Study of the Current of the Channel Stage of a Long Spark on a Model

Student N. A. Shcherbakova

Supervisor: Professor, Doctor I. S. Stekolnikov

(Presented to the scientific seminar on dielectrics)

The works of a number of authors have provided detailed material for the quantitative characterization of the process of the leader, channel, and arc stages of a high-voltage discharge obtained from a pulse-voltage generator.

The results obtained revealed a number of dependences of the current amplitude of the channel stage of a spark on the damping resistance, the length of the discharge gap, the grounding resistance of the current collector, and also the relation between the current amplitude and the maximum steepness of the current wave.

However, until recently no attempt had been made to calculate the current parameters of a long spark. To solve this question it proved necessary to have a computational circuit corresponding to the representation of the physical phenomena occurring in a high-voltage discharge.

In the laboratory of high-voltage gas discharge of the Power Engineering Institute of the Academy of Sciences of the USSR, a model of the equivalent circuit of the channel stage of a long spark, proposed by I. S. Stekolnikov, was implemented.

The calculation of the equivalent circuit had previously been carried out by a staff member of this laboratory, M. A. Tyumina. The numerical values of the model were chosen on the basis that the processes in the equivalent circuit of the channel stage can be described by the equations of a long line [1]:

\[ \frac{\partial U_{\mathrm{orig}}}{\partial x_{\mathrm{orig}}} = R_{\mathrm{orig}}\, i_{\mathrm{orig}} + L_{\mathrm{orig}}\, \frac{\partial i_{\mathrm{orig}}}{\partial t_{\mathrm{orig}}}, \tag{1} \]

\[ \frac{\partial i_{\mathrm{orig}}}{\partial x_{\mathrm{orig}}} = C_{\mathrm{orig}}\, \frac{\partial U_{\mathrm{orig}}}{\partial t_{\mathrm{orig}}}. \]

For the model of the equivalent circuit the same equations are valid:

\[ \frac{\partial U_{\mathrm{mod}}}{\partial x_{\mathrm{mod}}} = R_{\mathrm{mod}}\, i_{\mathrm{mod}} + L_{\mathrm{mod}}\, \frac{\partial i_{\mathrm{mod}}}{\partial t_{\mathrm{mod}}}, \tag{2} \]

\[ -\frac{\partial i_{\text{mod}}}{\partial x_{\text{mod}}} = C_{\text{mod}}\frac{\partial U_{\text{mod}}}{\partial t_{\text{mod}}}. \]

In order that the processes in the two cases be similar, the similarity conditions are necessary:

\[ \begin{gathered} \frac{U_{\text{orig}}}{U_{\text{mod}}}=U_0;\qquad \frac{x_{\text{orig}}}{x_{\text{mod}}}=x_0;\qquad \frac{R_{\text{orig}}}{R_{\text{mod}}}=R_0;\qquad \frac{i_{\text{orig}}}{i_{\text{mod}}}=i_0,\\[6pt] \frac{L_{\text{orig}}}{L_{\text{mod}}}=L_0;\qquad \frac{t_{\text{orig}}}{t_{\text{mod}}}=t_0;\qquad \frac{C_{\text{orig}}}{C_{\text{mod}}}=C_0, \end{gathered} \tag{3} \]

where \(R_{\text{orig}}, R_{\text{mod}}, C_{\text{orig}}, C_{\text{mod}}, L_{\text{orig}}, L_{\text{mod}}\) are the resistance, capacitance, and inductance per unit length of the original and of the model.

Using relations (3), we express all quantities entering equation (2) in terms of the quantities of equation (1):

\[ -\frac{\partial U_{\text{orig}}}{\partial x_{\text{orig}}} = \frac{R_{\text{orig}}U_0}{R_0x_0}\cdot \frac{i_{\text{orig}}}{i_0} + \frac{L_{\text{orig}}U_0t_0}{L_0x_0i_0}\cdot \frac{di_{\text{orig}}}{dt_{\text{orig}}}, \]
\[ -\frac{\partial i_{\text{orig}}}{\partial x_{\text{orig}}} = \frac{L_{\text{orig}}i_0}{L_0x_0}\cdot \frac{t_0}{U_0}\cdot \frac{\partial U_{\text{orig}}}{\partial t_{\text{orig}}}. \tag{4} \]

Equations (1) and (2) reflect similar processes only in the case when equation (4) coincides with equation (1); for this it is necessary that the following relations hold between the quantities characterizing the phenomenon and the model:

\[ \frac{U_0}{R_0x_0i_0}=1, \tag{5} \]

\[ \frac{U_0t_0}{L_0x_0i_0}=1, \tag{6} \]

\[ \frac{L_0x_0U_0}{i_0t_0}=1. \tag{7} \]

\(\dfrac{U_0}{i_0}=R_0x_0\) is eliminated from equations (6) and (7); then two similarity criteria are obtained:

\[ \frac{t_0}{C_0x_0^2R_0}=1, \tag{8} \]

\[ \frac{t_0R_0}{L_0}=1, \tag{9} \]

where \(C_0, t_0, x_0, R_0, L_0\) are similarity coefficients.

The obtained relations (8) and (9) are the sought similarity criteria. Since the number of unknown coefficients is five, and there are two equations, three coefficients may be assigned arbitrarily.

We set:

\[ C_0=1,\quad L_0=1,\quad x_0=\frac{1}{k}=\frac{1}{3500}. \tag{10} \]

Then, from the similarity criteria, \(R_0\) and \(t_0\) are found:

\[ R_0=\frac{1}{x_0}=k;\quad t_0=\frac{1}{R_0};\quad t_0=x_0=\frac{1}{3500}. \tag{11} \]

Proceeding from conditions (10), (11) and the similarity conditions (3), we find the relation between the total resistance of the spark channel of the original and of the model:

\[ R_{\text{orig total}}=R_{\text{orig}}x_{\text{orig}} =R_{\text{mod}}R_0x_{\text{mod}}x_0 =R_{\text{mod}}kx_{\text{mod}}\frac{1}{k} =R_{\text{mod}}x_{\text{mod}} =R_{\text{mod total}}. \]

Similarly:

\[ C_{\text{orig total}}=C_{\text{orig}}x_{\text{orig}} =C_{\text{mod}}C_0x_{\text{mod}}x_0 =\frac{C_{\text{mod}}x_{\text{mod}}}{k} =\frac{1}{k}C_{\text{mod total}}; \quad \text{and} \quad L_{\text{orig total}}=\frac{1}{k}L_{\text{mod total}}. \]

The circuit shown in Fig. 1 approximately models the processes in a discharge gap of the “rod–plane” type [2].

The spark channel is represented here by a chain circuit of \(n\) cells, each of which consists of a self-inductance \(L\), representing the inductance of the spark channel, \(C\)—the capacitance of the channel relative to the grounded plane \((P_2)\) and the measuring plane \((P_1)\), and \(R_l\)—the active resistance of the channel in the leader stage (Fig. 2). In parallel with \(R_l\), thyratrons are connected, whose purpose is to shunt, at the corresponding instants of time, \(R_l\) by other resistances characteristic of the end of the channel or arc stage of the spark.

Fig. 1. Circuit for modeling a rod–plane discharge.

Fig. 2. Schematic distribution of current in the main channel.

The capacitance \(C_p\) and the two self-inductances \(L_p\), connected at the beginning of the chain, model the capacitance and inductance of the upper electrode and of the wires leading to it. The successive switching-on of the thyratrons imitates the advance of the head of the channel stage of a long spark. Through the resistance \(R_{\text{sh}}\) and the capacitor \(C_1\), which models the capacitance between the planes \(P_1\) and \(P_2\), current will flow only from the second group of cells, which corresponds to the current component \(I_1\) (Fig. 2). The presence of \(C_1\) substantially reduces the current in \(R_{\text{sh}}\).

The purpose of the present study was to obtain, on the model, a wave of the channel current coinciding in its parameters with the wave of the channel current of the original for given conditions in the spark gap, and to investigate a number of relationships between individual parameters for a spark. Among the latter are the following relationships:

1. The dependence of the amplitude of the channel current on the magnitude of the grounding resistance (or the resistance of the current collector).

2. Study of the dependence of the amplitude of the channel current on the magnitude of the capacitance of the cells of the chain circuit.

3. The dependence of the front time of the channel-current wave on the capacitance of the cells.

4. The dependence of the amplitude of the channel current and the front time of the wave on the magnitude of the capacitance of the high-voltage electrode \((I_k = f(C_p))\), and \(I_k = f(n_l/n_p)\).

5. Determination of the ratio \(I_{R_1}/I_{R\text{sh}}\). The study of these relationships on the model is valuable above all because most of them either cannot be investigated at all on the original, or present great experimental difficulty. On the other hand, their investigation is of interest in connection with clarifying ideas about the process of the main discharge itself. Clarifying the ratio \(I_{R_1}/I_{R\text{sh}}\) (\(I_{R_1}\) is the total current taken from the resistance \(R_1 = 10\ \Omega\)) is of interest in connection with the considerations expressed by I. S. Stekolnikov concerning the division of the main-discharge current into several components.

To determine the dependence of the amplitude of the channel current on the magnitude \(C_p\), the capacitance of the high-voltage electrode, also does not appear possible for a spark, just as the dependence of the front time of the channel-current wave on \(C_p\) does not.

Initially, parameters were chosen on the model corresponding to the following data for the original: \(S_0 = 2\ \text{m}\); \(U_0 = 1100\ \text{kV}\); \(R_T = 1500\ \Omega\); \(L = 1.3\ \mu\text{H}/\text{m}\); \(C = 1.43 \cdot 10^{-11}\ \text{F}/\text{m}\); \(R_k = 200\ \Omega/\text{m}\) (channel resistance); the gap under investigation was rod—plane, with positive polarity of the rod. The waveform of the channel current for the original was chosen from an oscillogram obtained on the original with the above parameters (Fig. 3).

The parameters of the current wave for the original are: \(I_{l\max} = 335(375)\ \text{A}\); \(I_{R\text{sh}} = 648\ \text{A}\); \(I_{\text{dug}} = 570(575)\ \text{A}\); \(T_f = (0.076)\ \mu\text{s}\); \(T_{\text{imp}} = (0.22)\ \mu\text{s}\).

In parentheses are given the average values of the parameters obtained from processing a large number of oscillograms; moreover, the current \(I_{R\text{sh}}\) is taken as the total current, i.e., the sum of the channel and leader currents.

On the basis of the similarity criteria, the parameters of the model were chosen:

\[ C_{\text{mod}} = 3500\, C_{\text{orig}};\quad L_{\text{mod}} = 3500\, L_{\text{orig}};\quad R_{\text{mod}} = R_{\text{orig}};\quad t_{\text{mod}} = 3500\, t_{\text{orig}}, \]

where \(C_{\text{orig}}, R_{\text{orig}}, L_{\text{orig}}\) are, respectively, the capacitance, resistance, and inductance of the original. The model represents the discharge from the moment when the head of the leader touches the plane.

The capacitance \(C_p\) for the original was calculated by the formula for a cylindrical capacitor. If it is assumed that the rod is the inner cylinder of radius \(r_1 = 0.5\ \text{cm}\) and length \(S = 2\ \text{m}\), the walls of the hall will be...

by an external cylinder of radius \(r_2 = 7\ \text{m}\). The capacitance is calculated by the formula:

\[ C_{p\,orig}=\frac{\varepsilon S}{2\ln \dfrac{r_2}{r_1}} =\frac{200}{2\ln \dfrac{7000}{5}} =1.53\cdot 10^{-11}\ \text{F}. \]

Fig. 3. Oscillogram of the channel-current wave for the original.

For the model, \(C_p = 0.053\ \mu\text{F}\); we chose \(C_p = 0.05\ \mu\text{F}\). The inductance of the rod was determined as the inductance of a straight wire of diameter \(d = 1\ \text{cm}\) and length \(S = 2\ \text{m}\)

\[ L_p=\frac{\mu_0 S}{2\pi}\left(\ln\frac{2S_0}{r}-0.75\right)\ \text{H};\qquad \mu_0=4\pi\cdot 10^{-7}; \]

\[ L_p=2.4\cdot 10^{-6}\ \text{H}. \]

For the model \(L_{p\,mod}=8400\ \mu\text{H}\); we chose \(L_{p\,mod}=9000\ \mu\text{H}\).

Thus, the parameters of the model for the above-indicated parameters of the original are:

\[ L_{\text{cell}}=900\ \mu\text{H};\qquad C_{\text{cell}}=0.01\ \mu\text{F};\qquad R_l=150\ \Omega; \]

\[ R_k=40\ \Omega;\qquad R_T=1500\ \Omega;\qquad R_{\text{sh}}=51\ \Omega; \]

\[ R_l=10\ \Omega;\qquad C_p=0.05\ \mu\text{F};\qquad L_p=9000\ \mu\text{H};\qquad C_1=2.1\ \mu\text{F}, \]

where \(L_{\text{cell}}, C_{\text{cell}}, R_l, R_k\) are quantities calculated per one cell of the ladder network. The resistance \(R_k\) consists of the ohmic resistance \(R'_k\), the resistance \(R_{\text{tir}}=f(I)\), and the ohmic resistance of the inductance \(R_L=10\ \Omega\).

The channel resistance for the original can be calculated by the formula:

\[ R_k=\frac{U_0}{I_0}-R_T =\frac{1100\cdot 10^3}{578}-1500 =400\ \Omega, \]

where

\(R_k\) is the total resistance of the channel in the arc stage. The triggering of the thyratrons occurred successively from bottom to top; the “K” switch was closed, but the average switching speed was greater than the average velocity of propagation of the channel head in the long spark.

With the circuit parameters \(U_0=5.25\ \text{kV}\), \(U_n=6.36\ \text{V}\) (thyratron filament voltage), \(C_p=0.05\ \mu\text{F}\), \(C_{ya}=0.01\ \mu\text{F}\), \(R_{\sh}=51\ \Omega\), \(R_1=10\ \Omega\), \(R_k=17\ \Omega\); \(n_1/n_{\mathrm{II}}=5/5\), a wave was obtained on the model with the following parameters: \(I_l=1.57\ (330)\ \text{A}\), \(I_k=1.57\ (330)\ \text{A}\), \(I_d=2.69\ (565)\ \text{A}\). The quantities recalculated to the original are given in parentheses. The discrepancy with the original does not exceed \(2\%\).

As was established on the original, only a component of the total current passes through \(R_{\sh}\). On the model, the total current taken from \(R_1=10\ \Omega\) is 2.25 times greater than the channel current flowing through the shunt.

For the selected wave on the model, a number of dependences were obtained:

1. The dependence of the current of the main channel on \(C_p\) at various \(C_{ya}\) (Fig. 4). The dependence \(I_k=f(C_p)\) is in fact impossible to obtain on the original, since a slight increase in the geometrical dimensions of the high-voltage electrode, which leads to an increase in the capacitance \(C_p\), produces almost no change in the channel current. Since the wave front of the channel current is formed at the first moment of development of the channel, the discharge of this capacitance should not affect the front time, as was also proved on the model. With a considerable increase in the channel capacitance \((C_{ya})\) (Fig. 4), the amplitude of the channel current increases sharply, which likewise cannot be established on the original.

Fig. 4. Dependence \(I_k=f(C_p)\) at various \(C_{ya}\).

1. With the aid of the model, certain features of the phenomenon of the channel stage that were found on the original were reproduced. Thus, for example, the model displays the dependence \(I_k=f(R_{\sh})\) (Fig. 5), which agrees well with the original.

The model makes it possible to establish a more accurate dependence of the channel current (lightning current) on the magnitude of the grounding resistance, which must be known in order to solve the problem of lightning protection of transmission lines. The second curve (Fig. 5) depicts the dependence of the voltage drop across \(R_{\sh}\), from which it is possible to determine what percentage of the total voltage falls on the shunt.

These experiments make it possible to consider that, when lightning strikes a transmission-line support, the coefficient of voltage transfer to the top of the support, due to the voltage drop from the discharge current across the grounding resistance of the support (in our case \(R_{\sh}\)), amounts to \(3\text{--}4\%\).

With simultaneous switching on of all the thyratrons (the switch “\(K\)” closed), the magnitude of the amplitude of the channel current under synchronous switching exceeds the magnitude of the amplitude of the channel current under sequential switching of the thyratrons by \(12\%\), while the rise time is reduced

Fig. 5. Dependence \(I_k = f(R_{\mathrm{sh}})\).

by a factor of 2. The front of the channel-current wave is determined by the displacement of the current pulses in the resistance \(R_{\mathrm{sh}}\), which occurs because of the “delay” effect inherent in the ladder circuit, determined by a quantity equal to \(\sqrt{LC}\) per cell (Fig. 6).

Fig. 6. Oscillogram of the channel-current wave for the model.

It is practically difficult to realize instantaneous switching on of all the thyratrons. However, determining the current in this case proved possible to carry out analytically. Therefore, having obtained the parameters

waves by calculation and reducing them by 12–15%, one can obtain values closest to the original.

Conclusions

1. Obtaining on the model a current wave whose parameters coincide with the channel-current wave for the spark once again confirms the validity of the selected equivalent circuit for the channel stage of a long spark, proposed by I. S. Stekolnikov.

2. A chain circuit with successively varied \(R_l\) reproduces the known relationships between the discharge current and the parameters of the actual circuit in which the long spark develops.

3. With the aid of the model described, it is possible to measure the distribution of the main-channel current in various elements of the measuring circuit and to make measurements of the current along the channel, which is inadmissible by direct methods under actual conditions.

4. The existence of circuits with quantitatively determined parameters makes it possible to determine by calculation the form of the channel-current curve.

References

1. Venikov V. A. Application of the Theory of Similarity and Physical Modeling in Electrical Engineering. Gosenergoizdat, 1949.

2. Stekolnikov I. S. Report of the Laboratory of High-Voltage Engineering, Energy Institute, Academy of Sciences of the USSR, on the topic: Modeling the Current of the Channel Stage of a Long Spark, 1953.