Assessment of Insulation Operating Conditions in a Coaxial Line During Pulse Generator Operation into a Spark
Abstract Generated abstract
This paper addresses the estimation of voltages imposed on the insulation of a high-voltage coaxial line when a pulsed-voltage generator discharges into a spark load. It develops a calculation procedure based on operational circuit equations, traveling-wave reflections, time-dependent spark resistance, and an approximate Duhamel-integral tabular method for determining voltages at the beginning, end, and intermediate points of the line. An example for a 30 m coaxial system is compared with model oscillograms, indicating that multiple reflections can produce complex alternating-sign voltage waveforms with several peaks exceeding the initial pulse amplitude.
Full Text
ON ASSESSING THE OPERATING CONDITIONS OF INSULATION IN A COAXIAL LINE DURING OPERATION OF A PULSED GENERATOR ON A SPARK
I. I. Kalyatskii, A. T. Chepikov, A. A. Dulzon
(Presented by the scientific seminar of the Research Institute of High Voltages)
At present the field of application of pulsed-voltage generators (PVG) is very broad. They are used in radiolocation, for supplying pulsed light sources, for generating shock waves in a liquid medium, for the kinetics of chemical reactions, etc.
When high-voltage pulses with amplitudes up to 200–300 kV, intended for the technological use of their energy, are transmitted, there arises an acute need to assess the voltage acting on the insulation of the coaxial line. This is necessary for a well-founded design of the insulation and for determining its service life.
On calculating the voltage on the insulation of a coaxial line
The calculation scheme is presented in Fig. 1. We have adopted the following principal assumptions:
- The active resistance of the conductors and of the spark gaps of the PVG is \(R_1 = \text{const}\). This is adopted because it has a comparatively small effect
Fig. 1: \(a\) — calculation scheme; \(b\) — equivalent circuit; \(C\) — capacitance of the PVG in the discharge; 1 — parasitic inductance of the PVG; \(R_1\) — total resistance of the circuit (taking into account the resistance of the spark gaps); \(R_2(t)\) — resistance of the spark load; \(Z_{\mathrm{в}}\) — wave impedance of the lines; 1 — length of the line.
on the transient process up to the instant of arrival of the wave reflected from the end of the line, and during this time, even for a line length of 10–20 m, the resistance of the spark gaps of the PVG becomes approximately constant.
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The resistance of the spark load \(R_2(t)\) is taken only as a function of time, and not of voltage.
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Losses in the line are not taken into account. Allowing for attenuation of the pulses in the line does not change the course of the calculation, but greatly complicates the computations.
The operational form of Kirchhoff’s law (according to Carson) for the circuit of Fig. 1,b is:
\[ U_C(p)+U_L(p)+U_{R1}(p)+U_{x=0}(p)=0. \tag{1} \]
The current at the beginning of a lossless line of length \(l\) [1] is
\[ I_{x=0}(p)=U_{x=0}(p)\cdot \frac{ R_2(p)\operatorname{sh}\frac{l}{v}+z_{\mathrm{в}}\operatorname{ch}\frac{l}{v} }{ z_{\mathrm{в}}\left[ R_2(p)\operatorname{ch}p\frac{l}{v}+z_{\mathrm{в}}\operatorname{sh}p\frac{l}{v} \right] }. \tag{2} \]
On the other hand,
\[ i(t)=C\cdot\frac{dU_C}{dt}\div CpU_C(p)-pCU_C(0)=I(p)=I_{x=0}(p), \tag{3} \]
\[ U_L(t)=L\frac{di}{dt}\div p^2LCU_C(p)-p^2CLU_C(p), \tag{4} \]
\[ U_{R1}=I(p)\cdot R_1. \tag{5} \]
On the basis of (1)—(5) one can obtain
\[ U_{x=0}(p)=-U_C(0)\cdot \frac{pCZ(p)} {p^2LC+pR_1C+pCZ(p)+1}, \tag{6} \]
where
\[ Z(p)= \frac{ z_{\mathrm{в}}\left[ R_2(p)\operatorname{ch}p\frac{l}{v} +z_{\mathrm{в}}\operatorname{sh}p\frac{l}{v} \right] }{ R_2(p)\operatorname{sh}p\frac{l}{v} +z_{\mathrm{c}}\operatorname{ch}p\frac{l}{v} }. \tag{7} \]
Then the voltage at any point \(x\) of the long line [1] is
\[ U_x(p)=U_{x=0}(p)\cdot \frac{ R_2(p)\operatorname{ch}p\left(\frac{l}{v}-\frac{x}{v}\right) +z_{\mathrm{в}}\operatorname{sh}p\left(\frac{l}{v}-\frac{x}{v}\right) }{ R_2(p)\operatorname{ch}p\frac{l}{v} +z_{\mathrm{в}}\operatorname{sh}p\frac{l}{v} }. \tag{8} \]
In the general case it is not possible to find the original of expression (8). When the PGI operates into a matched line, i.e., when
\[ R_2=z_{\mathrm{в}}, \]
\[ U_x(p)=U_{01}(p)=-U_C(0)\cdot\frac{z_{\mathrm{в}}}{L}\, \frac{p}{p^2+p\frac{R_1+z_{\mathrm{в}}}{L}+\frac{1}{LC}} \tag{9} \]
or, when
\[ \frac{1}{LC}>\frac{(R_1+z_{\mathrm{в}})^2}{4L^2}, \]
\[ U_{01}(t)=U_C(0)\cdot\frac{z_{\mathrm{в}}}{L}\cdot\frac{1}{\omega}\cdot e^{-\frac{a}{2}t}\cdot\sin\omega t, \tag{10} \]
where
\[ a=\frac{R_1+z_{\mathrm{в}}}{L},\qquad \omega=\sqrt{\frac{1}{LC}-\frac{a^2}{4}}. \]
To calculate the voltage across the insulation of a coaxial line of finite length, we used the traveling-wave method [2]. The reflection coefficient at the end of the line is
\[ \beta(t)=\frac{R_2(t)-z_{\mathrm{w}}}{R_2(t)+z_{\mathrm{w}}}. \tag{11} \]
The refraction coefficient at the beginning of the line is
\[ \alpha(p)=\frac{2Z_1(p)}{Z_1(p)+z_{\mathrm{w}}}, \qquad \text{where } Z_1(p)=pL+\frac{1}{pC}+R_1. \tag{12} \]
The time for a double passage of the wave along the line is
\[ 2t_{\mathrm{pr}}=\frac{2l}{v}, \]
where \(l\) is the line length,
\(v\) is the velocity of propagation of electromagnetic waves.
Fig. 2. Voltage across the insulation of the coaxial line at point 1
The voltage at the beginning of the line (point 1) is determined as the sum of the refracted waves
\[ U'_1(t)=U'_{01}(t)+U^{\mathrm{pr}}_{21}(t), \tag{13} \]
where \(U^{\mathrm{pr}}_{21}(t)\) is the wave refracted at point 1 after reflection from point 2.
\[ U^{\mathrm{pr}}_{21}(t)=U'_{21}(p)\cdot \alpha(p), \tag{14} \]
where
\[ U'_{21}(p)\div U'_{21}(t)=U_{21}(t-2t_{\mathrm{pr}}), \]
and
\[ U'_{21}(t)=\beta(t)\cdot U_{12}(t), \tag{15} \]
where \(U_{12}(t)\) is the total wave traveling from point 1 to point 2.
The quantity \(U_{21}^{\mathrm{pr}}\) can be calculated approximately on the basis of the following considerations.
If we set \(U'_{21}(p)=1\), then \(U_{21}^{\mathrm{pr}}(p)=\alpha(p)\). Consequently, \(\alpha(p)\) is the transient function for point 1. Then, having found \(h(t)\doteq \alpha(p)\), \(U_{21}^{\mathrm{pr}}(t)\) can be calculated using the Duhamel integral [1].
\[ U_{21}^{\mathrm{pr}}(t)=\int_{0}^{t}\frac{dU'_{21}(\tau)}{d\tau}\cdot h(t-\tau)\cdot d\tau . \tag{16} \]
Replacing the integral by a sum, we obtain approximately [3]
\[ U_{21}^{\mathrm{pr}}(t)\approx \sum_{i=1}^{N}\Delta U'_{21i}\cdot h(t-\tau_i), \tag{17} \]
where \(N\) is the number of intervals into which the part of the wave \(U'_{21}(t)\) of interest to us is divided,
\(\Delta U'_{21i}\) is the increment of the wave during the \(i\)-th interval.
Having found \(U_1\) from (13)—(17), we can obtain
\[ U_{12}(t)=U_1(t)-U'_{21}(t). \tag{18} \]
Then the voltage at the end of the line (point 2) is
\[ U_2(t)=U_{12}(t)+U_{21}(t). \tag{19} \]
The voltage in the middle of the line (point 3) is
\[ U_3(t)=U_{12}(t)+U''_{21}(t), \]
where
\[ U''_{21}(t)=U_{21}(t-t_{\mathrm{pr}}). \]
Analogously, the voltage at any intermediate point can be found.
Fig. 3. The same, at point 2.
Example. Find the voltage on the insulation of a coaxial system of length \(l=30\ \text{m}\), with wave impedance \(Z_3=54\ \Omega\). The capacitance of the impulse-voltage generator in the discharge is \(C=20000\ \text{pF}\), the parasitic inductance is \(L=25\ \mu\text{H}\), and the active resistance of the conductors and spark gaps is \(R_1\approx 2\ \Omega\), \(nU_3=240\ \text{kV}\) (\(n\) is the number of stages, \(U_3\) is the charging voltage).
In accordance with (10),
\[ U_{01}(t)=605\cdot e^{-1.12t}\cdot \sin 0.857t, \]
where \(t\) is time in \(\mu\text{s}\).
For impulse-generator capacitances in the discharge of the order of \(10000\text{–}30000\ \text{pF}\) and voltages of \(100\text{–}300\ \text{kV}\), the time dependence of the resistance of the spark in a liquid or solid dielectric can be expressed approximately by the formula [4]:
\[ R(t)=10+\frac{0.6}{t-t_0},\quad (t\geq t_0), \tag{21} \]
where \(t_0\) is the instant of dielectric breakdown, \(\mu\text{s}\).
We assume that breakdown occurs at the instant when the voltage at point 2 reaches its maximum. In this case the largest overvoltages occur. The refraction and reflection coefficients are
\[ \beta(t)=\frac{0.6-44(t-t_0)}{0.6+64(t-t_0)};\quad \alpha(t)=2\left(1-2.54\cdot e^{-1.12t}\cdot \sin 0.857t\right). \]
Table 1
| $t$, μsec | $U_{01}$ Formula (10) $A_2$ |
$U_{21}^{\mathrm{pr}}$ Table 2 $A_3$ |
$U_1$ $A_1 + A_2$ $A_4$ |
$U'_{21}$ $A_8(t-0,3)$ $A_5$ |
$U_{12}$ $A_4 - A_5$ $A_6$ |
$\beta$ Formula (11) $A_7$ |
$U_{21}$ $A_6 \cdot A_7$ $A_8$ |
$U_2$ $A_6 + A_8$ $A_9$ |
$U''_{21}$ $A_8(t-0,15)$ $A_{10}$ |
$U_3$ $A_9 + A_{10}$ $A_{11}$ |
|---|---|---|---|---|---|---|---|---|---|---|
| 0 | 0 | — | 0 | — | 0 | 1 | 0 | 0 | — | 0 |
| 0,15 | 65 | — | 65 | — | 65 | 1 | 65 | 130 | 0 | 65 |
| 0,3 | 110 | 0 | 110 | 0 | 110 | 1 | 110 | 220 | 65 | 175 |
| 0,45 | 137 | 117 | 254 | 65 | 189 | 1 | 189 | 378 | 110 | 299 |
| 0,6 | 153 | 165 | 318 | 110 | 208 | 1 | 208 | 416 | 189 | 397 |
| 0,75 | 156 | 266 | 422 | 189 | 233 | −0,59 | −138 | 95 | 208 | 441 |
| 0,9 | 153 | 234 | 387 | 208 | 179 | −0,64 | −114 | 65 | −138 | 41 |
| 1,05 | 145 | −388 | −243 | −138 | −105 | −0,65 | 68,2 | −37 | −114 | −219 |
| 1,2 | 135 | −221 | −86 | −114 | 28 | −0,66 | −18,5 | 9,5 | 68,2 | 96 |
| 1,35 | 123 | 176 | 299 | 68 | 231 | −0,67 | −154 | 77 | −18,5 | 212,5 |
| 1,5 | 108 | −21 | 87 | −18,5 | 106 | −0,67 | −71 | 35 | −154 | −48 |
| 1,65 | 94 | −231 | −137 | −154 | 17 | −0,67 | −11,4 | 5,6 | −71 | −54 |
| 1,8 | 80 | 0,8 | 80,8 | −71 | 151,8 | −0,67 | −102 | 50 | −11,4 | 140,4 |
Table 2
| \(i\) | \(\tau_i\) | \(\Delta U'_{21i}\) | \multicolumn{11}{c}{Transition function \(h(t-\tau_i)\) at \(t\) equal to} | \multicolumn{11}{c}{\(\Delta U'_{21i}\,h(t-\tau_i)\) at \(t\) equal to} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | | | 0.75 | 0.6 | 0.75 | 0.9 | 1.05 | 1.2 | 1.35 | 1.5 | 1.65 | 1.8 | | 0.45 | 0.6 | 0.75 | 0.9 | 1.05 | 1.2 | 1.35 | 1.5 | 1.65 | 1.8 |
| 1 | 0.35 | 24 | 1.62 | 1.2 | 0.91 | 0.76 | 0.69 | 0.69 | 0.74 | 0.84 | 0.94 | 1.06 | | 38.9 | 28.8 | 21.8 | 18.2 | 16.5 | 16.5 | 17.8 | 20.2 | 22.6 | 25.4 |
| 2 | 0.40 | 21 | 1.8 | 1.31 | 0.99 | 0.80 | 0.70 | 0.69 | 0.72 | 0.80 | 0.90 | 1.02 | | 37.8 | 27.5 | 20.8 | 16.8 | 14.7 | 14.5 | 15.1 | 16.8 | 18.9 | 21.4 |
| 3 | 0.45 | 20 | 2.0 | 1.44 | 1.08 | 0.85 | 0.72 | 0.69 | 0.70 | 0.83 | 0.87 | 0.98 | | 40.0 | 28.8 | 21.6 | 17.9 | 15.1 | 14.5 | 14.7 | 16.6 | 17.4 | 19.6 |
| 4 | 0.50 | 17 | | 1.62 | 1.2 | 0.91 | 0.76 | 0.69 | 0.69 | 0.74 | 0.84 | 0.94 | | | 27.6 | 20.4 | 15.5 | 12.9 | 11.7 | 11.7 | 12.6 | 14.3 | 16 |
| 5 | 0.55 | 15 | | 1.8 | 1.31 | 0.99 | 0.80 | 0.70 | 0.69 | 0.72 | 0.80 | 0.90 | | | 27 | 19.7 | 14.9 | 12.0 | 10.5 | 10.4 | 10.8 | 12 | 13.5 |
| 6 | 0.60 | 13 | | 2.0 | 1.44 | 1.08 | 0.85 | 0.72 | 0.69 | 0.70 | 0.83 | 0.87 | | | 26 | 18.7 | 14.0 | 11.1 | 9.4 | 9.0 | 9.1 | 10.8 | 11.3 |
| 7 | 0.65 | 28 | | | 1.62 | 1.2 | 0.91 | 0.76 | 0.69 | 0.69 | 0.74 | 0.84 | | | | 45.3 | 33.6 | 25.4 | 21.2 | 19.3 | 19.3 | 20.7 | 23.5 |
| 8 | 0.70 | 27 | | | 1.8 | 1.31 | 0.99 | 0.80 | 0.70 | 0.69 | 0.72 | 0.80 | | | | 49.6 | 35.4 | 26.7 | 21.6 | 18.9 | 18.6 | 19.4 | 21.6 |
| 9 | 0.75 | 24 | | | 2.0 | 1.44 | 1.08 | 0.85 | 0.72 | 0.69 | 0.70 | 0.83 | | | | 48 | 34.6 | 26 | 20.4 | 17.3 | 16.6 | 16.8 | 19.9 |
| 10 | 0.80 | 9 | | | | 1.62 | 1.2 | 0.91 | 0.76 | 0.69 | 0.69 | 0.74 | | | | | 14.6 | 10.8 | 8.2 | 6.9 | 6.2 | 6.3 | 6.7 |
| 11 | 0.85 | 5 | | | | 1.8 | 1.31 | 0.99 | 0.80 | 0.70 | 0.69 | 0.72 | | | | | 9.0 | 6.6 | 4.9 | 4.0 | 3.5 | 3.5 | 3.8 |
| 12 | 0.90 | 5 | | | | 2.0 | 1.44 | 1.08 | 0.85 | 0.72 | 0.69 | 0.70 | | | | | 10.0 | 7.2 | 5.4 | 4.2 | 3.6 | 3.5 | 3.5 |
| 13 | 0.95 | -283 | | | | | 1.62 | 1.2 | 0.91 | 0.76 | 0.69 | 0.69 | | | | | | -458 | -340 | -257 | -215 | -195 | -195 |
| 14 | 1.00 | -53 | | | | | 1.8 | 1.31 | 0.99 | 0.80 | 0.70 | 0.69 | | | | | | -95.3 | -69.4 | -52.5 | -42.4 | -37.1 | -36.6 |
| 15 | 1.05 | -10 | | | | | 2.0 | 1.44 | 1.08 | 0.85 | 0.72 | 0.69 | | | | | | -20 | -14.4 | -10.8 | -8.5 | -7.2 | -6.9 |
| 16 | 1.10 | 6 | | | | | | 1.62 | 1.2 | 0.91 | 0.76 | 0.69 | | | | | | | 9.7 | 7.2 | 5.5 | 4.6 | 4.1 |
| 17 | 1.15 | 9 | | | | | | 1.8 | 1.31 | 0.99 | 0.80 | 0.70 | | | | | | | 16.2 | 11.8 | 8.9 | 7.2 | 6.3 |
| 18 | 1.20 | 10 | | | | | | 2.0 | 1.44 | 1.08 | 0.85 | 0.72 | | | | | | | 18.0 | 13.0 | 9.7 | 7.7 | 6.5 |
| 19 | 1.25 | 100 | | | | | | | 1.62 | 1.2 | 0.91 | 0.76 | | | | | | | | 162 | 120 | 91 | 76 |
| 20 | 1.30 | 54 | | | | | | | 1.8 | 1.31 | 0.99 | 0.80 | | | | | | | | 97.2 | 70.7 | 53.4 | 43.2 |
| 21 | 1.35 | 28 | | | | | | | 2.0 | 1.44 | 1.08 | 0.85 | | | | | | | | 56 | 40.3 | 30.2 | 23.8 |
| 22 | 1.40 | -18 | | | | | | | | 1.62 | 1.2 | 0.91 | | | | | | | | | -29.2 | -21.6 | -16.4 |
| 23 | 1.45 | -25 | | | | | | | | 1.8 | 1.31 | 0.99 | | | | | | | | | -45 | -32.8 | -24.8 |
| 24 | 1.50 | -45 | | | | | | | | 2.0 | 1.44 | 1.08 | | | | | | | | | -90 | -64.8 | -48.6 |
| 25 | 1.55 | -40 | | | | | | | | | 1.62 | 1.2 | | | | | | | | | | -65 | -48 |
| 26 | 1.60 | -50 | | | | | | | | | 1.8 | 1.31 | | | | | | | | | | -90 | -65.5 |
| 27 | 1.65 | -39 | | | | | | | | | 2.0 | 1.44 | | | | | | | | | | -78 | -56.1 |
| 28 | 1.70 | 19 | | | | | | | | | | 1.62 | | | | | | | | | | | 30.8 |
| 29 | 1.75 | 30 | | | | | | | | | | 1.8 | | | | | | | | | | | 54 |
| 30 | 1.80 | 34 | | | | | | | | | | 2.0 | | | | | | | | | | | 68 |
| \(U_{21}^{\mathrm{pr}}(t)=\displaystyle\sum_{i=1}^{N}\Delta U'_{21i}\,h(t-\tau_i):\) | | | | | | | | | | | | | | 117 | 165 | 266 | 234 | -388 | -221 | 176 | -21 | -231 | 0.8 |
Double transit time along the line
\[ 2t_{\mathrm{tr}}=\frac{2l}{v}=\frac{2\cdot 30}{200}=0.3\ \mu\mathrm{s}. \]
The calculation of the voltages \(U_1(t)\), \(U_2(t)\), and \(U_3(t)\) is carried out by a tabular method (Table 1), with the entries filled in successively, line by line.
Fig. 4. Oscillograms of voltage on the line insulation:
\(a\)—at the beginning of the line, \(b\)—at the end of the line, \(c\)—at the beginning of the line in the absence of breakdown at the end. Calibration frequency \(11\ \mathrm{MHz}\).
by line. The calculation of the value \(U_{21\mathrm{tr}}\) by means of the Duhamel integral is given in Table 2.
The voltage curves at the beginning and at the end of the coaxial line obtained as a result of the calculation are shown in Fig. 2 (curves 1). The voltage (curves 2) is also presented there for the case in which the spark gap at the end of the line did not break down. In the case of no-load operation of the line, when \(2t_{\text{pr}}\) is much smaller than the duration of the initial pulse \(U_{01}(t)\), the voltage can be estimated approximately if, in the circuit of Fig. 1,b, the line is replaced by a lumped capacitance of about \(100\ \text{pF/m}\) (curve 3 for \(R_1 = 2\ \Omega\) and curve 4 for \(R_1 = 30\ \Omega\)).
To check the calculations, the circuit of Fig. 1,b was modeled, with \(U_c(0)\) taken equal to \(18\ \text{kV}\). A spark gap in insulating oil was connected at the end of the line. The oscillograms obtained are shown in Fig. 4. The difference between the voltage curves at the end of the line (Fig. 4b and 4c) is explained by the fact that the spark gap in the modeled circuit is many times smaller than in the actual installation, and consequently the residual voltage on the spark is also smaller.
Conclusions
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A method has been considered for determining the voltages acting on the insulation at various points of a high-voltage coaxial transmission system, taking into account multiple reflections of the pulse.
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For aperiodic pulses with a wavelength comparable with the double transit time, the insulation of the coaxial system is subjected to voltages of a complex alternating-sign form with a large number of peaks exceeding the amplitude of the initial pulse.
References
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I. I. Teumin. Experimental Analysis of Transient Processes in Linear Electrical Circuits. “Sov. Radio,” 1956.
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A. I. Gavrilin, I. I. Kalyatskii, A. G. Sinebryukhov. Investigation of the energy characteristics of pulsed breakdown of solid dielectrics. Breakdown of Dielectrics and Semiconductors. Proceedings of the Fourth Interuniversity Conference, Tomsk. “Energiya,” 1964.