STUDY OF A CHARGER WITH SATURABLE REACTORS

F. P. Zverev, A. B. Tsukublin, S. A. Sipailov

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

In the development of autonomous systems, the problem often arises of simultaneously supplying power from a single generator to loads of different types, among which low-frequency pulse sources occupy a special place. Operating in a cyclic “charge–discharge” mode, these sources create interference for other consumers that require voltage stability.

The use of capacitance as the storage element in pulse sources leads to the need to select a charging device that would ensure the absence of sharp current surges and voltage dips of the main generator at the moment of charging. Approaching the solution of this problem with allowance for the fact that

1) the main source of energy is a three-phase synchronous generator,

2) the power consumed by the charging device constitutes only part of the total generator power,

3) during charging the circuit must ensure the absence of sharp current surges, and also taking into account that the circuit of the charging device must be as simple and reliable as possible, from the whole variety of charging devices the present article considers a rectifier circuit controlled by saturable reactors connected in the break of the neutral point of the transformer (Fig. 1). In addition, in the controlled mode such a circuit ensures constancy of the generator current regardless of the load magnitude, which meets the requirements listed above.

For simplicity of analysis, let us assume that the voltage on the primary winding of the transformer (Fig. 1) is maintained constant in amplitude, which corresponds to its operation from a generator with a voltage regulator, and that the magnetization characteristic of the saturable reactors is approximated by two straight lines, i.e., the inductive reactance of the unsaturated reactor is infinitely large \((z_{\mathrm{dr}} \approx \infty)\), while that of the saturated reactor is represented by a finite quantity \((z_{\mathrm{dr}} = jx_{\mathrm{dr}} + r_{\mathrm{dr}})\). With allowance for the latter assumption, the reactor can conduct current only when it flows in agreement with the bias current.

From the theory of the operation of rectifiers controlled by saturable reactors [1], it is known that the regulating action of the reactors is observed only at load currents equal to the bias current \((I_d = I_{\mathrm{p}})\), while at \(I_d < I_{\mathrm{p}}\) the rectifier operates as an uncontrolled bridge rectifier.

Here one can distinguish two modes of reactor biasing: forced, when a large in-

inductive reactance, and free, when the resistance of the bias source is close to zero. Regardless of the type of load at the rectifier output in such a system, the rectified current in the controlled

Fig. 1. Circuit of the charging device

mode \((I_d = I_{\mathrm{n}})\) fully corresponds to the bias current not only in magnitude but also in waveform. Therefore, of greatest interest for a charging device is forced biasing, which, during capacitor charging, provides a practically smoothed rectified current.

Considering the possible operating modes of the charging device, it should be noted that, with the exception of the short-circuit point, alternating two- and three-valve operating modes of the rectifier are observed in the circuit. With forced biasing in the two-valve mode, only two reactors corresponding to the conducting valves (Fig. 1) are saturated in the circuit, and therefore

\[ i_1 = i_6,\quad i_{1g} = i_{6g} = i_{\mathrm{n}},\quad \text{i.e., in this mode} \]

\[ i_1 = i_6 = i_{1g} = i_{6g} = i_d = i_{\mathrm{n}}(t). \]

In the three-valve mode an analogous picture is observed:

\[ i_1 + i_2 = i_6 = i_d;\quad i_{1g} + i_{2g} = i_{\mathrm{n}}(t);\quad i_d(t) = i_{\mathrm{n}}(t) \]

and if \(i_{\mathrm{n}}(t) = I_{\mathrm{n}} = \mathrm{const}\), then \(i_d(t) = I_d = \mathrm{const}\),

i.e., it is ideally smoothed. Taking the above into account, the circuit with saturating reactors (Fig. 1) can be represented by a certain equivalent circuit with unchanged anode parameters and fully controlled valves, whose firing angle \((\alpha)\) is a variable function and depends on the magnitude of the voltage on the storage capacitor. Since the time of energy storage significantly exceeds (by \(10^2\)–\(10^3\) times) the period of variation of the source voltage, the circuit can be calculated at discrete points, assuming that over one period of the electromotive force the voltage on the capacitor remains constant, i.e., replacing the real charging curve by a stepwise one. This technique makes it possible to carry out the calculation at a particular instant of time both when the circuit operates under counter-emf and with an ideally smoothed rectified current.

Using the equivalent-generator method [2], we determine the voltage across the capacitance at time \(t_1\) as

\[ u_c(t_1) = \frac{3}{4}\sqrt{3}E_m \cos \alpha(t_1) - \frac{3}{\pi} i_d x_{\phi} - 2 i_d R_{\phi}, \tag{1} \]

where \(E_m\) is the amplitude value of the phase emf on the secondary winding of the transformer; \(x_{\phi}=x_t+x_d\) is the sum of the inductive resistances of the referred transformer and the saturable reactor; \(r_{\phi}=r_t+r_d\) is the sum of the active resistances of the referred transformer and the saturable reactor.

It should be noted that when saturable reactors are connected in the primary circuit, the transformer parameters must be referred to the primary winding.

Taking into account that, under forced magnetization in the controlled mode, the charging current is stable, \(i_d=I_d=I_{\mathrm{p}}\), we determine the valve firing angle

\[ \alpha=\arccos \frac{4}{3\sqrt{3}}\, \frac{I_{\mathrm{p}}x_{\phi}}{E_m} \left( \frac{t}{C_{\mathrm{n}}x_{\phi}}+\frac{3}{\pi} +2\frac{r_{\phi}}{x_{\phi}} \right). \tag{2} \]

Thus, as the capacitor \(C_{\mathrm{n}}\) is charged, the angle \(\alpha\) varies according to a cosine law from \(90^\circ\) to \(0^\circ\) (the latter is observed upon transition to the uncontrolled mode).

For a given magnetization current, the time of controlled charging can be determined from the expression

\[ T_{\mathrm{c}}=t_{\max}=C_{\mathrm{n}}x_{\phi} \left( \frac{3\sqrt{3}}{4}\, \frac{E_m}{I_{\mathrm{p}}x_{\phi}} -2\frac{r_{\phi}}{x_{\phi}} -\frac{3}{\pi} \right) \tag{3} \]

and is the time of the storage-unit charging cycle.

Considering the three-valve operating mode of the circuit, it may be noted that, with a stable charging current, the commutation circuit retains its parameters unchanged throughout the entire time of controlled charging. This makes it possible to conclude that, in the regulation zone, the valve commutation angle remains constant and does not depend on the magnitude of the voltage on the capacitor. Taking into account ideal smoothing of the current, the commutation angle can be determined by the expression

\[ \gamma \simeq \arccos \left(1-\frac{2I_{\mathrm{p}}x_{\phi}}{\sqrt{3}E_m}\right). \tag{4} \]

The presence of the regulation angle \((\alpha)\) and the commutation angle \((\gamma)\) causes a time shift of the transformer phase current relative to the voltage and determines the power factor of the charging device

\[ \cos \varphi_1 \simeq \cos \left(\alpha+\frac{\gamma}{2}\right). \tag{5} \]

Analysis of this expression with allowance for (2) and (4) shows that, as energy accumulates in the capacitor, the power factor varies over wide limits practically according to a linear law. However, the most characteristic value for evaluating the charging device as a whole is the mean-cycle value of the power factor. Taking into account the constancy of the commutation angle and of the charging-cycle time, the mean-cycle value \(\cos\varphi_{1\mathrm{cp}}\) is determined by

\[ \cos \varphi_{1\mathrm{cp}} = \frac{1}{T_{\mathrm{c}}}\int_{0}^{T_{\mathrm{c}}}\cos\varphi_1(t)\,dt = 0.5[1+b]\, \frac{\cos\left(\frac{\gamma}{2}+\xi\right)}{\cos \xi}, \tag{6} \]

where

\[ \tg \xi= \frac{1}{1-b^2} \left( \frac{\pi}{2} -b\sqrt{1-b^2} -\arcsin b \right), \qquad b= \frac{4I_{\mathrm{p}}x_{\phi}}{3\sqrt{3}E_m} \left( \frac{3}{\pi} +2\frac{r_{\phi}}{x_{\phi}} \right). \]

Calculations carried out for real circuits of charging devices in accordance with expression (6) show that the mean-cycle co-

power factor does not exceed values of \(\cos\varphi_{\mathrm{av}} \simeq 0.5\), which is a serious drawback of devices of this type.

The second energy factor determining the charging device is the efficiency. Considering the efficiency as the ratio of the energy accumulated to the energy expended, it can be determined that the instantaneous value of the efficiency depends on the circuit parameters and the charging time

\[ \eta=\frac{W_c}{W_c+w_R}=\frac{1}{1+6\,\frac{p_\phi C_n}{t}\,k_i^2}, \tag{7} \]

where

\[ k_i=\frac{I_{\phi\mathrm{eff}}}{I_d}=\sqrt{\frac{2}{3}-\frac{\gamma}{3\pi}} \]

is the current coefficient of the three-phase bridge circuit [3].

Thus, as energy is accumulated in the capacitor, the efficiency increases, reaching values close to unity at the end of charging. The mean-cycle value of the efficiency, determined analogously to the power factor, can be found from the expression

\[ \eta=1-\frac{6C_n p_\phi k_i^2}{T_c}\ln\left(1-\frac{T_c}{6C_n p_\phi k_i^2}\right). \tag{8} \]

Analysis of this expression makes it possible to conclude that in low-power systems (\(W_c \simeq 10^3\) J), at a capacitor voltage of the order of \(U_c \simeq (1.5—2)\cdot 10^3\) V, the mean-cycle efficiency values reach 0.7—0.75, which is substantially higher than in systems with current-limiting resistances (\(\eta_{\max}<0.5\)).

Fig. 2. Energy characteristics of the charging device from the side of a) the generator, b) the charging circuit

Experimental investigations of the charging device, carried out according to the circuit of Fig. 1, confirmed the correctness of the assumptions made above and showed that charging devices with saturable reactors are quite effective in creating low-frequency pulsed sources. Figure 2 gives the energy characteristics of the charging device from the generator side (Fig. 2, a) and from the side of the storage capacitance (Fig. 2, b). As can be seen from the figures, the charging current and the generator current are sufficiently stable, which, on the one hand, corresponds to a practically linear increase of the voltage on the capacitance, and on the other hand ensures a uniform current load on the generator. The mean-cycle values of the efficiency and of the power factor

respectively, were here: \(\cos \varphi_{\mathrm{av}} = 0.47\); \(\eta_{\mathrm{av}} = 0.72\). The invariability of the waveform of the transformer phase current in the regulation zone is confirmed by the stability of the amplitudes of the fundamental harmonic components (Fig. 3) and indicates the constancy of the commutation angle \((\gamma)\) throughout the controlled charging period.

Fig. 3. Dependence of the relative values of the amplitudes of the higher harmonics of the phase current on the magnitude of the voltage across the capacitance

Thus, the analytical and experimental studies carried out make it possible to draw the following main conclusions:

1. A circuit with saturable reactors inserted in the break of the transformer neutral point ensures stability of the charging current, which, with forced biasing, can be ideally smoothed.

2. The circuit provides a sufficiently high value of the average cyclic efficiency \((\eta_{\mathrm{av}} = 0.7—0.75)\); however, the values of the power factor do not exceed 0.5.

3. The circuit ensures the absence of sharp current surges at the moment of charging and a uniform current load on the main source.

REFERENCES

1. Yu. G. Tolstov, G. P. Mostkova, F. I. Kovalev. Three-Phase Power Semiconductor Rectifiers. Publishing House of the Academy of Sciences of the USSR, 1963.

2. A. V. Posse. Justification for replacing a rectifier by an equivalent generator for calculating transient processes. Publishing House of the Academy of Sciences of the USSR, “Energetika i transport,” 1965, No. 4.

3. V. A. Zorin et al. Some questions concerning the operation of a low-power synchronous generator with a rectifier load. “Izvestiya TPI,” vol. 145, 1966.