On the Possibility of Using High-Voltage Transformers as a Power Supply for Electronic Accelerator Tubes
Abstract Generated abstract
This paper examines whether a high-voltage transformer powered from a 50 Hz industrial-frequency network can serve as a pulsed power supply for electron accelerator tubes requiring very small voltage variation during beam formation. Using an equivalent transformer circuit with leakage inductance, winding resistance, output capacitance, and load resistance, the authors derive differential equations for the load voltage and analyze both aperiodic and oscillatory transient regimes as functions of circuit parameters and switching phase. The analysis indicates that, for suitable ratios of transformer and load parameters and for load connection during a small part of the sinusoidal period, the transient and forced voltage components can combine to produce an approximately constant load-voltage pulse, with limiting and optimal parameter relations identified for maximum pulse duration.
Full Text
ON THE POSSIBILITY OF USING HIGH-VOLTAGE TRANSFORMERS AS A POWER SUPPLY FOR ELECTRON ACCELERATOR TUBES
D. I. ANDERMAN, I. D. KUTYAVIN
At present, various electron-beam installations have been developed that use the properties of the electron beam as a tool.
As a source of power for an accelerator tube, various circuits may be used. These include: transformers with a linear potential distribution, sources based on cascade voltage-multiplication circuits, pulse-voltage generators based on parallel charging of capacitors with switching to series connection, and others. One possible solution is the use of a high-voltage transformer operating from an industrial-frequency 50 Hz network in a pulsed mode at a secondary voltage \(u_2 = 80\text{--}100\ \mathrm{kV}\) [1]. The main requirement imposed on such a transformer is the following: in order to form a beam with a diameter on the order of \(10\text{--}50\ \mu\), the electron energy spread must not exceed \(0.01\%\). Consequently, during the period of pulsed switching-on of the load, the change in the secondary voltage must not exceed \(0.01\%\). In this case it is desirable to make the load switching-on time maximal. It is obvious, and this is confirmed by experiment, that as
Fig. 1. Oscillogram of the voltage across the load (half-period) and of the current in the load in the case of an oscillatory transient process.
when the load is switched on, as well as when it is switched off, transient processes arise in the circuit; their duration and character depend on the parameters of the transformer itself, on the magnitude and character of the load, on the capacitance connected at the transformer output, and on the voltage phase at the moment of switching.
Depending on the ratio of the parameters, the transient process may be either oscillatory in character (Fig. 1) or aperiodic.
For an analytical study of the voltage as a function of time, we shall use the equivalent circuit of the transformer shown in Fig. 2. Here
\(L_k\) — leakage inductance of the transformer windings, referred to the secondary winding, \(L_k = L'_1 + L_2\);
Fig. 2.
\(r_k\) — active resistance of the windings, referred to the secondary winding, \(r_k = r_2 + r'_1\);
\(C\) — capacitance, including the capacitance of the high-voltage winding to ground and an additionally connected capacitance;
\(R\) — resistance of the accelerator tube. In the case where the voltage applied to the cathode and the blocking signal on the grid remain constant, \(R\) may also be considered an unchanged quantity.
\(u'_1 = U_{1m}\sin(\omega t + \psi_1)\) — sinusoidal voltage with mains frequency \(f_1 = \dfrac{\omega}{2\pi}\), applied to the transformer input and referred to the secondary winding;
\(\psi_1\) — voltage phase at the moment the load is switched on. The equations of electrical equilibrium of the circuit in differential form are:
\[ \left. \begin{aligned} u'_1 &= r_k i_1 + L_k \frac{di_1}{dt} + u_c,\\ i_2 &= \frac{u_c}{R},\\ i_c &= C\frac{du_c}{dt},\\ i_1 &= i_2 + i_c \end{aligned} \right\} \tag{1} \]
This system of differential equations can be reduced to a single second-order differential equation:
\[ \frac{d^2 u_c}{dt^2} + \frac{du_c}{dt} \left( \frac{r_k}{L_k} + \frac{1}{CR} \right) + u_c \left( \frac{1}{L_k C} + \frac{r_k}{L_k CR} \right) = \frac{u'_1}{L_k C}. \tag{2} \]
Integrating equation (2), we obtain an expression for the transient voltage across the load. The character of the transient process is determined by the roots of the characteristic equation (3) and depends on the circuit parameters.
equivalent
\[ p^2+p\left(\frac{r_k}{L_k}+\frac{1}{CR}\right)+\frac{1}{L_k C}+\frac{r_k}{L_k CR}=0 . \tag{3} \]
\[ p_{1,2}=-\frac{1}{2}\left(\frac{r_k}{L_k}+\frac{1}{CR}\right)\pm \sqrt{\frac{1}{4}\left(\frac{r_k}{L_k}+\frac{1}{CR}\right)^2- \left(\frac{1}{L_k C}+\frac{r_k}{RCL_k}\right)} . \]
If the roots \(p_1\) and \(p_2\) are real distinct numbers, then the transient process is aperiodic, and the equation for the voltage \(u_c(t)\) can be obtained in the form
\[ u_c(t)=u_{c\mathrm{pr}+}+u_{c\mathrm{sv}}=u_{c\mathrm{pr}+}+A_1 e^{p_1 t}+A_2 e^{p_2 t}, \tag{4} \]
where \(u_{c\mathrm{pr}+}\) is the forced component of the voltage across the connected load,
\[ u_{c\mathrm{pr}+}= \frac{ U_{1m}R\sin\left(\omega t+\psi_1-\frac{\pi}{2}-\operatorname{arctg}\frac{\omega^2 L_k C R-r_k-R}{\omega C R r_k+\omega L_k}\right) }{ \sqrt{(\omega C R r_k+\omega L_k)^2+(\omega^2 L_k C R-r_k-R)^2} }, \tag{5} \]
\[ u_{c\mathrm{sv}}=A_1 e^{p_1 t}+A_2 e^{p_2 t}, \tag{6} \]
\(u_{c\mathrm{sv}}\) is the free component of the voltage.
\(A_1\) and \(A_2\) are constants of integration, which are determined on the basis of the laws of commutation from the initial conditions of the circuit at the instant of commutation,
\[ A_1=\frac{b_1p_2-\omega b_2}{p_3-p_1}. \tag{7} \]
\[ A_2=\frac{\omega b_2-b_1p_1}{p_2-p_1}, \tag{8} \]
where
\[ b_1=U_{1m}\left[ \frac{ \sin\left(\psi_1-\frac{\pi}{2}-\operatorname{arc\,tg}\frac{\omega L_k-\frac{1}{\omega C}}{r_k}\right) }{ \sqrt{r_k^2\omega^2 C^2+(\omega^2 L_k C-1)^2} } - \frac{ R\cdot\sin\left(\psi_1-\frac{\pi}{2}-\operatorname{arc\,tg}\frac{\omega^2 L_k C R-r_k-R}{r_k R\omega C+\omega L_k}\right) }{ \sqrt{(r_k R\omega C+\omega L_k)^2+(\omega^2 L_k C R-r_k-R)^2} } \right], \tag{9} \]
\[ b_2=\frac{U_{1m}}{\omega C}\left[ \frac{ \sin\left(\psi_1-\operatorname{arc\,tg}\frac{\omega L_k-\frac{1}{\omega C}}{r_k}\right) }{ \sqrt{r_k^2+\left(\omega L_k-\frac{1}{\omega C}\right)^2} } - \frac{ \sin\left(\psi_1-\frac{\pi}{2}-\operatorname{arc\,tg}\frac{\omega L_k-\frac{1}{\omega C}}{r_k}\right) }{ \sqrt{r_k^2\omega^2 C^2+(\omega^2 L_k C-1)^2} } - \frac{ R\omega C\sin\left(\psi_1-\operatorname{arc\,tg}\frac{\omega^2 L_k C R-r_k-R}{r_k R\omega C+\omega L_k}\right) }{ \sqrt{(r_k R\omega C+\omega L_k)^2+(\omega^2 L_k C R-r_k-R)^2} } \right]. \tag{10} \]
In the case when the roots of the characteristic equation (3) are conjugate complex numbers,
\[ p_{1,2}=-b\pm j\omega_{\mathrm{sv}}, \]
where
\[ b=\frac{1}{2}\left(\frac{r_k}{L_k}+\frac{1}{CR}\right), \tag{11} \]
\[ \omega_{\mathrm{sv}}= \sqrt{\left(\frac{1}{L_k C}+\frac{r_k}{RCL_k}\right) -\frac{1}{4}\left(\frac{r_k}{L_k}+\frac{1}{CR}\right)^2}, \tag{12} \]
the transient process has an oscillatory character, and the equation for the voltage \(u_c(t)\) can be obtained in the form
\[ u_c(t)=u_{c\mathrm{pr}+}+A_3 e^{-bt}\sin(\omega_{\mathrm{sv}}t+\varphi). \tag{13} \]
where
\[ u_{c\,\mathrm{free}} = A_3 e^{-bt}\sin(\omega_{\mathrm{free}}t+\varphi) \tag{14} \]
is the free component of the voltage;
\(A_3\) and \(\varphi\) are constants of integration determined from the initial conditions,
\[ \varphi=\arc\operatorname{tg}\frac{b_1\omega_{\mathrm{free}}}{\omega b_2+bb_1}. \tag{15} \]
\[ A_3=\frac{b_1}{\sin\varphi}. \tag{16} \]
The constants of integration and the forced component of the voltage \(u_{c\,\mathrm{forced}+}\) depend not only on the parameters of the equivalent circuit, but also on the phase of the voltage \(u_1\) at the instant of switching—on \(\psi_1\). Figure 3 shows the curve of the voltage \(u_c(t)\) for a certain combination of parameters. Here the free component of the voltage is oscillatory in character; the frequency of the free oscillations \(\omega_{\mathrm{free}}\) is close to the line frequency \(\omega\).
Fig. 3. Forced, free, and transient voltage across the load when it is switched into the circuit of Fig. 2, supplied by the voltage \(u_1\)
After commutation, the forced component of the voltage \(u_{c\,\mathrm{forced}+}\) increases in absolute value, while the free component \(u_{c\,\mathrm{free}}\) decreases. The resultant voltage remains constant in magnitude for a certain time.
From the foregoing one may draw the following conclusion: with the proper ratio of the parameters of the transformer and the load, switched on for a small part of the period of an alternating sinusoidal voltage, it is possible to obtain a constant voltage across the load. As a result of the investigation carried out by the authors, limiting and optimal ratios of the parameters of the transformer and the load have been obtained, making it possible to obtain a pulse of constant voltage of maximum duration.
REFERENCES
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D. A. Noskov, A. T. Sokolov, “Application of alternating voltage for supplying accelerator tubes,” Electron Accelerators, Proceedings of the Fifth Interuniversity Conference, Tomsk, March 17–21, 1964, Atomizdat, Moscow, 1966.
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G. V. Zeveke, P. A. Ionkin, A. V. Netushil, S. V. Strakhov, Fundamentals of Circuit Theory, Gosenergoizdat, 1963.