SIMPLIFIED SCHEMES FOR INJECTION AND DISPLACEMENT OF ELECTRONS IN A BETATRON

V. M. Razin

(Presented by the scientific seminar of the Faculty of Physics and Technology)

If the supply frequency of the charging transformer in injection and displacement circuits is equal to (or a multiple of) the frequency of the current feeding the betatron electromagnet, then it appears possible to construct the charging part of the circuits without using a valve element (Fig. 1). This achieves a considerable simplification of the circuits. The calculation of such circuits can be carried out on the basis of the following theoretical considerations.

Fig. 1

We shall assume that the repetition frequency of the pulses in the discharge part of the circuit is equal to the supply frequency of the entire circuit. Then the processes in the charging part of the circuit will proceed as follows. During each period of the charging current, the voltage on the capacitance reaches the value \(U_{cx}\) at a certain phase. At this moment, as a result of pulse generation in the discharge circuit of the circuit, the voltage on the capacitance becomes equal to \(-kU_{cx}\), where \(k\) is a coefficient characterizing the losses of the discharge circuit. Evidently \(0 < k < 1\).

Thus, during each period there occurs a transient process in the circuit consisting of a series-connected capacitance, switched under the voltage of the charging transformer

\(u = U_m \sin(\omega t + \psi)\)

at one and the same phase \(\alpha\), and a resistance serving as a current-limiting and isolating element. At the beginning of the transient process the voltage on the capacitance is equal to \(-kU_{cx}\), and at the end it is equal to \(+U_{cx}\). The transient process in the circuit of series-connected resistance \(R\) and cap—

capacitance \(C\) when, at the instant \(t=0\), the voltage \(u=U_m\sin(\omega t+\psi)\) is switched on, as is known [1], is described by the equation:

\[ U_c=-\frac{I_m}{\omega C}\cos(\omega t+\psi-\varphi)+Ae^{-\frac{t}{\tau}}, \tag{1} \]

where \(u_c\) is the voltage across the capacitance,
\(\omega\) is the angular frequency of the supply voltage,

\[ I_m=\frac{U_m}{z};\quad z=\sqrt{R^2+\frac{1}{\omega^2 C^2}};\quad \varphi=\operatorname{arctg}\left(\frac{1}{\omega C}\right);\quad \tau=RC. \]

\(A\) is the constant of integration, determined from the boundary conditions.
In our case, at the instant \(\omega t=\alpha,\ u_c=-kU_{cx}\), i.e.,

\[ -kU_{cx}=-\frac{I_m}{\omega C}\cos(\alpha+\psi-\varphi)+Ae^{-\frac{\alpha}{\omega\tau}}, \tag{2} \]

at the instant \(\omega t=\alpha+2\pi,\ u_c=U_{cx}\), i.e.,

\[ U_{cx}=-\frac{I_m}{\omega C}\cos(\alpha+\psi-\varphi)+Ae^{-\frac{\alpha+2\pi}{\omega\tau}}. \tag{3} \]

Solving (2) and (3) simultaneously, we obtain:

\[ U_{cx}=-\frac{I_m}{\omega C}\cos(\alpha+\psi-\varphi) \frac{1-e^{-\frac{2\pi}{\omega\tau}}}{1+ke^{-\frac{2\pi}{\omega\tau}}}, \tag{4} \]

\[ A=\frac{I_m}{\omega C}\cos(\alpha+\psi-\varphi) \frac{1+k}{\left(1+ke^{-\frac{2\pi}{\omega\tau}}\right)e^{-\frac{\alpha}{\omega\tau}}}. \tag{5} \]

Consequently,

\[ u_c=-\frac{I_m}{\omega C}\cos(\omega t+\psi-\varphi)+ \]

\[ +\frac{I_m}{\omega C}\cos(\alpha+\psi-\varphi) \frac{1+k}{\left(1+ke^{-\frac{2\pi}{\omega\tau}}\right)e^{-\frac{\alpha}{\omega\tau}}} e^{-\frac{t}{\tau}}. \tag{6} \]

Expression (4) makes it possible to determine the voltage across the capacitance at the instant when the pulse is generated in the discharge circuit, while expression (6) determines the character of the variation of the voltage at the anode of the switching thyratron. Let us find the character of the variation of the current in the charging circuit of the scheme:

\[ i=C\frac{du_c}{dt}=I_m\sin(\omega t+\psi-\varphi)- \]

\[ -\frac{I_m}{\omega\tau}\cos(\alpha+\psi-\varphi) \frac{1+k}{1+ke^{-\frac{2\pi}{\omega\tau}}e^{-\frac{\alpha}{\omega\tau}}} e^{-\frac{t}{\tau}}. \tag{7} \]

Average value of the current:

\[ I_0=\frac{1}{2\pi}\int\limits_{\alpha}^{\alpha+2\pi} i\,d(\omega t) =\frac{I_m}{2\pi}\cos(\alpha+\psi-\varphi)\, \frac{1+k}{1+k e^{-\frac{2\pi}{\omega\tau}}} \times \left(e^{-\frac{2\pi}{\omega\tau}}-1\right). \tag{8} \]

Effective value of the current:

\[ I=\sqrt{\frac{1}{2\pi}\int\limits_{\alpha}^{\alpha+2\pi} i^2\,d(\omega t)} \]

\[ = \frac{I_m}{\sqrt{2}} \left[ 1+ \frac{\cos^2(\alpha+\psi-\varphi)}{2\pi\omega\tau} \left( \frac{1+k}{1+k e^{-\frac{2\pi}{\omega\tau}}} \right)^2 \left(1-e^{-\frac{4\pi}{\omega\tau}}\right) \right]^{1/2}. \tag{9} \]

On the basis of the relations obtained, the calculation of the charging transformer is carried out by known methods [2]. The efficiency of the charging circuit is determined as follows. The power dissipated in the active resistance of the charging circuit is

\[ P_1=I^2R. \tag{10} \]

The useful power is

\[ P_2=\frac{\omega}{2\pi}\,\frac{C U_{cx}^{2}(1+k)^2}{2}. \tag{11} \]

The efficiency is

\[ \eta=\frac{P_2}{P_1+P_2}\leq 0.5. \tag{12} \]

The magnitude of the resistance \(R\) is determined on the basis of the following considerations. After the end of the pulse in the discharge circuit, the voltage on the capacitance will vary according to the law

\[ u_c=U_m+A_1 e^{-\frac{t}{\tau}}. \tag{13} \]

At the moment

\[ t=0; \qquad u_c=-kU_{cx}, \]

i.e.,

\[ -kU_{cx}=U_m+A_1, \]

whence

\[ A_1=-kU_{cx}-U_m. \]

Consequently,

\[ U_c=U_m\left(1-e^{-\frac{t}{\tau}}\right)-kU_{cx}e^{-\frac{t}{\tau}}. \]

or

\[ U_c=U_m\left[1-\left(1+\frac{kU_{cx}}{U_m}\right)e^{-\frac{t_x}{\tau}}\right]. \]

The instant of time when the voltage on the capacitance becomes equal to zero is determined from the relation:

\[ U_m\left[1-\left(1+\frac{kU_{cx}}{U_m}\right)e^{-\frac{t_x}{\tau}}\right]=0, \]

whence

\[ t_x=\tau\ln\left(1+\frac{kU_{cx}}{U_m}\right). \tag{14} \]

The last relation makes it possible to calculate \(\tau\), if \(t_x\) is known. To ensure normal operation of the thyratron in pulsed mode, it is necessary, with some margin, to satisfy the approximate inequality \(t_x>t_g\), where \(t_g\) is the deionization time of the thyratron. This is necessary so that, during the action of the negative voltage on the anode of the thyratron, the control action of the grid has time to be restored.

A certain calculated margin is present in relation (13), where it is assumed that after the pulse in the discharge circuit the voltage on the capacitance will tend to \(U_m\). In reality, this steady-state voltage, for certain phase relations in the circuit, will be less than \(U_m\), and, consequently, the actual value of \(t_x\) will be greater than the calculated value determined from (14).

Examples of calculation. Injection circuit

For the injection circuit, the following values of the quantities entering into the calculation equations are characteristic:

\[ C=0.1\div0.2\ \mu\mathrm{F};\quad U_{cx}\simeq3000\ \mathrm{V},\quad k\simeq0.1\div0.2. \]

Taking for an approximate calculation \(t_x\simeq10^{-4}\ \mathrm{s}\), \(U_{cx}\simeq U_m\), we find from (14):

\[ \tau=\frac{t_x}{\ln\left(\frac{1+kU_{cx}}{U_m}\right)} \simeq \frac{t_x}{\ln(1+k)} \simeq \frac{t_x}{k}=10^{-3}\ \mathrm{s}. \]

Then

\[ R=\frac{\tau}{C}=\frac{10^{-3}}{0.1\div0.2}\,10^{-6}=(5\div10)\,10^3\ \Omega. \]

The quantity \(e^{-\frac{2\pi}{\omega\tau}}\), entering into the calculation equations, for \(\omega=314\), \(\tau=10^{-3}\ \mathrm{s}\) will be equal to

\[ e^{-\frac{2\pi}{0.314}}=e^{-20}\simeq0. \]

For this reason the calculation formulas are considerably simplified. Taking into account that

\[ I_m=\frac{U_m\omega C}{\sqrt{1+\omega^2\tau^2}}, \]

We write equation (4) in the form

\[ U_{cx} \simeq -\frac{U_m}{\sqrt{1+\omega^2\tau^2}}\cos(\alpha+\psi-\varphi). \]

If the charging transformer is supplied by a voltage in phase with the voltage on the betatron electromagnet, then the equality

\[ \alpha+\psi=\frac{\pi}{2}, \]

must be satisfied, since injection of electrons into the betatron occurs at the maximum value of the voltage on the electromagnet winding.

Consequently,

\[ U_{cx}=-\frac{U_m}{\sqrt{1+\omega^2\tau^2}}\sin\varphi. \]

Taking into account that \(\varphi=\operatorname{arctg}\left(-\frac{1}{\omega\tau}\right)\), we obtain \(\sin\varphi=-\frac{1}{\sqrt{1+\omega^2\tau^2}}\) and

\[ U_{cx}=\frac{U_m}{1+\omega^2\tau^2} \]

for \(\omega=314\), \(U_m\simeq 1.1\,U_{cx}\), i.e. \(U_m\simeq 3300\ \text{V}\).

The average value of the current in the charging circuit, taking into account the relations written above, is determined from (8):

\[ I_o=\frac{I_m}{2\pi}\frac{1+k}{\sqrt{1+\omega^2\tau^2}}\simeq \frac{I_m}{2\pi}. \]

Since

\[ I_m=\frac{U_m\omega C}{\sqrt{1+\omega^2\tau^2}} =\frac{3300\cdot314\cdot0.2\cdot10^{-6}}{1.05}\simeq 0.2\ \text{A}, \]

then

\[ I_o=\frac{0.2}{2\pi}\cdot\frac{1.1}{1.05}\simeq 0.033\ \text{A}. \]

The effective current of the charging circuit from (9):

\[ I\simeq \frac{I_m}{\sqrt{2}}\sqrt{1+\frac{(1+k)^2}{2\pi\omega\tau(1+\omega^2\tau^2)}}\simeq 0.9\,I_m=0.18\ \text{A}. \]

Transformation ratio:

\[ n=\frac{U_m}{U_{m1}}=\frac{3300}{220\sqrt{2}}=10.6. \]

Effective current of the line winding:

\[ I_1=n\sqrt{I^2-I_o^2}\simeq 1.9\ \text{A}. \]

Calculated power of the charging transformer:

\[ P=\frac{U_{m1}I_1+U_m I}{2\sqrt{2}}\simeq 420\ \text{VA}. \]

Power of the losses:

\[ P_1 = I^2 R = 0.18^2 \cdot 5000 \simeq 160\ \text{W}. \]

Useful power:

\[ P_2 = \frac{\omega}{2\pi}\,\frac{C U_{c.x}^{2}(1+k)^2}{2} \simeq 55\ \text{W}. \]

Efficiency:

\[ \eta=\frac{P_2}{P_1+P_2}\simeq 0.25. \]

The efficiency can be increased if the capacitance is charged through an inductance; however, the use of such a method for the injection circuit should not be considered expedient in view of the possibility of overvoltages arising at the anode of the switching thyratron under abnormal operation of the circuit. The power consumed by the charging transformer of the injection circuit from the mains is usually small; therefore the low value of the efficiency in this case proves quite permissible.

Displacement Circuit

The charging circuit of the displacement scheme differs from the corresponding circuit of the injection scheme by a different phase relation between the charging and discharging processes, and also by different values of the circuit parameters. Thus, in most practical cases, for displacing electrons from the equilibrium orbit onto the target in betatrons of energy \(15\text{--}25\ \text{MeV}\), a capacitance discharged into the turns of the displacing winding of the order of several \(\mu\text{F}\) is required, at a voltage across it of the order of \(2\text{--}3\ \text{kV}\). For charging the capacitance, the same circuit (Fig. 1) may be used as in the injection scheme.

For the purpose of charging the capacitance in this case, it is expedient to use a voltage lagging by 60 electrical degrees behind the voltage on the electromagnet. Such a phase relation is not difficult to obtain when a three-phase supply network is available. Consequently, the voltage of the charging transformer must vary according to the law:

\[ u=U_m \sin(\omega t - 60^\circ),\quad \text{i.e. } \psi=-60^\circ. \]

Electron dumping is usually carried out at the instant corresponding to an angle of 75 electrical degrees after the instant of injection, i.e. \(\alpha=90^\circ+75^\circ=165^\circ\).

To simplify the calculations, let us shift the origin of coordinates to the point where \(\psi=0\) and \(\alpha=165^\circ-60^\circ=105^\circ\).

Using the relations given above, let us carry out an approximate calculation of the displacement circuit for the parameters

\[ C=2\ \mu\text{F};\quad U_{c.x}=2000\ \text{V};\quad k=0.6;\quad a=1.83. \]

From condition (14):

\[ \tau=\frac{t_x}{\ln\left(1+\frac{kU_{c.x}}{U_m}\right)}. \]

Still taking, for the approximate calculation,

\[ t_x=10^{-4}\ \text{s};\quad U_{c.x}\simeq U_m, \]

we obtain \(\tau\simeq 2.14\cdot 10^{-4}\ \text{s}\), whence

\[ R=\frac{\tau}{C}\simeq 100\ \Omega. \]

The parameter $\omega\tau = 0.067$; consequently, in this case too one may use an approximate expression, as in the calculation of the injection scheme.

Voltage across the capacitance:

\[ U_{cx}=-\frac{U_m}{\sqrt{1+\omega^2\tau^2}}\cos(\alpha-\varphi). \]

In our case

\[ \varphi=\operatorname{arctg}\left(\frac{-1}{\omega\tau}\right)\simeq-\frac{\pi}{2}. \]

Consequently,

\[ \cos(\alpha-\varphi)=-\sin\alpha=-\sin105^\circ=-0.97 \]

\[ U_{cx}\simeq0.97\,U_m,\quad \text{since } \sqrt{1+\omega^2\tau^2}\simeq1. \]

We obtain

\[ U_m=\frac{U_{cx}}{0.97}\simeq2060\ \text{V}. \]

In an approximate calculation one may take $U_{cx}\simeq U_m=2000\ \text{V}$.

Mean value of the current:

\[ I_o=-\frac{I_m}{2\pi}\cos(\alpha-\varphi)(1+\kappa)\simeq0.3\ \text{A}, \]

where

\[ I_m=\frac{U_m\omega C}{\sqrt{1+\omega^2\tau^2}}\simeq U_m\omega C=1.25\ \text{A}. \]

Effective current of the charging circuit:

\[ I=\frac{I_m}{\sqrt{2}}\sqrt{1+\frac{\cos^2(\alpha-\varphi)(1+\kappa^2)}{2\pi\omega\tau}}\simeq2.3\ \text{A}. \]

Transformation coefficient:

\[ n=\frac{U_m}{U_m}\simeq6.4. \]

Effective current of the grid winding:

\[ I_1\simeq nI=14.7\ \text{A}. \]

Useful power:

\[ P_2\simeq\frac{\omega}{2\pi}\frac{C U_{cx}^2(1+\kappa)^2}{2}\simeq512\ \text{W}. \]

Power losses:

\[ P_1\simeq I^2R\simeq536\ \text{W}. \]

Efficiency: $\eta\simeq0.48$.

The rated power of the transformer is

\[ P \simeq I_1 U_1 \simeq 3250\ \text{VA}. \]

For the charging circuit in the scheme using a gasotron, with the same values of the quantities \(C\), \(U_m\), \(R\), and \(k\), we obtain:

\[ P = 670\ \text{VA},\quad P_1 = 27.5\ \text{W},\quad P_2 + P_1 = 512\ \text{W},\quad \eta = 0.95. \]

Consequently, dispensing with the use of a gasotron in this case leads to a considerable increase in the dimensions of the charging transformer and to a reduction in efficiency, which should not be considered expedient.

References

1. Kalantarov P. L. and Neiman L. R. Theoretical Foundations of Electrical Engineering, Gosenergoizdat, 1951.

2. Kaganov I. L. Electronic and Ionic Converters, Part 1, Gosenergoizdat, 1950.