On the Practical Feasibility of Transmitting High-Power High-Voltage Pulses Over Long Distances
Abstract Generated abstract
The paper examines whether powerful high-voltage pulses can be transmitted over kilometer-scale distances with acceptable attenuation and limited waveform distortion in a coaxial transmission system. It refines earlier distortionless-line calculations by including skin effect, frequency-dependent inductance and resistance, and increased dielectric conductance for steep pulse fronts, then evaluates attenuation, phase change, and propagation velocity for representative geometries. The calculations indicate that although real losses substantially increase attenuation relative to an idealized model, the distortionless-line condition remains nearly satisfied, with reported deviations of about 0.1 to 2 percent, and pulse attenuation can remain practically acceptable over several kilometers. Model experiments on refraction into media of varying conductivity and on front broadening support the calculated trends, suggesting that sufficiently steep pulses can still be delivered over distances of roughly 4 to 5 km.
Full Text
ON THE PRACTICAL FEASIBILITY OF TRANSMITTING POWERFUL HIGH-VOLTAGE PULSES OVER LONG DISTANCES
T. Yu. Mogilevskaya
(Presented by the scientific seminar of the Department of Theoretical Foundations of Electrical Engineering)
The search for the possibility of transmitting powerful high-voltage pulses over long distances with minimal attenuation and distortion of shape is one of the essential aspects of the general set of problems involved in the use of these pulses in high-current engineering. We have shown \([1, 2, 3]\) the possibility of creating a nondistorting line for transmitting such pulses. However, at the first stage of the work, in choosing the parameters of the nondistorting transmission system, we did not take into account their variation with increasing steepness of the pulse front as a result of the influence of the skin effect and the increase in the conductivity of the dielectric.
Fig. 1. Diagram of a coaxial transmission system
The attenuation coefficient of a pulse in the transmission system, under the idealization allowed, turned out to be negligibly small. The assumptions adopted are justified by the fact that changes in active resistance, conductance, and inductance with frequency are different \([4, 5, 6, 7]\) and to a considerable degree compensate one another, as is evident from the expressions given below.
The active resistance and inductance of a coaxial system (Fig. 1), per unit length and taking into account the skin effect, can be determined from the expressions \([6, 7]\):
\[ \left. \begin{aligned} R_0 &= 2\sqrt{\frac{f}{10}}\left(\frac{1}{d}\sqrt{\frac{\mu_{r_1}}{\gamma_1}}+\frac{1}{D}\sqrt{\frac{\mu_{r_2}}{\gamma_2}}\right)\cdot 10^{-3}\ \Omega/\mathrm{m},\\[6pt] L_0 &= \left[2\ln\frac{D}{d}+\frac{10}{\pi\sqrt{f\cdot 10^{-5}}}\left(\frac{1}{d}\sqrt{\frac{\mu_{r_1}}{\gamma_1}}+\frac{1}{D}\sqrt{\frac{\mu_{r_2}}{\gamma_2}}\right)\right]\cdot 10^{-7}\ \mathrm{H}/\mathrm{m}, \end{aligned} \right\} \tag{1} \]
where \(f\) is the frequency, determined from the length of the front \(\tau_\phi\) as
\[ f=\frac{1}{4\tau_\phi}; \]
\(\gamma_1\) and \(\gamma_2\) are the specific conductivities of the central conductor and the casing pipe;
\(\mu_{r_1}\) and \(\mu_{r_2}\) are their relative magnetic permeabilities.
The magnetic permeability of the ferromagnetic pipe \((\mu_2)\) has been taken as constant in the calculations, since the nonconstancy of \(\mu\) for any materials and in any fields has practically no effect on the magnitude of the internal reactance, and affects the magnitude of the active resistance only insignificantly, and moreover approximately equally for all materials [8].
Expressions (1) give the same result as the exact formulas in which the active resistance and inductance are expressed through special functions [4, 9].
The conductance in the case of high frequencies is determined from the expression [6, 7]
\[ g_0 = g_{\text{dc}} + g_{\sim} \approx g_{\sim} = \omega C_0 \operatorname{tg}\delta, \tag{2} \]
where
\(g_{\text{dc}}\) is the conductance at direct current;
\(g_{\sim}\) is the component of conductance determined by dielectric losses;
\(C_0\) is the capacitance of the system per unit length, calculated from the formula
\[ C_0 = \frac{\pi \varepsilon}{\ln \dfrac{D}{d}} . \tag{3} \]
The wave attenuation coefficient, under the condition \(R_0 \ll \omega L_0,\ g_0 \ll \omega C_0\), is determined by the formula [7, 10]
\[ \beta = \frac{R_0}{2}\sqrt{\frac{C_0}{L_0}} + \frac{g_0}{2}\sqrt{\frac{L_0}{C_0}}, \tag{4} \]
and the phase-change coefficient under the same condition [10] is
\[ \alpha = \omega \sqrt{L_0 C_0} \left[ 1 + \frac{1}{4} \left( R_0 \sqrt{\frac{C_0}{L_0}} - g_0 \sqrt{\frac{L_0}{C_0}} \right)^2 \right]. \tag{5} \]
Calculations we carried out for a number of variants showed that, under the influence of the surface effect and the increase in dielectric conductance with increasing pulse steepness, the attenuation coefficient, in comparison with the idealized value, increases considerably; however, the condition for a distortionless line,
\[ \frac{R_0}{L_0} = \frac{g_0}{C_0} \tag{6} \]
remains almost undisturbed (the error amounted, in the various variants, to \(0.1\)--\(2\%\)).
Figure 2 shows the dependence of the pulse amplitude on the distance it has traveled. Curve 1 corresponds to a system with electrode and pipe diameters of 50 mm and 200 mm, respectively, and curve 2 to diameters of 50 mm and 250 mm, i.e., with a larger ratio \(D/d\).
It follows from the curves presented that, when a pulse with front duration \(\tau_{\phi} = 0.5\ \mu\text{s}\) is transmitted along the first line over a distance of 5 km, the amplitude decreases by a factor of 3.76. When the same pulse is transmitted along the second line, its amplitude decreases over a length of 5 km by approximately a factor of two, and over a length of 10 km by a factor of 5.
Thus, the attenuation of a pulse in a real system whose parameters are chosen in accordance with the condition of a distortionless li-
of (6), proves to be, although not negligibly small, nevertheless quite acceptable in practice, especially since upon incidence on a dielectric or semiconductor the voltage amplitude is almost doubled [3].
We carried out experiments with specimens whose specific conductivity varied over a very wide range. In accordance with the magnitude of the conductivity, the coefficient of voltage refraction also changed.
\[ k_u=\frac{U_m}{U_{\phi m}}, \tag{7} \]
Fig. 2. Dependence of the relative attenuation of the pulse amplitude on the length of the transmission system.
Fig. 3. Dependence of the refraction coefficient \(K_u\) on the specific conductivity of the medium at the output of the coaxial system.
where \(U_{\phi m}\) is the voltage amplitude of the incident wave,
\(U_m\) is the amplitude of the refracted wave at the output of the coaxial system.
Figure 3 gives the curves \(k_u(\gamma)\), obtained by calculation for the experimental modeling setup (curve 1) and as a result of processing oscillograms (curve 2). The discrepancy between these curves lies within the limits of measurement error.
The quantity
\[ \left(R_0\sqrt{\frac{C_0}{L_0}}-g_0\sqrt{\frac{L_0}{C_0}}\right), \]
which enters into (5), proved in all cases to be of order \(10^{-8}\), i.e., \(\ll 1\). Therefore the phase-change coefficient, even with real conditions taken into account, is obtained the same as for a distortionless line and a lossless line:
\[ \alpha=\omega\sqrt{L_0C_0}, \tag{8} \]
and the velocity of wave propagation
\[ v=\frac{\omega}{\alpha}=\frac{1}{\sqrt{L_0C_0}} \]
does not depend on frequency, i.e., there should be no distortion of the wave shape.
As experiments carried out on the modeling setup showed, the wave front lengthens, when passing along a cable over a distance equal to the wavelength, by 10–15%. Consequently, when pulses with a front duration \(\tau_\phi=0.6\ \mu\text{s}\) are transmitted over a distance of 4–5 km, the front will lengthen by a factor of 2.3–3, that is, it will remain sufficiently steep.
References
-
T. Yu. Mogilevskaya. Calculation of the motion of a pulse along a single wire and of its transition into a semiconducting medium. Izv. vuz., Energetika, No. 6, pp. 24–29, 1959.
-
A. A. Vorob’ev, T. Yu. Mogilevskaya. Motion of a single unipolar voltage pulse along a coaxial cable with a ferromagnetic sheath and transition of the pulse into a semiconducting medium. Izv. vuz., Elektromekhanika, No. 7, pp. 3–9, 1959.
-
T. Yu. Mogilevskaya. Investigation of the motion of a unipolar voltage pulse along a single wire inside a ferromagnetic surface and of its transition into a dielectric or semiconducting medium. Dissertation, Sverdlovsk, 1960.
-
L. R. Neiman and P. L. Kalantarov. Theoretical Foundations of Electrical Engineering, Part III, GEI, 1954.
-
V. V. Kogen-Dolin. Propagation of pulses along a wire laid in laminated steel. Author’s abstract of Candidate’s dissertation, MEI, 1950.
-
I. I. Grodnev, R. M. Lakernik, and D. L. Sharle. Fundamentals of the Theory and Production of Communication Cables, GEI, 1956.
-
N. I. Belorussov, I. I. Grodnev. Radio-Frequency Cables. GEI, 1959.
-
L. R. Neiman. The Skin Effect in Ferromagnetic Bodies. GEI, 1949.
-
K. Kupfmüller. Fundamentals of Theoretical Electrical Engineering, GEI, 1960.
-
K. A. Krug. Fundamentals of Electrical Engineering, GEI, 1936.