STATISTICAL PROPERTIES OF CERTAIN ELEMENTS IN THE BETATRON CONTROL SYSTEM

V. M. Razin

(Presented by the Joint Scientific Seminar of the Faculty of Automation and Computer Engineering)

It has been established experimentally that such an important output parameter of the betatron as the radiation dose rate is very sensitive to the method of phasing of the injection. In connection with this, it is of interest to formulate and consider the question of choosing the optimal ratio of parameters in the circuit of Fig. 1, intended for transforming the signal from a permalloy sensor of the zero magnetic field in the working zone.

Fig. 1.

At the moment when the magnetic field at the location of the sensor passes through zero, the permalloy sensor generates voltage pulses \(l_{\mathrm{p}}\), the form of which can be represented by the equation [1]:

\[ l_{\mathrm{p}} = -\frac{H_{\mathrm{p.t.}}^{\omega} W_{\mathrm{p}} S_c}{\left(a+\dfrac{N}{4\pi}\right)} \frac{10^{-8}} {\left[ 1+6.75 \dfrac{b\omega^2}{\left(a+\dfrac{N}{4\pi}\right)^3} H_{\mathrm{p.sh.}}^2 t^2 \right]}, \tag{1} \]

where \(H_{\mathrm{p.t.}}\) is the amplitude of the sinusoidal magnetic field at the location of the sensor;

\(\omega\) is the angular frequency of the magnetic field;

\(W_{\mathrm{p}}\) is the number of turns of the signal winding on the permalloy core;

\(S_c\) is the area of the sensor core;

\(a\) and \(b\) are coefficients depending on the magnetic properties of the material;
\(N\) is the demagnetizing coefficient, depending on the dimensions of the core and winding;
\(t\) is time.

The voltage amplitude \(l_{\text{p.t.}}\) and the pulse duration at the level \(l = 0.5\,l_{\text{p.t.}}\), in accordance with formula (1), are determined by the relations:

\[ l_{\text{p.t.}} = -\frac{\omega W_{\text{p}} S_{\text{c}} H_{\text{p.t.}}}{\left(a+\dfrac{N}{4\pi}\right)}\,10^{-8}, \tag{2} \]

\[ \tau_{0.5} = \frac{1.5}{\omega H_{\text{p.t.}}} \sqrt{\frac{\left(a+\dfrac{N}{4\pi}\right)^3}{b}}. \tag{3} \]

In the circuit of Fig. 1, the voltage pulse described by equation (1) is first filtered in order to weaken the influence of fluctuation disturbances caused by remagnetization of the core and by other causes, and then, after amplification in an ideal amplifier with gain \(K\), is differentiated. After differentiation the pulse is detected near the zero level and is fed to the comparing device for generation of the injection phasing signal. Thus, the circuit of Fig. 1, on the one hand, transforms a certain deterministic signal, whose mathematical description is given by formula (1), regarded as the mathematical expectation of this signal; moreover, the parameter \(\omega H_{\text{p.t.}}\) in this formula is a random variable depending on random changes in the line voltage. On the other hand, a disturbance in the form of white noise is superposed on this signal, whose statistical properties are represented by the relation

\[ R_{\text{f.n.}}(\tau)=D_0\delta(\tau), \tag{4} \]

where \(R_{\text{f.n.}}(\tau)\) is the autocorrelation function of the fluctuation disturbances;
\(D_0\) is the total variance of the fluctuation noises;
\(\delta(\tau)\) is the delta function.

To simplify the calculations, we shall replace the signal whose form is specified by equation (1) by a cosine-shaped voltage pulse

\[ U=\frac{U_{\tau}}{2}(1-\cos \omega_{\text{p}}t), \tag{5} \]

where the pulse amplitude \(U_{\tau}\) must be equal to the voltage amplitude \(l_{\text{p.t.}}\) from relation (2), and the reduced frequency \(\omega_{\text{p}}\), on the basis of equality (3), can be approximately determined by the relation:

\[ \omega_{\text{p}}= \frac{2\pi}{T} = \frac{2\pi}{2\tau_{0.5}} = \frac{\pi}{1.5}\,\omega H_{\text{p.t.}} \sqrt{\frac{b}{\left(a+\dfrac{N}{4\pi}\right)^3}}. \tag{6} \]

We note in passing that the pulse amplitude \(U_{\tau}\) from the zero-field sensor and its duration \(\tau_{0.5}\) at the level \(U=0.5\,U_{\tau}\) can easily be measured experimentally, for example by oscillographic methods, and used in the calculations given below.

We assume that equation (5) is valid for \(0 \leq \omega_{\text{p}}t \leq 2\pi\), while for all other \(\omega_{\text{p}}t\) the voltage \(U\) is identically equal to zero. It is easy to see that at \(\omega_{\text{p}}t=\pi\) the voltage \(U\) reaches its maximum.

Passage of a pulse of the form under consideration through the first-order filter in the left-hand part of the circuit in Fig. 1 is determined by the superposition of the transients produced by switching on an \(RC\) circuit simultaneously to a constant voltage \(U_= = U_{\tau}/2\) and an alternating voltage

\[ U_{\sim}=-\frac{U_{\tau}}{2}\sin\left(\omega t-\frac{\pi}{2}\right) \]

in the form of equations for the voltage \(u_c\) across the capacitance and the current \(i\) through the resistance \(r\).

\[ u_c=\frac{U_{\tau}}{2\sqrt{1+(\omega_{\mathrm{p}}T_1)^2}} \sin\left(\omega_{\mathrm{p}}t-\frac{\pi}{2}-\arctan \omega_{\mathrm{p}}T_1\right) -\frac{U_{\tau}}{2\sqrt{1+(\omega_{\mathrm{p}}T_1)^2}}\times \]

\[ \times \sin\left(-\frac{\pi}{2}-\arctan \omega_{\mathrm{p}}T_1\right)e^{-t/T_1} +\frac{U_{\tau}}{2}\left(1-e^{-t/T_1}\right); \tag{7} \]

\[ i=C_1\frac{du_c}{dt}= \frac{CU_{\tau}\omega_{\mathrm{p}}}{2\sqrt{1+(\omega_{\mathrm{p}}T_1)^2}} \cos\left(\omega_{\mathrm{p}}t-\frac{\pi}{2}-\arctan \omega_{\mathrm{p}}T_1\right)+ \]

\[ +\frac{CU_{\tau}}{2T_1\sqrt{1+(\omega_{\mathrm{p}}T_1)^2}} \sin\left(-\frac{\pi}{2}-\arctan \omega_{\mathrm{p}}T_1\right)e^{-t/T_1} +\frac{CU_{\tau}}{2}\frac{1}{T_1}e^{-t/T_1}. \tag{8} \]

The time constant \(T_1\) in the equations can be made sufficiently small. Thus, for example, with a coil resistance on the permalloy core of the sensor of \(100\ \Omega\) and a capacitance of the lead-out wires of \(1000\ \mathrm{pF}\), the time constant is \(T_1=0.1\ \mu\mathrm{s}\). This value of \(T_1\) can be increased by including in the circuit, in parallel with the transmitted signal, an additional noninductive capacitance.

In analyzing the passage of the signal, it is of interest to consider only the modifications of the signal in its middle part, i.e., in the neighborhood of the value \(\omega_{\mathrm{p}}t \simeq \pi\), or \(t=\tau_{0.5}=\pi/\omega_{\mathrm{p}}\). In practically attainable cases the value \(t=\tau_{0.5}\) may be of the order of \(1\text{–}10\ \mu\mathrm{s}\). Hence it follows that in equations (7) and (8) one may neglect terms \(Ce^{-t/T_1}\to 0\) for the time instants under consideration. The time instant at which the voltage on the capacitance \(C_1\) reaches a maximum corresponds to the instant at which the current \(i\) passes through zero, and is determined from the equation:

\[ \cos\left(\omega_{\mathrm{p}}t_1-\frac{\pi}{2}-\arctan \omega_{\mathrm{p}}T_1\right)=0. \tag{9} \]

In practice, \(\omega_{\mathrm{p}}T_1\simeq 0.03\) and \(\arctan \omega_{\mathrm{p}}T_1\simeq \omega_{\mathrm{p}}T_1\); therefore

\[ \omega_{\mathrm{p}}t_1=\pi+\arctan \omega_{\mathrm{p}}T_1=\pi+\omega_{\mathrm{p}}T_1. \tag{10} \]

Putting

\[ t_1=\tau_{0.5}+\delta_1, \tag{11} \]

we find

\[ \delta_1=T_1, \tag{12} \]

and the amplitude of the voltage on the capacitance is

\[ u_{c.\tau}= \frac{U_{\tau}}{2\sqrt{1+(\omega_{\mathrm{p}}T_1)^2}} +\frac{U_{\tau}}{2} = U_{\tau}\left(1-\frac{1}{4}\omega_{\mathrm{p}}^2T_1^2\right). \tag{13} \]

Thus, as a result of the passage of a pulse of cosine form with frequency \(\omega_{\mathrm{p}}\) and amplitude \(U_m\) through a first-order filter with sma-

... time constant, the pulse shape is basically preserved. The changes in the middle part of the pulse reduce to the appearance of a constant delay \(\delta_1 = T_1\) and a decrease in the pulse amplitude by a factor of \(n\), where

\[ n=\frac{2\sqrt{1+(\omega_n T_1)^2}}{1+\sqrt{1+(\omega_n T_1)^2}} \simeq 1+\frac{\omega_n^2T_1^2}{4}. \tag{14} \]

From relation (13) it is seen that, as the time constant \(T_1\) increases, a corresponding decrease in the pulse amplitude is observed. This decrease in the pulse amplitude after differentiation will lead to a certain decrease in the steepness of the differentiated pulse and to an increase in the variance; i.e., in order to decrease the variance it is necessary to decrease the time constant \(T_1\).

On the other hand, in order to decrease the variance it is necessary to increase the time constant \(T_1\). This is evident from the fact that, after a random process with correlation function (4) passes through a first-order filter, the variance will have the form

\[ D_{\phi}=\frac{D_0}{2T_1}, \tag{15} \]

i.e., as the time constant \(T_1\) is increased, the variance at the filter output will decrease.

At the output of the differentiating circuit of Fig. 1, the variance due to fluctuation noise will be equal to

\[ D_{\text{out}}=\frac{D_0K^2}{T_1}\,\frac{1}{1+\dfrac{T_1}{T_2}}. \tag{16} \]

These contradictory requirements can be satisfied by one or another optimization method. For this purpose it is necessary to formulate some optimality criterion.

As is known, the problem of choosing a quality criterion subject to optimization cannot be solved within the framework of the theory of optimal systems itself [2]. In our case, for solving the optimization problem, as a quality criterion one may take the sum of the noise variance at the filter output and the square of the mathematical expectation of the loss in pulse amplitude in the vicinity of the maximum. The quality criterion formulated in this way, on the basis of relations (15) and (13), takes the form:

\[ Q\simeq \frac{D_0}{2T_1} +U_t^2\left(1-\frac{1}{\sqrt{1+(\omega_nT_1)^2}}\right)^2\cdot\frac{1}{4}. \tag{17} \]

By known methods one should find the minimum of the function \(Q(T_1)\) for some optimal value \(T_1\), which then gives the solution of the posed problem. The analytical solution is quite cumbersome; therefore, for finding the optimal value of \(T_1\) from equation (17), the use of electronic digital computers should be recommended.

For some ratios of the parameters the function \(Q(T_1)\) may have no minimum. In this case some other quality criterion must be formulated, allowing the optimization problem to be solved to completion and in a somewhat different way.

If it is assumed that the product \(\omega_nT_1\) is small, i.e. \(\omega_nT_1 \ll 1\),

then from equation (17) the approximate condition for minimizing the function \(Q\) will take the form:

\[ T_1 \simeq \sqrt[5]{\frac{2D_0}{U_\tau^2 \omega_{\mathrm{p}}^4}} . \tag{18} \]

Calculation shows that for practical values of the quantities in relation (18), indeed \(T_1 < 0.1\ \mu\text{sec}\) and \(\omega_{\mathrm{p}} T_1 \ll 0.03\). We shall consider the transformation of the voltage pulse from the zero magnetic-field sensor by the differentiating circuit \(v_2 c_2\) in the right-hand part of Fig. 1 under the assumption that the change of the pulse in the filter \(v_1 c_1\) is negligibly small and, consequently, it may be assumed that at the input of the differentiating circuit there again acts a pulse whose shape is determined by equation (5), while the amplitude along the ordinate axis is increased by a factor \(K\), where \(K\) is the gain of the ideal amplifier in the circuit of Fig. 1.

Multiplying term by term both sides of equation (8) by the resistance \(v_2\) and replacing the time constant by \(T_2\), after simple transformations we obtain an expression for the voltage at the output of the differentiating circuit

\[ u_{t_2} = \frac{K U_\tau \omega_{\mathrm{p}} T_2}{2} \left[ \frac{\sin\left(\omega_{\mathrm{p}} t - \operatorname{arctg}\omega_{\mathrm{p}}T_2\right)} {\sqrt{1+\omega_{\mathrm{p}}^2 T_2^2}} + \frac{\omega_{\mathrm{p}}T_2}{1+\omega_{\mathrm{p}}^2 T_2^2} e^{-\frac{t}{T_2}} \right] . \tag{19} \]

With ideal differentiation of the signal under consideration by means of a differentiating element having the time constant \(T_2\), the voltage obtained should have been

\[ u_r = \frac{K U_\tau \omega_{\mathrm{p}} T_2}{2}\sin \omega_{\mathrm{p}} t . \tag{20} \]

Let us calculate the total delay that occurs in the shaping circuits of the phasing channel, in order to obtain the statistical dependence of the injection phase on random variations of the line voltage.

It was established above that the delay in the filter is \(\delta_1 \simeq T_1\), i.e., it does not depend on the line voltage.

In the literature [3] it is indicated that the value of the parameter \(\omega_{\mathrm{p}}T_2\), optimal in the sense of ensuring the best differentiation and retaining sufficient amplitude after differentiation of approximately rectangular pulses, is determined by the inequality \(\omega_{\mathrm{p}}T_2 \leqslant 0.25\). This circumstance still means that, when considering transient processes in the differentiating circuit for the value of the argument \(\omega_{\mathrm{p}}t \simeq \pi\) that is of interest to us, the exponential component in equation (19) may be neglected. In this case, the instant \(t_2\) at which the differentiated voltage passes through zero is determined, according to relation (19) with \(u_{t_2}=0\), from the solution of the equation:

\[ \sin\left(\omega_{\mathrm{p}}t_2 - \operatorname{arctg}\omega_{\mathrm{p}}T_2\right)=0 . \tag{21} \]

From equation (21) we obtain

\[ \omega_{\mathrm{p}}t_2 = \pi + \operatorname{arctg}\omega_{\mathrm{p}}T_2 . \tag{22} \]

Putting \(T_2 = \tau_{0.5} + \delta_2\), we find, taking into account that \(\omega_{\mathrm{p}}\tau_{0.5}=\pi\),

\[ \delta_2 = \frac{1}{\omega_{\mathrm{p}}}\operatorname{arctg}\omega_{\mathrm{p}}T_2 . \tag{23} \]

Applying the approximate formula \(\operatorname{arctg} x \simeq x - \dfrac{x^3}{3}\), valid—

with an accuracy of up to \(0.1\%\) for \(-0.29 < x < 0.29\), we determine the time delay \(\delta_2\) in differentiation by a real \(RC\)-circuit as

\[ \delta_2 \simeq T_2\left(1-\frac{\omega^2 T_2^2}{3}\right). \tag{24} \]

The finite steepness of the voltage rise at the output of the differentiating circuit after passage through zero and the presence of a definite threshold \(U_0\) for the operation of the comparing device determine the delay \(\delta_3\), which we determine in the following way. The slope of the voltage curve as a function of time at the output of the differentiator is equal to

\[ \frac{dU_{r_2}}{dt} = \frac{K U_\tau \omega_{\mathrm{p}}^2 T_2}{2} \frac{\cos\left[\omega_{\mathrm{p}} t-\operatorname{arctg}\omega_{\mathrm{p}}T_2\right]} {\sqrt{1+\omega_{\mathrm{p}}^2T_2^2}}, \tag{25} \]

For \(\omega_{\mathrm{p}}t=\pi+\operatorname{arctg}\omega_{\mathrm{p}}T_2\) we have

\[ \frac{dU_{r_2}}{dt} = -\frac{K U_\tau \omega_{\mathrm{p}}^2 T_2}{2} \frac{1}{\sqrt{1+\omega_{\mathrm{p}}^2T_2^2}} . \tag{26} \]

Consequently, near the passage through zero and after this instant,

\[ U_{r_2}\simeq -\frac{K U_\tau \omega_{\mathrm{p}}^2 T_2}{2} \frac{t}{\sqrt{1+\omega_{\mathrm{p}}^2T_2^2}} . \tag{27} \]

At the moment of comparison \(u_{\gamma_2}=-U_0\) and \(t=\delta_3\); consequently,

\[ \delta_3= \frac{2U_0\sqrt{1+\omega_{\mathrm{p}}^2T_2^2}} {K U_\tau \omega_{\mathrm{p}}^2 T_2}. \tag{28} \]

The total delay is equal to

\[ \delta=\delta_1+\delta_2+\delta_3 = T_1+T_2\left(1-\frac{\omega_{\mathrm{p}}^2T_2^2}{3}\right) + \frac{2U_0\sqrt{1+\omega_{\mathrm{p}}^2T_2^2}} {K U_\tau \omega_{\mathrm{p}}^2T_2}. \tag{29} \]

From equation (6) it is evident that the angular frequency \(\omega_{\mathrm{p}}\) is proportional to the parameter \(\omega H_{\mathrm{p.t.}}\) and, consequently, to the line voltage and, correspondingly, to the pulse amplitude \(U_\tau\); therefore we write

\[ \omega_{\mathrm{p}}=\gamma U_\tau, \tag{30} \]

where \(\gamma\) is the proportionality coefficient.

Substituting (30) into (29) and assuming, for \(\omega_{\mathrm{p}}^2T_2^2\ll 1\),

\[ \sqrt{1+\omega_{\mathrm{p}}^2T_2^2}\simeq 1+\frac{1}{2}\omega_{\mathrm{p}}^2T_2^2 . \]

we obtain

\[ \delta = T_1+T_2+ \frac{2U_0}{K U_\tau^3\gamma^2T_2} + \frac{u_0T_2}{K U_\tau} - \frac{\gamma^2U_\tau^2T_2^3}{3}. \tag{31} \]

As a result, we have obtained a certain nonlinear function of the random argument \(U_\tau\), proportional to the corresponding random variations of the line voltage. In order to linearize the equation obtained, let \(U_\tau=U_{\tau0}+\Delta U_\tau\), where \(U_{\tau0}=\mathrm{const}\) will correspond to a certain nominal value of the line voltage, while \(\Delta U_\tau\) will represent purely random deviations from the nominal value. Taking into account that

\[ U_\tau=U_{\tau0}\left(1+\frac{\Delta U_\tau}{U_{\tau0}}\right) \tag{32} \]

and assuming \(\dfrac{\Delta U_\tau}{U_{\tau0}}\ll 1\), one can substitute (32) into (31) and ob-

drop the small terms from consideration in the equation thus obtained. The indicated transformations make it possible to write

\[ \delta \simeq T_1+T_2+\frac{2U_0}{K\gamma^2 U_{\tau0}^3 T_2} +\frac{U_0T_2}{K U_{\tau0}} -\frac{\gamma^2 T_2^3 U_{\tau0}^2}{3} - \frac{\Delta U_{\tau}}{U_{\tau0}} \left( \frac{6U_0}{K\gamma^2 U_{\tau0}^3T_2} +\frac{U_0T_2}{K U_{\tau0}} +\frac{2}{3}\left(\gamma^2T_2^3U_{\tau0}^2\right) \right). \tag{33} \]

Now one can compute the dependence of the variance of the centered random variable \(\delta'=\delta-\delta_0\), determined by the last term on the right-hand side of equation (33), on the variance of the centered random variable \(\Delta U_{\tau}\) in the form of the relation

\[ D(\delta')= \frac{D(\Delta U_{\tau})}{U_{\tau0}^{2}} \left( \frac{6U_0}{K\gamma^2U_{\tau0}^{3}T_2} +\frac{U_0T_2}{KU_{\tau0}} +\frac{2\gamma^2T_2^3U_{\tau0}^2}{3} \right)^2 . \tag{34} \]

Next it is necessary to take into account the variance due to the action of fluctuation noises. Taking relation (27) into account, the instant of zero crossing of the total curve of the transformed signal and of the random noise voltage is determined from the solution of the equation

\[ u_{z2}\simeq -\frac{KU_{\tau}\omega_{\mathrm{p}}^2T_2}{2}\,t+a(t)=0 \tag{35} \]

with respect to the variable \(t\). In equation (35), in order to simplify the calculations, it is assumed that \(\sqrt{1+\omega_{\mathrm{p}}^2T_2^2}\simeq 1\), while the signal \(a(t)\) is the random component due to the action of thermal and magnetic fluctuation noises, whose variance is determined by relation (16). Performing substitutions by means of equalities (30) and (32) and solving equation (35) with respect to the variable \(t\), we obtain, in the first linear approximation, the equation

\[ t\simeq \frac{2a(t)}{K\gamma^2U_{\tau0}^{3}T_2} - \frac{6a(t)\Delta U_{\tau}}{K\gamma^2U_{\tau0}^{4}T_2}. \tag{36} \]

Assuming that the random quantities \(\Delta U_{\tau}(t)\) and \(a(t)\) are not correlated, we compute the resulting variance from the action of the fluctuation signals \(a(t)\) and the random variations of the line voltage \(\Delta u_{\tau}(t)\). It is known that the variance \(D_{xy}\) of the product of uncorrelated random variables \(x\) and \(y\) can be determined by the formula

\[ D(xy)=D(x)D(y)+D(x)m_y^2+D_y m_x^2, \tag{37} \]

where \(D(x)\), \(D(y)\), \(m_x\), and \(m_y\) are the variances and mathematical expectations of the random variables \(x\) and \(y\), respectively.

Since we consider centered random variables \(\Delta U_m\) and \(a(t)\), \(m_{\Delta U_m}=0\) and \(m_{a(t)}=0\).

Consequently, the variance of the instant at which the differentiated signal passes through the zero value, due to the action of fluctuation effects and the influence of random oscillations of the line voltage on the values of the coefficients multiplying the variable \(t\) in equation (35), is determined as follows:

\[ D_{[t]}= \frac{4D[a(t)]}{\left(K\gamma^2U_{\tau0}^{3}T_2\right)^2} + \frac{36D[a(t)]D[\Delta U_{\tau}]}{\left(K\gamma^2U_{\tau0}^{4}T_2\right)^2}. \tag{38} \]

Now let us take into account that, according to (16),

\[ D_{\mathrm{out}}=D[a(t)]= \frac{D_0K^2}{T_1}\, \frac{T_2}{T_1+T_2}. \tag{39} \]

and that the total resulting variance due to the random variables \(\delta'\) from equation (33) and \(t\) from equation (36) is equal to

\[ D = D[\delta'] + D[t]. \tag{40} \]

In writing relation (40) it is taken into account that, in practice, the random variables \(\delta'\) and \(t\) are almost uncorrelated.

In its final form, the function constituting the quality criterion for determining the time constant \(T_2\) and the gain coefficient \(K\), taking into account (40), (37), (38), and (34), is written as follows:

\[ D = \frac{4D_0}{T_1}\, \frac{T_2}{T_1 + T_2}\, \frac{1}{\left(\gamma^2 U_{\tau0}^{3} T_2\right)^2} + \frac{36D_0}{T_1}\, \frac{T_2}{T_1 + T_2}\, \frac{D[\Delta U_\tau]}{\left(\gamma^2 U_{\tau0}^{4} T_2\right)^2} + \frac{D(\Delta U_\tau)}{U_{\tau0}^{2}} \left[ \frac{6U_0}{K\gamma^2 U_{\tau0}^{3} T_2} + \frac{U_0 T_2}{K U_{\tau0}} + \frac{2\gamma^2 T_2^{3} U_{\tau0}^{2}}{3} \right]^2 . \tag{41} \]

Analysis of equation (41) makes it possible, in our opinion, to draw important preliminary conclusions from the practical standpoint. First, the obtained relation has a minimum with respect to the time constant \(T_2\). By investigating this relation for a minimum by known methods, one can find a certain optimal value of the time constant \(T_2\) that provides the minimum variance at the output of the channel for forming the control pulse in the injection phasing circuit. Second, an arbitrarily small value of the variance can be achieved by increasing the gain coefficient \(K\) of an ideal amplifier, i.e., an amplifier of the inertialess type with an ideal linear amplitude characteristic and an infinite passband. It is clear that, in a second approximation, it is necessary to take into account the real characteristics of the amplifier, and then, apparently, a certain optimal value of the amplifier gain coefficient may appear. We note that an analytical investigation of expression (41) for a minimum presents insurmountable difficulties, and the problem must be solved graphically or with the aid of electronic digital computers. The solution of the stated problem becomes even more complicated when the real parameters of the amplifier and of the other elements in the injection phasing channel are taken into account.

In conclusion, we note that in a complete calculation of the statistical properties of the injection phasing channel it is necessary to take into account such statistical properties of thyratrons in the final modulating stage and in the submodulator, if one is present.

It is known [4] that in hydrogen thyratrons, which are predominantly used in injection circuits, the ignition time is of the order of tenths of a \(\mu\)sec, while the instability of the ignition instant reaches several hundredths of a \(\mu\)sec.

Two types of instability are distinguished. Periodic instability manifests itself in changes in the thyratron firing instant from pulse to pulse. Aperiodic instability manifests itself in a gradual change of the ignition time, for example, as a consequence of a change in the temperature of the hydrogen generator and other causes. Under the operating conditions of a thyratron in an injection circuit, if the power supply of the charging device is not stabilized, there may be instability caused by a change in the anode voltage from cycle to cycle. Instability of the latter type can, in a first approximation, be determined in the form of a change in the ignition time as a function of the anode voltage.

\[ t_3 = A - BU_a = A - BU_{a0}\left(1 + \frac{\Delta U_a}{U_{a0}}\right). \tag{42} \]

Here \(A\) and \(B\) are constants, and the anode voltage is \(U_a = U_{a0} + \Delta U_a\), where \(\Delta U_a\) is a random quantity.

The variance of the ignition time due to random changes in the voltage \(\Delta U_a\) will be equal to

\[ D(t_3)_{\Delta u_a} = B^2 D[\Delta U_a]. \tag{43} \]

If it is assumed that the periodic instability and the random changes in the quantity \(\Delta U_a\) are uncorrelated, then the resulting variance of the firing time \(t_{\mathrm{cp}}\) of the thyratron will be equal to

\[ D(t_{\mathrm{cp}}) = D(t_3)_{\Delta u_a} + D(t'_3), \tag{44} \]

where \(D(t'_3)\) is the variance of the periodic instability.

Aperiodic instability of thyratron ignition will introduce a slow drift with respect to the value of the mathematical expectation of the thyratron ignition time. Consequently, appropriate measures must be taken to reduce the “internal” instability of thyratron ignition by forming a control pulse with suitable parameters on the thyratron grid.

Recently, thyratrons have been developed with an overall instability on the order of thousandths of a microsecond. For such thyratrons, the principal destabilizing factor thus becomes the effect of the nonconstancy of the anode voltage in accordance with equation (42).

It should be especially noted that the introduction of various kinds of electronic delay circuits into the injection phasing channel is highly undesirable, since such circuits introduce significant fluctuations in the delay time due to their internal instabilities and thereby substantially degrade the statistical characteristics of the channel. Providing a certain functional flexibility in the injection channel in this case leads to unjustified losses in the productivity of the betatron installation.

References

1. L. M. Anan’ev, A. A. Vorob’ev, V. I. Gorbunov. Electron Induction Accelerator—the Betatron. Atomizdat, 1961.
2. Fundamentals of Automatic Control, ed. V. S. Pugachev. Fizmatgiz, 1963.
3. Ya. S. Itskhoki. Pulse Devices. Sov. Radio, 1959.
4. G. A. Vorontsev. Pulse Thyratrons. Sov. Radio, 1958.