CALCULATION OF THE ELEMENTS OF THE ELECTRON BIAS CIRCUIT IN A BETATRON

V. M. Razin

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

In most betatrons the bias circuit has the form shown in Fig. 1. Let us note some features in the operation of this circuit. The main purpose of the bias circuit is to dump electrons from the equilibrium orbit onto the target at the end of acceleration. The dumping of electrons is accomplished by one or another change in the distribution of magnetic fields in the inter-pole space of the betatron. This change, which leads to disruption of the acceleration process and to the dumping of the accelerated electrons onto the target, is achieved by generating current pulses in the bias winding \(L\) when the capacitance \(C\) is discharged through the switching thyratron TG. Depending on the method of biasing, the bias winding has a definite geometrical configuration and is arranged in the corresponding manner in the inter-pole space of the betatron.

Fig. 1

In practice, the following methods may be used for biasing electrons from the equilibrium orbit onto a target located outside the orbit, i.e., onto the so-called external target.

Additional increase at the end of acceleration of the magnetic flux in the circle of the equilibrium orbit

This increase leads to violation of the betatron condition \(2:1\). Due to the vortex electric field from the additional increase of magnetic flux inside the orbit, the electrons acquire additional energy, the radius of the equilibrium orbit increases, and the electrons strike the external target.

The increment of the magnetic flux \(\Delta \Phi_o\) in the orbit circle, necessary for increasing the orbit radius from \(r_o\) to \(r_1\) at the place where the target is installed, is calculated by the formula [1]

\[ \Delta \Phi_o = 2\pi H_o r_o^n \, \frac{2(1-n)}{2-n} \left( r_1^{\,2-n} - r_o^{\,2-n} \right), \]

where \(H_o\) is the magnetic-field strength on the orbit at the moment of dumping, and \(n\) is the exponent of the decrease of the magnetic field with radius. Taking into account that \(r_1 = r_o + \Delta r\), \(\Delta r \ll r_o\), and \(\Phi_o = 2\pi r_o^2 H_o\), it is easy to obtain from the preceding expression the simplified calculation formula:

\[ \frac{\Delta \Phi_o}{\Phi_o} \simeq (1-n)\,\frac{\Delta r}{r_o}. \]

If the turns of the displacing winding are arranged directly on the cheeks, i.e., on the central inserts, then one should use an analogous expression for determining the increment of induction \(\Delta B_o\) required to increase the radius by the amount \(\Delta r\), at the induction, at the moment of dumping, equal to \(B_o\),

\[ \frac{\Delta B_o}{B_o} \simeq (1-n)\,\frac{\Delta r}{r_o}. \]

Reduction of the magnetic field on the equilibrium orbit and in its vicinity [2]

In this case the energy of the electrons remains practically constant, and the increase in the orbit radius occurs at the expense of weakening the field in the region of the equilibrium orbit.

The magnitude of the field change \(\Delta H_o\) required to increase the radius by \(\Delta r = r_1 - r_o\) is determined by the expression:

\[ \Delta H_o = H_o \left[ \frac{r_o}{r_1} - \left(\frac{r_o}{r_1}\right)^n \right] \]

or, approximately, for \(\Delta r \ll r_o\),

\[ \frac{\Delta H_o}{H_o} \simeq -(1-n)\,\frac{\Delta r}{r_o}. \]

The minus sign indicates the circumstance that, when the orbit radius is increased, the field in the region of the orbit decreases.

Azimuthal disturbance of the field on the equilibrium orbit [3]

With this method the equilibrium orbit is distorted, deformed, i.e., it acquires a shape different from a circle. At some azimuth the electron beam approaches the target and is decelerated in it. The azimuthal size \(\varphi_1\) of the displacing winding is determined from the condition of ensuring the minimum reactive energy required to displace the orbit by the amount \(\Delta r\)

\[ \varphi_1 \sqrt{1-n} = 133^\circ 34' = 2.331 \text{ radians}. \]

It should be noted that the optimal value of the parameter \(\varphi_1\) established experimentally may differ significantly from the calculated one. Thus, for example,

for the betatrons available to us, on the basis of experimental data, the optimum value was found to be \(\varphi_1 \simeq 120^\circ\), whereas calculation gave \(\varphi_1 = 267^\circ\).

The relative change in the magnetic field on the equilibrium orbit in the region of the displacing winding, required for displacing the orbit by an amount \(\Delta r\), is determined from the condition

\[ \frac{\Delta H_0}{H_0} = (1-n)\,\frac{\Delta r}{r_0}\, \frac{\sin\!\left(\pi\sqrt{1-n}\right)} {\sin\!\left(\dfrac{\varphi_1\sqrt{1-n}}{2}\right)} . \]

Increase of the magnetic field on one half of the orbit and decrease of the field on the other half \([4]\)

The physical aspect of the process is analogous to the preceding method. In this case the relative change of the field on the orbit is positive on one side of the pole and negative on the other; the value required for displacing the orbit by \(\Delta r\) is calculated from the formula

\[ \frac{\Delta H_0}{H_0} \simeq -\frac{\pi n}{4}\,\frac{\Delta r}{r_0} \]

or, more accurately,

\[ \frac{\Delta H_0}{H_0} = (1-n)\,\frac{\Delta r}{r_0}\, \frac{1}{\sec\!\left(\dfrac{\pi\sqrt{1-n}}{2}\right)-1} . \]

On the basis of theoretical calculations it has been established \([1,5]\) that in most practical cases the more advantageous method, from the standpoint of ensuring a minimum of reactive energy, is that of azimuthal perturbation of the magnetic field on the equilibrium orbit. It should be noted that in practice combinations of the above-considered methods of displacement may also be used.

The choice of the type of switching device in the discharge circuit of the displacement circuit of Fig. 1 is made on the basis of the following considerations.

On the basis of the law of conservation of energy, if the active losses in the discharge circuit are not taken into account,

\[ \frac{CU_m^2}{2} = \frac{10^{-8}}{0.8\pi} \int_V \frac{B_m^2\,dV}{\mu} = A_m, \]

where
\(C\) — storage capacitance in farads,
\(U_m\) — voltage on the capacitance in volts,
\(B_m\) — amplitude of induction in gausses \((B_m=\Delta B_0)\),
\(V\) — volume in \(\text{cm}^3\),
\(\mu\) — relative magnetic permeability; for air \(\mu=1\),
\(A_m\) — energy of the magnetic field in joules.

In an approximate calculation one may neglect the active losses in the discharge circuit and write

\[ I_m = \omega_0 C U_m = 2\pi f_0 C U_m, \]

where
\(I_m\) — current amplitude in the discharge circuit,
\(\omega_0 = 2\pi f_0\) — angular frequency of the natural oscillations of this circuit.

Eliminating the capacitance \(C\) from the equations written above, we obtain

\[ U_m I_m = \frac{10^{-7}}{2}\, f_0 \int_V \frac{B_m^2}{\mu}\, dV = 4\pi f_0 A_M . \]

The expression obtained makes it very simple to justify the choice of the switching device in the discharge circuit, proceeding directly from the parameters of the pulsed magnetic field. If, for the switching device, the permissible anode voltage \(U_{a\max}\), the permissible current amplitude \(I_{a\max}\), and the permissible pulse duration \(\tau_u\) are known, then, obviously, the choice should be made from the conditions:

\[ U_m \ll U_{a\max}, \qquad I_m \ll I_{a\max}, \]

\[ t_u = \frac{T_0}{2} = \frac{1}{2 f_0} \ll \tau_u . \]

The calculation of the ampere-turns of the biasing winding is carried out by ordinary methods.

The expression we have obtained for \(U_m I_m\) should be recommended for the corresponding calculations also for any other pulsed magnetic fields with a sinusoidal law of variation in time.

If there is the possibility of a comparatively simple calculation of the active losses \(A_n\) in the discharge circuit during the time in which the magnetic-field induction changes from zero to maximum, then for practical calculations one may use the more exact expression (with allowance for losses):

\[ U_m I_m = f_0 \left( \frac{10^{-7}}{2} \int_V \frac{B_m^2}{\mu}\, dV + 4\pi A_n \right) = 4\pi f_0 (A_M + A_n), \]

where \(A_n\) is the energy of losses in joules.

Thus, in our calculation the amplitude and duration of the current pulse in the biasing windings are determined, ensuring the complete dumping of the accelerated electrons onto the target for one or another method of biasing. In practical cases, for betatrons of 15 MeV, the amplitude of the current pulses reaches values of the order of hundreds of amperes, and the duration is of the order of tens of \(\mu\)sec. The switching thyratron is selected according to these parameters, taking into account the magnitude of the voltage to which the storage capacitance must be charged before the beginning of the pulse.

The calculation of the biasing winding can be carried out by a somewhat different method [2]. The value of the capacitance \(C\) and the voltage across it may be determined approximately, without allowance for active losses in the discharge circuit, on the basis of the law of conservation of energy from the relation:

\[ \frac{C U_{cm}^2}{2} = \frac{L_0 I_m^2}{2}, \tag{1} \]

where
\[ \begin{aligned} C &\text{— capacitance of the capacitor,}\\ U_{cm} &\text{— maximum voltage across the capacitance,}\\ I_m &\text{— amplitude of the current pulse, determined from the calculation of the biasing winding,}\\ L_0 &\text{— total inductance of the entire discharge circuit, i.e., taking into account all leakage fields.} \end{aligned} \]

It should be noted that calculating \(L_0\) is very difficult, since the inductance of the leakage fields and of the connecting leads may in some cas-

may exceed the inductance of the bias winding itself [1]. For this reason, it is advisable to recommend a more careful accounting of the inductances of all elements of the discharge circuit, or else to resort to direct measurement of the circuit inductance on a model. If the quantities \(L_o\) and \(I_m\) are specified, then, as is evident from relation (1), there remains some freedom in the choice of the parameters \(C\) and \(U_{cm}\).

As was already indicated above, the amplitude of the current pulse in the discharge circuit may reach a considerable value. For switching such a pulse in the bias circuit, the use of special pulsed thyratrons of the TGI-200, TGI 1—400/16 type, or ignitrons is required.

The permissible pulse duration for pulsed thyratrons is established by the manufacturer’s data. Consequently, this condition imposes a restriction on the choice of the capacitance value

\[ C \leq \frac{\tau_u^2}{\pi^2 L_o}, \tag{2} \]

where \(\tau_u\) is the permissible pulse duration according to the manufacturer’s data for the thyratron used.

If simultaneous fulfillment of conditions (1) and (2) proves impossible for a thyratron, then an ignitron must be used in the circuit; in that case the pulse duration will no longer be a limiting factor.

Here we do not consider the calculation of the control circuit in the grid circuit of the switching thyratron, since this question is sufficiently covered in the literature [6, 7].

Let us note the distinguishing features of the operation of the charging circuit of the bias circuit.

1. The voltage on the capacitance after the end of the pulse in the discharge circuit, owing to the comparatively high \(Q\) of the circuit, will differ from zero. The magnitude of this negative voltage on the capacitance after the pulse can be determined by ordinary methods after calculating the damping coefficient in the discharge circuit. Oscillographic measurements in practical circuits show that the negative voltage on the capacitance after the pulse amounts, on average, to about 60–80% of the initial positive voltage before the beginning of the pulse.

2. The phase of generation of the current pulse relative to the supply voltage may be varied within 90 electrical degrees, if there is a need to vary the maximum energy of the accelerated electrons from zero to the maximum. Here, as in the case of considering the charging circuit of the injection circuit [8], we assume that the frequency of pulse generation is equal to the frequency of the supply voltage. The need to vary the phase relationships considerably complicates the proper phasing of the generation of current pulses and the supply voltage. However, in the presence of a three-phase voltage network, such a region of the pulse-generation phase can be selected in the corresponding phase of the network that, in this region, the voltage on the anode of the charging gasotron will be negative and significant in magnitude. How this is accomplished in practice is explained in Fig. 2. With such phasing, the mutual influence of the charging and discharging circuits of the bias circuit can be completely eliminated. This leads to increased stability of operation of the circuit and also considerably facilitates calculation of the processes in the charging circuit.

The transient process when a circuit consisting of a series-connected resistance \(R\) and capacitance \(C\) is switched on to a sinusoidal voltage
\(u=U'_m\sin(\omega t+\psi)\) is described by the equations [9]:

\[ u_c=-\frac{I'_m}{\omega C}\cos(\omega t+\psi-\varphi)+Ae^{-\frac{t}{\tau}}, \tag{3} \]

\[ i=I'_m\sin(\omega t+\psi-\varphi)-\frac{CA}{\tau}e^{-\frac{t}{\tau}}, \tag{4} \]

where \(u_c\) is the voltage across the capacitance,
\(i\) is the charging current,
\(A\) is the integration constant, determined from the initial conditions.

Fig. 2

Here the following notation has also been adopted:

\[ \left. \begin{aligned} I'_m&=\frac{U'_m}{z} =\frac{U'_m}{\sqrt{R^2+\dfrac{1}{\omega^2 C^2}}} =\frac{U'_m\omega C}{\sqrt{1+(\omega\tau)^2}},\\[6pt] \tau&=RC,\qquad \varphi=\operatorname{arc\,tg}\left(\frac{-1}{\omega\tau}\right). \end{aligned} \right\} \tag{5} \]

The angle \(\psi\) will be determined, according to Fig. 3, in the following manner. The voltage across the capacitance \(C\) before the start of the pulse in the discharge circuit will be equal to \(U_{cm}\). After the pulse, the voltage across this capacitance will be taken as equal to \(-kU_{cm}\), where the coefficient \(0<k<1\). The value of the coefficient \(k\) can be determined by calculation if the quality factor \(Q\) of the discharge circuit is known:

\[ k=e^{-\frac{\pi}{2Q}}. \]

Ignition of the charging gasotron occurs when the voltage at the anode exceeds the voltage at its cathode by the value of the ignition potential \(U_z\); at that instant the cathode voltage is \(U_k=-kU_{cm}\). Consequently, the angle \(\psi\), indicating the moment at which the voltage is switched on,

\(u = U'_m \sin(\omega t + \psi)\) in a circuit consisting of series-connected resistance \(R\) and capacitance \(C\), is determined as follows:

\[ \psi = \arc \sin \frac{U_b - k U_{cm}}{U'_m}. \tag{6} \]

The constant of integration \(A\) in equations (3) and (4) is determined from the initial condition: at the moment \(t = 0\) the voltage across the capacitance is equal to

\[ u_c(0) = - k U_{cm}. \]

From equation (3), under this condition we obtain:

\[ A = \frac{I'_m}{\omega C}\cos(\psi - \varphi) - k U_{cm}. \tag{7} \]

Fig. 3

The maximum voltage across the capacitance \(U_{cm}\) is found from equation (3). For this it is necessary to differentiate the right-hand side of equation (3), set the result of the differentiation equal to zero, and solve the resulting equation with respect to \(\omega t\).

Consequently, it is necessary to solve the equation

\[ \sin(\omega t_x + \psi - \varphi) - \frac{CA}{I'_m \tau} e^{-\frac{\omega t}{\omega \tau}} = 0. \tag{8} \]

The solution of this equation is considerably simplified if we take into account the practically possible values of the quantities entering this equation. In most practical cases, for the bias circuit the following relations are valid: \(R < 10\ \text{ohms}\); \(C < 20\ \mu\text{F}\). Consequently, \(\tau < 2 \cdot 10^{-5}\ \text{sec}\) and \(\omega \tau < 0.6\) for \(\omega = 314\). If we take into account that the expected value of the root \(\omega t_x = \theta\) will lie within the limits \(\frac{\pi}{2} < \theta < \pi\), then it is easy to verify by calculation the validity of the following inequality:

\[ \frac{CA}{I'_m \tau} e^{-\frac{\theta}{\omega \tau}} \ll 1. \]

It follows from this that, to determine the value of the root \(\omega t_x = \theta\), one may solve the approximate equation

\[ \sin(\theta + \psi - \varphi) \simeq 0. \]

Then, determining the positive value of the root \(\theta\), we obtain

\[ \theta = \pi - \psi + \varphi. \tag{9} \]

Taking into account the inequality \(e^{-\frac{\theta}{\omega \tau}} \ll 1\), from equation (3) we find, for \(\omega t = \theta = \pi - \psi + \varphi\), the maximum value of the voltage across the capacitance

\[ U_{cm} \simeq - \frac{I'_m}{\omega C}\cos(\theta + \psi - \varphi) = \frac{I'_m}{\omega C} = \frac{U'_m}{\sqrt{1 + (\omega \tau)^2}}. \tag{10} \]

The latter relation makes it possible to determine the required value of the voltage amplitude on the secondary winding of the charging transformer, if the quantities \(U_{cm}\) and \(\omega \tau = \omega RC\), determined from the calculation of the bias winding by formula (1), are known. The resistance \(R\) is determined from the same conditions as in the injection circuit [8]

\[ R=-\frac{U_z}{I_{a\max}} . \]

The reverse voltage \(U_{mo}\) on the charging gasotron, according to Fig. 2, is

\[ U_{mo}=U'_m+kU_{cm}. \]

Approximately, with a certain margin, one may take

\[ U_{mo}\simeq 2U_m . \tag{11} \]

The amplitude value of the anode current through the charging gasotron may approximately, also with a certain margin, be taken equal to

\[ I_{am}=I'_m . \tag{12} \]

The possibility of neglecting the transient components in determining \(U_{cm}\) and \(I_{am}\) is physically explained by the fact that the time constant \(\tau < 2\cdot 10^{-3}\ \mathrm{sec}\) is considerably smaller in magnitude than the period of the supply voltage \(T=2\cdot 10^{-2}\ \mathrm{sec}\).

The average value of the rectified current \(I_o\), required for recharging the capacitance \(C\) from the voltage \(-kU_{cm}\) to the voltage \(+U_{cm}\), can be calculated from the formula

\[ I_o\simeq (1+k)U_{cm}Cf, \tag{13} \]

where \(f\) is the frequency of the supply voltage.

The same result can be obtained, taking into account the practical values of the quantities \(R\) and \(C\), by evaluating the integral

\[ I_o=\frac{1}{2\pi}\int_0^\theta i\,d(\omega t)\simeq (1+k)U_{cm}Cf . \]

To determine the effective value of the charging current it is necessary to evaluate the integral

\[ I^2=\frac{1}{2\pi}\int_0^\theta i^2\,d(\omega t). \]

Substituting the expression for the current \(i\) from (4) and carrying out the integration, we obtain

\[ I=\frac{I'_m}{2\sqrt{\pi}} \left\{ \theta-\cos(\theta+2\psi-2\varphi)\sin\theta -\frac{4\omega CA}{I'_m\sqrt{1+(\omega\tau)^2}} \times \left[ \cos\psi-e^{-\frac{\theta}{\omega\tau}}\cos(\theta+\psi) \right] +\left(\frac{A}{I'_mR}\right)^2 \omega\tau \left(1-e^{-\frac{2\theta}{\omega\tau}}\right) \right\}^{1/2}. \tag{14} \]

Taking into account that in practical cases for the bias circuit

\[ e^{-\frac{2\theta}{\omega\tau}}<e^{-\frac{\theta}{\omega\tau}}\ll 1, \]

for calculations with sufficient accuracy one may use the simpler approximate formula

\[ I \simeq \frac{I'_m}{2\sqrt{\pi}} \left[ \theta-\cos(\theta+2\psi-2\varphi)\sin\theta +\left(\frac{4}{I'_m R}\right)^2 \omega\tau - \frac{4\omega C A}{I'_m\sqrt{1+(\omega\tau)^2}}\cdot \cos\psi \right]^{1/2}. \tag{15} \]

Substituting the value of the constant \(A\) and taking into account the other approximate relations given above, the last formula may be reduced to the form

\[ I \simeq \frac{I'_m}{2\sqrt{\pi}} \left\{ \theta-\cos(\theta+2\psi-2\varphi)\sin\theta + \frac{1}{\omega\tau} \left[ \cos(\psi-\varphi)-k \right]^2 - 4\left[ \frac{\cos(\psi-\varphi)-k}{\sqrt{1+(\omega\tau)^2}} \right]\cos\psi \right\}^{1/2}. \tag{16} \]

For a rough preliminary estimate one may recommend an even more approximate formula

\[ I \simeq \frac{I'_m}{2}\sqrt{\frac{\theta}{\pi}} = \frac{I'_m}{2}\sqrt{1+\frac{\varphi-\psi}{\pi}} . \tag{17} \]

The active, i.e. effective, value of the current \(I_1\) in the primary winding of the charging transformer is calculated from the formula

\[ I_1=n\sqrt{I^2-I_o^2}, \tag{18} \]

where \(n\) is the transformation ratio of the charging transformer. The calculated (rated) power \(P\) of the charging transformer is determined in the usual way:

\[ P=\frac{U_{m1}I_1+U'_m I}{2\sqrt{2}}\ \text{VA}. \]

Thus, the calculation presented above determines all the principal electrical parameters of the elements of the electron-bias circuit in a betatron.

References

1. Razin V. M. Candidate dissertation, TPI, 1953.
2. T. W. Dietze, T. M. Dickinson, PURE, 96, 1171—1178, 1949.
3. F. K. Goward, Proc. Phys. Soc., 61, 284, 1948.
4. J. S. Clark, J. A. Getting, J. E. Thomas, Phys. Rev., 70, 562, 1946.
5. F. K. Goward, J. Dain, Nature, 159, 639, 1947.
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7. Z. Katz, A. G. McNamara, P. A. Forsyth, R. N. H. Haslam, H. E. Johns, Canadian J. of Research, 28-A, 113, 1950.
8. Razin V. M. Calculation of the elements of the electron-injection circuit in a betatron. Izvestiya TPI, vol. 87, 1957.
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