On Estimating the Power in the Channel of a Pulsed Electric Discharge
Abstract Generated abstract
The paper addresses estimation of the maximum power developed in the channel of a pulsed electric discharge, a parameter linked to shock wave pressure and the destructive action of breakdown in dielectric media. Treating the post-breakdown circuit as an RLC discharge, the authors derive expressions relating peak channel power to stored pulse energy, inductance, capacitance, channel resistance, and the characteristic impedance, and identify a limiting power attained when the channel resistance equals the circuit impedance. An empirical approximation is proposed for practical estimates over different resistance ratios. Comparison with oscillographic measurements for pulsed breakdown of paraffin gives calculated peak power in satisfactory agreement with experiment, supporting the usefulness of the method for estimating discharge-channel power from circuit parameters.
Full Text
PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV
Vol. 149 1966
ON ESTIMATING THE POWER IN THE CHANNEL OF A PULSED ELECTRIC DISCHARGE
I. I. Kalyatskii, Yu. N. Leont’ev
(Submitted by the scientific seminar of the Research Institute
of High Voltages)
The processes that occur during pulsed electrical breakdown of a dielectric medium are similar to the processes arising in the explosion of charges of condensed explosives. Studies of pulsed discharge in gases [4, 6, 7, 8] have shown that the processes during expansion of the discharge channel can be described on the basis of the laws of hydrodynamics. Shock waves arising during electrical breakdown of a medium, under certain conditions, can lead to destruction of the solid medium. Shock waves arising during pulsed electrical breakdown are characterized by the pressure at the front of the moving wave.
In the work of A. S. Zingerman [3] it is shown that the magnitude of the pressure at the front of the shock wave in a pulsed discharge depends on the steepness of the pulse front, the specific energy content in the discharge channel, and the density of the medium. The power developed in the spark channel is one of the basic parameters that determine the destructive action of a pulsed discharge in a dielectric medium. The pressure at the front of the shock wave and, ultimately, the explosive action of the pulsed discharge depend on the magnitude of the power developed in the channel. During pulsed breakdown of solid dielectrics, a power develops in the discharge channel which, under certain conditions, can reach \(10^8 \div 10^9\) W [2].
According to the investigations of S. I. Andreev [1], the limiting power in the channel during a discharge in gases does not exceed 50 MW, i.e., it is an order of magnitude lower than during pulsed breakdown in solid dielectrics.
As is known, the power in the channel of a pulsed discharge is determined, under unchanged breakdown conditions \((L, C, U, W_0)\), by the magnitude of the resistance of the discharge channel—\(R_k(t)\). In [1] it is shown that, during pulsed breakdown of gases, the resistance of the discharge channel decreases very rapidly to a certain value and then remains almost unchanged until the end of the first half-period of the current oscillation. The active resistance of the discharge circuit may be several orders of magnitude smaller than the minimum resistance of the spark channel. The magnitude of the power in the discharge channel is determined by the resistance—\(R_k(t)\).
Let us consider the change in power in the channel of a pulsed discharge for different resistance—\(R_k(t)\).
Considering the oscillatory operating mode of the discharge circuit with parameters \((R, L, C)\) in the post-breakdown period, we have the following relations:
The magnitude of the current in the discharge channel
\[ I=\frac{U_0}{\omega L}\cdot e^{-at}\cdot \sin(\omega t), \tag{1} \]
when
\[ R<2\sqrt{\frac{L}{C}}, \]
where
\[ a=\frac{R}{2L}\quad \text{and}\quad \omega=\sqrt{\frac{1}{LC}-\frac{R^2}{4L^2}}, \]
\(L\) is the inductance of the discharge circuit,
\(R\) is the active resistance of the discharge circuit,
\(C\) is the discharge capacitance of the circuit.
The magnitude of the first current maximum can be determined as
\[ I_{\mathrm{m}}=U_0\sqrt{\frac{C}{L}}\cdot \exp\left[ -\frac{ m\cdot \arcsin \sqrt{1-\frac{m^2}{4}} }{ 2\sqrt{1-\frac{m^2}{4}} } \right], \tag{2} \]
where
\[ m=\frac{R}{R_{\mathrm{opt}}}, \]
\(R_{\mathrm{opt}}\) for a discharge circuit with parameters \((L, C, R)\) is a constant quantity and is equal to
\[ R_{\mathrm{opt}}=\sqrt{\frac{L}{C}}, \]
i.e., to the characteristic impedance of the circuit. The time at which the current in the discharge channel will be maximal is determined by the expression
\[ \tau_{\mathrm{m}}= \frac{ \sqrt{LC}\cdot \arcsin \sqrt{1-\frac{m^2}{4}} }{ \sqrt{1-\frac{m^2}{4}} }. \tag{3} \]
Let us note that the time of the power maximum and the time of the current maximum, as will be shown experimentally below, coincide:
\[ \tau_{\mathrm{m}}(i)=\tau_{\mathrm{m}}(N). \]
From this, the maximum power in the channel of a pulsed discharge is
\[ N_{\mathrm{m}}=I_{\mathrm{m}}^2\cdot R_{\mathrm{k}}, \tag{4} \]
where \(R_{\mathrm{k}}\) is the resistance of the pulsed-discharge channel. Using (2) and (4), we obtain
\[ N_{\mathrm{m}}=\frac{U_0^2 C}{L}\cdot R_{\mathrm{k}}\cdot f^2(m) \]
or
\[ N_{\mathrm{m}}=\frac{2W_0\cdot R_{\mathrm{k}}}{L}\cdot f^2(m), \tag{5} \]
where \(W_0\) is the total energy of the pulse and
\[ f^2(m)=\exp\left[-\frac{m\cdot \arcsin\sqrt{1-\dfrac{m^2}{4}}}{\sqrt{1-\dfrac{m^2}{4}}}\right]. \]
The curve of the dependence \(f^2(m)\) is shown in Fig. 1. The maximum possible (limiting) power that can occur in the channel of a pulsed discharge with resistance \(R_k\) will take place when
\[ R_k=R_{\mathrm{opt}}=\sqrt{\frac{L}{C}}, \]
i.e., when \(m=1\). In this case, as is seen from the graph in Fig. 1, \(f^2(m)=0.3\). Hence the limiting power in the discharge channel may be expressed as
\[ N_{\mathrm{p}}=0.6\,\frac{W_0}{\sqrt{LC}} . \tag{6} \]
Fig. 1. Graphical dependence of the function \(f^2(m)\).
Thus, with the known parameters of the discharge circuit \((R,\ L,\ C)\) and the pulse energy \(W_0\), it is possible to estimate the maximum power in the discharge channel and to follow its variation for different values of
\[ m=\frac{R}{R_{\mathrm{opt}}}. \]
The maximum power in the discharge channel can be expressed in relative units as
\[ \frac{N_{\mathrm{m}}}{N_{\mathrm{p}}} = \frac{2\cdot m}{0.6}\cdot f^2(m). \tag{7} \]
The curve of the dependence
\[ \frac{N_{\mathrm{m}}}{N_{\mathrm{p}}}=f(m) \]
is shown in Fig. 2. From the graph in Fig. 2 it is seen that, for any values \(m\ne 1\), the ratio of the maximum power in the channel to the limiting power is always less than unity,
\[ \frac{N_{\mathrm{m}}}{N_{\mathrm{p}}}<1. \]
Fig. 2. Variation of the relative power in the discharge channel for different ratios of the active resistance to the surge impedance of the circuit.
For any values \(R\gtrless R_{\mathrm{opt}}\), the power in the channel of the pulsed discharge is always less than the limiting value,
\[ N_{\mathrm{m}}<N_{\mathrm{p}}. \]
For \(m \leq 2\) we have
\[ R \leq 2R_{\mathrm{opt}}, \qquad R \leq 2\sqrt{\frac{L}{C}}, \]
i.e., the conditions for the oscillatory mode of the circuit; consequently, the left branch of the curve in Fig. 2 corresponds to the oscillatory mode of operation of the circuit.
For \(m \geq 2\) we have
\[ R \geq 2R_{\mathrm{opt}}, \qquad R \geq 2\sqrt{\frac{L}{C}}, \]
i.e., the right branch of the curve in Fig. 2 corresponds to the aperiodic mode of the circuit.
The obtained dependence
\[ \frac{N_{\mathrm{m}}}{N_{\mathrm{p}}}=f(m) \]
can, for practical estimates of the power in the channel of a pulsed discharge, be represented with satisfactory accuracy in the form of the empirical formula
\[ \frac{N_{\mathrm{m}}}{N_{\mathrm{p}}} = 1.18\,m\left(e^{-\frac{m}{8}}-e^{-3m}\right) \tag{8} \]
or
\[ N_{\mathrm{m}} = 0.707\,m\, \frac{W_0}{\sqrt{L\cdot C}} \left(e^{-\frac{m}{8}}-e^{-3m}\right). \tag{9} \]
Fig. 3. Change in the relative power in the discharge channel at different discharge capacitances in the circuit.
The data in the graph of Fig. 1 and Fig. 2 and formulas (6), (7), given the dependence of the channel resistance on the magnitude of the overvoltage at breakdown,
\[ R_{\mathrm{k}}(t)=f\left(\frac{U_{\mathrm{p}}}{U_{\mathrm{pr}}}\right), \]
can be used to estimate the power developed by a pulsed discharge in the channel during breakdown of a dielectric medium.
Figure 3 gives the dependence of the relative power on the magnitude of the discharge capacitance, $\dfrac{N_{\mathrm{m}}}{N_{\mathrm{p}}}=f(c)$, for various values of $R_{\mathrm{k}}$. From the data of Fig. 3 it is seen that the optimum value of the relative power shifts toward smaller $C$ as the resistance $R_{\mathrm{k}}$ increases. These curves were obtained by calculation for the conditions of pulsed breakdown: $U_0=240\cdot10^3\ \mathrm{V}$, $L=16.5\cdot10^{-6}\ \mathrm{H}$.
The procedure described above for estimating the maximum power developed in the channel of a pulsed discharge was used to compare calculated values with data obtained experimentally.
Fig. 4. Change in the parameters of the discharge channel during the first half-period of current oscillations:
a) voltage — $U(t)$, b) current — $I(t)$, c) channel resistance — $R_{\mathrm{k}}(t)$,
d) power in the discharge channel — $N(t)$.
Figure 4 shows the change in voltage, $U(t)$; current in the discharge channel, $I(t)$; channel resistance, $R_{\mathrm{k}}(t)$; and power in the channel, $N(t)$, during pulsed electric breakdown of paraffin of thickness $d=1.5\ \mathrm{cm}$. The capacitance in the discharge was $C=0.00166\cdot10^{-6}\ \mathrm{F}$, the inductance of the circuit $L=10.5\cdot10^{-6}\ \mathrm{H}$, the voltage $U_0=240\cdot10^3\ \mathrm{V}$, and the pulse energy $W_0=48\ \mathrm{W\,s}$. For these paraffin breakdown conditions the limiting power is $N_{\mathrm{p}}=218\ \mathrm{MW}$.
As is seen from the data of Fig. 4, the maximum power determined from the oscillograms $U(t)$ and $I(t)$ is, in this case, $N_{\mathrm{m}}=28\cdot10^6\ \mathrm{W}$, and the minimum channel resistance is $R_{\mathrm{k}}=3\ \Omega$. Figure 4 also shows that the time of the power maximum differs little from the time of the current maximum.
If one uses the minimum value of the resistance of the discharge channel \(R_{\mathrm{k}} = 3\ \Omega\) and estimates \(\dfrac{N_{\mathrm{m}}}{N_{\mathrm{p}}}\) from (7), then we obtain
\[ \frac{N_{\mathrm{m}}}{N_{\mathrm{p}}} = 0.12 \quad N_{\mathrm{m}} = 0.12 N_{\mathrm{p}}, \]
i.e.,
\[ N_{\mathrm{m}} = 26.1 \cdot 10^6\ \mathrm{W}, \]
which is in satisfactory agreement with the experimental data.
Conclusions
- A method has been considered for estimating the maximum power developed in the discharge channel from the known parameters of the discharge circuit. An empirical formula has been proposed for estimating the maximum power in the channel from the ratio of the minimum resistance of the discharge channel to the characteristic impedance of the circuit.
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