PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR
POLYTECHNIC INSTITUTE

Vol. 190
1971

STUDY OF A BETATRON ELECTROMAGNET AS A LOAD FOR A SEMICONDUCTOR FREQUENCY CONVERTER

L. M. ANAN'EV, S. G. CHERNYSHEVA

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

Introduction

Portable small-size betatrons for low energies up to 6 MeV are used for flaw detection of blanks and products made of various materials, in radiation therapy of malignant tumors, and for research in chemistry, biology, and other fields.

However, since betatrons at present are no longer unique devices, but are in fact produced in series, practice persistently raises the requirement of reducing their dimensions and increasing their efficiency.

Work in this direction is being carried out on the use of new, high-quality grades of transformer steel, improvement of the electromagnet design, and also the development of effective methods for cooling betatrons and new designs of high-voltage injectors. All this has made it possible to develop the design of a portable small-size betatron that is very convenient for flaw detection.

In addition, in order to increase the output of bremsstrahlung radiation, new types of betatrons are being created that operate at an increased frequency. The requirement of minimum weight and economy is met by choosing the most optimal operating mode of the electromagnet with maximum loading of the magnetic-core steel, i.e., the operating point is chosen on the bend of the magnetization curve.

This condition imposes certain restrictions in the selection and calculation of the power-supply circuit of the betatron electromagnet. In view of the considerable inductance of the electromagnet winding, supplying it with alternating current is associated with a low value of $\cos \varphi$. To increase $\cos \varphi$, a resonant circuit for supplying the windings is used. This circuit includes the magnetizing windings, into whose circuit a compensating capacitor is connected in parallel. The capacitance of the capacitors is selected so that the natural frequency of the resulting oscillatory circuit is equal to the frequency of the supply current. A feature of this circuit is the nonlinear nature of the inductance.

When increasing the efficiency of portable betatrons through the use of increased frequency and stabilization of the supply voltage, their power is supplied from converters and devices with limited output power. Therefore, a change in the operating mode of the circuit, associated with the nonlinearity of its elements and high selectivity, when the frequency, waveform, and magnitude of the supply voltage change, may lead to the appearance of a reactive component

current through the source. As a result, the installed power of the converting device will increase, and its operating mode will be disrupted.

General properties of the linear circuit of a betatron electromagnet

The power-supply circuit of the betatron electromagnet is shown in Fig. 1, where
\(I_1, U_1\) are the current and voltage on the primary winding of the electromagnet,
\(I_2, U_2\), respectively, are those on the secondary winding,
\(L\) is the inductance of the betatron electromagnet,
\(C\) is the compensating capacitance.

Fig. 1. Power-supply circuit of the betatron electromagnet

In the betatron circuit, the parallel circuit is connected to the output circuit of the frequency converter.

When calculating the circuit, for simplicity the internal resistance of the power source may be neglected, i.e., we assume that \(R_i \ll Z_{0e}\), where
\(R_i\) is the internal resistance of the source,
\(Z_{0e}\) is the equivalent resistance of the circuit.
In this case it may be assumed that the generator voltage acts at the terminals of the circuit.

\[ U_2 = k_e U_1, \]

where \(K_e\) is the transformation ratio of the electromagnet windings.

Since the voltage at the circuit terminals does not depend on the frequency of the supply voltage (neglecting leakage inductance), the behavior of the current in the supply circuit may serve as an indicator of the tuning of the resonant circuit:

\[ I_2 = \frac{U_2}{Z_{0e}} = I_1 k_e, \tag{1} \]

\[ |Z_{0e}| = \frac{R_{0e}}{\sqrt{1+\xi^2}}, \tag{2} \]

\[ \xi = Q\nu = 2Q\frac{\Delta\omega}{\omega_c}, \tag{3} \]

where \(R_{0e}\) is the complex value of the equivalent resistance of the circuit at resonance,
\(\dfrac{\Delta\omega}{\omega_c}\) is the relative detuning of the circuit,
\(\omega_c\) is the frequency of the supply voltage,
\(Q\) is the quality factor of the circuit.

Then, taking (2) and (3) into account, expression (1) is written as

\[ I_2 = \frac{U_2}{R_{0e}}\sqrt{1 + 4Q^2\left(\frac{\Delta\omega}{\omega_c}\right)^2} \tag{4} \]

or

\[ I_2 = I_p \sqrt{1 + 4Q^2 \left(\frac{\Delta \omega}{\omega_c}\right)^2}. \tag{5} \]

This dependence describes the equation of the resonance curve with respect to current (Fig. 2).

Formula (5) makes it possible to estimate the overload of the power-supply source when the frequency of the supply voltage \(\Delta \omega\) and the quality factor of the circuit \(Q\) change, and thereby to determine the required frequency stability.

Fig. 2. Equivalent circuit for supplying the electromagnet, taking into account the internal resistance of the power-supply source

Fig. 3. Resonance curve of a parallel circuit

In the case of a source of limited power, having some resistance \(Z_i\), a change in frequency will also cause a change in the voltage on the electromagnet.

Indeed, in accordance with Fig. 3,

\[ U_2 = k_{\mathrm{e}} \frac{U_1}{Z_i + Z_{oe}} \cdot Z_{oe}, \]

\[ U_2 = k_{\mathrm{e}} \frac{U_1}{1 + \dfrac{Z_i}{Z_{oe}}}. \tag{6} \]

For a purely active internal resistance this expression is written as

\[ U_2 = k_{\mathrm{e}} \frac{U_1}{1 + \dfrac{R_i}{R_{oe}} \sqrt{1 + 4Q^2 \left(\dfrac{\Delta \omega}{\omega_c}\right)^2}}. \tag{7} \]

Nonlinearity of the betatron electromagnet circuit

In connection with the use of a ferromagnetic magnetic core, the inductance of the magnetizing windings changes when the supply voltage changes.

It is necessary to estimate how the power supplied to the circuit will change as a function of the supply voltage. In doing so, it is necessary to take into account the nonlinear nature of the change in inductance, since, from considerations of maximum loading of the magnetic core and minimum weight of the apparatus, the operating inductance of the electromagnet is chosen at the boundary of the linear region, close to saturation.

To solve the problem posed, it is necessary to find an analytical expression for \(\mu_{\mathrm{e}}\), the effective magnetic permeability, i.e.

an analytical expression for the magnetization curve \(H=f(B)\). For this it is necessary to approximate the latter.

There are very many methods of approximation and analytical expression of the magnetization curve.

Let us represent the latter in the form

\[ H=\kappa B^p \]

or

\[ H=H_{\mathrm{b}}\left(\frac{B}{B_{\mathrm{b}}}\right)^p, \tag{8} \]

where

\[ \begin{aligned} &H_{\mathrm{b}}\ \text{and}\ B_{\mathrm{b}} &&\text{are the amplitude values of the field strength and induction,}\\ &&&\text{adopted as basic in approximating the curve;}\\ &B &&\text{is the current coordinate of induction;}\\ &p &&\text{is a coefficient determined by the method of successive}\\ &&&\text{selection until the best agreement of the calculated curve}\\ &&&\text{with the experimental one is obtained;}\\ &B_{\mathrm{p}} &&\text{here and in subsequent calculations is the amplitude value}\\ &&&\text{of the working induction in the steel of the magnetic circuit}\\ &&&\text{under nominal conditions.} \end{aligned} \]

It is most rational to take \(p=5\). Then

\[ H=H_{\mathrm{b}}\left(\frac{B_{\mathrm{p}}}{B_{\mathrm{b}}}\right)^5. \tag{9} \]

With the nonlinear character of the variation of the inductance \(L\), it is necessary to introduce into the calculations the value of the effective inductance \(L_{\mathrm{eff}}\), which, of course, introduces some error, but is quite justified from the standpoint of simplifying the calculations and the rational use of the mathematical apparatus.

\[ L_{\mathrm{eff}}=\sqrt{\frac{1}{T}\int_0^T L^2\,dt}, \tag{10} \]

where \(L\) is the instantaneous value of the inductance when the supply voltage varies over a period, determined from the calculation scheme of the electromagnet (Fig. 4).

Fig. 4. Calculation scheme of the electromagnet

\[ L=\frac{w\Phi}{I}=\frac{w^2 S B_{\mathrm{p}}\sin\omega t}{F}\cdot 10^{-8}. \]

In the presence of an air gap, the magnetomotive force is

\[ F=0.8\,\delta_{\mathrm{eff}}B+Hl, \]

then

\[ L=w^{2}s\,\frac{B_{0}\sin\omega t}{0.8\,\delta_{\mathrm{eff}}B_{\mathrm{p}}\sin\omega t+H(t)l}, \tag{11} \]

where

\(w\)—number of turns of the magnetizing winding (turns),
\(s\)—cross section of the magnetic core \((\mathrm{cm}^{2})\),
\(\delta_{\mathrm{eff}}\)—value of the effective air gap \((\mathrm{cm})\),
\(B_{\mathrm{p}}\)—maximum value of the induction \((\mathrm{G})\),
\(l\)—mean length of the line of force \((\mathrm{cm})\),
\(H\)—value of the magnetic-field intensity of the electromagnet \((\mathrm{A}/\mathrm{cm})\), determined according to formula (9).

Thus, replacing \(H(t)\) according to (9) and carrying out certain transformations, we have

\[ L=\frac{w^{2}s}{0.8\,\delta_{\mathrm{eff}}\left(1+\dfrac{H_{б}lB^{4}}{0.8\delta_{\mathrm{eff}}B_{0}^{5}}\sin^{4}\omega t\right)}\cdot 10^{-8}. \]

If we denote

\[ L_{0}=\frac{w^{2}s}{0.8\,\delta_{\mathrm{eff}}}\cdot 10^{-8} \]

—the initial inductance of the electromagnet, then

\[ L=L_{0}\frac{1}{1+\dfrac{H_{б}l}{B_{б}0.8\delta_{\mathrm{eff}}}\left(\frac{B_{\mathrm{p}}}{B_{б}}\right)^{4}\sin^{4}\omega t}. \tag{12} \]

To simplify the notation, introduce the coefficient

\[ m=\frac{H_{б}l}{B_{б}0.8\delta_{\mathrm{eff}}}, \]

then

\[ L=L_{0}\frac{1}{1+m\left(\frac{B_{\mathrm{p}}}{B_{б}}\right)^{4}\sin^{4}\omega t}. \tag{13} \]

Substituting (13) into (10), we have

\[ L_{\mathrm{eff}}= \sqrt{ \frac{1}{T} \int_{0}^{T} \frac{L_{0}^{2}} {\left[1+m\left(\dfrac{B_{\mathrm{p}}}{B_{б}}\right)^{4}\sin^{4}\omega t\right]^{2}} \,dt }. \tag{14} \]

Carrying out the corresponding operations, we obtain

\[ L_{\mathrm{eff}}=L_{0}\left[1-\frac{3}{8}m\left(\frac{B_{\mathrm{p}}}{B_{б}}\right)^{4}\right]. \]

To shorten the notation, introduce the coefficient

\[ \kappa=\frac{0.37m}{B_{б}^{4}}, \]

then

\[ L_{\mathrm{eff}}=L_{0}\left(1-\kappa B_{\mathrm{p}}^{4}\right). \tag{15} \]

Or, passing to the supply voltage,

\[ L_{\mathrm{eff}}=L_{0}\left(1-\kappa U_{2}^{4}\right). \tag{16} \]

Formulas (15) and (16) show, respectively, the dependence of the change in the inductance of the betatron electromagnet on the supply voltage and on the induction.

It is now possible to determine the change in the current in the supply circuit of the betatron circuit \(I_2\) as a function of the change in the inductance of this circuit, which in turn is a function of the supply voltage \(U_2\) (under the condition \(L_{\text{leakage}} = \mathrm{const}\)), i.e., to find \(I_2 = f(U_2)\).

As is known, \(I_2 = \dfrac{U_2}{Z_{0e}}\), or, passing to admittances,

\[ I_2 = U_2 Y = U_2 (y_{0e} + j y_L - j y_c), \]

where

\[ y_L = \frac{1}{\omega_c L} \text{ — inductive admittance,} \]

\[ y_c = \omega_c C \text{ — capacitive admittance,} \]

\[ y_{0e} = \frac{1}{R_{0e}} \text{ — admittance of the circuit at resonance.} \]

At resonance the inductive and capacitive reactances are equal,

\[ 1/\omega_c L_p = \omega_c C, \]

whence

\[ C = \frac{1}{\omega_c^2 L_p}, \]

therefore

\[ I_2 = \frac{U_2}{R_{0e}} + j U_2 \left( \frac{1}{\omega_c L} - \frac{1}{\omega_c^2 L_p} \right). \]

Substituting into this formula the value for \(L\) from (16) and performing some transformations, we finally obtain

\[ I_2 = \frac{U_2}{R_{0e}} \left( 1 - j Q \kappa \frac{U_0^4 - U_2^4}{1 - \kappa U_2^4} \right). \tag{17} \]

Formula (17), together with formula (15), makes it possible to estimate the overload of the power-supply source of a resonant parallel circuit under fluctuations of the frequency and magnitude of the supply voltage.

\(I_2 = f(U_c)\) — experimental.

Fig. 5. Volt-ampere characteristic of a parallel-resonant circuit (experimental).

Taking into account the internal resistance of the power-supply source, it is possible to estimate the possibility of using the parametric method of stabilizing the output voltage \(U_2\) when the supply voltage \(U_c\) changes, by analogy with ferroresonant stabilizers.

When the line voltage changes, the current \(I_2\) in the wire feeding the resonant circuit will vary according to the law described by equation (17), Fig. 6. However, in this case the voltage drop across the internal resistance of the power source will change correspondingly, so that the output voltage can remain constant.

Fig. 6. Volt-ampere characteristics of a parallel resonant circuit (calculated)

\[ U_2 = U_c - I_2 Z_i . \]

For such a system, an expression has been obtained for the stabilization coefficient in the form

\[ K_{\text{st.max}} = 1 + \frac{2Q}{2.1\,\dfrac{B_0 \delta_{\text{ef}}}{H_0 l}\left(\dfrac{B_0}{B_p}\right)^4 - 1}. \tag{18} \]

Conclusion

Portable small-size betatrons are powered by an alternating sinusoidal voltage of industrial frequency \(50\ \text{Hz} \pm 0.2\%\). In this case, no special frequency-stabilization circuit is required.

However, when betatrons are powered by a voltage of increased frequency, i.e., from frequency converters, the stability of the latter must be no less than \(1\%\), since otherwise the installed power will vary by more than \(25\%\). This will lead to excessive weight of the converter, since the specific weight of semiconductor power supplies is approximately \(15\ \text{kg/kVA}\).

The introduction into the source circuit of a stabilizing link, in the presence of a master oscillator, will increase the weight only very slightly.

A change in the magnitude of the supply voltage leads to a considerable increase in the installed power of the betatron power supply because of the nonlinearity of the resonant circuit.

As calculation shows, for the PMB-5 betatron, when the voltage changes by 10%, the current of the power supply may change by approximately 40% owing to the nonlinear inductance of the resonant circuit; without taking nonlinearity into account, the power of the supply changes by approximately 20% solely owing to the 10% change in supply voltage.

Thus, owing to the combined effect, the calculated power of the supply will increase by roughly one and a half times, which, when calculated for this power, will sharply increase the weight of the apparatus.

In addition, the need to stabilize the value of the supply voltage is determined by the permissible change in the radiation intensity of the betatron.

If the voltage stability is approximately 3%, then the increase in source power, without taking nonlinearity into account, will be 7%, and owing to nonlinearity approximately 10%. The total increase in power will be no more than 20%.

As the calculations have shown, to stabilize the voltage of the power supply when the latter operates into a nonlinear resonant circuit, the parametric method may be used. Moreover, the stabilization coefficient of this nonlinear system depends substantially on the parameters of the magnetic core and will be the higher, the greater the quality factor of the resonant circuit and the higher the operating induction of the betatron electromagnet.

It should be noted that the calculation was carried out for the industrial frequency of 50 cps. When the frequency of the output voltage of the power supply is increased to 400–500 cps in order to increase the efficiency of the betatron, the tolerances with respect to voltage stability remain the same.

REFERENCES

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3. L. A. Bessonov. Self-oscillations in electrical circuits with steel. Gosenergoizdat, 1958.

4. B. K. Bul’. Fundamentals of the theory and calculation of magnetic circuits. Izd. “Energiya,” 1964.

5. I. S. Gonorovskii. Radio-engineering circuits and signals. Izd. “Sov. radio,” 1964.

6. B. P. Aseev. Oscillatory circuits. Svyazizdat, 1955.