A Simplified Injection Scheme for a Pulsed Betatron

D. A. Noskov

(Presented by the scientific seminar of the Physicotechnical Faculty)

Principle of Operation of the Scheme

The pulsed mode of operation of betatrons, as of other accelerators, is characterized by the fact that when a current pulse passes through the coils of the electromagnet, the voltage across the latter changes, as a rule, according to a cosinusoidal law.

At the initial moment, the maximum voltage is instantaneously applied to the coils. We use this instantaneous voltage surge to generate injection pulses by means of a very simple circuit.

The primary winding of a transformer, a capacitance, and a resistance connected in series are switched in parallel with the coils of the electromagnet (Fig. 1, circuit a).

The parameters of this circuit are chosen so that, at the moment of the voltage surge on the coils of the betatron electromagnet, an oscillatory transient process arises in the injection circuit; the first half-wave of this process is transformed by a pulse transformer and applied to the electrodes of the electron gun. All subsequent oscillations, owing to the large damping, have a small amplitude and do not affect the operation of the accelerator. The amplitude and duration of the pulse are determined by the circuit parameters.

Fig. 1

Since the transformer parameters cannot be changed, the resistance $R$ and the capacitance $C$ play the main role in varying the duration and amplitude of the pulse.

A change in capacitance affects the width of the pulse and, to a considerable degree, changes its amplitude. The value of the resistance determines the amplitude of the pulse and substantially affects the pulse width. An exact calculation—

to calculate the course of transient processes in the circuit is practically very difficult because of the presence of a pulse transformer in the circuit. Therefore it is more convenient to analyze the circuit by considering a simplified equivalent circuit.

A pulse transformer, as is known, has relatively large parasitic capacitance and inductance, which exert a substantial influence on the transient process in a circuit with a pulse transformer. At the initial moment of the transient process, when the inductance of the transformer represents a large resistance, the course of the transient phenomena is determined chiefly by the parasitic capacitances of the transformer windings; at subsequent instants, when current begins to pass through the inductance, the entire course of the transient process is determined by the inductance of the circuit.

On the basis of this, we shall carry out the analysis of the transient process in the injection circuit in two stages. The phenomena occurring in the circuit at the first instant of time, i.e., on the leading edge of the injection pulse, will be considered using a simplified equivalent circuit in which the transformer is replaced by a circuit consisting of a capacitance and a resistance (Fig. 1, circuit b), where \(C_2\) is the parasitic capacitance of the windings of the pulse transformer, referred to the primary winding,

\(R_2\) is the load resistance, referred to the primary winding,

\(C_1 R_1\) are the circuit parameters determining the shape and amplitude of the pulse.

The transient processes occurring on the tail of the pulse (subsequent time intervals) will be considered using the equivalent circuit shown in Fig. 1 (circuit c), where:

\(C_1 R_1\) are the parameters of the forming circuit,

\(L\) is the apparent inductance of the transformer,

\(R_2\) is the load resistance of the transformer, referred to the primary winding.

Determination of the Circuit Parameters

The quantities \(R_2, C_2\), and \(L\)—the parameters of the pulse transformer referred to the primary winding—are introduced into the circuit by the pulse transformer. In the general case, each transformer has definite values of these parameters.

The value of the resistance \(R_2\)—the load resistance referred to the primary winding—is determined by the magnitude of the load. The load of the transformer is an electron gun, whose resistance may vary over rather wide limits, depending on the filament current. To simplify the calculation, some constant resistance \(R_э\) may be taken as the equivalent resistance of the gun.

In our particular case the following parameter values were adopted:

\[ R_э = 64\,000\ \text{ohms}; \qquad C_2 = 50\,000\ \mu\mu\text{F}; \]

\[ R_2 = \frac{R_э}{n^2} = 40\ \text{ohms}; \qquad L = 500\ \mu\text{H}. \]

To calculate the value of the capacitance \(C\), which determines the pulse duration, it is necessary to know the apparent inductance of the transformer and the duration of the injection pulse. The duration of the injection pulse in the given circuit must be calculated especially carefully, since the matching of the injection phase and of the pulse amplitude with the magnetic-field strength is performed only by changing the duration and amplitude of the pulse.

For our betatron, at an injection voltage of 25 kV, the duration of the injection pulse should be \(\sim 25\ \mu\text{sec}\).

Since we use only a quarter-wave of a cosine as the injection pulse, the oscillation period for this case will be \(T=4 \times 25 = 100\ \mu\text{sec}\). Hence the capacitance \(C\) is approximately determined by the formula:

\[ C=\frac{T^2}{4\pi L}. \]

For our case the capacitance is equal to \(0.25\ \mu\text{F}\). This approximate calculation makes it possible to determine \(C\) with sufficient accuracy without knowing any other circuit parameters except the apparent inductance.

The value of \(R_1\) must be such that the transient process is oscillatory, not aperiodic; i.e., \(R_1\) must be less than the critical resistance \(R_{\text{cr}}\) for the circuit of Fig. 1c. The critical resistance is determined from the condition that the roots of the characteristic equation be equal:

\[ \frac{1}{Cp}+R_1+\frac{LpR_2}{Lp+R_2}=0. \tag{1} \]

For the given equation this condition reduces to the following:

\[ R_{\text{cr}}=\frac{L}{R_2C}\pm 2\sqrt{\frac{L}{C}}. \tag{2} \]

Substituting the numerical values of \(R_2\), \(L\), and \(C\), we find the value of \(R_{\text{cr}}\):

\[ R_{\text{cr}1}=140\ \Omega,\qquad R_{\text{cr}2}=-40. \]

In the present case the negative value has no meaning; therefore the critical resistance is equal to \(140\ \Omega\), and consequently the resistance \(R_1\) must be less than \(140\ \Omega\).

For convenience in adjusting the amplitude of the injection voltage, it is better to use, as \(R_1\), a variable resistance from 0 to 140. In the subsequent calculation of the circuit and in the analysis of phenomena on the tail and on the front of the pulse we shall take \(R_1=20\ \Omega\).

Thus, using this calculation method and knowing the parameters of the pulse transformer, one can approximately determine all the parameters of the injection circuit for any pulsed betatron.

Analysis of the transient process on the front of the injection pulse

The character of the transient process on the front of the pulse is determined, as already indicated, by the parasitic capacitances of the transformer coils. We shall assume that the inductance of the pulse transformer has no effect on the course of the transient process in this time interval.

Since the parameters of the injection circuit are tens, even hundreds, of times smaller than the parameters of the electromagnet circuit, and, consequently, the duration of the transient process is considerably shorter than the duration of the electromagnet current pulse, the transient process in the injection circuit may be regarded as the transient process in the circuit of Fig. 1 (circuit b) when it is connected to a constant voltage.

To determine the dependence of the voltage between points \(ab\) on time during the transient process, we use the expansion formula

\[ U_{ab}=\frac{U}{H(0)}+\sum \frac{U e^{p_k t}}{p_k H'(p_k)}. \tag{3} \]

For our particular case the solution has the form

\[ U_{ab}=760 e^{-1.64t}-760 e^{-0.06t}, \tag{4} \]

where the time \(t\) is in microseconds.

The graph of the dependence \(U_{ab}=f(t)\) for this transient process is presented in Fig. 2 (curve \(a\)). In this process we are interested only in the beginning of the transient regime, i.e., the rise of the voltage, since the subsequent course of the transient process will be determined by other parameters of the pulse transformer. As is evident from the graph, the voltage at the output of the circuit rises rather rapidly to its maximum value (in about \(2\ \mu\text{s}\)) and then slowly decreases.

Fig. 2

By changing the pulse shape through connecting a capacitance to the secondary winding of the transformer, the leading edge of the pulse can be made more gradual. It does not appear possible to change the leading edge of the pulse in another way, since it depends mainly on the parasitic capacitance of the pulse transformer, which cannot be changed without changing the construction of the transformer.

Analysis of the transient process at the tail of the injection pulse

The transient process determining the phenomena at the tail of the pulse is governed mainly by the apparent inductance of the pulse transformer and by the values of the capacitance and resistance in the injection circuit. We shall analyze this process using the equivalent circuit of the injection circuit shown in Fig. 1 (circuit \(v\)). The circuit parameters are:

\[ R_1=20\ \text{ohms},\qquad C=0.25\ \mu\text{F},\qquad L=500\ \mu\text{H},\qquad R_2=40\ \text{ohms}. \]

We are interested in the character of the change in voltage between points \(a\) and \(b\) (Fig. 1, circuit c).

Using the operator method, let us write the expression for the voltage between points \(a\) and \(b\)

\[ U_{ab}(p)=\frac{U}{Z(p)}Z_{ab}(p), \tag{5} \]

where

\[ Z(p)=\frac{1}{Cp}+R_1+\frac{Lp\cdot R_2}{Lp+R_2}; \qquad Z_{ab}(p)=\frac{Lp\cdot R_2}{Lp+R_2}. \]

Substituting the values of \(Z(p)\) and \(Z_{ab}(p)\) into equation (5) and carrying out some transformations, we obtain:

\[ U_{ab}(p)= \frac{ U\frac{R_2}{R_1+R_2}p^2 }{ p^2+\frac{L+CR_1R_2}{LC(R_1+R_2)}p+\frac{R_2}{LC(R_1+R_2)} } = \frac{UR_2}{R_1+R_2}\cdot\frac{A(p)}{B(p)}. \tag{6} \]

The solution of this equation as a function of time has the form:

\[ U_{ab}(t)=\frac{UR_2\omega_0}{(R_1+R_2)\omega}\cdot e^{-\delta t}\cdot\cos(\omega t+\gamma), \]

where:

\[ \delta=\frac{L+CR_1R_2}{2LC(R_1+R_2)}; \qquad \omega_0=\sqrt{\frac{R_2}{LC(R+R_2)}}; \tag{7} \]

\[ \omega=\sqrt{\omega_0-\delta^2}; \qquad \gamma=\operatorname{arctg}\frac{\delta}{\omega}. \]

Substituting numerical values into formula (7), we obtain the expression for the voltage between points \(a\), \(b\) in the following form:

\[ U_{ab}=1040\,e^{-4.67\cdot10^4t}\cdot\cos(5.62\cdot10^4t+0.68). \tag{8} \]

Using this expression, one can construct the curve of the voltage variation with time.

In Fig. 2 (curve b) the dependence of the voltage between points \(a\) and \(b\) on time is shown. As already indicated, only the first voltage peak is used for injection, i.e., one quarter of a half-wave. It is evident from the figure that the amplitudes of all the remaining oscillations are much smaller than the amplitude of the first oscillation and do not have any substantial effect on the injection process.

The shape of the voltage pulse is such that capture of electrons into acceleration is possible only on the tail of the injection pulse, since the leading edge of the pulse is very short (2–4 μsec), while the beginning of the pulse and the beginning of the required-voltage curve coincide, and matching the injection voltage to the required-voltage curve is possible only on the tail of the pulse.

The duration of the injection pulse, determined by the frequency of the damped oscillations, depends chiefly on the value of the capacitance \(C\) (Fig. 1). This is clearly seen from equation (7) for the frequency \(\omega\). With some approximation it may be assumed that the pulse duration is directly proportional to the square root of the capacitance. A more accurate dependence can be obtained from formula (7).

A change in the resistance \(R_1\) affects mainly the amplitude of the pulse. From formula (7) we see that, as the resistance is increased, the voltage decreases, and as it is decreased, it rises.

A change in \(R_1\) has little effect on the pulse duration.

Operation of the Injection Circuit

From the results of an analysis of the transient processes determining the leading and trailing edges of the injection pulse, a theoretical pulse shape was obtained; it is shown in Fig. 2 (curve \(b\)).

Figure 2 (curve \(2\)) shows an oscillogram of the actual shape of the pulse generated by the injection circuit. Comparison of the experimental and calculated pulses shows that the assumptions adopted by us in analyzing the transient processes are close to reality.

Both pulses have the same shape; the discrepancy in duration is of the order of \(5 \div 7\%\); the amplitudes also differ only insignificantly.

During operation of the betatron, the injection circuit supplied a voltage of about \(27\ \mathrm{kV}\). In this case the radiation intensity differed by \(10\text{–}15\%\) from the radiation intensity when the betatron was operated with the usual injection circuit using a TG 15/3000 thyratron, at the same filament current and injection voltage.

This difference can be explained by the fact that in the present circuit there is no precise adjustment of the injection phase.

Adjustment of the injection phase in this circuit can be replaced by changing the duration and amplitude of the pulse. By varying the capacitance \(C\) and the resistance \(R_1\), the duration and amplitude of the injection pulse can be varied over wide limits until the optimum matching of the injection instant with the magnetic-field strength (or with the curve of the required voltage) is achieved.

Fig. 3

The experimental and calculated dependences obtained by us for the duration and amplitude of the pulse on the value of the resistance are given in Fig. 3. Figure 4 shows the experimental dependences of the amplitude and width of the pulse on the value of the capacitance \(C\).

As already indicated above, the pulse shape can be changed by connecting a small capacitor to the secondary winding of the pulse transformer.

Measurements of the radiation intensity, made with various capacitances from 70 to 2500 \(\mu\mu\mathrm{F}\) connected, at the same injection voltage, showed that changing the pulse shape within these limits does not lead to any significant increase in the radiation intensity.

Thus, the calculations and the results of experiments with the simplified injection scheme show its suitability for practical use in pulsed electron accelerators.

This simplified scheme has a number of advantages over the thyratron-based scheme.

Fig. 4

1. The principal advantage is the simplicity of the circuit. Our entire circuit consists of a pulse transformer, a capacitor, and a variable resistor. It should be noted that the conventional circuit contains two gas-discharge devices—a gasotron and a thyratron—as well as filament and plate transformers.

2. The circuit is more reliable in operation than electronic circuits.

3. The circuit is not susceptible to current overloads.

A drawback of the circuit is the difficulty of synchronizing the moment of injection with the magnetic-field strength on the orbit, as well as the impossibility of operating on the leading edge of the pulse.