CALCULATION OF ELEMENTS OF THE ELECTRON INJECTION CIRCUIT IN A BETATRON

V. M. Razin

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

As the standard injection circuit in betatrons of the TPI designs, the circuit shown in Fig. 1 has been adopted. Such a circuit is used when the repetition frequency of the pulses in the discharge circuit is equal to the frequency of the current feeding the charging transformer \(T_z\).

The circuit operates as follows. The capacitors of the pulse-forming line \(FL\) are charged by the positive half-wave of the voltage from the charging transformer \(T_z\) through the rectifier \(VG\), the charging resistance \(R_z\), and the primary winding of the pulse transformer \(IT\). Then, when at the anode of the rectifier \(VG\) there is a negative half-wave of voltage with respect to the ground potential, the pulse-forming line \(FL\) discharges into the primary winding of the pulse transformer \(IT\) through the switching thyratron \(TG\) (when a control pulse is applied to its grid).

Fig. 1

At the same time, in the secondary winding of the pulse transformer \(IT\) a short high-voltage pulse arises, which is applied to the electrodes of the electron gun of the betatron or synchrotron.

With proper phasing (described above), the processes in the charging and discharging circuits occur at different times and can be considered separately. The possibility of separate consideration is also justified by the fact that the durations of the processes in the two circuits differ sharply. The variation of the voltage on the pulse-forming line, with proper phasing, has the form shown in Fig. 2.

Fig. 2. \(J_M\) — current in the electromagnet coils. \(U_M\) — voltage on the electromagnet. \(U_C\) — voltage across capacitor \(C\).

Calculation of the Elements of the Discharge Circuit

The discharge circuit includes the following elements: the pulse-forming line FL, the pulse transformer PT with resistance \(R\) for matching the wave impedance of the line with the load, and the switching thyratron TG. It should be noted that here we shall not consider the calculation of the grid control circuit of the thyratron TG, since the literature [1, 2, 3] contains sufficiently complete information on this question; however, we point out that in the calculation one important requirement for the injection circuit must be satisfied, namely: the scatter in the firing time of the thyratron must not exceed hundredths of a microsecond.

The problem of injecting electrons into the accelerating chamber of a betatron is one of the most important among the problems of induction-acceleration engineering. In the first betatrons, various methods were used for forming the voltage on the electrodes of the electron gun.

In recent accelerator installations, an air-core transformer was used for this purpose; a charged capacitor was discharged through a switching discharge device into its primary winding. At the terminals of the secondary winding of the air-core transformer, a pulse of sufficient amplitude and definite duration was obtained.

In 1948, at the Department of High-Voltage Engineering of the Tomsk Polytechnic Institute, the idea was proposed of using, for forming the voltage pulse, an artificial pulse-forming line and a pulse transformer with an iron core, similar to the way this is done in radar engineering in modulating devices. In 1949–1950, information appeared on the application of a similar circuit for feeding the electrodes of the electron gun of a betatron [4, 5].

At the department this problem was also solved. Below is given the calculation of a pulse transformer with an iron core and of an artificial pulse-forming line for forming a voltage pulse on the electrodes of an electron gun.

It should be noted that the method for calculating and designing the IT has been developed in a number of works [6–11, etc.].

In the present article we use the results of these works, on the basis of which we have developed a method for calculating the pulse transformer for the electron gun of a betatron. The following requirements were taken as the basis of the calculation:

1. The amplitude of the voltage pulse must be not less than 60 kV; the current in the pulse should be of the order of 1–3 A.

2. The pulse shape must be such that, at voltages close to the maximum, there are portions with rising voltage of small steepness.

3. The pulse duration must be the minimum possible for obtaining the specified voltage, of the order of several microseconds.

The considerably less stringent requirements on the shape of the transformed voltage pulse, as compared with the requirements in radar engineering, made it possible to design and manufacture an IT with a large transformation ratio. For fabrication of the core it proved possible to use ordinary transformer steel of grade E-42, 0.35 mm thick. As the switching device, a TGI-200 thyratron was used, with an anode voltage of 3.5 kV and an allowable current amplitude of 200 A.

The calculation of the IT with an iron core was carried out in the following order.

Initial data for the calculation

1. For calculating the IT core it is necessary to have the dependence of the magnetic permeability $\mu_{\Delta}$ on the partial magnetization cycle on the increment of induction $\Delta B_c$ under unipolar magnetization. Since the magnetic characteristic depends to a great extent on the magnitude of the air gap in the magnetic circuit, the measurements were made directly on one of the cores manufactured for the IT. The dependence obtained from the measurements is shown in Fig. 3. To reduce the size of the core, large increments of induction per pulse should be chosen. In magnetic measurements it was established that the residual induction can reach a value of 3000 G.

Fig. 3

The demagnetizing action of eddy currents leads to a nonuniform distribution of induction over the cross section of the steel sheet; as a result, at considerable increments of induction per pulse, the edge of the sheet quickly becomes saturat—

decrease. This phenomenon has a harmful effect on the operation of the core; therefore very large increments of induction cannot be accepted. In the present calculation, taking into account the presence of residual induction and the possibility of slight saturation at large increments of induction, \(\Delta B_c \leqslant 6000\) gauss has been adopted.

1. The primary voltage is determined by the permissible voltage on the anode of the TGI-200 thyratron, equal to \(3.5\ \mathrm{kV}\). When the artificial forming line is discharged into a resistance equal to the characteristic impedance of the line, the pulse voltage is equal to one half of the voltage to which the line was charged. On this basis \(U_1' = 1750\ \mathrm{V}\) has been adopted.

2. The secondary voltage is \(U_2 = 60\ \mathrm{kV}\), i.e. the transformation ratio is

\[ n \simeq \frac{U_2}{U_1} = 34 . \]

1. The current amplitude in the load is \(I_n \leqslant 3\ \mathrm{A}\). The referred value of the load current is \(I_n' = I_n n \leqslant 100\ \mathrm{A}\).

2. The duration of the transformed pulse is taken as \(t_i = 10\ \mu\mathrm{s}\).

3. The process of establishment of eddy currents has a substantial influence on the operation of the PT. The duration of establishment of the eddy currents depends on the specific resistance of the steel. The electrical resistivity of silicon steel \(\rho_{\mathrm{st}}\), depending on the percentage content of silicon \((x_{\mathrm{Si}},\%)\), is expressed rather accurately by the formula:

\[ \rho_{\mathrm{st}} = (13.25 + 11.3\,x_{\mathrm{Si}}\%) \cdot 10^{-6}\ \Omega\cdot\mathrm{cm}. \]

According to State Standard data, for E42 steel \(x_{\mathrm{Si}}\% = 4.6\%\); then \(\rho_{\mathrm{st}} = 65 \cdot 10^{-6}\ \Omega\cdot\mathrm{cm}\).

1. The thickness of the steel sheets is \(\delta_{\mathrm{st}} = 0.035\ \mathrm{cm}\).

2. When rectangular pulses are transformed, the pulse on the secondary side of the PT must not have a large decrease of the flat portion (Fig. 4) toward the end of the pulse. The decrease of the flat portion is characterized by the coefficient of decrease of the flat portion of the pulse

\[ \lambda_i = \frac{U' - U''}{U'} . \]

Fig. 4

In radar engineering, usually

\[ \lambda_i = 0.05 \div 0.1 . \]

Since, when the circuit is used in a betatron, larger distortions can be permitted, \(\lambda_i = 0.3\) has been adopted in the calculation.

All the initial data for the calculation are given below:

Calculation of the Transformer

As was already indicated above, the demagnetizing action of eddy currents at high rates of change of induction in PT cores causes a nonuniform distribution of induction over the cross section of the steel sheet. This effect is taken into account by introducing the so-called apparent permea-

…permeability \(\mu_k\), a quantity that depends on the eddy-current time constant:

\[ \Theta_{\mathrm{в}}=\frac{\mu \Delta \delta_{\mathrm{ж}}^{2}}{1000 \tau \rho_{\mathrm{ж}}}\ \mu\mathrm{sec}. \]

The relative pulse duration:

\[ \tau_u=\frac{t_u}{\Theta_{\mathrm{в}}}. \]

The apparent permeability is calculated from the formula:

\[ \mu_k=\frac{\mu_\Delta}{1+\dfrac{\tau^{2}}{3\tau_u}}. \]

The value of the apparent inductance of the primary winding depends on the magnitude \(\mu_k\); its value can be found from the conditions for distortion of the flat part of the pulse \([9, 10]\) in the following way. To examine the processes during transformation of this part of the pulse, the equivalent circuit of Fig. 5 is used, where \(\Gamma\) is a rectangular-pulse generator (an artificial shaping line) having an internal resistance \(R_1\) (equal to the characteristic impedance of the line), \(L_k\) is the apparent inductance of the primary winding of the IT (allowing for the demagnetizing action of eddy currents), and

\[ R_{\mathrm{н}}'=\frac{U_2}{I_2 n^2} \]

is the load resistance in the secondary circuit of the IT, referred to the primary winding.

Fig. 5

For proper operation of the line, the characteristic impedance of the line must be matched to the referred load resistance, i.e., the equality must be fulfilled:

\[ R_1 \simeq R_{\mathrm{н}}. \]

When a rectangular voltage pulse is applied to the input of the circuit of Fig. 5, the voltage across the load will vary according to the law

\[ u_{\mathrm{н}}=U e^{-\dfrac{R_1 R_{\mathrm{н}}'}{L_k\left(R_1+R_{\mathrm{н}}'\right)}\,t}, \]

where \(U\) is the initial voltage.

Since usually in an IT

\[ t_u \ll \frac{L_k\left(R_1+R_{\mathrm{н}}'\right)}{R_1 R_{\mathrm{н}}'}, \]

then by the end of the pulse the voltage across the load will have the value:

\[ U_h'' \simeq \left[1-\frac{t_u R_1 R_h'}{L_k(R_1+R_h')}\right]U. \]

Since at \(t=0\), \(U_h'=U\), then

\[ \lambda_u=\frac{U_h'-U_h''}{U_h'}=\frac{t_u R_1 R_h'}{L_k(R_1+R_h')}. \]

When the matching condition \(R_1=R_h'\) is satisfied,

\[ \lambda_u=\frac{t_u R_1}{2L_k}. \]

Hence

\[ L_k=\frac{t_u R_1}{2\lambda_u}. \]

The value of the apparent inductance \(L_k\) is directly related to the design parameters of the pulse transformer in the following way:

\[ L_k=\frac{4\pi w_1^2 S_{\mathrm{Fe}}\mu_k}{l_{\mathrm{Fe}}}\,10^{-3}, \]

where \(w_1\) is the number of turns of the primary winding,
\(S_{\mathrm{Fe}}\) is the cross section of the core in \(\mathrm{cm}^2\),
\(l_{\mathrm{Fe}}\) is the length of the mean line in \(\mathrm{cm}\),
\(L_k\) is in \(\mu\mathrm{H}\).

On the other hand, the increment of induction per pulse is expressed through some of these design parameters:

\[ \Delta B_c=\frac{100 k_u U_h' t_u}{w_1 S_{\mathrm{Fe}}}. \]

Here the coefficient \(k_u=1-0.5\lambda_u\) takes into account the small decrease of the flat part of the pulse caused by the action of the magnetizing current of the core. We use the available formulas as follows. Eliminating \(w_1\) from the equations given above, we obtain:

\[ L_k=4\pi\left(\frac{100 k_u U_h' t_u}{\Delta B_c S_{\mathrm{Fe}}}\right)^2 \frac{S_{\mathrm{Fe}}}{l_{\mathrm{Fe}}}\mu_k\,10^{-3}. \]

From this equation we find the volume of iron:

\[ S_{\mathrm{Fe}}l_{\mathrm{Fe}}=V_{\mathrm{Fe}}= \left(\frac{100 k_u U_h' t_u}{\Delta B_c}\right)^2 4\pi\frac{\mu_k}{L_k}\,10^{-3}. \]

Taking into account the filling factor \(k_{\mathrm{Fe}}<1\), the core volume is:

\[ V=\frac{V_{\mathrm{Fe}}}{k_{\mathrm{Fe}}}. \]

Next we choose the window dimensions for accommodating the windings, on the basis of data on the electric strength of insulating materials in transformer oil:

1) permissible voltage gradient along the winding — \(4\,\mathrm{kV/cm}\),
2) strength over the surface of varnished cloth in oil — \(20\,\mathrm{kV/cm}\),

3) winding strength—10 kV/mm,
4) transformer-oil strength—40 kV/cm.

At high voltages it is advantageous to make the winding frame cylindrical (round), and the cross section of the central limb of a shell-type core square.

Fig. 6

Knowing the window dimensions, it is not difficult to write an equation relating the volume of the core to the cross section of the central limb (Fig. 6):

\[ \frac{V}{2S} - (c + 0.207\sqrt{S} + d) = \sqrt{S}, \]

where

\[ \begin{aligned} V &\text{ is the volume of the core in } \mathrm{cm}^3,\\ S &\text{ is the cross section of the central limb in } \mathrm{cm}^2,\\ c \text{ and } d &\text{ are the window dimensions in cm, found from the conditions of electrical strength,} \end{aligned} \]

\(0.207\sqrt{S}\) is a term that takes into account the increase in the window width due to the use of a round cylindrical frame for the windings (Fig. 7).

\[ a=\sqrt{S} \]

\[ b = a\frac{\sqrt{2}-1}{2} \simeq 0.207a = 0.202\sqrt{S}. \]

Fig. 7

We write the equation given above in the following form:

\[ \frac{V}{2S} - (c+d) = 1.207\sqrt{S}. \]

Solving this cubic equation by trial, we find one real root \(S\) and determine \(S_{\mathrm{ж}} = k_{\mathrm{ж}}S\). Knowing the value \(S_{\mathrm{ж}}\), we calculate the number of turns of the primary winding:

\[ w_1 = \frac{100 k_u U_k' t_u}{S_{\mathrm{ж}}\Delta B_c} \]

and the number of turns of the secondary winding \(w_2 = n w_1\). The size of the central limb of the core is \(a=\sqrt{S}\).

The cross section of the winding conductors is determined by known methods from the conditions of a thermal calculation on the basis of the following data:
1) pulse current in the primary winding—\(I_1\), 2) pulse current in the secondary winding—\(I_n\), 3) pulse duration—\(t_u\), 4) repetition frequency—\(f\).

After determining the design dimensions of the pulse transformer, one should calculate the rise time of the transformed pulse and make sure that a pulse \(t_u = 10\ \mu\mathrm{s}\) will be transformed, i.e., that the rise time \(t_\phi < t_u\).

For analysis of the pulse front, a simplified equivalent circuit of the transformer circuit may be used (Fig. 8), where
\(L_{\mathrm{п}}\) is the leakage inductance of the primary winding,
\(R_1\) is the internal resistance of the rectangular-pulse generator (the wave impedance of the line),
\(R_2'\) is the load resistance referred to the primary winding,

\[ R_2' = R_{\mathrm{н}}' = \frac{U_2}{I_2 n^2}; \]
with proper matching \(R_1 = R_2'\).

\(C_{\mathrm{п}}'\) is the parasitic capacitance of the windings, referred to the primary winding.

Fig. 8

The leakage inductance may be calculated by formula [12]:

\[ L_{\mathrm{п}} = \frac{1.14\, p\, w_1^2}{h} \left( \Delta + \frac{\alpha_1 + \alpha_2}{3} \right)\cdot 10^{-2}\ \mu\text{H}, \]

where
\(w_1\) is the number of turns of the primary winding,
\(p\) is the perimeter of the mean turn in cm,
\(\Delta\) is the distance between the I and II windings in cm,
\(\alpha_1\) and \(\alpha_2\) are the thicknesses of the I and II windings in cm,
\(h\) is the height of the coils in cm.

The equivalent dynamic capacitance of the secondary winding, referred to its terminals [13]:

\[ C_0 = \frac{0.118\, p\, h\, \varepsilon}{\Delta_2 N}\ \text{pF}, \]

where
\(\Delta_2\) is the distance between the copper of adjacent layers of the secondary multilayer coil (up to 10 layers and more),
\(N\) is the number of layers,
\(\varepsilon\) is the average dielectric constant of the insulation between layers.

The capacitance of the high-voltage lead wire may be taken as \(C_1 = 0.6\ \text{pF}\). The capacitance of the output insulator is \(C_2 = 8 \div 10\ \text{pF}\). The total capacitance in the secondary circuit is

\[ C = C_0 + C_1 + C_2. \]

The parasitic capacitance referred to the primary winding is:

\[ C_{\mathrm{п}}' = C_2' = C n^2. \]

To determine the shape of the pulse front at the terminals of the secondary winding, we find the value of the oscillatory constant for the circuit (Fig. 8):

\[ \vartheta_k = \frac{1}{2\sqrt{1+\gamma_R}} \left( \frac{\sqrt{\dfrac{L_{\mathrm{п}}}{C_{\mathrm{п}}'}}}{R_2'} + \frac{R_1}{\sqrt{\dfrac{L_{\mathrm{п}}}{C_{\mathrm{п}}'}}} \right), \]

where

\[ \gamma_R = \frac{R_1}{R_2'} = 1 \]
(with matching).

From Fig. 58 [8] we find that the curve for \(\delta=\delta_k\) reaches its maximum at [[unclear: value]]. The duration of the front (when the voltage pulse on the secondary side changes from zero to maximum) in this case is equal to:

\[ t_{\phi}=\frac{\tau_k\sqrt{L_n C_n}}{\sqrt{1+\gamma_R}} . \]

It should be expected that on the secondary winding the voltage pulse will have the form shown in Fig. 4 (when a strictly rectangular pulse is applied to the primary winding). An oscillogram of the pulse is presented in Fig. 9.

Fig. 9

For a complete calculation of the pulse transformer it is necessary to estimate the magnitude of the losses in the transformer and to consider the nature of the transient process at the cut-off (tail) of the pulse.

The calculation of the losses showed that in this design the main share of the losses is due to eddy-current losses (this is understandable, since ordinary transformer steel E42 of thickness \(0.035\ \text{cm}\) is used, whereas for these purposes special alloys with rolled thickness down to \(0.005\ \text{cm}\) and less are used), and the total magnitude of the losses does not exceed \(15\%\) of the transformed power.

For a large pulse duty factor

\[ Q=\frac{T}{t_u}=\frac{0.02}{10^{-5}}=2000, \]

which is required for operation of the betatron at a frequency \(f=50\ \text{Hz}\), the power dissipated in the PT is very small and cannot lead to overheating of the transformer; therefore the details of the loss calculation are not given here.

The nature of the transient process at the pulse cut-off is not of any substantial importance for the operation of the betatron and therefore was not subjected to special analysis by calculation (but was checked by oscillograms).

Investigations of the PT showed that all calculated quantities that can be measured agree well with experiments.

We calculate the artificial forming line, using the design formulas given in [8]. The total inductance of the line is:

\[ L_0=\frac{R_1 t_u}{2}. \]

The total capacitance of the line is:

\[ C_0=\frac{t_u}{2R_1}. \]

We take the number of sections:

\[ m\simeq 8\text{--}10. \]

The inductance of one section is:

\[ L=\frac{L_0}{m}. \]

The capacitance of one section is:

\[ C=\frac{C_0}{m}. \]

Calculation of the Elements of the Charging Circuit

From the calculation of the pulse transformer and the forming line, the value of the total capacitance of the line and the maximum voltage to which the line must be charged become known. Knowledge of these quantities makes it possible to calculate the charging circuit of the scheme comparatively simply.

When considering the processes in the charging circuit, we may neglect the influence of the line inductances and the inductance of the primary winding of the pulse transformer (Fig. 1), since these quantities have no substantial effect on the nature of the transient process during charging of the line. For this reason we may represent the total capacitance of the line as a single lumped capacitance. The charging circuit, simplified in this way, will have the form shown in Fig. 10. In this circuit the value of the capacitance \(C\) is known, as is the voltage to which this capacitance must be charged during each period of the alternating current. To simplify the calculation, let us assume that after each discharge of the capacitor \(C\) the voltage on it will be zero. This assumption is close to reality,

Fig. 10

since, with proper matching of the line to the load, the voltage on it falls to zero by the end of the pulse, and only a small oscillatory process on the tail of the pulse can cause the appearance of a small negative voltage on the capacitance.

On the basis of the assumptions made above, we may consider that the series-connected charging resistance \(R\) and capacitance \(C\) (Fig. 10) are switched on at the moment \(t=0\) to the voltage \(u = U_m \sin(\omega t + \psi)\), where \(U_m\) is the amplitude voltage of the secondary winding of the charging transformer, \(\omega\) is the angular frequency of the alternating current, and \(\psi\) is the angle determined by the presence of a definite value of the ignition potential of the gas-filled rectifier VG,

\[ \psi = \arcsin \frac{U_z}{U_m}. \]

We take the initial voltage on the capacitor \(C\) to be zero. It is known that the ignition potential \(U_z\) of a gas-filled rectifier is of the order of \(20 \div 50\ \mathrm{V}\), whereas the amplitude voltage of the secondary winding of the charging transformer may reach values of the order of \(500 \div 3000\ \mathrm{V}\). For this reason, in practice the values of \(\psi\) will be \(\psi \simeq 0.01 \div 0.1\) radian \(\simeq 0.5 \div 5\) electrical degrees. To simplify the calculations one may take \(\psi \simeq 0\). The transient conditions in this case [14] are described by the equations:

\[ u_c = -\frac{I_m}{\omega C}\cos(\omega t - \varphi) + \frac{I_m}{\omega C}\cos\varphi\, e^{-t/\tau}, \tag{1} \]

\[ i = I_m \sin(\omega t - \varphi) - \frac{I_m}{\omega \tau}\cos\varphi\, e^{-t/\tau}. \tag{2} \]

Here the notation of [14] is adopted:

\(U_c\)—the voltage across the capacitance \(C\),

\(i\)—the charging current,

\[ \left. \begin{aligned} I_m &= \frac{U_m}{z}, \qquad z=\sqrt{R^2+\frac{1}{\omega^2 C^2}},\\ \tau &= RC, \qquad \varphi=\operatorname{arc\,tg}\left(-\frac{1}{\omega\tau}\right). \end{aligned} \right\} \tag{3} \]

Since \(i=C\dfrac{du_c}{dt}\), it is evident that the voltage across the capacitance \(C\) reaches a maximum at the instant when the charging current becomes zero. Consequently, it is necessary to solve equation (2) with respect to \(\omega t\) for \(i=0\):

\[ \sin(\omega t-\varphi)-\frac{1}{\omega\tau}\cos\varphi\, e^{-\frac{\omega t}{\omega\tau}}=0. \tag{4} \]

Before solving this equation, let us establish the possible practical values of the quantities entering into it. The value of the resistance \(R\) can be determined from the following considerations. As is known, a thyratron begins to conduct current in the forward direction when the voltage on its electrodes exceeds the firing potential. Consequently, at the moment of firing a current \(i_m=\dfrac{U_z}{R}\) will pass through the thyratron. The magnitude of this current must not exceed the permissible peak value of the anode current for the given type of thyratron. Thus, for the thyratron VG-1/8500, \(I_{ma}=1\) A, and, taking \(U_z\simeq 50\) V, we obtain \(R=\dfrac{U_z}{I_{ma}}=50\ \Omega\). For other low-power high-voltage thyratrons used in the charging circuit of the injection scheme, the value of \(R\) in any case does not exceed several hundred ohms.

The value of the capacitance \(C\) is determined in calculating the pulse-forming line. If one calculates for the limiting operating modes of the thyratrons TG 15/3000, TGI-200, and TGI-1 400/16 used in the injection circuit, and assumes that the pulse duration in all cases does not exceed \(10\ \mu\text{s}\), then the possible values of the capacitance \(C\) will have the magnitude presented in the table.

Thus, in the majority of practical cases \(R\leq 200\ \Omega\), \(C\leq 0.6\ \mu\text{F}\). Consequently, for expressions (3) we may write the relations:

\[ \left. \begin{aligned} \tau &\ll 180\cdot 10^{-6}\ \text{s}; \qquad \omega\tau \ll 0.057; \qquad \frac{1}{\omega\tau}\gg 1;\\ \varphi &\simeq -\frac{\pi}{2}; \qquad R \ll \frac{1}{\omega C}; \qquad z\simeq \frac{1}{\omega C}. \end{aligned} \right\} \tag{5} \]

On the basis of these relations, expressions (1) and (2), with a small error (not more than 5%), can be represented in simplified form as:

\[ u_c \simeq U_m \sin \omega t, \tag{6} \]

\[ i \simeq U_m \omega C \sin \left( \omega t + \frac{\pi}{2} \right) - \frac{U_m}{R} \cos \varphi \, e^{-\frac{\omega t}{\omega \tau}} . \tag{7} \]

It follows from these expressions that the voltage on the capacitance reaches a maximum, approximately equal in magnitude to the amplitude value of the voltage on the secondary winding of the charging transformer, at the instant

\[ \omega t_1 \simeq \frac{\pi}{2}. \]

A more exact value of the root \(\omega t_1\) can be found from the solution of equation (4). If one takes into account the practically possible values of the quantities entering into this equation (5), then the quantity \(\omega t_1\) in this case can be determined from the approximate equation:

\[ \sin(\omega t_1 - \varphi) \simeq 0. \]

Hence

\[ \omega t_1 - \varphi = \pi n, \qquad \text{where } n = 0,\ 1,\ 2,\ 3,\ldots \]

Finding one positive value of the root \(\omega t_1\), we obtain

\[ \omega = \omega t_1 = \varphi + \pi = \pi + \operatorname{arc\,tg}\left(\frac{-1}{\omega \tau}\right). \]

For selecting the type of gasotron in the circuit of Fig. 10 it is necessary to establish, in addition to the amplitude value of the anode current, which we took into account above when determining the value of the charging resistance \(R\), also the possible magnitude of the inverse voltage and the mean value of the rectified current. In this circuit the inverse voltage \(U_{mo}\) will be measured by the quantity \(U_{mo} \simeq 2U_m\), if it is assumed that the capacitance \(C\) is not discharged owing to the absence of control signals on the grid of the switching thyratron (such cases are quite possible in practice, especially during adjustment of the circuit).

The mean value of the rectified current can be found from the following simple considerations. In calculating the charging circuit we proceed from the assumption that, at the end of each injection pulse, the voltage on the capacitance falls to zero. Consequently, to charge the capacitance \(C\) to the voltage \(U_m\) before the beginning of each pulse, a charge \(Q = C U_m\) is required. Hence the mean value of the charging current is \(I_o = Q f\), where \(f\) is the frequency of the supply current. If it is taken into account that \(\omega = 2\pi f\), it is not difficult to obtain the expression

\[ I_o = \frac{I_m}{2\pi}, \tag{8} \]

well known from the theory of rectifier devices [15, 16].

For calculating the resistance (the permissible power dissipation) and the charging transformer, it is necessary to determine the rms (effective) value of the current in the charging circuit. As is known, the rms value \(I\) of a nonsinusoidal current \(i\) is determined by the relation:

\[ I = \sqrt{\frac{1}{T}\int_0^T i^2 dt}. \]

Substituting into this relation the exact value of the transient current \(i\) from expression (2) and integrating over the limits from 0 to \(\Theta\) (when the current decreases to zero), one can obtain:

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

Taking into account that in practical cases one may assume

\[ \Theta \simeq \frac{\pi}{2} \quad \text{and} \quad \varphi \simeq -\frac{\pi}{2}, \]

it is not difficult to find from the last expression the approximate value of the effective current:

\[ I \simeq \frac{I_m}{2\sqrt{2}}. \]

Consequently, in the resistance \(R\) the power \(P_1=I^2R\) will be dissipated, and this resistance must be rated for it. The effective value of the current in the primary winding is determined from the expression:

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

where \(n\) is the transformation ratio of the charging transformer.

It is now not difficult to determine the efficiency of the charging device. The useful power here may be taken as:

\[ P_2=\frac{CU_m^2}{2}f, \]

where \(f\) is the frequency of the alternating current.

Then, defining the efficiency as

\[ \eta=\frac{P_2}{P_1+P_2}, \]

after simple transformations we obtain the following expression:

\[ \eta=\frac{1}{1+\pi^2\frac{\tau}{T}}, \]

where \(\tau=RC\) and \(T=\frac{1}{f}\) is the period of the supply alternating current. It was indicated above that, in practical cases for the injection circuit, \(\tau < 180\cdot 10^{-6}\) sec. For industrial frequency \(T=0.02\) sec. Consequently, the possible practical values of the efficiency of the charging circuit of the injection scheme are: \(\eta>0.9\).

The calculated (rated) power of the charging transformer is determined, as usual, by the formula:

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

Thus, in the calculation presented above, all the principal electrical parameters of the elements of the electron injection circuit in a betatron have been determined.

References

1. A. L. Gorelik. Industrial Electronics, GEI, 1951.
2. A. A. Bulgakov. Electronic Devices for Automatic Control, GEI, 1951.
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