Toward an Engineering Calculation of a Sinusoidal Oscillation Generator Based on Gas-Discharge Devices
Abstract Generated abstract
This paper develops engineering relations for calculating a sinusoidal oscillation generator using gas discharge switching devices for loads that primarily consume reactive energy, with emphasis on excitation of a betatron electromagnet at frequencies other than 50 Hz. The analysis models the generator as coupled resonant circuits, evaluates energy input from a direct current source against losses in the tank circuit, and derives practical formulas for circuit voltage amplitude, valve peak and average currents, reverse valve voltage, anode reactor inductance, and reactor stored energy. An experimental generator for a PMB type betatron operating at 200 Hz was built using the derived relations, and measured oscillograms and operating values were reported. The experiments supported the applicability of the theoretical assumptions for engineering calculation and indicated that the supplied energy can be regulated by changing the valve ignition phase.
Full Text
On the Engineering Calculation of a Sinusoidal Oscillation Generator Using Gas-Discharge Devices
M. V. Kim, Yu. P. Yarushkin
(Presented by the scientific seminar of the Research Institute of Nuclear Physics, Electronics, and Automation at TPI and of the Department of Industrial Electronics)
For loads that are mainly consumers of reactive energy, when it is necessary to supply them with a voltage at a frequency other than 50 Hz, it is possible to construct a sinusoidal-oscillation generator based on pulsed energy input.
One of the consumers of reactive energy is the betatron. Figure 1 shows the schematic and equivalent circuits of a generator for exciting the electromagnet of a betatron at an increased frequency.
Fig. 1
In the equivalent circuit the following notation is adopted:
\(L_1\) — inductance of the anode reactor,
\(R_1\) — equivalent resistance of the anode reactor and valve \(T\),
\(L_2 C\) — inductance and capacitance of the resonant circuit for exciting the betatron,
\(R_c\) — loss resistance in the capacitance \(C\) for the parallel equivalent circuit of the tank circuit.
In the steady-state regime, the energy supplied from the direct-current circuit compensates for the active losses in the \(L_2 C\) circuit. The energy of losses in the oscillatory circuit \(L_2 C\) is equal to
\[ W_1 = \frac{U_{km}^{2} C}{2Q}, \tag{1} \]
where \(U_{km}\) is the amplitude value of the voltage across the circuit,
\(Q\) is the quality factor of the circuit.
Substituting the value of the quality factor of the \(L_2C\) circuit into (1), we obtain:
\[ W_1=\frac{U_{km}^{2}C\omega_0L_2}{2R_CR_L}(R_C+R_L). \tag{2} \]
Considering the voltage diagram in Fig. 2, one may conclude that the maximum input of energy into the circuit from the DC circuit corresponds to the phase of opening the valve at the zero value of the voltage on the circuit, i.e., \(\psi=0\).
Fig. 2
We take the instant \(\psi=0\) as the beginning of the input of energy into the capacitance. The amount of energy stored in the capacitance, without taking into account the alternating voltage \(U_k\) (since \(L_2 \gg L_1\)), is equal to
\[ W_2=\frac{U_C^2C}{2}, \tag{3} \]
where \(U_C\) is the amplitude of the voltage on the capacitance \(C\) to which it is charged when the valve is open.
Considering the series oscillatory circuit \(L_1C\), obtained from the instant the valve opens (closing of switch \(K\)), we determine the amplitude of the voltage on the capacitor at \(\omega_{\mathrm p}t=\pi\):
\[ U_C=\frac{U_0R_C}{R_1+R_C}\left(1-e^{-\delta\frac{\pi}{\omega_{\mathrm p}}}\right). \]
After transformations we obtain
\[ U_C=\frac{U_0R_C}{R_1+R_C}\left(2-\delta\frac{\pi}{\omega_{\mathrm p}}\right). \tag{4} \]
Equating (2) and (3), with (4) taken into account, we write the amplitude of the voltage on the circuit as:
\[ U_{km}=\frac{U_0R_C}{R_1+R_C}\left(2-\delta\frac{\pi}{\omega_{\mathrm p}}\right) \sqrt{\frac{R_CR_L}{\omega_0L_2(R_C+R_L)}}. \tag{5} \]
Taking into account that \(R_1 \ll R_C\) and \(\delta \ll \omega_{\mathrm p}\), after simplification we obtain
\[ U_{km}=2U_0\sqrt{Q}. \tag{6} \]
Thus, the amplitude of the voltage on the circuit depends on the parameters of the circuit at the given frequency and may considerably exceed \(U_0\). The maximum value of the current through the valve at the instant energy is fed into the \(L_2C\) circuit is also found by considering the series oscillatory circuit and setting \(\omega_{\mathrm p}t=\frac{\pi}{2}\).
\[ I_{\mathrm{vt}}=U_0\frac{\omega_{\mathrm p}L_1+R_C}{\omega_{\mathrm p}L_1R_C}. \tag{7} \]
Calculation of the average value of the current through the valve, taking into account the sinusoidal form of the current pulses following with \(\omega_{\mathrm p}\), gives
\[ I_0=\frac{f_0}{\pi f_{\mathrm p}}\cdot \frac{\omega_{\mathrm p}L_1+R_C}{\omega_{\mathrm p}L_1R_C}\,U_0. \tag{8} \]
As can be seen from Fig. 2, the reverse voltage applied to the valve is equal to
\[ U_{\text{rev}}=U_{k m}-U_c=U_0(2\sqrt{Q}-1). \tag{9} \]
Normal operation of the generator is ensured under the condition \(f_{\text{p}}\gg f_0\); in practice \(f_{\text{p}}=(7\div 8)f_0\) must be satisfied. Hence the inductance of the anode reactor is
\[ L_1 \approx \frac{L_2}{50}. \tag{10} \]
The design parameters of the choke \(L_1\) are determined by the amount of energy stored in it, which in the present case depends on the losses in the \(L_2 C\) circuit and can be calculated from the formula
\[ W_{L_1}=\frac{f_0^2 W_{L_2}}{f_{\text{p}}^2 Q^2}. \tag{11} \]
Thus, for given \(U_k\), \(Q\), and \(f_0\) of the oscillatory circuit, expressions (6), (7), (8), (9), (10), and (11) make it possible to carry out certain elements of the calculation of the generator circuit.
Using the relations given above, an experimental generator setup was calculated and assembled for exciting the electromagnet of a PMB-type betatron at a frequency of 200 Hz, with the following data: \(Q \approx 10\), \(L_2=0.08\) H, \(C=8.25\ \mu\text{F}\), \(L_1=1.7\) mH.
Fig. 3
The voltage across the circuit in this case was 1300 V, the current in the circuit was 14 A, and the average value of the rectifier current was 1.5 A. Figure 3 shows oscillograms of voltages and currents: 1 — current in the inductance \(L_2\), 2 — current in the capacitance \(C\), 3 — anode-current pulses, 4 — voltage across the capacitance \(C\), 5 — voltage across the storage capacitance.
As a result of the experimental investigations, it was established, first, that the theoretical assumptions in the analysis of the generator-
and the suitability of the mathematical relations obtained for engineering calculation of the circuit; secondly, the experiments confirmed the possibility of regulating the magnitude of the energy introduced by means of the ignition phase of the valve.
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