GENERATOR OF SINUSOIDAL OSCILLATIONS BASED ON SEMI-CONTROLLED VALVES FOR SUPPLYING ACCELERATOR ELECTROMAGNETS

V. M. Osipov, M. V. Kim, Yu. P. Yarushkin

(Presented by the scientific seminar of the Department of Industrial Electronics)

When a betatron is supplied with sinusoidal current of industrial or elevated frequency, a resonant circuit for supplying the windings is used. Owing to this, the power source compensates only the energy losses in the circuit. Since the quality factor of the resonant circuit is comparatively high, excitation of the circuit at an elevated frequency is possible by pulsed introduction of energy into the oscillatory circuit.

In the circuit presented, the oscillatory circuit is connected in series with a thyratron, which plays the role of a switch (Fig. 1).

A feature of this circuit is the use of a semi-controlled device as the switching element. This requires that certain conditions be observed for the operation of the circuit.

First, the instant of ignition of the thyratron must be in a definite phase with the voltage on the circuit; second, the duration of energy input must be such as to compensate the losses in the circuit; third, the circuit elements \(C_d\) and \(L_d\) must ensure extinction of the thyratron.

These conditions can be chosen so that energy input will take place either every period or after several periods. In the latter case we obtain a generator of damped oscillations, which can be used in those cases when pulsations of the alternating voltage are of no importance.

Fig. 1

Let us consider the conditions for exciting undamped oscillations in the generator circuit shown in Fig. 1.

Passing to the equivalent circuit of the oscillatory circuit and thyratron in the steady-state regime, we obtain the circuit shown in Fig. 2.

Let the capacitor \(C\) be charged from a dc-voltage source to the value \(U_0\). When \(k_2\) is closed, damped oscillations of current and voltage on the capacitance will arise in the circuit \(\left(r_2 < r_{\mathrm{cr}} = 2\sqrt{\dfrac{L}{C}}\right)\).

The process is characterized by the following expressions:

\[ U_C(t)=\frac{U_0}{\omega_0\sqrt{LC}}\sin(\omega_0t+\beta_0)e^{-\alpha t}; \tag{1} \]

\[ i(t)=C\frac{dU_C}{dt}=-\frac{U_0}{\omega_0 L}e^{-\alpha t}\sin\omega_0t; \]

where

\[ \omega_0=\sqrt{\frac{1}{LC}-\alpha^2};\qquad \alpha=\frac{r_2}{2L};\qquad \sin\beta_0=\omega_0\sqrt{LC}; \]

At some instant of time \(t_1\), by means of switch \(k_1\), the circuit is connected to a dc-voltage source \(U_0\). The capacitor is charged up to the value \(U_0\), after which switch \(k_1\) opens and the cycle repeats. The next closing of the switch will occur at the instant \(t=t_1+t_2\), where \(t_2\) is the instant at which \(k_1\) opens (from the beginning of the time count).

Manipulations with switch \(k_1\) are equivalent in their action to a pulsed-voltage source producing rectangular pulses of height \(U_0\) and duration \(\tau=t_2-t_1\). The pulses follow one another with the natural-oscillation frequency of the circuit. An approximate diagram of the voltage and current pulses at the input of the circuit is shown in Fig. 3. We shall assume that during the interval in which the pulses act the internal resistance of the source is equal to zero, and in the intervals between pulses—to infinity.

Fig. 2

Fig. 3

Let us determine under what conditions the described operating mode of the circuit is possible. In other words, let us determine under what conditions the excitation of undamped oscillations in the circuit is possible.

Since the operating mode of the circuit of interest to us begins from the instant switch \(k_1\) is closed (instant \(t_1\)), it is expedient to take it as the origin of the time \(\tau\).

Thus, let at the instant \(\tau=0\), or \(t=t_1\), the circuit be connected to the voltage \(U_0\), which acts until the voltage on the capacitance reaches \(U_0\) and the current in the circuit becomes equal to zero. At this instant \((\tau=t_2)\) the action of the voltage ceases and a free mode arises in the circuit, determined by expression (1). In the interval \(0<\tau\le t_2\) the process in the circuit will still have an oscillatory character, but its parameters will be different. Obviously,

\[ U_C=\frac{U_0r_2}{r_1+r_2}+(A_1\sin\Omega\tau+A_2\cos\Omega\tau)e^{-\beta\tau}. \tag{2} \]

Solution (2) gives

\[ U_c(\tau)=\frac{U_0r_2}{r_1+r_2} +\left\{ \frac{[U_0-U_c(t_1)](\beta L-r_2)-i(t_1)r_1(2\beta L-r_2) +\dfrac{U_0r_1}{r_1+r_2}\beta L}{\Omega L} \sin\Omega\tau +\left[U_c(t_1)-\frac{U_0r_2}{r_1+r_2}\right]\cos\Omega\tau \right\}e^{-\beta\tau}; \tag{3} \]

According to the condition of the problem it is necessary to ensure the following equalities

\[ U_c(t_2)=U_0, \tag{4} \]

\[ \left.\frac{dU_c(\tau)}{d\tau}\right|_{\tau=t_2}=0. \]

These conditions mean that at the instant \(\tau=t_2\) the circuit is in the initial state: the voltage on the capacitance is equal to \(U_0\), and the currents in all branches are absent. The latter follows from relation (3). From conditions (4) we shall find the duration of the voltage pulse \(\tau_i=t_2-t_1\) and the value of the resistance. Direct use of the exact expression (3) for this purpose makes an analytical solution of the problem impossible and greatly complicates the analysis. It is therefore expedient, allowing a certain error, to replace the exact expression (3) by an approximate but simpler one.

Let us expand expression (3) in a power series at \(\tau=0\) (i.e., \(t=t_1\)) and take the first 3 terms, i.e.

\[ U_c(t_1\tau)\simeq U_c(t_1)+\frac{dU_c(t_1\tau)}{d\tau}\tau+ \frac{d^2U_c(t_1\tau)}{d\tau^2}\cdot\frac{\tau^2}{2}. \]

Calculations give

\[ \frac{dU_c(t_1\tau)}{d\tau}= \frac{U_0-U_c(t_1)-i(t_1)r_1}{L}(2\beta L-r_2)=\varphi(t_1), \tag{5} \]

\[ -f(t_1)=\frac{d^2U_c(t_1\tau)}{d\tau^2} =-\left\{(\Omega^2+\beta^2)\left[U_c(t_1)-\frac{U_0r_2}{r_1+r_2}\right]+\right. \]

\[ \left. +\frac{U_0-U_c(t_1)-i(t_1)r_1}{L}\,2\beta(2\beta L-r_2)\right\}. \]

Thus,

\[ U_c(t_1\tau)\simeq U_c(t_1)+\varphi(t_1)\tau-f(t_1)\frac{\tau^2}{2}. \]

Conditions (5) give

\[ \left.\frac{dU_c(t_1\tau)}{d\tau}\right|_{\tau=t_2}=0 =\varphi(t_1)-f(t_1)t_2, \]

whence the pulse duration \(\tau_i=t_2-t_1\) will be equal to

\[ \tau_i=\frac{\varphi(t_1)}{f(t_1)}=F(t_1), \]

i.e., the pulse duration depends on the instant of switching on. Further we have

\[ U_c(t_1t_2)=U_0=U_c(t_1)+\frac{\varphi(t_1)^2}{2f(t_1)}, \tag{6} \]

whence

\[ f(t_1)=\frac{\varphi(t_1)^2}{2[U_0-U_c(t_1)]}. \]

Finally we obtain

\[ U_c(t_1\tau)\simeq U_c(t_1)+\varphi(t_1)\tau- \frac{\varphi^2(t_1)}{2[U_0-U_c(t_1)]}\cdot\frac{\tau^2}{2}, \]

\[ \tau_i=\frac{2[U_0-U_c(t_1)]}{\varphi(t_1)}. \tag{7} \]

Let us find the value of the resistance \(r_1\) that ensures satisfaction of conditions (4). Introduce the quantity \(x\), related to \(r_1\) by the relation

\[ \frac{L}{r_1 C}=x \quad \text{or} \quad r_1=\frac{L}{xC}. \]

Bearing in mind that

\[ 2\beta L-r_2=\frac{r_1 r_2 C+L}{r_1 C}-r_2=\frac{L}{r_1 C}=x, \]

\[ L(\Omega^2+\beta^2)=\frac{(r_1+r_2)}{r_1 C}=\frac{1}{C}+\frac{r_2}{L}x, \tag{8} \]

\[ 2\beta(2\beta L-r_2)=\frac{r_2}{L}x+\frac{1}{L}x^2, \]

we obtain

\[ \varphi(t_1)= \frac{[U_0-U_c(t_1)]x-i(t_1)\dfrac{L}{C}}{L}, \]

\[ f(t_1)=\frac{1}{L} \left\{ \frac{1}{C}U_c(t_1)+[U_0-U_c(t_1)]\frac{x^2}{L} -i(t_1)\frac{1}{C}(r_1+x) \right\}. \tag{9} \]

From equation 6 we find

\[ U_0-U_c(t_1)= \frac{ [U_0-U_c(t_1)]^2 x^2-[U_0-U_c(t_1)]\,i(t_1)\dfrac{2L}{C}x+i(t_1)^2\left(\dfrac{L}{C}\right)^2 }{ \dfrac{2L}{C}U_c(t_1)+[U_0-U_c(t_1)]2x^2-i(t_1)\dfrac{2L}{C}(r_2+x) }, \]

whence

\[ x(t_1)= \frac{ \sqrt{ i^2(t_1)\left(\dfrac{L}{C}\right)^2 -\dfrac{2L}{C}[U_c(t_1)-i(t_1)r_2][U_0-U_c(t_1)] } }{ U_0-U_c(t_1) }. \tag{10} \]

Thus, the value of the resistance \(x\) (or \(r_1\)) that ensures the required operating mode depends on the instant at which the source is switched on. For a real circuit, obviously, \(x\) is an essentially positive quantity; however, the form of the functional dependence \(x(t_1)\) gives grounds for asserting the appearance of purely imaginary values of \(x\). The latter indicates the impossibility of exciting undamped oscillations in the circuit for the given value of \(t_1\).

Let us turn to an analysis of this question.

From equation (9) we find

\[ x(t_1)= \frac{\dfrac{L}{C}[\;C\varphi(t_1)+i(t_1)\;]}{U_0-U_c(t_1)}. \tag{11} \]

The simultaneous solution of (9) and (10) gives

\[ \varphi(t_1)= -i(t_1)\frac{1}{C} + \sqrt{ i^2(t_1)\frac{1}{C^2} -\frac{2}{LC}[U_c(t_1)-i(t_1)r_2][U_0-U_c(t_1)] } \tag{12} \]

The average power delivered by the source during the action of the pulse will be equal to

\[ P_{\mathrm{av}}= \frac{1}{t_2}\int_0^{t_2} U_0 i_0(\tau)\,d\tau = \frac{U_0}{t_2}\int_0^{t_2} \frac{U_0-U_c(t_1,\tau)}{r_1}\,d\tau = \]

\[ = U_0^2 \frac{C}{L} x(t_1) - \frac{U_0}{t_2}\frac{C}{L} x(t_1)\int_0^{t_2} U_c(t_1\tau)\,d\tau . \]

Taking (7) into account, we obtain

\[ P_{\mathrm{av}} = U_0^2\frac{C}{L}x(t_1) - U_0\frac{C}{L}x(t_1) \left[ U_c(t_1)\frac{1}{3}+\frac{2}{3}U_0 \right] = \]

\[ = \frac{1}{3}U_0\frac{C}{L}x(t_1)\,[U_0-U_c(t_1)] . \]

Taking (1) into account, we shall finally have

\[ P_{\mathrm{av}}=\frac{1}{3}U_0[C\varphi(t_1)+i(t_1)] \]

or

\[ P_{\mathrm{av}}= \frac{U_0}{3} \sqrt{ i^2(t_1)-2\frac{C}{L}[U_c(t_1)-i(t_1)r_2][U_0-U_c(t_1)] }. \tag{13} \]

Since the zeros and poles, as well as the extrema points, of the functions \(P_{\mathrm{av}}(t_1)\) and \(P_{\mathrm{av}}^2(t)\) coincide, we shall consider the function

\[ P_{\mathrm{av}}^2(t_1)= \left(\frac{U_0}{3}\right)^2 \left\{ i^2\left(t_1-2\frac{C}{L}[U_c(t_1)-i_1(t_1)r_2][U_0-U_c(t_1)]\right) \right\}. \tag{14} \]

After transformations we obtain

\[ P_{\mathrm{av}}^2 = -\frac{U_0^4}{18(\omega_0L)^2} \left\{ 3+\cos 2\omega_0t_1 -3\frac{C}{L}r_2^2\cos 2\omega_0t_1 +4\omega_0Cr_2\sin 2\omega_0t_1 + \right. \]

\[ \left. +\frac{C}{L}r_2^2 \right\}e^{-2\alpha t_1} - \frac{U_0^4}{9}\,2\frac{C}{L} \left[ \frac{3}{2}\frac{r_2}{\omega_0L}\sin\omega_0t_1 +\cos\omega_0t_1 \right]e^{-\alpha t_1}. \]

Taking into account that, at frequencies of the order of \(1000\) Hz and higher,

\[ \frac{r_2}{\omega_0L}\ll 1;\qquad \frac{C}{L}r_2^2\ll 1;\qquad \omega_0Cr_2\ll 1; \]

with a high degree of accuracy we obtain

\[ P_{\mathrm{av}}^2(t_1)\simeq \frac{U_0^4}{18(\omega_0L)^2} [3+\cos 2\omega_0t_1]e^{-2\alpha t_1} - \frac{U_0^4}{9}\,2\frac{C}{L}\cos\omega_0t_1 e^{-\alpha t_1}. \]

Let us find the intervals of values of \(t_1\) for which \(P_{\mathrm{av}}^2(t_1)<0\) and, consequently, excitation of oscillations is impossible. The zeros of the function \(P_{\mathrm{av}}^2(t_1)\) are found from the equation \(P_{\mathrm{av}}^2(t_1)=0\). It has the form

\[ \frac{1}{2(\omega_0L)^2} [3+\cos 2\omega_0t_1]e^{-2\alpha t_1} = 2\frac{C}{L}\cos\omega_0t_1e^{-\alpha t_1} \]

or, taking into account that

\[ \cos 2\omega_0t_1 = 2\cos^2\omega_0t_1 - 1 \]

and that, at the frequencies under consideration, with high accuracy one may take \(\omega_0\simeq \dfrac{1}{\sqrt{LC}}\), we obtain the simpler equation

\[ (1+\cos^2\omega_0t_1)e^{-\alpha t_1}=2\cos\omega_0t_1 \]

or

\[ \varphi e^{-\alpha t_1}= \frac{2\cos\omega_0t_1}{1+\cos^2\omega_0t_1}. \]

This equation can be solved graphically. An approximate form of the construction is shown in Fig. 4. The interval of values \(t_{11}<t_1<t_{12}\) is the region of negative \(P_{\mathrm{cp}}^2(t_1)\). In this region, excitation of undamped oscillations is impossible.

An approximate form of the dependence \(P_{\mathrm{cp}}^2(t_1)\) is shown in Fig. 4. The curve has a maximum. Let us find the value \(t_{1m}\) at which this maximum is observed.

Fig. 4

\[ \frac{d\left[P_{\mathrm{cp}}^2(t_1)\right]}{dt_1} = \frac{U_0^4\,\dfrac{C}{L}}{9} \left\{ -2\cos\omega_0 t_1 \sin\omega_0 t_1 e^{-2\alpha t_1} -2\alpha\left(1+\cos^2\omega_0 t_1\right)e^{-2\alpha t_1} + \right. \]
\[ \left. +\,2\omega_0\sin\omega_0 t_1 e^{-\alpha t_1} +2\alpha\cos\omega_0 t_1 e^{-\alpha t_1} \right\} =0 \]

or

\[ e^{-\alpha t_{1m}} = \frac{ \omega_0\sin\omega_0 t_{1m}+\alpha\cos\omega_0 t_{1m} }{ \omega_0\cos\omega_0 t_{1m}\sin\omega_0 t_{1m} +\alpha\left(1+\cos^2\omega_0 t_{1m}\right) }. \]

This equation can be rewritten in another form:

\[ e^{\alpha t_{1m}} = \frac{ \cos\omega_0 t_{1m}\left(\omega_0\sin\omega_0 t_{1m}+\alpha\cos\omega_0 t_{1m}\right)+\alpha }{ \omega_0\sin\omega_0 t_{1m}+\alpha\cos\omega_0 t_{1m} } = \]

\[ = \cos\omega_0 t_{1m} + \frac{\alpha}{ \omega_0\sin\omega_0 t_{1m}+\alpha\cos\omega_0 t_{1m} }. \]

Since \(\alpha\ll\omega_0\), approximately we have

\[ e^{\alpha t_{1m}} = \cos\omega_0 t_{1m} + \frac{\alpha}{\omega_0\sin\omega_0 t_{1m}}. \]

The graphical solution of this equation is presented in Fig. 5, from which it is seen that the maximum is reached at \(t_{1m}\). If it is assumed that

\[ t_{1m}\simeq \frac{\pi}{\omega_0}, \]

then the value of the maximum itself will be

\[ P_{\mathrm{av}}^{2}\simeq \frac{2U_0^{4}\dfrac{C}{L}}{9} \left[ e^{-2\alpha \frac{\pi}{\omega_0}} + e^{-\alpha \frac{\pi}{\omega_0}} \right]. \]

The operating value of the instant \(t_1\) at which the source is switched on should be chosen near the forbidden zone, for example, at the point \(t_1=\dfrac{3\pi}{2\omega_0}\), when \(P_{\mathrm{av}}\) is small.

Fig. 5

\[ P_{\mathrm{av}\,t_1}= \frac{U_0^{2}}{3} \sqrt{\frac{C}{L}}\, e^{-\alpha \frac{3\pi}{2\omega_0}}. \]

In this case we have

\[ U_c(t_1)= -\frac{U_0\alpha}{\omega_0} e^{-\alpha \frac{3\pi}{2\omega_0}}, \]

\[ i(t_1)= \frac{U_0}{\omega_0 L} e^{-\alpha \frac{3\pi}{2\omega_0}}. \]

From (13) we find

\[ \varphi(t_1)= \frac{\dfrac{3P_{\mathrm{av}}}{U_0}-i(t_1)}{C} = \frac{ U_0\sqrt{\dfrac{C}{L}}+\dfrac{U_0}{\omega_0 L} }{C} e^{-\alpha \frac{3\pi}{2\omega_0}}. \]

If we take \(\omega_0\simeq \dfrac{1}{\sqrt{LC}}\), then \(\omega_0L=\sqrt{\dfrac{L}{C}}\), and

\[ \varphi\!\left(\frac{3\pi}{2\omega_0}\right) = 2\frac{U_0}{\sqrt{LC}} e^{-\alpha \frac{3\pi}{2\omega_0}}. \]

The pulse duration in this case will be equal to

\[ \tau_i=t_2= \frac{2\left[U_0-U_c(t_1)\right]}{\varphi(t_1)} = \sqrt{LC} \left[ 1+\frac{\alpha}{\omega_0} e^{-\alpha \frac{3\pi}{2\omega_0}} \right] e^{\alpha \frac{3\pi}{2\omega_0}}. \]

Finally, let us find the value of \(x\). From (11) and (13) it follows that

\[ x(t_1)= \frac{\dfrac{L}{C}\dfrac{3P_{\mathrm{av}}}{U_0}} {U_0-U_c(t_1)} = \frac{\sqrt{\dfrac{L}{C}}\,e^{-a\frac{3\pi}{2\omega_0}}} {1+\dfrac{a}{\omega_0}e^{-a\frac{3\pi}{2\omega_0}}}. \]

Conclusions

1. Pulse excitation of undamped oscillations in the circuit under consideration is possible provided that the corresponding synchronization between the exciting pulses and the oscillations in the circuit is ensured.

2. There exist time intervals into which, if the exciting pulses fall, an undamped oscillatory process is impossible.

3. The value of the resistance \(r_1\) depends substantially on the instant at which the exciting pulse is applied. In the forbidden zone it becomes imaginary.

4. From the standpoint of the magnitude of the power supplied to the circuit from the exciting source, it is advisable to switch on at a time close to the value

\[ \frac{3\pi}{2\omega_0}. \]

The pulse duration depends substantially on this instant.