Calculation of Transient Processes in Two-Loop Pulse Circuits

V. S. Melikhov, I. P. Chuchalin

To create pulsed magnetic fields, the discharge of a previously charged capacitor bank is often used. The discharge circuit usually consists of a capacitance, an inductance, and a controlled switching device (thyratron, ignitron, etc.).

The capacitors are charged from a direct-current source, most often from a multiphase rectifier. In order to save the power consumed, the limiting element included in the charging circuit is not an active resistance, but a sufficiently large inductance (a choke). The equivalent circuit of the circuits under consideration is shown in Fig. 1.

Here: \(U_b\) is the average value of the rectifier voltage; \(L_1\) and \(r_1\) are the inductance and active resistance of the charging circuit; \(C\) is the capacitance of the capacitor bank; \(L_2\) and \(r_2\) are the inductance and active resistance of the discharge circuit; \(K. P.\) is the switching device.

Fig. 1.

Let us consider the transient processes occurring in this circuit during periodic charging and discharging of the capacitance \(C\).

At the initial instant of time the rectifier is disconnected, and the voltage across the capacitors is zero. When a circuit possessing inductance, capacitance, and active resistance is connected to a constant voltage, in the case where the charging of the capacitors is oscillatory in character, the current in the charging circuit and the voltage across the capacitors vary according to the well-known laws:

\[ i_{\text{ch}}=-\frac{U_{\text{r}}e^{-\delta_1 t}}{\omega'_1 L_1}\,\sin \omega'_1 t, \tag{1} \]

\[ U_c=U_{\text{r}}-U_{\text{r}}\frac{\omega_1}{\omega'_1}e^{-\delta_1 t}\cdot \cos(\omega'_1 t-\xi_1), \tag{2} \]

where

\[ \omega_1=\frac{1}{\sqrt{L_1 C}} \]
is the angular frequency of the undamped oscillations of the charging circuit;

\[ \omega'_1=\sqrt{\omega_1^2-\delta_1^2} \]
is the angular frequency of the damped oscillations of the charging circuit;

\[ \delta_1=\frac{r_1}{2L_1}; \]

\[ \xi_1=\operatorname{arctg}\frac{\delta_1}{\omega'_1} \]
lies within the limits \(0\leq \xi_1\leq \frac{\pi}{2}\).

Since in the charging circuit the source of constant voltage is a rectifier, the process of charging the capacitor will cease after a time equal to one half-period of the natural damped oscillations of the charging circuit. The voltage on the capacitors at this instant will be

\[ U_1=U_{\text{r}}\left(1+e^{-\frac{\pi}{2Q_1}}\right), \tag{3} \]

where

\[ Q_1=\frac{\omega_1 L_1}{r_1} \]
is the quality factor of the charging circuit.

After the capacitors have been charged, the switching device \(K.\ P.\) is turned on and the capacitors begin to discharge through the circuit \(L_2, r_2\). As soon as the voltage on them in the course of the discharge becomes less than the rectifier voltage, a recharge current will flow from the latter. Thus, discharge and charge of the capacitors will occur simultaneously. For an exact analysis of the transient processes occurring in the circuit (Fig. 1) during this time interval, it is necessary to set up a system of equations whose solution will lead to a linear differential equation of the third order. In general form, the solution of a third-order differential equation is complicated.

In the pulsed mode of operation, when the time interval between pulses is many times greater than the pulse duration, it is advisable to choose the period of the natural oscillations of the charging circuit to be much greater than the period of the natural oscillations of the discharge circuit. This will make it possible to reduce the amplitude of the charging current and to use less powerful valves in the rectifier. With a sharp difference between the frequencies of the charging and discharge circuits, the influence of the charging circuit on the discharge circuit is weakened to such a degree that the discharge of the capacitor may approximately be considered without taking into account the influence of the charging circuit. This latter circumstance greatly simplifies the analysis of transient processes.

If, for the discharge circuit, the condition

\[ \delta_2<\omega_2, \]

where

\[ \omega_2=\frac{1}{\sqrt{L_2 C}} \]

is the angular frequency of the natural undamped oscillations of the discharge circuit;

\[ \delta_2=\frac{r_2}{2L_2}, \]

then the capacitor discharge will have an oscillatory character. The current in the discharge circuit and the voltage across the capacitors will be described by the equations:

\[ i_{\text{disch.}}=\frac{U_0}{\omega_2' L_2}\, e^{-\delta_2 t}\,\sin \omega_2' t, \tag{4} \]

\[ U_c=U_0\frac{\omega_2}{\omega_2'}\, e^{-\delta_2 t}\,\cos(\omega_2' t-\xi_2), \tag{5} \]

where \(U_0\) is the voltage across the capacitance at the beginning of its discharge;

\[ \omega_2'=\sqrt{\omega_2^2-\delta_2^2} \]

is the angular frequency of the natural damped oscillations of the discharge circuit;

\[ \xi_2=\operatorname{arctg}\frac{\delta_2}{\omega_2'} \]

and lies within the limits \(0<\xi_2\le \dfrac{\pi}{2}\).

Fig. 2.

At the moment when the current passes through zero, the switching device extinguishes and the process ceases. Setting \(t=T_2/2\), we obtain the value of the voltage across the capacitors at this moment of time (Fig. 2):

\[ U_1^{1}=-U_0 e^{-\frac{\pi}{2Q_2}}, \tag{6} \]

where

\[ Q_2=\frac{\omega_2 L_2}{r_2} \]

is the quality factor of the discharge circuit.

Substituting into formula (6), in place of \(U_0\), the value of the voltage from formula (3), we obtain the following expression for the magnitude of the voltage across the capacitors after the first discharge:

\[ U_1^{1}=-U_v\left(1+e^{-\frac{\pi}{2Q_1}}\right)e^{-\frac{\pi}{2Q_2}}. \tag{7} \]

After the end of the discharge, the capacitors are again charged from the rectifier. The voltage to which they are charged the second time can be determined from the following expression:

\[ U_2=U_v+(U_v-U_1^{1})e^{-\frac{\pi}{2Q_1}}. \]

Substituting for \(U_1'\) its value from (7), we obtain

\[ U_2 = U_{\mathrm{v}}\left(1+e^{-\frac{\pi}{2Q_1}}\right) \left(1+e^{-\frac{\pi}{2Q_3}}\right), \tag{8} \]

where

\[ Q_3=\frac{Q_1\cdot Q_2}{Q_1+Q_2}. \]

Reasoning analogously, we obtain an expression for the voltage on the capacitors after the \(n\)-th discharge:

\[ U_n = \frac{ U_{\mathrm{v}}\left(1+e^{-\frac{\pi}{2Q_1}}\right) \left(1-e^{-\frac{n\pi}{2Q_3}}\right) }{ 1-e^{-\frac{\pi}{2Q_3}} }. \tag{9} \]

In the steady-state regime the capacitors are charged to the voltage

\[ U_{\mathrm{st}} = U_{\mathrm{v}}\, \frac{ 1+e^{-\frac{\pi}{2Q_1}} }{ 1-e^{-\frac{\pi}{2Q_3}} }. \tag{10} \]

From formula (10) it is evident that the greater the quality factors of the circuits, the higher the value of the steady-state voltage on the capacitors. The value of the steady-state voltage depends to a greater extent on the quality factor of the charging circuit than on that of the discharging circuit. The charging current in the steady-state regime is determined by the expression

\[ i_{\mathrm{ch}} = \frac{ U_{\mathrm{v}}\left(1+e^{-\frac{\pi}{2Q_2}}\right)e^{-\delta_1 t} }{ \omega_1' L_1\left(1-e^{-\frac{\pi}{2Q_3}}\right) } \sin \omega_1' t . \tag{11} \]

The power drawn from the rectifier can be found from the formula

\[ P_{\mathrm{v}}=U_{\mathrm{v}} i_{\mathrm{av}} = U_{\mathrm{v}}\cdot \frac{1}{T} \int_{0}^{T_1/2} i_{\mathrm{ch}}\,dt, \]

where \(T\) is the repetition period of the current pulses in the discharge circuit. Substituting the value of \(i_{\mathrm{ch}}\) from (11) and integrating, we obtain the following expression for the power consumed by the circuit:

\[ P_{\mathrm{v}}= \frac{ \dfrac{C U_{\mathrm{v}}^{2}}{T} \left(1+e^{-\frac{\pi}{2Q_1}}\right) \left(1+e^{-\frac{\pi}{2Q_2}}\right) }{ 1-e^{-\frac{\pi}{2Q_3}} }. \]

\[ \frac{C U_{\mathrm{st}}^{2}}{T}\, \frac{ \left(1-e^{-\frac{\pi}{2Q_{2}}}\right) \left(1-e^{-\frac{\pi}{2Q_{3}}}\right) }{ 1-e^{-\frac{\pi}{2Q_{1}}} }. \tag{12} \]

Figure 3 shows an oscillogram of the rise of the voltage across the capacitors for a particular case \((Q_{1}=6.5\) and \(Q_{2}=11.3)\). For the quality-factor values \(Q_{1}=6.5\) and \(Q_{2}=11.3\), the value of the steady-state voltage, calculated from formula (10), is equal to

\[ U_{\mathrm{st}}=5.63\,U_{v}. \]

As can be seen, in order to charge the capacitors to the specified voltage value, in this case a rectifier unit with an average value of the rectified voltage almost six times smaller is required. This circumstance is a definite advantage of the given circuit, making it possible to charge capacitors to high voltages from low-voltage rectifiers.

Fig. 3.

However, a rather large current is drawn from the rectifier unit. In the pulsed mode of operation, in the case of a comparatively large duty factor, the choice of valves is determined mainly by the maximum value of the charging current. To reduce its magnitude, it is necessary to increase the inductance of the choke in the charging circuit, and this may lead to the dimensions of the choke becoming unreasonably large. It is also easy to note that at the moment when the switching device is extinguished, a large negative voltage is applied almost instantaneously to its anode, which, at a low deionization rate, may lead to the occurrence of reverse ignition.

Another drawback of the circuit considered is the sharp dependence of the voltage across the capacitors in the steady state on the quality factor of the charging and discharging circuits, which leads to increased instability of operation.

The indicated drawbacks are eliminated when the capacitors are recharged through an additional valve connected in inverse-parallel to the main switching device.

Let us consider the operation of the circuit with recharging of the capacitors. In practical cases, the quality factor of the discharge circuit is higher than the quality factor of the charging circuit. With small damping in the discharge circuit, i.e., for \(\delta_{2}\ll\omega_{2}^{1}\), the voltage across the capacitors during discharge will vary according to the law

\[ U_{C}=U_{0}e^{-\delta_{2}t}\cos \omega_{2}^{1}t. \tag{13} \]

Find the expression for the charging current during the discharge and recharge of the capacitors.

In this case,

\[ U_{\mathrm{v}} = i_1 r_1 + L_1 \frac{d i_1}{d t} + U_c . \]

Solving the above equation with allowance for the initial conditions (at \(t=0\), \(i_1=0\)) and assuming \(\omega_2^1 \gg (2\delta_1-\delta_2)\), we obtain the expression for the current in the form

\[ i_1^1 = \left[ \frac{U_{\mathrm{v}}}{r_1}\left(e^{2\delta_1 t}-1\right) - \frac{U_0}{L_1 \omega_2^1}\, e^{(2\delta_1-\delta_2)t}\cdot \sin \omega_2^1 t \right] e^{-2\delta_1 t}. \tag{14} \]

The value of the charging current at the instant when the discharge and recharge of the capacitors end is

\[ I_1^1 = \frac{U_{\mathrm{v}}}{r_1}\left(1-e^{-2\delta_1 T_2}\right). \tag{15} \]

We have considered the process of charging the capacitors during their simultaneous discharge. For this interval of time we found the law of variation of the charging current. After the switching devices cease operating, the discharge of the capacitors will stop and the charging current will vary according to another law, different from (14).

In this case, as before, we have:

\[ U_{\mathrm{v}} = i_1 r_1 + L_1 \frac{d i_1}{d t} + \frac{1}{c}\int i_1\,dt . \]

The solution of this equation, with allowance for the initial conditions (at

\[ t=0 \qquad i_1=I_1,\quad U_c=U_0 e^{-\frac{\pi}{Q_2}} \]

), will be:

\[ i_1'' = \left[ \frac{1}{\omega_1^1} \left( \frac{U_{\mathrm{v}}-U_0 e^{-\frac{\pi}{Q_2}}}{L_1} -\delta_1 I_1^1 \right)\cdot \sin \omega_1^1 t + I_1^1 \cos \omega_1^1 t \right]\cdot e^{-\delta_1 t}. \tag{16} \]

Approximately (in practice, with an accuracy of up to 5%), the preceding formula may be written as

\[ i_1'' \simeq I_1^1 e^{-\delta_1 t}\cdot \cos \omega_1^1 t. \tag{17} \]

Substituting in (17), instead of \(I_1^1\), its value from (15), the charging current after the switching devices cease operating is finally written as:

\[ i_1'' \simeq \frac{U_{\mathrm{v}}}{r_1} \left(1-e^{-2\delta_1 T_2}\right)\cdot e^{-\delta_1 t}\cdot \cos \omega_1^1 t . \tag{18} \]

For recharging a capacitor bank, a rectifier unit is required that is capable of delivering a current smaller in magnitude than the current required for charging the capacitors when they are not recharged. The average value of the rectifier voltage in the case with recharging of the capacitor bank must be higher than in the case without recharging.

In the steady-state operating regime of the circuit with recharging of the capacitors, the latter are charged each time to the maximum value of the rectifier voltage.

It is not difficult to see that the maximum current from the rectifier during operation of the circuit with recharging of the capacitors will occur at the time the rectifier is switched on, at the first instant of capacitor charging, when the voltage across them is zero. To limit the initial current wave from the rectifier, a limiting active resistance may be provided in the charging circuit, which is shunted after the capacitors have been charged.

Figure 4 presents oscillograms of the voltage \(U_c\) on the capacitor bank, of the discharge current \(i_{\text{disch}}\), and of the charging current \(i_{\text{ch}}\), taken during operation of the circuit with recharging of the capacitors and for the following numerical parameter values: \(r_1 = 3.7\ \Omega\), \(L_1 = 0.87\ \text{H}\), \(C = 1417\ \mu\text{F}\), \(r_2 = 0.19\ \Omega\), \(L_2 = 6.8\ \text{mH}\), \(U_v = 234\ \text{V}\).

Fig. 4.

In the same Fig. 4 a graph is plotted of the dependence of the charging current \(i_{\text{ch}}\) on time, calculated from formulas (14) and (18). As can be seen, the agreement of the experimental curve \(i_{\text{ch}}\) with the calculated one is sufficiently good. This shows that, in calculations of a circuit with recharging of the capacitor bank, one may use the approximate formulas given here.

We have considered the operation of the pulse circuit in two cases: without recharging and with recharging of the capacitors. In the discussion it was assumed that there is a sharp difference between the frequencies of the charging and discharge circuits, which practically occurs in the pulsed mode of operation. In both cases the rectifier was assumed to be uncontrolled, i.e., it was not disconnected during the period of discharge and recharging of the capacitors.

When considering operation of the pulse circuit “without recharging,” we meant the absence of recharging of the capacitors through the discharge circuit. In reality, however, the capacitors were recharged in the first case as well. Only their recharging occurred in the charging circuit through the rectifier valves. In the second case, when an additional device is connected in the discharge circuit in antiparallel with the main switching device, recharging of the capacitors could take place simultaneously through two circuits: the charging and the discharge circuits. But since the inductive and active resistances of the charging circuit are much greater than the analogous resistances of the discharge circuit, in practice, in the second case the capacitors were recharged only through the discharge circuit.