STABILIZATION OF THE POWER SUPPLY OF THE ELECTROMAGNET OF A PULSED BETATRON UNDER QUASI-TRIANGULAR CURRENT PULSES

B. A. Baginskii, Yu. A. Otrubyannikov

(Submitted by the Department of Industrial and Medical Electronics)

Studies carried out in recent years in the field of designing small-sized betatrons [1, 2, 3] have demonstrated the advantages of various methods of pulsed power supply in comparison with continuous supply. At the same time, the structure of stabilization systems for supplying electromagnets has changed substantially and has required special investigations and developments.

The present work is devoted to stabilization of the magnetic field of an accelerator when the current in the electromagnet windings has a quasi-triangular form [2]. For such circuits, the known method of field stabilization may be applied, consisting in regulating the magnitude of the energy introduced. However, such stabilization systems are automatic regulation systems with respect to the deviation relative to the maximum value of the induction. In this case [4], in order to obtain high stability, a large gain factor of the feedback circuit is required. In the present case, stabilization of the maximum value of the magnetic field can be accomplished by the required switching instant of the corresponding valves. Linking the switching instant to a prescribed field level determines the energy \(W_{\text{p}}\) consumed by the electromagnet circuit (the energy of losses). The amount of energy introduced, \(W_{\text{in}}\), depends on the supply-line voltage, the parameters of the input circuit, and, without stabilization, may vary over wide limits. If the energy losses during a current pulse are less than the introduced energy, this will lead to an increase in the voltage on the circuit capacitance, an increase in the rate of current rise in the electromagnet winding into which the capacitance is discharged, and to an increase in losses. The transient process will end when \(W_{\text{in}} = W_{\text{p}}\). An increase in voltage may lead to an emergency operating condition. To determine the magnitude of the overvoltage, it is necessary to carry out an energy analysis and to formulate requirements for the method of stabilization. Energy losses in a betatron consist of losses in the capacitors and losses in the electromagnet. Energy losses in the capacitors, in the ranges of frequencies and voltages in which most pulsed betatrons operate, are determined mainly by the magnitude of the alternating component of the voltage [3]. When the voltage on the circuit capacitance is increased, the alternating component at constant induction \(B_m\) decreases; therefore the losses in the capacitors do not increase.

The total losses in the magnet consist of losses in the copper and losses in the steel:

\[ P=P_{\mathrm{m}}+P_{\mathrm{st}}. \tag{1} \]

For quasi-triangular current pulses in the magnetizing winding [5],

\[ P_{\mathrm{m}}= \frac{2\pi R}{\sqrt{3S}}\rho j I_m w \left[ 1+\frac{n^2-1}{9}\left(\frac{d}{\Delta}\right)^4 \right]. \tag{2} \]

where \(R\) is the mean radius of the winding;
\(w\) is the number of turns of the winding;
\(d\) is the radial dimension of the winding conductor;
\(n\) is the number of winding layers;

\[ \Delta=\frac{68}{\sqrt{f_y}} \]

is the depth of current penetration in copper;
\(f_y\) is the equivalent frequency of the current pulses;

\[ S=\frac{f_y}{f_{\mathrm{p}}} \]

is the duty factor of the current pulses;
\(f_{\mathrm{p}}\) is the repetition frequency of the acceleration cycles.

The quantity that depends on the voltage on the circuit capacitance is, in our case, the duty factor \(S\). For analysis we simplify expression (2):

\[ P_{\mathrm{m}}=l\frac{1}{\sqrt{3S}}\left(1+mS^2\right). \tag{3} \]

The coefficient \(m\) takes into account the influence of additional losses due to the skin effect, the proximity effect, and also eddy currents. Such a notation is possible because all the indicated components depend on the square of the frequency and, consequently, on the duty factor as well. Expanding (3) in a Taylor series and retaining two terms, we find the relative changes of the quantities:

\[ \frac{\Delta P_{\mathrm{m}}}{P_{\mathrm{m}}} = K_s\frac{\Delta S}{S}; \qquad K_s= \frac{2mS^2}{1+mS^2}-\frac{1}{2}. \tag{4} \]

The losses in steel consist of losses due to eddy currents and hysteresis:

\[ P_{\mathrm{st}}=P_{\mathrm{st.v}}+P_{\mathrm{st.g}} = \frac{1}{3S}\gamma f_y^2\delta^2 B_m^2 V + \frac{1}{2}\eta f_r B_m^2 \tag{5} \]

where

\(\gamma\) is the specific conductivity of the material,
\(\delta\) is the sheet thickness,
\(V\) is the sheet volume,
\(\eta\) is a coefficient,
\(B_m\) is the maximum induction in the electromagnet.

The hysteresis losses are determined by the area of the loop and by the number of remagnetization cycles per unit time and, consequently, do not depend on the duty factor. Analyzing (5), we have:

\[ \frac{\Delta P_{\mathrm{st}}}{P_{\mathrm{st}}} = \frac{\Delta S}{S'}. \tag{6} \]

Thus, with an increase in the input energy, its compensation at \(B_{\max}=\mathrm{const}\) must occur through a change in the losses in copper and the eddy-current losses in steel:

\[ \Delta P_{\mathrm{vv}}=\Delta P_{\mathrm{m}}+\Delta P_{\mathrm{st.v}}. \]

Using expressions 4 and 6, we obtain

\[ \frac{\Delta W_{\mathrm{vv}}}{W_{\mathrm{vv}}} = \frac{\Delta P_{\mathrm{vv}}}{P_{\mathrm{vv}}} = (aK_s+b)\frac{\Delta S}{S}, \tag{7} \]

where

\[ a=\frac{P_{\mathrm{m}}}{P_{\Sigma\mathrm{nom}}}; \qquad b=\frac{P_{\mathrm{st.v}}}{P_{\Sigma\mathrm{nom}}}. \]

Taking the tank capacitance to be a voltage source (which does not introduce a significant error when operating with a low ripple level and a small commutation angle), one may write

\[ I_m=\frac{U_{\mathrm{sk}}}{L_k}\cdot t_y; \tag{8} \]

hence

\[ \frac{\Delta U_{\mathrm{sk}}}{U_{\mathrm{sk}}} = \frac{\Delta S}{S}. \tag{9} \]

From equations (7,9)

\[ \frac{\Delta U_{\mathrm{sk}}}{U_{\mathrm{sk}}} = \frac{\Delta W_{\mathrm{vv}}/W_{\mathrm{vv}}}{aK_s+b}. \tag{10} \]

It follows from this that overvoltages on the tank capacitance may be substantial. Thus, at \(m=0.1\), \(a=0.4\), \(b=0.2\) (sample data for the MIB-6), for \(\frac{\Delta W_{\mathrm{vv}}}{W_{\mathrm{vv}}}=25\%\), the voltage on the tank capacitance \(C_k\) will double. The smaller the fraction of \(P_\Sigma\) accounted for by copper losses and the smaller \(P_{\mathrm{st.v}}\), the greater will be the possible excess of \(U_{\mathrm{sk}}\) for this method of stabilizing \(B_{\max}\). The approximate calculations and the results obtained above make it possible to conclude that it is practically impossible to build an equalization system for a current-pulse generator with an unstabilized input and precise fixation of the field maximum. This may lead to inadmissibly large overvoltages on the tank circuit and to failure of the switching devices. Even if an increase in losses compensates for the excess of input energy at a voltage level permissible for the switching devices used (thyristors and diodes), such a regime is undesirable because of the unjustified increase in losses.

To obtain high field stability and to exclude possible emergency modes, it is necessary, as indicated, to form the pulse for triggering the commutation thyristors according to the field; however, in doing so it is necessary to stabilize the magnitude of the input energy.

REFERENCES

1. Yu. A. Otrubyannikov. Some problems of pulsed power supply for electromagnets of induction accelerators. Dissertation. Tomsk, 1967.

1. V. V. Ivashin. Current commutation in circuits for producing magnetic fields and in electrical machines. Dissertation. Tomsk, 1968.

2. E. G. Furman. Development and investigation of circuits for pulsed power supply of accelerator electromagnets. Dissertation. Tomsk, 1972.

3. M. V. Meerov. Synthesis of structures of high-precision automatic control systems. Moscow, “Nauka,” 1967.

4. Yu. P. Yarushkin et al. Development and investigation of pulsed power-supply systems for a small betatron. Scientific report of the Research Institute of Nuclear Physics, Electronics, and Automation at TPI, 1972.