Calculation of Attenuation and Distortion of High-Voltage Unipolar Pulses in a Coaxial Transmission System on the “Minsk-1” Electronic Digital Computer

I. I. Kalyatskii, A. T. Chepikov, A. A. Dulzon, V. A. Mal’tsev

(Presented by the scientific seminar of the Research Institute of High Voltages)

In a previous work we considered a graphical-analytical calculation of attenuation and distortion in a coaxial transmission system. Owing to the unwieldiness of the calculations, it is difficult to analyze different variants of the design of a coaxial system from the point of view of pulse distortion and attenuation. In this connection, we used the “Minsk-1” electronic digital computer for the calculations. A calculation was carried out for six principal variants (Table 1) with lengths of the coaxial system from 1 to 6000 m.

The resistance, inductance, and capacitance of the coaxial system for all the cases considered were determined from the formulas given in the preceding work; moreover, the specific resistance of nonmagnetic tubes was taken as \(\rho = 0.03\ \Omega\cdot \mathrm{mm}^2/\mathrm{m}\), which represents a certain average value for possible tube materials (duralumin, copper, brass, etc.).

The active conductance of the coaxial system was determined from the formula

\[ G=\frac{2\pi}{\rho_v \ln \dfrac{r_2}{r_1}}+\omega C \operatorname{tg}\delta . \tag{1} \]

For the case in which the insulator is technical water, it was assumed that \(\rho_v = 3\cdot 10^3\ \Omega\cdot\mathrm{cm}\). The dependence of \(\operatorname{tg}\delta\) on frequency for water, given in [1], in the frequency range \(10^5\)–\(10^7\) Hz was approximated by the formula

\[ \operatorname{tg}\delta=\frac{4\cdot 10^4}{f}. \]

For mixtures of transformer oil with water, the known relations were used

\[ \varepsilon=\frac{\varepsilon_1 V_1+\varepsilon_2 V_2}{V_1+V_2}, \tag{2} \]

\[ \operatorname{tg}\delta= \frac{\varepsilon_1 \operatorname{tg}\delta_1 V_1+\varepsilon_2 \operatorname{tg}\delta_2 V_2} {\varepsilon_1 V_1+\varepsilon_2 V_2}, \tag{3} \]

where \(V_1\) and \(V_2\) are the volumes of the dielectrics,

\(\varepsilon, \varepsilon_1\), and \(\varepsilon_2\) are the dielectric permittivity of the mixture and of its components,

\(\operatorname{tg}\delta, \operatorname{tg}\delta_1, \operatorname{tg}\delta_2\) are the loss tangents of the mixture and of its components.

Table 1

As shown in [2], to calculate the specific conductivity of a mixture of two dielectrics one may use a relation analogous to (2):

\[ \gamma=\frac{\gamma_1 V_1+\gamma_2 V_2}{V_1+V_2}, \tag{4} \]

where \(\gamma, \gamma_1, \gamma_2\) are the specific conductivities of the mixture and of its components.

On the calculation of the transient characteristics of a coaxial system

For calculating the transient characteristics on the computer, we used the most general expressions for the attenuation coefficient and the phase coefficient:

\[ \beta=\sqrt{\frac{1}{2}(RG-\omega^2LC)+\frac{1}{2}\sqrt{(R^2+\omega^2L^2)(G^2+\omega^2C^2)}}, \tag{5} \]

\[ \alpha=\sqrt{-\frac{1}{2}(RG-\omega^2LC)+\frac{1}{2}\sqrt{(R^2+\omega^2L^2)(G^2+\omega^2C^2)}}. \tag{6} \]

The transient characteristics were computed from the expression

\[ h(l,t)=\frac{2}{\pi}\int_0^\infty e^{-\beta l}\cos\left(\alpha l-l\omega\sqrt{LC}\right)\cdot\frac{\sin\omega t}{\omega}\,d\omega. \tag{7} \]

The computation of integral (7) was carried out on the “Minsk-1” electronic digital computer

Fig. 1. Integrand function \(f(\omega)\). Calculation case 3, \(l=300\) m, \(t=10^{-7}\) sec; \(\omega_1, \omega_2, \omega_3\) are even roots of the function.

by Simpson’s method. The form of the integrand function \(f(\omega,t)\) is shown in Fig. 1.

The calculation of the integral

\[ I=\int_0^\infty f(\omega,t)\cdot d\omega \]

is reduced to the calculation of the sum

\[ I=\int_0^{\omega_1} f(\omega,t)\,d\omega+ \int_{\omega_1}^{\omega_2} f(\omega,t)\,d\omega+ \int_{\omega_2}^{\omega_3} f(\omega,t)\,d\omega+\cdots, \]

where \(\omega_1,\omega_2,\omega_3\), etc., are even roots of the integrand \(f(\omega)\) (Fig. 1).

The roots of \(f(\omega)\) are found from the expression \(f(\omega,t)=0\), or \(\sin\omega t=0\), whence

\[ \omega_{i+1}=\frac{2\pi(i+1)}{t}, \]

where \(i=0,1,2,\ldots\).

The accuracy of computing integral (7) was set equal to \(\xi=0.005\). To ensure such accuracy, three or four roots \(\omega_{i+1}\) are sufficient, i.e., \(i\leqslant 3\).

For calculating an integral of the form

\[ I=\int_a^b f(x)\cdot dx \]

Simpson’s formula [3] is used:

\[ I=\frac{h}{3}\{(y_0+y_{2n})+4(y_1+y_3+\cdots+y_{2n-1})+2(y_2+y_4+\cdots+y_{2n-2})\}, \]

where \(y_k=f(a+kh)\), where \(k=0,1,2,\ldots,2n\).

Here the step is

\[ h=\frac{b-a}{2n} \]

(\(2n\) is the number of subdivisions).

The initial number of subdivisions of the interval \([ab]\) was set equal to 8, i.e., \(n=4\).

Next, the integral \(I(h)\) and \(I\left(\frac{h}{2}\right)\) were computed successively, and the check

\[ \left|\frac{I(h)-I x}{x\left(\frac{h}{2}\right)}\right|\leqslant \xi \]

was performed.

Fig. 2. Flowchart of the program for calculating transient functions

Visible labels in the flowchart:

• \(L=\mathrm{const}\)
• \(R,C,\xi,l,t\) initial
• Input from console
• Calculation \(t_0=2\sqrt{LC}\)
• \(t_{\mathrm{init}}\Rightarrow \Delta t\)
• \(\Delta t\Rightarrow t\)
• print \(t\)
• \(0\Rightarrow J,i\)
• \(i\Rightarrow \omega_i\)
• \(\omega_i=\dfrac{2\pi}{t}\Rightarrow \omega_{i+1}\)
• \(\dfrac{\omega_i-\omega_{i-1}}{8}\Rightarrow h\)
• \(J_i=\displaystyle\int_{\omega_i}^{\omega_{i+1}} f(\omega,t)\,d\omega=h(\omega,t)\)
• \(i+1\Rightarrow i\)
• \(i\leq 3\)
• yes
• no
• \(J_i+J\Rightarrow J\)
• \(\omega_i+1\Rightarrow \omega_i\)
• \(\dfrac{2\pi(i+1)}{t}\Rightarrow \omega_{i+1}\)
• \(J_i+J\Rightarrow J\)
• print \(J\)
• \(t+\Delta t\Rightarrow t\)
• \(t\leq t_{\max}\)
• yes
• no
• stop
• Transition to next case

If this inequality is not satisfied, the step is automatically halved and the procedure is repeated until the required accuracy is attained.

The computation program was compiled according to the flowchart shown in Fig. 2.

The principal results of the calculations are given in Figs. 3–6.
Figure 3 presents the transition functions \(h(l,t)\) for a coaxial system made of steel pipes filled with transformer oil, for system lengths from 30 to 6000 m. As can be seen from Fig. 3, when pulses several microseconds long are transmitted, they will undergo substantial attenuation, and their shape will be considerably distorted.

Fig. 3. Transition functions \(h(l,t)\) for computational case 1 (steel pipes filled with transformer oil).
1—\(l=30\) m, 2—\(l=300\) m, 3—\(l=1000\) m,
4—\(l=1500\) m, 5—\(l=2000\) m, 6—\(l=3000\) m.

In Fig. 4 the transition functions \(h(l,t)\) for computational cases 1, 2, and 3 are compared for system lengths of 300 m and 3000 m. From the comparison

Fig. 4. Transition functions \(h(l,t)\) for computational cases 1, 2, and 3 at lengths of 300 and 3000 m.
1, 2, 3—respectively computational cases 1, 2, and 3 for \(l=300\) m;
4, 5, 6—respectively computational cases 1, 2, and 3 for \(l=3000\) m.

it is clear that replacing the inner pipe by a nonmagnetic one substantially reduces the attenuation and distortion of the pulse. Replacing the second pipe by a nonmag-

... into nonmagnetic already gives a considerably smaller effect and, for lengths up to hundreds of meters, is hardly expedient.

Figure 5 shows the transient function for the fourth calculated case (insulation—water) for a coaxial-system length \(l=1\ \text{m}\). It is obvious that the use of technical water as insulation in a coaxial transmission system is impossible even when the system length is several meters.

Fig. 5. Transient function \(h(l,t)\) for calculated case 4 (steel tubes, insulation—water).

Figure 6 shows the transient function for the fifth calculated case (insulation—transformer oil with \(0.1\%\) water) for a system length \(l=1\ \text{m}\).

It follows from Fig. 6 that, when transmitting a rectangular pulse over a distance \(l=1\ \text{m}\), the pulse-front length will be more than \(10\ \mu\text{sec}\). However, this contradicts some indirect experimental data. In this connection, we checked the correctness of the calculation on the electronic digital computer by the graphical-analytical method described in our previous work. The calculated points obtained are plotted in Fig. 6. They agree well with the results of the calculation on the electronic digital computer. Hence it follows that, apparently, substituting in (1) the values of \(\rho\nu\) measured at constant voltage is not justified. They should be determined under pulsed action. In addition, the validity of using expression (4) for pulses is doubtful.

Fig. 6. Transient functions \(h(l,t)\) for calculated case 5 (steel tubes, insulation—a mixture: “\(0.1\%\) water in transformer oil”). The solid curve is the calculation on the electronic digital computer.

On the Calculation of Energy Losses in a Coaxial System

Figure 7 gives the pulse shape at the beginning of the coaxial system (curve 1). With the aid of Duhamel’s integral and the transient characteristics of the system given above, one can calculate the pulse shape at the end of the coaxial system. Figure 7 gives the results of the calculation for a coaxial-system length \(l=300\ \text{m}\) with both tubes made of steel (curve 2), and for the case in which the inner tube is replaced by a nonmagnetic one (curve 3).

Determining the pulse energy by the graphical-analytical method from the formula

\[ \mathcal{E}=\int_{0}^{t}\frac{U^{2}(t)}{z_b}\cdot dt, \]

we find that the energy losses of the pulse in the coaxial system with both steel pipes amount to 20%, while when the inner pipe is replaced by a nonmagnetic one they amount to 8% of the energy of the initial pulse.

Fig. 7. Attenuation and distortion of a pulse in a coaxial system. 1 — pulse shape at the beginning of the coaxial system; 2 — pulse shape after traveling 300 m through a coaxial system of steel pipes filled with transformer oil; 3 — the same, with the inner pipe replaced by a nonmagnetic one.

Conclusions

1. When transmitting powerful high-voltage pulses over distances of hundreds and thousands of meters, from the standpoint of reducing distortion and energy losses it is expedient to use, instead of the inner ferromagnetic pipe, a pipe made of nonmagnetic material. Replacing the outer pipe by a nonmagnetic one gives an insignificant effect.

2. The use of technical water with a dc resistivity of the order of \(\rho_v = 3 \cdot 10^3\ \Omega \cdot \text{cm}\) as insulation in the transmitting coaxial system is impossible even when transmitting a pulse over several meters.

3. In order to obtain reliable calculated data on pulse attenuation for cases in which mixtures of insulating oil with water are used as insulation, it is necessary to carry out experimental studies of the pulsed conductivity of water and of mixtures of insulating oils with water.

References

1. A. R. Hippel, Dielectrics and Their Applications. GEI, 1959.

2. V. I. Obukhov, T. Yu. Mogilevskaya. On determining the specific conductivity of a mixture of dielectrics. Izv. TPI, vol. 149, 1966.

3. I. S. Berezin, N. P. Zhidkov. Methods of Computation. Fizmatgiz, 1959.