INDUCTIVE PARAMETRIC GENERATOR AS AN EXCITER OF THE RESONANT CIRCUIT OF A BETATRON

V. M. VYSOTSKAYA

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

In the field of physical sciences, inductive accelerators of charged particles—betatrons—have in recent years become widespread, giving Soviet researchers a powerful means for obtaining high-energy electrons.

The efficiency of accelerator installations depends to a considerable degree on the frequency of the alternating current that feeds them. The best use of an installation can be obtained if it is provided with a special generator whose alternating-current frequency corresponds to the given installation.

It is known that the power circuit of the betatron electromagnet constitutes a resonant circuit. The active power consumed by the circuit, due to losses in the winding of the electromagnet, in the steel of the magnetic core, and in the capacitor bank, amounts to no more than 5% of the reactive power of the circuit. Consequently, the betatron electromagnet in practice consumes only reactive power.

For compensation of the reactive power, a corresponding capacitance is connected into the supply circuit of the electromagnet. The capacitor bank and the magnetizing winding form an oscillatory circuit tuned to resonance at the frequency of the alternating current feeding the electromagnet.

Fig. 1

The investigations we carried out [2] showed that it is convenient and economically advantageous to excite the resonant circuit of the betatron electromagnet by means of an inductive parametric generator system; for this purpose, a variable inductance, which performs the role of the exciting generator, is sufficiently included in the resonant circuit of the betatron electromagnet.

The electrical circuit of an inductive-type parametric generator is shown in Fig. 1. Here \(L_o\) is a parametric generator, representing

its variable inductance. In the circuit of a parametric generator, in addition to the active resistance \(R\), a constant capacitance \(C\) is included.

One of the most important design features of such a generator is the absence of a winding on the rotor, which permits the use of higher peripheral speeds. In this case the machines are obtained with a small machine constant \(C\), i.e., with a small weight of active materials per unit of power, and with a very simple design. The significance of this most important advantage of the parametric induction generator is reduced by the presence of the obligatory capacitance in the generator circuit, necessary for tuning the circuit into resonance. Therefore, the region of higher frequencies is more advantageous for parametric generators, since in this case the dimensions and cost of the capacitors required to create resonance in the circuit of the parametric generator decrease.

The chief disadvantage of parametric generators is the low power factor of the circuit. In addition, the generator voltage varies with changes in load, and under emergency conditions in the load circuit the parametric generator completely loses excitation.

The listed disadvantages, and chiefly the low power factor, hinder the use of parametric generators for supplying industrial installations.

But in some special cases, when the reduced power factor is of no importance or is naturally unavoidable and capacitors have to be used to improve \(\cos \varphi\), i.e., to create a resonant circuit, parametric generators can be used successfully, especially at increased frequency, owing to the simplicity of their construction.

In this case the operating conditions of the parametric generator differ substantially from those occurring in ordinary power installations.

The presence of a ready resonant circuit, in which the parametric generator can be used as the exciter of the circuit, makes insignificant one of the serious disadvantages of parametric generators—the reduced power factor. On the contrary, in installations of this kind, which require reactive energy, the smallest possible power factor is desirable, since losses are thereby reduced.

When the parametric generator rotates, an alternating current arises and is maintained in its circuit, which is an oscillatory circuit, under certain conditions, with a frequency half as large as the frequency of variation of the parameter \(L\). For simplicity it is convenient to make the assumption, close to the truth, that the variable part of the inductance of the parametric generator varies according to a sinusoidal law.

Fig. 2 shows the curve of harmonic variation of the inductance of the parametric generator. Here \(L_0\) is the mean value of the inductance, \(L_n\) is the amplitude of its variable part. As is seen from Fig. 2,

\[ L=\frac{L_{\max}+L_{\min}}{2} \quad \text{and} \quad L_n=\frac{L_{\max}-L_{\min}}{2}. \]

Let us denote by \(m_e\) the depth of modulation of the inductance of the parametric generator, i.e., the relative change of its parameter \(L\)

\[ m_e=\frac{L_{\max}-L_{\min}}{L_{\max}+L_{\min}}=\frac{2L_n}{2L_0}. \tag{1} \]

When the winding of the betatron electromagnet is included in the circuit of an inductive parametric generator, being in this case an additional

(with the operating) inductance \(L_{\mathrm{work}}\) (Fig. 3), the modulation-depth coefficient of the entire circuit \(m_k\) will decrease, namely:

\[ m_k = \frac{(L_{2\,\max}+L_{\mathrm{work}})-(L_{2\,\min}+L_{\mathrm{work}})} {(L_{2\,\max}+L_{\mathrm{work}})+(L_{2\,\min}+L_{\mathrm{work}})} = \frac{L_{2\,\max}-L_{2\,\min}} {L_{2\,\max}+L_{2\,\min}+2L_{\mathrm{work}}}; \]

using equation (1), we find that

\[ m_k=\frac{2L_p}{2L_o+2L_{\mathrm{work}}} = \frac{L_p}{L_o+L_{\mathrm{work}}}. \]

Taking the ratio of the modulation-depth coefficient of the parametric generator \(m_2\) to the modulation-depth coefficient of its entire circuit \(m_k\), when the additional inductance \(L_{\mathrm{work}}\) is included, we obtain

\[ \frac{m_2}{m_k} = \frac{2L_p(L_o+L_{\mathrm{work}})}{2L_o \cdot L_p} = \frac{L_o+L_{\mathrm{work}}}{L_o}. \tag{2} \]

Fig. 2

Fig. 3

Thus,

\[ L_o = L_{\mathrm{work}}\cdot \frac{m_k}{m_2-m_k}. \]

This equation (2) makes it possible to determine the necessary inductance that the parametric generator must have when a specified operating inductance \(L_{\mathrm{work}}\) is connected, so that, at the minimum permissible value of \(m_k\), the possibility of exciting its circuit is preserved. The experiments we carried out showed that if an operating inductance \(L_{\mathrm{work}}\), reducing the circuit modulation-depth coefficient to the value \(m_k=0.12\), is included in the circuit of the parametric generator, then the parametric generator is still capable of self-exciting its operating circuit. If, with the generator in the excited state, the operating inductance is smoothly increased, then the generator continues to operate also at \(m_k=0.1\), but after it is switched off and switched on again it does not self-excite. In order to make it possible to excite the parametric generator also at \(m_k=0.1\), we proposed an electrical connection circuit for the parametric-generator circuit (Fig. 4), which, by means of a switch \(p\), makes it possible at the moment the parametric generator is switched on to shunt the operating inductance. In this case the parametric generator self-excites easily, after which the shunting conductor is opened and the parametric generator continues to operate, since the indicated circuit makes it possible to preserve resonance conditions when the circuit parameters change.

It is clear that, to excite a circuit with inductance \(L=L_o+L_{\mathrm{work}}\), the capacitance required by the resonance conditions will be smaller.

The difference between the values \(m_k=0.1\) and \(m_k=0.12\) is substantial, since in the case \(m_k=0.1\) it is possible to excite ra-

the side circuit of the parametric generator with a total inductance 20% greater than in the case \(m_k = 0.12\).

The investigations we carried out [2] showed that it is advisable to build inductive-type parametric generators with a modulation-depth coefficient \(m_z\) of no more than 0.5. It was indicated above that the smallest value of the modulation-depth coefficient \(m_k\) at which it is still possible to excite the working circuit of the parametric generator is 0.1, or 10%.

Fig. 4

The ratio of the inductance of the parametric generator \(L_o\) to the working inductance \(L_{\text{work}}\) for the limiting conditions, i.e., for \(m_z = 0.5\) and \(m_k = 0.1\), will be

\[ \frac{L_o}{L_{\text{work}}} = \frac{m_k}{m_z - m_k} = \frac{0.1}{0.5 - 0.1} = 0.25. \]

Consequently, under the stated conditions the inductance of the parametric generator \(L_o\) is 25% of the working inductance \(L_{\text{work}}\).

Thus, the nominal apparent power of the parametric generator, provided stable operation is ensured, must amount to 25% of the reactive power consumed by the working inductance, i.e., 25% of the reactive power of the betatron electromagnet.

References

1. Mandelshtam L. I. and Papaleksi N. D. On parametric excitation. Journal of Technical Physics, vol. 4, 1934.

2. Vysotskaya V. M. Development and study of a parametric generator for use in supplying the winding of a betatron electromagnet. Candidate dissertation, TPI, 1953.