STATIC CHARACTERISTICS OF A THYRISTOR CONVERTER WITH COMBINED COMMUTATING INDUCTANCE

V. P. LAIER, V. N. MISHIN

(Presented by the scientific-technical seminar of the Research Institute of AEM)

A significant drawback of pulsed DC thyristor converters with a saturation reactor DN in the commutating circuit [1, 2] is the passage of current pulses of considerable amplitude through the thyristor during recharging of the capacitor after “positive” saturation of the DN; to eliminate these pulses, in [1] it was proposed to shunt the DN with a linear inductance and a diode (Fig. 1a). The use of a combined inductance in the unit of forced commutation UPK makes it possible to reduce the amplitude of current pulses, which is especially effective when operating at elevated frequencies. The analysis available in the literature of the operation of similar converters and the recommendations for calculation are insufficient [3], [4]. The purpose of the present article is to attempt to fill the indicated gap and to supplement the investigations presented in [5], without taking into account energy losses in the UPK.

Fig. 1. Thyristor interrupter: a) schematic diagram, b) and c) equivalent circuits.

The equivalent circuits of the converter for two magnetic states of the DN core, \(|B| < B_s\) and \(|B| = B_s\), are shown in Fig. 1b and 1c.

The initial equations for the circuit of Fig. 1b at \(|B| < B_s\) are

\[ \left. \begin{gathered} u_L + u_C + iR_1 = 0;\quad u_L = NS\frac{dB}{dt} = L_1\frac{di_1}{dt};\\[6pt] u_C = \frac{1}{C}\int_0^t i(t)\,dt;\quad i = i_1 + I_0 + \frac{u_L}{R(B)} \end{gathered} \right\} \tag{1} \]

and the differential equation of the process

\[ L_1\frac{di_1}{dt} + \frac{1}{C}\int_0^t \left( i_1 + I_0 + \frac{u_L}{R(B)} \right)dt + \left( i + I_0 + \frac{u_L}{R(B)} \right)R_1 =0. \]

Taking into account that \(L_1 \gg L_s\) when \(|B| = B_s\), the differential equation of the process for the circuit of Fig. 1c is described by the expression

\[ L_s C \frac{d^2 u_c}{dt^2} + R_2 C \frac{d u_c}{dt} + u_c = 0, \tag{2} \]

where

\(N\) and \(S\) are the number of turns and the cross section of the core of the saturable reactor;

\(R(B)\) is the nonlinear resistance of the saturable reactor;

\(L_s\) and \(L_1\) are the leakage inductances of the windings of the saturable reactor and the linear inductance;

\(R_1\) and \(R_2\) are the equivalent resistances of the circuits, respectively, for Figs. 1b and 1c;

\(B_s\) is the saturation induction of the core of the saturable reactor.

The analysis of quasi-steady electromagnetic processes is carried out by the method of matching, taking into account the initial conditions of the preceding time interval, under the assumption that the valves are ideal, the load current is constant over the interval of one operating cycle of the converter, and the squareness coefficient of the hysteresis loop \(\Pi_B = 1\).

The process of variation of voltage and current in the commutating circuit after the thyristor is switched on is described by expressions obtained in solving (1), taking into account that

\[ I_0 + \frac{u_L}{R(B)} = 0 \quad \text{and} \quad i = i_1 = C \frac{d u_c}{dt} \]

and the initial conditions

\[ u_c(0) = -U_0 \quad \text{and} \quad i(0) = 0; \]

\[ u_c = -U_0 e^{-\alpha_1 t} \left( \cos \omega_1 t + \frac{\alpha_1}{\omega_1}\sin \omega_1 t \right) \tag{3} \]

and

\[ i = \omega_1 C U_0 e^{-\alpha_1 t} \left( 1 + \frac{\alpha_1^2}{\omega_1^2} \right) \sin \omega_1 t, \tag{4} \]

where

\[ \alpha_1 = \frac{R_1}{2L_1} \]

is the damping coefficient;

\[ \omega_1 \approx \frac{1}{(L_1 C)^{1/2}} \]

is the angular frequency of the recharge circuit formed by the shunt circuit of the saturable reactor.

The law of variation of the induction of the saturable-reactor core is determined from (1), taking into account (4), with the initial condition \(B(0)=B_0\):

\[ B = B_0 + \frac{(L_1 C)^{1/2}}{NS} \left( 1 + \frac{\alpha_1^2}{\omega_1^2} \right) U_0 e^{-\alpha_1 t}\sin \omega_1 t, \tag{5} \]

The solution of the obtained transcendental equation with respect to the time required to remagnetize the saturable-reactor core by the induction increment \(\Delta B = B - B_0\) will be found under the condition of neglecting damping in the circuit, i.e., for \(\alpha_1 = 0\):

\[ t_1 = (L_1 C)^{1/2} \arcsin \frac{\Delta B\,NS}{(L_1 C)^{1/2}U_0}. \tag{6} \]

It follows from (6) that, depending on the parameters of the saturable-reactor core and of the shunting circuit with linear inductance, two operating modes of the saturable reactor are possible:

\[ 1)\quad \text{for } \frac{\Delta B\,NS}{U_0} > (L_1 C)^{1/2} \text{ — remagnetization along partial loops} \]

hysteresis, i.e., without the core entering the region of “positive” saturations (Fig. 2b);

2) when

\[ \frac{\Delta BNS}{U_0} \leq (L_1C)^{1/2} \]

— remagnetization along the full hysteresis loop (Fig. 2b) at \(\Delta B = 2B_s\).

The best utilization of the core material occurs in the second regime, provided that

\[ \frac{t_0}{(L_1C)^{1/2}} = 1, \tag{7} \]

where

\[ t_0=\frac{2B_sNS}{U_0} \]

is the time of remagnetization of the core along the full hysteresis loop in the absence of the shunting circuit \(D3, L1\).

This condition corresponds to the maximum duration of the first time interval

\[ t_{1\max}=\frac{\pi}{2}(L_1C)^{1/2}, \tag{8} \]

The required value of the inductance \(L_1\) is determined from expression (8) and is several orders of magnitude greater than the leakage inductance of the winding of the saturable reactor at \((B)=B_s\).

In this connection, the quality factor of the additional recharge circuit of the commutating capacitance \(C\) exceeds by more than an order of magnitude the quality factor of the circuit formed for recharge when the saturable-reactor core is saturated; therefore the earlier assumption made in determining the time interval \(t_1\) from (5) is sufficiently justified. Thus, the use of combined inductance in the thyristor converter increases the utilization of the saturable reactor by a factor of \(\pi/2\) owing to the broadening of the pulse duration, while simultaneously reducing direct current surges as a result of the increase in the characteristic impedance of the circuit, since

\[ \left(\frac{L_1}{C}\right)^{1/2} \gg \left(\frac{L_s}{C}\right)^{1/2}, \]

which is not difficult to observe when solving (8) with allowance for (7), i.e.,

\[ \frac{t_{1\max}}{t_0}=\frac{\pi}{2}=1.57. \]

The first regime, i.e., remagnetization along partial hysteresis loops, is not characteristic of auxiliary power supplies, since invariance of the voltage at the interrupter output is not ensured when the supply-source voltage changes.

The second regime at \(|B|=B_s\) corresponds to oscillatory recharge of the capacitor with frequency \(\omega \gg \omega_1\).

The process of variation of the voltage and current of the capacitor is described by formulas obtained as a result of solving (2) under the initial conditions \(U_c(0)=U_c(t_1)\) and \(i_s(0)=0\), while the duration of recharge (the second interval) is \(\Delta t_2 \simeq \pi(L_sC)^{1/2}\).

In the third interval the capacitor is additionally charged by the energy of the inductor \(L_1\). The process is described by the original equations (1). The additional charging of the capacitor ends when the inductor current is equal to zero, and the charging time is

\[ \Delta t_3 \approx t_1, \]

since \(\Delta t_3\) and \(t_1\) are connected by the common energy reserve of the inductor.

Accordingly, during this time the core of the DN will be remagnetized to the induction

\[ B(t_3)=B_s-\frac{(L_1C)^{1/2}\cdot K_3\cdot U_0}{NS}\sin \omega_1 t_1, \tag{9} \]

where

\[ K_3=e^{-\frac{\pi}{2Q}} \quad \text{is the half-period damping decrement;} \]

\[ Q=\frac{(L_sC)^{1/2}}{R_2} \quad \text{is the quality factor of the recharge circuit.} \]

Because of the presence of damping in the recharge circuit, the induction \(B(t_3)<B_s\); consequently, an additional time \(\Delta t'_3\) is required, during which the DN core reaches “negative” saturation,

\[ \Delta t'_3=\frac{[B_s-B(t_3)]NS}{K_3U_\theta}. \tag{10} \]

The voltage on the capacitor at the instant of “negative” saturation is

\[ U_{c3}'=U_{c3}\simeq K_3U_0. \]

Shunting the DN winding with a linear inductance leads to the appearance of an additional time interval \(\Delta t'_3\), whose duration depends on the value of the half-period damping decrement.

The subsequent course of the electromagnetic processes is analogous to that considered in [2].

The equation of the external characteristic of the converter has the form

\[ U=\left\{ E\left[ 2(L_1C)^{1/2}\arcsin\frac{2B_sNS}{(L_1C)^{1/2}U_0} +(L_sC)^{1/2}\left(2\pi-\frac{1}{K_i}\right) +\frac{2B_sNS}{K_3U_0} -(L_1C)^{1/2}\sin\omega_1t_1 \right] +\frac{CE^2}{2I}\left(1-mK_3^2\cos\frac{1}{K_i}\right)^2 \right\}f, \tag{11} \]

where

\[ K_i=\frac{K_3U_0(C/L_s)^{1/2}}{I_{\max}} \quad \text{is the current ratio in the commutating circuit.} \]

The time diagrams of the converter when the DN core is remagnetized along the limiting and partial hysteresis loops are shown in Fig. 2.

Figure 3 gives the external characteristics of the converter, constructed in accordance with (11), for the relative time constant of the DN and \(L_1\) circuits equal to \(\tau=1\) at \(E_{\min}=0.7E_n\).

The relative time constant is determined from the expression

\[ \tau=\frac{t_0}{(L_1C)^{1/2}}=\frac{2B_sNS}{(L_1C)^{1/2}E_{\min}}. \tag{12} \]

To compare the influence of the linear inductance on the external characteristics of the converter with the DN in the UPC, in Fig. 3 the characteristic for \(\tau=0\) \((L_1=\infty)\) is shown by a dashed line.

A comparison of the external characteristics shows that the average voltage at the output of the converter with combined inductance is higher for this particular example by a factor of 1.4; however, the stabilization coefficient \(K_{\text{st}}\approx 2\), i.e., it decreased by a factor of 3.5.

Fig. 2. Timing diagrams.

Fig. 3. External characteristics of the interrupter.

The dependence of the saturation-current amplitude during oscillatory recharging of the capacitor over the time interval \(\Delta t_2\)

\[ I_{\max s} = - \frac{U_0 \cos \omega_1 t_1}{\rho_s}, \]

where

\[ \rho_s = \left(\frac{L_s}{C}\right)^{1/2} \]

is the characteristic impedance of the recharging circuit when \((B) = B_s\),

of the relative constant \(\tau\) is shown in Fig. 4 for two limiting values of the supply-source voltage \(E=0.7E_{\mathrm{n}}\) and \(E=1.3E_{\mathrm{n}}\). Thus, at \(\tau=1\) and a source voltage of \(0.7E_{\mathrm{n}}\), the amplitude of the recharge current \(I_{\max s}=0\).

The value \(\tau=1\) corresponds to the minimum stabilization coefficient \(K_{\mathrm{st}}=1.95\), whose dependence \(K_{\mathrm{st}}=F(\tau)\) is also given in Fig. 4. The extremum of \(K_{\mathrm{st}}\) of the regulator with combined inductance is observed in the range of values \(\tau=0.6\div0.7\), which is more than twice the stabilization coefficient of the circuit without shunting linear inductance. However, the effect from reducing the amplitude of the current \(I_{\max s}\) at the indicated values of \(\tau\) is insignificant.

Fig. 4. On determining the influence on the stability of the output voltage and the amplitude of the recharge current.

The ratio of the recharge currents of the circuits for \(\tau=0\) and with combined inductance for \(\tau=1\) in the second interval at \(E=0.7E_{\mathrm{n}}\)

\[ \frac{I_{\max s}(\tau=0)}{I_{\max s}(\tau\ne0)} = \left(\frac{L_1}{L_s}\right)^{1/2} \]

is determined by the ratio of the linear inductance and the leakage inductance of the DN winding.

Thus, in the converter under study with the parameters of the UPK: core of grade 79NM, \(B_s=1.06\ \text{T}\), \(S=3.4\cdot10^{-4}\ \text{m}^2\), \(N=100\), \(C=4\cdot10^{-6}\ \text{F}\), \(L_s=64\cdot10^{-6}\ \text{H}\), and \(L_1=0.118\ \text{H}\), the current amplitude decreased by more than 40 times, which corresponded to the calculated data.

Figure 5 gives the control characteristics of the converter for two values \(\tau=0.8\) and \(1.0\) and for a load-current variation range \(I=(0.1\div1.0)I_{\mathrm{n}}\). The minimum range of frequency variation occurs at values \(\tau=0.65\div0.7\), which agrees with the maximum value of the stabilization coefficient in Fig. 4.

Conclusions

1. The use of combined inductance in an ITPLN operating in an open-loop output-voltage regulation system,

Fig. 5. Control characteristics of the interrupter.

makes it possible to increase the voltage stabilization coefficient at \(\tau = 0.65 \div 0.7\) by more than a factor of 2, with a corresponding reduction in the range of frequency variation.

1. The use of a combined inductance with a relative constant \(\tau = 1\) is expedient in networks with a stable supply-line voltage as a component element of the commutation unit of pulse-width converters [6], since in this case the dimensions, weight, and cost of the commutation unit are reduced by approximately 30%.

REFERENCES

1. B. Bedford, R. Hoft. Theory of Autonomous Inverters. Moscow, “Energiya,” 1969.

2. V. V. Toppelberg. A thyristor DC voltage stabilizer with a saturating choke in the commutation circuit. Thyristor Converters. Leningrad, “Nauka,” 1970.

3. R. E. Morgan. Time ratio control with combined SCR and SR commutation. IEEE Transactions on Communication and Electronics, No. 7, 1967.

4. V. S. Rudenko, A. I. Denisov. Pulse Converters and Stabilizers on Thyristors. Kiev, “Tekhnika,” 1972.

5. V. N. Mishin, V. P. Laier, Yu. N. Buchenko. An interrupter on a thyristor with combined inductance. Izvestiya TPI, vol. 262, 1972, 20–23.

6. R. E. Morgan. Time ratio control and inverter power circuits; U.S. Patent No. 3,360,712, 1967.