On the Development of a Computing Device for Regulating the Injection Phase of a Betatron
Abstract Generated abstract
This paper examines the design requirements for a computing device to regulate the injection phase of a betatron in order to improve bremsstrahlung intensity stability. Starting from the injection condition, it analyzes how errors in injection voltage, magnetic field, and supply frequency affect the permissible timing error, and shows that the problem can be reduced to measuring injection voltage and the voltage induced in a magnetic-field sensing winding. The authors argue that analog solution methods are insufficiently accurate for the required phase precision and propose a digital computing device comprising input conversion, arithmetic, output timing, and control units, with detailed variants and experimental results deferred to later work.
Full Text
On the Development of a Computing Device for Regulating the Injection Phase of a Betatron
V. M. Razin, V. L. Ryabukhin
(Presented by the scientific seminar of the Department of Mathematical and Computing Instruments and Devices)
The most important parameters of a betatron are the magnitude of the bremsstrahlung intensity and its stability. Radiation stability is determined by a number of factors [1], which cannot be taken into account when the operating mode of the betatron is regulated manually. One of the main causes affecting radiation stability is the exact observance of the injection condition, which is defined as
\[ t_i=\frac{1}{\omega}\arcsin\left[\frac{10^4\sqrt{U_i(U_i+2U_0)}}{3r_0H_m}\right], \tag{1} \]
where
\(t_i\) is the time in seconds;
\(\omega\) is the angular frequency of the current feeding the electromagnet;
\(U_i\) is the injection voltage in megavolts;
\(H_m\) is the amplitude of the magnetic-field strength on the equilibrium orbit, in oersteds;
\(r_0\) is the radius of the equilibrium orbit in cm;
\(U_0 = 0.511\) megavolt.
Expression (1) shows that, depending on the design parameter of the betatron \(r_0\), at the moment of injection it is necessary to coordinate the values of four quantities: the injection instant, the voltage on the injector, the magnitude of the magnetic field, and the frequency of variation of the betatron magnetic field. Most devices currently in existence for setting the injection instant are based on the principle of manual or automatic variation of the injection phase as a function of the output parameter, i.e., of the magnitude of the radiation [3]. However, these devices have low accuracy and considerable inertia in tracking a specified or maximum radiation level. Indeed, with abrupt changes in \(U_i\), \(H_m\), or \(\omega\), determined by jumps in the voltage and frequency of the supply network feeding the betatron electromagnet, these devices pass from the mode of tracking the specified level into the mode of searching for it, usually carried out by finding the injection phase optimal for the given conditions. The duration of the search mode depends on the response speed of the automatic-control system, the magnitude and rate of change of the supply-network voltage, and takes a time ranging from several periods of the supply network to several seconds, which in a number of cases of betatron operation proves inadmissible. The accuracy and speed of the tracking system can be increased by means of a controlling computing device that determines, from the current values of \(U_i\),
$H_m$ and $\omega$ of the injection phase according to expression (1) and generating, at the required instant, a triggering pulse for the injection circuit. Such a computing device can regulate the operating mode of the betatron both independently and in combination with a tracking device, performing disturbance-based regulation.
Let us consider the requirements imposed on a computing device operating as an autonomous compensating regulator of a betatron.
For the series of betatrons manufactured at the Tomsk Polytechnic Institute, the injection phase may vary from 15 to 50 $\mu\text{s}$ at an injection voltage of about 40 kV. The permissible error of the injection phase, determined on the basis of experimental studies for radiation oscillations of 5%, is measured by a value of the order of $0.1\ \mu\text{s}$. Consequently, for $t_i = 50\ \mu\text{s}$ the error in determining $t_i$ must be no more than 0.2%.
On the basis of the obtained error $\delta t$, we shall find the permissible errors in the measurement of $U_i$, $H_m$, and $\omega$.
Taking into account that for most betatrons injection is carried out at the moment $\omega t_i \leqslant 1^\circ$, it is possible, with a high degree of accuracy (up to 0.025%), to adopt the approximation
$$ \sin \omega t_i \simeq \omega t_i . $$
Assuming that for the given betatron $r_0 = \text{const}$, we write expression (1) in the form
$$ t_i = \frac{k\sqrt{U_i(U_i+2U_0)}}{\omega H_m}, \tag{2} $$
where
$$ k=\frac{10^4}{3r_0}. $$
Differentiating (2) and dividing by the original equality, we obtain
$$ \frac{dt_i}{t_i} = \frac{(U_i+U_0)\,dU_i}{U_i(U_i+2U_0)} - \frac{d\omega}{\omega} - \frac{dH_m}{H_m}. $$
or, in finite increments,
$$ \frac{\Delta t_i}{t_i} = \frac{\Delta U_i}{U_i}\cdot \frac{1+\dfrac{U_i}{U_0}}{2+\dfrac{U_i}{U_0}} + \frac{\Delta \omega}{\omega} + \frac{\Delta H_m}{H_m}. $$
For
$$ U_i \ll U_0 $$
$$ \frac{1+\dfrac{U_i}{U_0}}{2+\dfrac{U_i}{U_0}} \simeq \frac{1}{2}, $$
For
$$ U_i = U_0 $$
$$ \frac{1+\dfrac{U_i}{U_0}}{2+\dfrac{U_i}{U_0}} = \frac{2}{3}. $$
Thus,
\[ \frac{1}{2} \leq \frac{1+\dfrac{U_i}{U_0}}{2+\dfrac{U_i}{U_0}} \leq \frac{2}{3}. \]
Then, for \(\delta t_i \leq 0.2\%\), the errors of the components will be determined by the inequalities
\[ \frac{\Delta \omega}{\omega} \leq 0.05\%,\quad \frac{\Delta U_i}{U_i} \leq 0.1\% \quad \text{and} \quad \frac{\Delta H_m}{H_m} \leq 0.1\%. \]
Thus, the computing device solving expression (2) must measure the instantaneous values of three parameters: \(U_i\), \(H_m\), and \(\omega\). The construction of such a device is obviously inexpedient, since regulation by three parameters can be carried out with the aid of a device that measures the instantaneous values of only two quantities.
Indeed, the magnetic-field sensor is usually an additional winding on the yoke of the electromagnet, in which a voltage \(U_m = k\omega H_m\) is induced.
Therefore expression (2) is written as
\[ t_i = \frac{k\sqrt{U_i(U_i+2U_0)}}{U_m}. \tag{3} \]
It is obvious that the error in measuring \(U_i\) and \(U_m\) will be determined by the inequalities
\[ \frac{\Delta U_i}{U_i} \leq 0.2\% \quad \text{and} \quad \frac{\Delta U_m}{U_m} \leq 0.1\%. \]
The solution of equation (3) can be carried out with the aid of a computing device [2]; however, the error of the solution will exceed the permissible value. The required accuracy in determining the injection instant can be provided by a digital computing device, whose block diagram is shown in Fig. 1, where 1 is the converting device, 2 the arithmetic device, 3 the output device, and 4 the control device.
Fig. 1. Block diagram of the computing device for regulating the injection phase of a betatron: 1—converting device, 2—arithmetic device, 3—output device, 4—control device
The input device converts the input voltages \(U_i\) and \(U_m\) into digital code. The arithmetic device is intended for solving equation (3). The output device performs the conversion of the digital code of the injection phase into a time interval. The control device ensures coordination of the operation of the individual units by issuing control pulses at the appropriate instants and into the required circuits.
Depending on the algorithm adopted for solving expression (3), several variants of the construction of the computing device are possible; their consideration and the results of experimental studies will be given in subsequent papers.
REFERENCES
- V. M. Razin. Stabilization of betatron radiation. Izvestiya TPI, vol. 87, 1957.
- A. A. Naumov; V. I. Pil’skii, L. I. Yudin. USSR Author’s Certificate No. 175153.
- V. I. Gorbunov, O. I. Nedavnii. Stabilizer of the intensity level of the bremsstrahlung radiation of a betatron. Proceedings of the Fourth Interuniversity Conference on Electron Accelerators, “Vysshaya shkola” Publishing House, Moscow, 1964.