeply regulated output voltage and load current, and also in low-voltage high-current converters, where the switching conditions require a large value of the commutating capacitance.

In the latter case, the use of inductor-based power extraction from the power circuit into the switching device is highly effective [3].

An important and useful property of the proposed circuits is their ability to continue operating with the power valve continuously switched on. Owing to this, independent restoration of commutation becomes possible after its failure.

CONCLUSIONS

1. The independence of the charging and discharging processes of the commutating capacitance from the state of the power valve (open or closed) and from the magnitude of the load current makes it possible to regulate its conduction interval from zero to a value equal to the commutation period.

2. In the developed circuits, the natural frequency of the charging oscillatory circuit can be reduced to one-half of the commutation frequency, which makes it possible to construct the circuit with a high quality factor and to reduce the losses in it to a negligible value.

3. The autonomous operation of the switching device makes it possible to restore commutation of the power valve independently after its failure.

REFERENCES

1. Neville W. Mapham, John C. Hey. The control of Battery Powered DC motors Using SCR’S in the Jones circuit. IEEE International Convention Record, 1964, NT-4.

2. Gurwicz D. Apulsed d. c. motor control system. Electrical Review, 1964, v. 175, No. 19.

3. Lapin B. A., Zaitsev A. I. Circuit for artificial commutation with inductor charging of the commutating capacitance. Izvestiya TPI, vol. 153, 1965.