Formation of Voltage Pulses in a Modulator with a Combined Storage Unit

V. A. Moskalev, Yu. M. Skvortsov, B. A. Ryzhkov

For high-voltage injection it is necessary to introduce an electron beam with an increased charge density [1]. In order to obtain a certain spread in the phase of the velocities of the injected electrons, improving their capture into acceleration, it is possible to use, for powering the injector, a voltage whose pulse shape must vary within specified limits.

The proposed and implemented [2] modulator circuit combines the positive qualities of a capacitive storage unit [3], designed for a specified generator current, and an artificial line [4], which makes it possible to correct the pulse shape by connecting it in parallel with the load (Fig. 1). To obtain the necessary correspondence

Fig. 1. Generator of the injection circuit at 300 kV

between the voltage $U_{\mathrm{i}}$ and the injection current $I_{\mathrm{i}}$, in the general case determined, according to [5], by a quantity proportional to $((\beta \gamma)^3$, where $\beta$ and $\gamma$ are relativistic factors depending on the velocity and energy of the electrons, for the selected injection voltage the storage capacitance $C_0$ of the indicated modulator was calculated as

\[ C_0 = 2 \frac{I_{\mathrm{i}}}{U_{\mathrm{i}}} g^2 \tau_{\mathrm{i}}. \tag{1} \]

The pulse-voltage generator feeding the injection system of a high-current betatron at 25 MeV [6], at a rated voltage \(U_i=300\) kV, was to provide a current \(I_i=60\) A. The pulse preliminarily formed in the modulator, with amplitude \(E_0=50\) kV and duration of the order of \(\tau_i=3.5\ \mu\text{s}\), was raised to the required voltage by means of a pulse transformer with a transformation ratio \(g \ge 6\).

Fig. 2. Waveform diagrams of pulsed processes in a modulator with a combined storage unit

The process of forming voltage pulses in the modulator can be traced by considering its operation. A direct-current source charges the capacitance \(C_0=50\cdot10^3\) pF of the main storage unit, calculated according to (1), and the distributed capacitance \(C_{\mathrm{cor}}=22.4\cdot10^3\) pF of the correcting forming line (CFL) with a certain characteristic impedance \(\rho_{\mathrm{cor}}\), to the source voltage \(E_c\). During discharge, the instant of which is determined

Fig. 3. Equivalent circuit of the pulse generator studied on the MPG-9 setup

by the application of a control pulse to the switch \(\Lambda_4\), a voltage pulse from the main storage unit \(U_0\) and a voltage wave \(U_{\mathrm{cor}}\) in the CFL act in winding \(W_1\) of the PT transformer (Fig. 2). The reactive parameters of this line, \(L_{\mathrm{cor}}\) and \(C_{\mathrm{cor}}\), determine the propagation time of the incident and reflected waves along it,

\[ t = 2\sqrt{L_{\mathrm{cor}}\cdot C_{\mathrm{cor}}}, \]

from \(t_{\min}\) to \(t_{\max}\). Owing to the finite value of the inductance introduced by the connecting leads, and the appreciable inductance of the cathode circuits

in the form of filament transformers of thyratrons \(L_3\) and \(L_4\) (type TGI-2500/35), the interaction of the pulses from the main storage unit and the correcting line takes place in the primary winding of the pulse transformer. Depending on the instant at which the reflected wave of the correcting pulse \(U_{\mathrm{cor}}\), whose polarity is opposite to that of the main pulse \(U_0\), arrives, a reduction of the pulse voltage occurs on the leading edge, at the top, or on the trailing edge; this corresponds to the delivery of voltage pulses of different shapes to the load.

An analytical study of the process of voltage-pulse formation in the circuit of a modulator with a combined storage unit is associated with the solution of a system of differential equations of at least the 11th order, compiled, for example, for the simplified equivalent circuit (Fig. 3). Nevertheless, in order to clarify the influence of individual parameters of such a circuit on the shape of the generated pulse, it was necessary, in particular, to establish the qualitative dependence between the pulse shape and the value of the inductance \(L_1\) of the correcting line. For this purpose an electronic linear modeling installation of the MPT-9 type was used.

For a mathematical description of the formation process, necessary for drawing up the analog circuit of the operational amplifiers of the modeling installation, the system of differential equations was reduced to the following form:

Fig. 4. Structural diagrams for determining the coefficients of the operational amplifiers of the MPT-9 installation

\[ \begin{aligned} \frac{d^2 U_{c1}}{dt_m^2} &= -\frac{\kappa_{11}}{M_t^2} U_{c1} -\kappa_{12}\frac{d^2 U_{c2}}{dt_m^2} -\kappa_{13}\frac{d^2 U_{c0}}{dt_m^2} \\ &\quad -\frac{\kappa_{14}}{M_t}\frac{dU_{c1}}{dt_m} -\frac{\kappa_{15}}{M_t}\frac{dU_{c2}}{dt_m} -\frac{\kappa_{16}}{M_t}\frac{dU_{c0}}{dt_m}; \\[0.4em] \frac{d^2 U_{c2}}{dt_m^2} &= \frac{\kappa_{21}}{M_t^2} U_{c1} -\frac{\kappa_{22}}{M_t^2} U_{c2}; \\[0.4em] L_\mu \frac{di_5}{dt_m} &= U_{c1}-U_{c0} -L_p C_0 \frac{d^2 U_{c0}}{dt_m^2} M_t^2; \\[0.4em] \frac{d^4 U_{c0}}{dt_m^4} &= \frac{\kappa_{31}}{M_t^2}\frac{d^2 U_{c1}}{dt_m^2} +\frac{\kappa_{32}}{M_t^2}\frac{dU_{c1}}{dt_m} +\frac{\kappa_{33}}{M_t^4}U_{c1} - \end{aligned} \tag{2} \]

\[ -\frac{K_{34}}{M_t}\frac{d^3 U_{C_0}}{d t_{\mathrm{M}}^3} -\frac{K_{35}}{M_t^2}\frac{d^2 U_{C_0}}{d t_{\mathrm{M}}^2} -\frac{K_{36}}{M_t^3}\frac{d U_{C_0}}{d t_{\mathrm{M}}} -\frac{K_{37}}{M_t^4}U_{C_0}. \]

In accordance with the equivalent circuit of the generator with a combined-type storage unit, and also taking into account the applied notation for the ratios of the circuit parameters in the form of coefficients \(K_{mn}\), with allowance for machine time \(t_{\mathrm{M}}=tM_t\), where \(M_t=3\cdot10^{-6}\) is the chosen scale,

Fig. 5. Shapes of the voltage pulses \(U_a\) on the screen of the I4-M oscillograph of the MPT-9 installation

the equations (2) made it possible to construct structural diagrams (Fig. 4). To simulate the unidirectional conductivity of the switch, diode \(D\) is included in the circuit under study.

The solution obtained concerning the influence of the wave impedance of the correcting network confirmed our assumptions [7] regarding the possibility

adjustment of the pulse shape by changing the inductance of the PFL, which determines its characteristic impedance. The effectiveness of this correction depends to a considerable extent on the complex impedance of the switch circuit \(Z_k(R_k, L_k)\), corresponding to the internal differential resistance of high-power thyratrons of the TGI-2500/35 type and their cathode circuits. It is precisely because of the inequality \(L_k \gg 0\) that

Fig. 6. Pulse shapes in the actual generator of the 300 kV injection system

the interaction of the pulses occurs in the load, whose role is played by the resistance of the pulse transformer referred to the primary circuit.

Adjustment of the PFL inductance within the limits \(L_1 = (0.8—1.7)L_{1opt}\), where \(L_{1opt}\) is the conventionally accepted value of the inductance corresponding to obtaining a pulse shape with a rising top, po-

made it possible to trace the influence, on the propagation time of the wave along the CΦL up to the moment of interaction, from the start of switching. The established instants of superposition of the pulses from the main storage capacitor \(U_0\) and the correcting line \(U_{\mathrm{cor}}\) (Fig. 2), for the adopted adjustment range \(L_1\), lie in the interval

\[ t_1+\frac{t_{\mathrm{cor}}}{2}<t_1+\left(\tau_0-t_{\mathrm{cor}}\right)<t_2-\frac{t_{\mathrm{cor}}}{2}. \tag{3} \]

The results of varying \(L_{\mathrm{cor}}=L_1=\mathrm{var}\) at \(Z_k=\mathrm{const}\) and \(C_{\mathrm{cor}}=C_1+C_2=\mathrm{const}\) for the adopted generator parameters were obtained in the form of oscillograms of the voltage \(U_m'=f(L_{\mathrm{cor}})\) (Fig. 5). Moreover, the oscillograms in Fig. 5 were obtained on the modulating installation, while Fig. 6 corresponds to voltage pulses obtained under real conditions when varying the inductance \(L_{\mathrm{cor}}=(35\text{—}75)\,\mu\mathrm{H}\).

Operation of the modulator with the indicated combined storage unit showed that the stability of electron capture in a 25-MeV high-current betatron [8], recorded from the gamma-radiation intensity with a standard “Kaktus” röntgenometer, improved on average by 10%, and the reliability of operation of the injection circuit increased.

REFERENCES

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7. V. A. Moskalev, Yu. M. Skvortsov, A. M. Slutsky. Author’s Certificate No. 207979 of October 16, 1967.

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