UDC 537.52

TORCH DISCHARGE AS A LINE WITH DISTRIBUTED PARAMETERS

I. A. TIKHOMIROV, V. V. TIKHOMIROV, V. S. LEVASHOV

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

An equivalent circuit of a torch discharge is considered. From an analysis of the equations for a line with distributed parameters, a method is proposed that makes it possible, from experimentally measured input resistance, frequency, and discharge length, as well as from the attenuation coefficient of the current in the discharge, to determine the capacitance, inductance, and active resistance of the discharge. Satisfactory agreement between the results of electrical and microwave measurements is shown.

Table 1, references 12.

At present, industrial rf self-oscillators with oscillation frequencies of 10–40 MHz [1–2] are used to supply power to high-frequency plasmatrons of the torch type. In this case, direct connection of the plasmatron to the rf generator leads to detuning of its oscillatory circuit, because the electrical parameters of the torch discharge are added to the parameters of the generator circuit. Therefore, for effective use of the generator–plasmatron system it is necessary to know the equivalent electrical parameters of the discharge.

A shielded high-frequency torch discharge (HFTD) in molecular gases is a plasma formation with a clearly defined channel, whose temperature is considerably higher than that of the envelope [3] and whose length is much greater than its transverse dimensions. Developing the thesis of [4], where it is asserted that the HFTD channel may be regarded as a thin wire guiding an electromagnetic wave and absorbing power from this wave, and that the spatial distribution of fields in the HFTD is well described by Bessel functions in the channel and Hankel functions outside it, we shall represent the equivalent load of the rf generator in the form of a shielded HFTD as a line with distributed parameters; i.e., the active resistance, capacitance, and inductance are not concentrated at one point, but are distributed along the length of the HFTD channel. In this case, at high powers supplied to the discharge, it is necessary to take into account, in contrast to Neumann [5], not only the active resistance and capacitance of the HFTD, but also the inductance of the discharge, since the geometry of the discharge in a metallic plasmatron is comparable with the geometry of the anode inductance of the rf generator.

If one uses the equations [6] for a coaxial line describing the variation of current and voltage in space and time, then, as applied to the conditions under consideration, we obtain the following relations:

\[ \frac{\partial^2 U}{\partial x^2} = L_0 C_0 \frac{\partial^2 U}{\partial t^2} + (C_0 R_0 + G_0 L_0)\frac{\partial U}{\partial t} + G_0 R_0 U, \]

\[ \frac{\partial^2 i}{\partial x^2} = L_0 C_0 \frac{\partial^2 i}{\partial t^2} + (C_0 R_0 + G_0 L_0)\frac{\partial i}{\partial t} + G_0 R_0 i, \tag{1} \]

the solutions of which, in the case of sinusoidal time dependences, will be the complexes

\[ \dot U = U_0 e^{j\omega t-\gamma x} \quad \text{and} \quad \dot I = I_0 e^{j\omega t-\gamma x}. \tag{2} \]

Here \(R_0\) is the active resistance of the HFD channel, \(L_0\) is the inductance, and \(C_0\) is the capacitance of the “discharge channel—plasmatron” system; \(G_0\) is the active conductance between the discharge channel and the plasmatron body. The quantities \(R_0\), \(L_0\), \(C_0\), and \(G_0\) are taken per unit length of the coaxial line formed by the plasmatron body and the HFD channel. The entire line may be regarded as a set of small elements \(dx\) connected in series.

The propagation constant of the electromagnetic wave, \(\gamma\), may be written in the form

\[ \gamma = \pm(\alpha + i\beta) = \pm \sqrt{(R_0 + j\omega L_0)(G_0 + j\omega C_0)}, \tag{3} \]

where \(\omega\) is the field frequency, \(\alpha\) is the attenuation coefficient, and \(\beta\) is the phase constant, or phase coefficient.

If the torch discharge is excited in a metallic plasmatron inside which a quartz tube is placed, then with sufficient accuracy it may be asserted that

\[ G_0 = 0. \tag{4} \]

In addition, from a comparison of the results of probe and microwave measurements carried out by us [7, 8] and the results of works [9, 10], one may conclude that

\[ R_0^2 \gg (\omega L_0)^2. \tag{5} \]

Using expressions (4) and (5), after some transformations one can obtain the following expressions for the attenuation coefficient:

\[ \alpha = \pm \sqrt{\frac{1}{2}(\omega C_0 R_0 - \omega^2 L_0 C_0)} \tag{6} \]

and for the wave resistance of the torch

\[ Z_b = \sqrt{L_0/C_0}. \tag{7} \]

A section of coaxial line of length \(l\) must have a natural frequency which, when conditions (4) and (5) are satisfied, will have the form

\[ \omega_{\mathrm{p}} = \frac{\pi n}{l} \sqrt{\frac{1}{L_0 C_0} - \frac{l^2 R_0^2}{2\pi n^2 L_0^2}}. \tag{8} \]

From expression (8) it may be noted that for a shielded HFD \(l \sim \omega^{-1}\), in contrast to an unshielded one, where \(l \sim \omega^{-1/2}\); i.e., at constant power considered in the HFD plasma, the length of its channel is established such that the natural frequency of the HFD in the plasmatron is equal to the generation frequency, i.e., condition (8) is satisfied, where \(\omega_{\mathrm{p}} = \omega_{\mathrm{gen}}\).

From a comparison of equations (6), (7), and (8), one may conclude that, in the presence of a sufficient amount of experimental data, by solving these equations with respect to \(L_0\), \(R_0\), and \(C_0\), one can obtain infor-

information on the magnitude of the active resistance, inductance, and capacitance of the HFD.

Thus, the new method we propose for determining \(R\), \(C\), and \(L\) is as follows.

The frequency of the rf field, the length of the HFD channel, and also its input resistance are measured; according to [4], the latter is equal to its wave impedance (the ratio of the potential difference between the discharge channel and the shielding cylinder to the current in the discharge). The attenuation coefficient \(\alpha\) is determined from expression (2) for a known distribution of the current along the length of the discharge, which is found by the procedure described in our paper [11]. The desired parameters are determined from the solution of the system of equations (6), (7), (8).

To verify the proposed method, microwave probing of the discharge was carried out simultaneously with electrical measurements, i.e., the values of the electron concentration \(n_e\), averaged over the discharge diameter, and of the effective collision frequency \(\nu_{\mathrm{eff}}\) were determined. Knowing \(n_e\) and \(\nu_{\mathrm{eff}}\), for \(\omega_{\mathrm{gen}}^2 \ll \nu_{\mathrm{eff}}^2\), one can determine the specific electrical conductivity

\[ \sigma = \frac{e^2 \cdot n_e}{m \cdot \nu_{\mathrm{eff}}}, \tag{9} \]

and then, for the known discharge geometry, calculate the specific active resistance of the discharge.

The setup we developed for carrying out microwave measurements, as well as the method for processing the experimental data, are described in [12].

The measurements were performed at atmospheric pressure in an air plasma.

The results of the measurements and calculations are presented in Table 1.

Table 1

Capacitance, inductance, and active resistance of the torch discharge

From a comparison of the values of \(R_0\) calculated from the results of electrical measurements and \(R_{0\ \mathrm{svch}}\) from microwave measurements, one may conclude that the discrepancy does not exceed 20%; i.e., the method proposed by us is suitable for determining the electrical parameters of the HFTD. The overestimated values of \(R_{0\ \mathrm{svch}}\) are explained by the fact that in microwave measurements the parameters determined are those averaged over the probing path, rather than those pertaining directly to the discharge channel.

As can be seen from the table, not only the equivalent capacitance of the torch discharge but also its inductance are comparable with the parameters of the oscillatory circuit of an h.f. generator; therefore they must be taken into account in calculating and designing torch plasmatrons for industrial h.f. generators.

REFERENCES

1. G. N. Kolpakov, V. L. Teploukhov, I. A. Tikhomirov, K. N. Yugai. In: Generators of Low-Temperature Plasma. Moscow, Energiya, 1969, p. 341.

2. I. A. Tikhomirov, V. M. Savel’ev, V. L. Teploukhov, G. N. Kolpakov, V. V. Tikhomirov, K. N. Yugai, A. P. Kutlin, V. V. Tsoi, V. P. Dryamov. In: Physics, Technology, and Application of Low-Temperature Plasma. Alma-Ata, 1970, p. 702.

3. A. Grigorovici, D. Cristescu. Optics and Spectroscopy, 6, 129 (1959).

4. A. V. Kachanov, E. S. Trekhov, E. P. Fetisov. In: Physics of Gas-Discharge Plasma. Issue 1, Moscow, Atomizdat, 1968, pp. 39, 48.

5. M. S. Neiman. Proceedings of the Electrical Industry of Weak Current, No. 7, 1 (1935).

6. K. Simonyi. Theoretical Electrical Engineering. Translated from German, edited by K. M. Polivanov. Moscow, Mir, 1964.

7. I. A. Tikhomirov, V. V. Tikhomirov, V. Ya. Fedyanin. Proceedings of TPI, vol. 225, Tomsk, TPI Publishing House, 1972, p. 89.

8. I. A. Tikhomirov, V. V. Tikhomirov, V. V. Marusin. In: Problems in the Physics of Low-Temperature Plasma. Minsk, Nauka i Tekhnika, 1970, p. 189.

9. A. A. Kuzovnikov, N. A. Kaptsov. Proceedings of Higher Educational Institutions, Physics, No. 6, 64 (1960).

10. A. Talsky. Czechoslovak Journal of Physics, SB. 14, 594 (1964).

11. I. A. Tikhomirov, V. V. Tikhomirov, N. A. Moshnenko, V. Ya. Fedyanin. In: Physics, Technology, and Application of Low-Temperature Plasma. Alma-Ata, 1970, p. 709.

12. I. A. Tikhomirov, V. V. Tikhomirov, A. A. Solov’ev, V. I. Shishkovskii. Proceedings of TPI, vol. 276. TSU Publishing House, 1975, p. 26.