Analysis of the Betatron Displacement Scheme
Abstract Generated abstract
The paper analyzes a betatron biasing circuit designed to produce nearly rectangular radiation pulses of increased duration for counting-method registration. The proposed circuit uses a premagnetized saturable choke in the bias-winding discharge circuit, and its behavior is treated by a matching method with a piecewise-linear approximation of the reactor magnetization curve and simplified lossless equivalent circuits. The analysis derives relations connecting the storage capacitance, bias winding, choke parameters, induced emf, and remagnetization time, which largely determines the radiation pulse duration. Experimental testing on a 6 MeV betatron produced approximately rectangular pulses lasting 400 to 600 microseconds, and the derived formulas are presented as a basis for approximate circuit design.
Full Text
ANALYSIS OF THE BETATRON BIASING SCHEME
G. I. Burov, M. M. Shtein, V. L. ChakhloV
(Submitted by the Scientific-Research Institute
of Nuclear Physics, Electronics, and Automation)
To increase the effectiveness of recording betatron radiation by the counting method, radiation pulses of rectangular shape and sufficiently long duration are required. The generation of such radiation pulses is possible if current pulses of such a shape are produced in the bias winding that, at the beginning of the interaction of the electron beam with the target, the rate of rise of the current decreases with time, then remains constant for a certain time, and at the end of the interaction process rises again (Fig. 1). To form a current pulse of the required shape, a biasing circuit has been proposed [1], in which a saturable choke with premagnetization is included in the bias-winding circuit (Fig. 2). The biasing circuit is powered from an alternating-voltage source and includes a charging resistor \(R\), a rectifier diode \(D\), a storage capacitance \(C\), a switching device—thyristor \(T\)—and the bias winding, in whose circuit a saturable choke \(D_{\mathrm{rn}}\) with premagnetization is included. The choke premagnetization circuit is powered from a direct-voltage source through the filter \(D_{\mathrm{rf}}, C_{\mathrm{f}}\).
Fig. 1. Required shape of the current pulse in the bias winding
Fig. 2. Biasing circuit
A test of this biasing circuit was carried out on a betatron with a maximum energy of 6 MeV; radiation pulses of nearly rectangular shape with a duration of \((400 \div 600)\ \mu\text{s}\) were obtained. It should be noted that known slow-biasing circuits provide radiation-pulse durations of \((100 \div 300)\ \mu\text{s}\) [3, 2]. Therefore,
it is advisable to carry out an analysis of the proposed circuit, which will make it possible to take a well-founded approach to the choice of its elements.
An exact analysis of the electromagnetic processes in the circuit (Fig. 2) encounters considerable mathematical difficulties. For the analysis we used the matching method, which assumes a piecewise-linear approximation of the magnetization curve of the saturating-reactor core. In doing so, we make a number of assumptions which, on the one hand, will not significantly affect the accuracy of the analysis and, on the other, will greatly simplify it.
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We neglect active losses in the discharge circuit.
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We assume that, during the process of remagnetization of the reactor, the emf induced in the bias winding varies with time according to a linear law, while during beam transport to the target the emf is constant, since this time must be much shorter than the duration of remagnetization of the saturating reactor; therefore the change in emf may be neglected.
Fig. 3. Approximation of the magnetization curve of the saturating reactor.
- We approximate the magnetization curve of the saturating reactor by three straight-line segments, and assume that in the saturated state the reactor corresponds to the inductance \(L_1\) (Fig. 3).
Fig. 4. Equivalent circuit of the discharge circuit before the start of reactor remagnetization
Fig. 5. Equivalent circuit of the discharge circuit during reactor remagnetization
Taking into account the assumptions made, the equivalent circuit of the discharge circuit before the start of reactor remagnetization is shown in Fig. 4, and during remagnetization in Fig. 5.
The differential equation for the circuit (Fig. 4) has the form
\[ (L_1+L_{\mathrm{cm}})C\frac{d^2U_c}{dt^2}+U_c=E_1, \tag{1} \]
whose solution, as is known, is
\[ U_c(t)=(U_0-E_1)\cos \omega_0 t+E_1, \tag{2} \]
\[ i(t)=-\frac{U_0-E_1}{\rho}\sin \omega_0 t, \tag{3} \]
where \(\omega_0=\dfrac{1}{\sqrt{(L_1+L_{\mathrm{cm}})C}}\) is the natural frequency of the circuit,
\[
\rho=\sqrt{\frac{L_1+L_{\mathrm{cm}}}{C}}
\]
is the characteristic impedance of the circuit.
At \(t=t_1\), when the beam approaches the target, the remagnetization of the choke begins, which corresponds to the current
\[ i_1(t)=-\frac{U_0-E_1}{\rho}\sin \omega_0 t_1=\frac{A}{W_{\mathrm{cm}}}=i_0 . \tag{4} \]
and to the voltage across the capacitance
\[ U_c(t_1)=U_0'=-(U_0-E_1)\cos \omega_0 t_1+E_1=K_1U_0 . \tag{5} \]
Here
\(A\) is the ampere-turns required to bring the beam to the target,
\(W_{\mathrm{cm}}\) is the number of turns of the bias winding.
From the instant \(t=t_1\), according to the assumption adopted, the entire voltage will drop across the choke, i.e.
\[ U_{\mathrm{dr}}=U_c(t)-E_2 . \]
Taking into account that
\[ U_c(t)=K_1U_0-\frac{i_0t}{C}, \]
we obtain
\[ U_{\mathrm{dr}}(t)=K_1U_0-\frac{i_0(t)}{C}-(E_1-Kt). \tag{6} \]
Since
\[ \frac{dB}{dt}=\frac{U_{\mathrm{dr}}(t)}{S_{\mathrm{dr}}W_p}, \]
where
\(S_{\mathrm{dr}}\) is the cross section of the magnetic core of the saturation choke;
\(W_p\) are the choke turns included in the bias-winding circuit,
then
\[ \int_{-B_{\mathrm{sat}}}^{+B_{\mathrm{sat}}} dB = \int_{0}^{t_{\mathrm{rem}}} \frac{U_{\mathrm{dr}}(t)}{S_{\mathrm{dr}}W_p}\,dt . \tag{7} \]
Substituting its value for \(U_{\mathrm{dr}}(t)\) and integrating, we obtain
\[ 2B_{\mathrm{sat}}S_{\mathrm{dr}}W_p = (K_1U_0-E_1)t_{\mathrm{rem}} + \left(\frac{K}{2}-\frac{i_0}{2C}\right)t_{\mathrm{rem}}^2 . \tag{8} \]
From expression (8), the remagnetization time of the saturation choke, by which the duration of the radiation pulse is mainly determined, is defined as
\[ t_{\mathrm{rem}}= \frac{ \sqrt{C^2(K_1U_0-E_1)^2+4CB_{\mathrm{sat}}W_pS_{\mathrm{dr}}} - C(K_1U_0-E_1) }{ KC-\dfrac{A}{W_{\mathrm{cm}}} }. \tag{9} \]
It follows from expression (9) that the remagnetization time of the choke increases with an increase in the number of turns of the bias winding, the value of the storage capacitance, the number of turns and the cross section of the magnetic core of the choke, and also with an increase in the saturation induction of the core material. Therefore, to increase the duration of the radiation pulse it is necessary to increase the indicated parameters.
Thus, the expressions obtained as a result of the analysis, (4, 5, 6, 8, 9), relate the parameters of the bias circuit to one another and may serve as the basis for an approximate calculation of the proposed circuit.
REFERENCES
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L. M. Anan'ev, G. I. Burov, M. M. Shtein. Registration of betatron radiation by the counting method. Proceedings of the Scientific Research Institute of Nuclear Physics, Electronics, and Automation (in press).
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S. Oines, N.J.M. 21, No. 2, 345—346, 1963.
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K. Abe, H. Kawamura, M. Sugawara. 26, No. 2, 250—252, 1964.