On the Stabilization of Radiation Parameters in Pulsed Betatrons
Abstract Generated abstract
This paper analyzes factors affecting the stabilization of radiation intensity and pulse timing in pulsed betatrons, with emphasis on changes in magnet circuit parameters caused by thermal operating conditions. It compares power supply systems without artificial commutation using inductive and capacitive energy input, deriving approximate expressions and sensitivity coefficients for the maximum magnet current and the interval from accelerator triggering to the radiation pulse. The results indicate that inductive energy input is more sensitive to variations in circuit parameters, especially input conditions and magnet inductance, while capacitive input is mainly affected by supply voltage, capacitance, and circuit inductance. The study concludes that conventional stabilization by regulating energy input time is unsuitable when pulse timing must also be stable, and suggests separately stabilized or parameter controlled power sources for such cases.
Full Text
On the Stabilization of Radiation Parameters of Pulsed Betatrons
B. A. BACHINSKY, Yu. A. OTRUBYANNIKOV, M. M. SHTEIN
(Presented by the Department of Industrial and Medical Electronics)
It has been established that, for betatrons, among the most promising are power-supply systems that provide unipolar current pulses under a unipolar mode of operation with respect to the voltage of the storage capacitor bank. In works [1, 2, 3], various variants of such circuits were considered and studied, as well as certain questions concerning stabilization of magnet power supply. As the principal, and often the sole, destabilizing factor, variation of the supply-network voltage was considered. However, the severe thermal operating conditions of betatrons [4] lead to a substantial change in the parameters of the magnet circuit during operation. Therefore, the question of studying the influence of the indicated parameters on the stability of radiation intensity is quite topical.
In addition, up to the present time the main attention has been paid to the stabilization of the radiation intensity itself (average value, pulse value, etc.). New areas of application of pulsed betatrons, associated with monitoring the relative position of parts of mechanisms that are in periodic motion, impose stringent requirements on the stability of the time intervals of operation and, in particular, on the stability of the intervals between the start of the accelerator and the moment of receipt of the radiation pulse.
When studying stabilization problems, one should distinguish between pulsed power-supply systems with inductive and capacitive energy input, as well as the method of current commutation, since they largely determine the quantitative influence of destabilizing factors and the principles for constructing stabilization circuits.
In the present article, the influence of destabilizing factors is evaluated for power-supply systems without artificial commutation, with energy input into the inductance and capacitance of the betatron circuit. The excitation system of the betatron electromagnet with inductive energy input is shown in Fig. 1 [5]. In the initial state, the voltage of the circuit capacitance \(C_k\) is greater than the voltage on the filter capacitance \(C_f\). With the arrival of the control pulse at the energy-input thyristor \(T_1\), the current in the electromagnet \(L_k\) begins to rise. When the magnitude of this current becomes sufficient—
sufficient to compensate the losses per pulse, the thyristors of the forward branch \(T_2\) and \(T_3\) are switched on, the capacitor \(C_k\) is discharged into the electromagnet, and the input thyristor is switched off. When the voltage on \(C_k\) reaches a certain negative value (\(U_1\)), which is determined by the volt-seconds of the commutating choke \(\mathrm{Dr}_1\), the current is transferred from the circuit of the thyristors to the circuit of the diodes.
Fig. 1
The current flowing through the recharge circuit \(L_k, D_2, C_k, D_1\) charges the capacitor bank \(C_k\) with the initial polarity. The accelerated electrons are dumped at the moment the current is transferred to the diode circuit [6]. Figure 2 shows a pulsed-power circuit with capacitive energy input [1]. Its difference from the circuit considered above is that the input thyristor, connected to the tank capacitor \(C_k\), opens at the instant when current commutation has occurred in the magnet circuit. In this case the current from the source begins to recharge the capacitance \(C_k\), and when it becomes equal to zero the energy-input thyristor closes.
Fig. 2
The requirements for stabilizing the radiation intensity, as applied to the magnetic-field formation system, reduce to ensuring a stable value of the induction at the instant of electron dumping and, consequently, a stable value of \(B_{\max}\) in the circuits under consideration.
For comparison of different power-supply systems, in a first approximation one may assume that the magnitude of the induction in the electromagnet winding is directly proportional to the current. In the analysis we adopt the following assumptions:
- The filter capacitance \(C_{\phi}\) is a voltage source.
- The reverse resistance of the valve is infinitely large.
- Losses in the energy-injection circuit are neglected.
Taking the adopted assumptions into account, the maximum value of the current for inductive and capacitive injection, respectively, is written as
\[ I_m^L = e^{ -\frac{R}{2\omega L_k}\operatorname{arctg} \frac{2}{ R\sqrt{\frac{C}{L_k}}+ 2\sqrt{1-e^{-\frac{\pi R}{\omega L_k}}} } } \cdot \frac{U_c t_v}{L_k} \sqrt{ \frac{1}{ 1-e^{-\frac{\pi R}{\omega L_k}} } +1 }, \tag{1} \]
\[ I_m^c = \left[ 2C \sqrt{ \frac{ U_c\left(U_c-U_1\right)^3 }{ 3L_pL_k\left(e^{\frac{\pi R}{2\omega L_k}}-e^{-\frac{\pi R}{2\omega L_k}}\right) } } \right]^{\frac12}, \tag{2} \]
where
\(L_k\) is the magnetizing inductance of the magnet;
\(C\) is the capacitance of the contour capacitor bank;
\(R\) is the equivalent resistance of losses in the contour;
\(t_v\) is the time of energy injection for inductive injection.
The time interval between the triggering pulse and the current maximum is determined by
\[ t^L = \frac{I_mL_k}{U_c} e^{ \frac{R}{2\omega L_k} \operatorname{arctg} \frac{1}{ \frac{R}{2}\sqrt{\frac{C}{L_k}} + \sqrt{1-e^{-\frac{\pi R}{\omega L}}} } } \cdot \sqrt{ \frac{ 1-e^{-\frac{\pi R}{\omega L}} }{ 2-e^{-\frac{\pi R}{\omega L}} } } + \frac{1}{\omega} \operatorname{arctg} \frac{1}{ \frac{R}{2}\sqrt{\frac{C}{L_k}} + \sqrt{1-e^{-\frac{\pi R}{\omega L}}} }, \tag{3} \]
\[ t^c=\frac{1}{\omega}\operatorname{arctg}\frac{2\omega L}{R}. \tag{4} \]
To investigate the influence of various destabilizing factors, whose relative variations are small, we find an expression relating the direct increment of a function to the increment of its arguments. The change in current and time when the parameters \((R,L_k,C,U_c,t_v,U_1)\) deviate can be found by expansion in a Taylor series:
\[ \frac{\Delta I_m^L}{I_m^L} = K_u^L\frac{\Delta U_c}{U_c} + K_t^L\frac{\Delta t}{t} + K_R^L\frac{\Delta L_k}{L_k} + K_C^L\frac{\Delta C_k}{C_k}. \tag{5} \]
Here the coefficients are
\[ K_u^L=K_t^L=1;\quad K_R^L=-\frac{1}{2Q}\gamma;\quad K_L^L=-\frac{1}{4Q}\gamma;\quad K_c^L=-1+\frac{1}{4Q}\gamma; \]
\[ \text{where }\ \gamma=m-n+p;\quad m=\operatorname{arctg}\frac{1}{\dfrac{1}{2Q}\sqrt{1-e^{-\pi/Q}}}; \]
\[ n= \frac{ 1+\pi\dfrac{e^{-\pi/Q}}{\sqrt{1-e^{-\pi/Q}}} }{ 2Q\left[1+\left(\dfrac{1}{2Q}+\sqrt{1-e^{-\pi/Q}}\right)^2\right] }; \quad p=\frac{\pi e^{-\pi/Q}}{\left(1-e^{-\pi/Q}\right)\left(2-e^{-\pi/Q}\right)}; \]
\[ \frac{\Delta t^L}{t^L} = K_u^L\frac{\Delta U_c}{U_c} + K_I^L\frac{\Delta I_m}{I_m} + K_R^L\frac{\Delta R}{R} + K_L^L\frac{\Delta L_k}{L_k} + K_c^L\frac{\Delta C_k}{C_k}. \tag{6} \]
Coefficients
\[ K_u^L=-K_I^L=1;\quad K_R^L=\frac{1}{2Q}\left(\gamma+\frac{1}{2Q}-2Q\frac{n}{m}\right), \]
\[ K_L^L = 1-\frac{1}{4Q}\left(\gamma-2Q+\frac{1}{Q}-4Q\frac{n}{m}\right); \quad K_c^L = \frac{1}{4Q}\left(\gamma+2Q-4Q\frac{n}{m}\right); \]
For capacitive input
\[ \frac{\Delta I_m^c}{I_m^c} = K_1^c\frac{\Delta U_c}{U_c} + K_2^c\frac{\Delta U_1}{U_1} + K_R^c\frac{\Delta R}{R} + K_L^c\frac{\Delta L_k}{L_k} + K_c^c\frac{\Delta C}{C}. \tag{7} \]
Coefficients
\[ K_1=1-K_2= \frac{4\dfrac{U_{c0}}{U_1}+1}{4\left(\dfrac{U_{c0}}{U_1}+1\right)}; \quad K_R^c=-\frac{\pi}{8Q}\beta; \]
\[ K_L^c=\frac{\pi}{8Q}\beta-\frac{1}{4}; \quad K_c^c=\frac{1}{2}-\frac{\pi}{16Q}\beta, \]
where
\[ \beta=\frac{e^{\pi/Q}+1}{e^{\pi/Q}-1}; \]
\[ \frac{\Delta t_c}{t_c} = K_R^c\frac{\Delta R}{R} + K_L^c\frac{\Delta L}{L} + K_c^c\frac{\Delta C}{C}. \tag{8} \]
Coefficients
\[ K_R^c=\frac{1}{4Q^2}-2Qd;\quad K_L^c=\frac{1}{2}+4Q^2d;\quad K_c^c=\frac{1}{2}-Qd; \]
where
\[ d=\frac{1}{(1+4Q^2)\operatorname{arctg}2Q}. \]
The expressions obtained make it possible to determine the possible influence of each of the destabilizing factors, and also to calculate the required control laws for the selected stabilization method. Figures 3 and 4 show the dependences of the coefficients characterizing the effect of changes in the circuit parameters on the quality factor \(Q\). Analyzing the results obtained for inductive and capacitive energy input, one may note the following:
- The stability of the time intervals of the radiation pulses, as well as the stability of the current maximum when a pulsed-power system with energy input into the magnet inductance is used, depends most strongly on changes in the parameters of the input circuit \((U_c,t_3)\), as well as on changes in the inductance of the electromagnet; moreover, the stability of the time interval depends to the greatest extent on \(L_k\).
Fig. 3
Fig. 4
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When energy is input into the circuit capacitance, the main influence on the stability of the current pulses is exerted by the supply-line voltage and the magnitude of the capacitance of the capacitor bank, while the stability of the time intervals is determined by the constancy of the inductance and capacitance of the betatron circuit.
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Inductive energy input, in stabilizing both the time intervals and the current maximum, imposes substantially more stringent requirements on the stability of the circuit parameters than capacitive input.
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Since the time of energy input into the magnet inductance is included in the interval between the start of the accelerator and the moment at which the radiation pulse is obtained, the most developed method of field stabilization by regulating this time is unacceptable for systems that impose requirements on the stability of the time intervals. Simultaneous stabilization of the indicated parameters for power-supply systems with energy input into the inductance is possible only by using a separate stabilized power source; moreover, it may prove expedient to use a stabilized dc-voltage source controlled as a function of changes in the parameters of the circuit and of the source.
References
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