Graphical-Analytical Calculation of Attenuation and Distortion of High-Voltage Unipolar Pulses in a Coaxial Transmission System
Abstract Generated abstract
This paper develops a graphical and analytical procedure for estimating attenuation and waveform distortion of high-voltage unipolar pulses transmitted through oil-filled coaxial lines made from steel tubes. It derives primary line parameters, propagation coefficients, frequency characteristics, and transient responses, using trapezoidal approximation of the real transfer characteristic and Duhamel integration to compute the output pulse shape from a known input pulse. Calculations for a transformer-oil-filled coaxial system show that, for pulses with fronts of 0.1 microsecond or longer and amplitudes up to 300 kV, distortion over several tens of meters is negligible, while transmission over hundreds of meters can increase the pulse-front duration by a factor of about 2 to 5. An approximate transient-response estimate using Kramp’s function is also shown to be feasible.
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Graph-Analytical Calculation of the Attenuation and Distortion of High-Voltage Unipolar Pulses in a Coaxial Transmission System
I. I. Kalyatskii, A. T. Chepikov, A. A. Dulzon
(Presented by the scientific seminar of the Research Institute of High Voltages)
The field of application of high impulse voltages is at present continuously expanding. They are used in radiolocation, for generating shock waves in a liquid medium, for obtaining powerful flashes of light, etc. For technical and technological reasons, there quite often arises the need to transmit powerful high-voltage pulses over distances of tens and hundreds of meters. In this connection, assessment of the attenuation and distortion of pulses when they are transmitted through coaxial systems is of substantial interest.
On the calculation of the primary parameters of a coaxial system
As the transmission system we shall consider a coaxial line consisting of two steel tubes filled with insulating oil. As is known [1], in the case when the wall thickness of the tubular conductor
\[ t>\frac{5}{\sqrt{2}\cdot \kappa}, \tag{1} \]
its resistance is equal to the resistance of a solid conductor. Here
\[ \kappa=\sqrt{\frac{\omega\mu_r}{\rho}}, \]
where \(\mu_r\) is the relative magnetic permeability of the conductor material,
\(\rho\) is the specific resistance of the conductor material.
In the frequency range of interest to us, \(10^5\)—\(10^8\) Hz, in all cases the wall thickness of the inner conductor, for reasons of mechanical strength, will considerably exceed the value \(t\). Taking this into account, it can be shown that for calculating the resistance of a coaxial system the formula
\[ R=3.16\cdot 10^{-4}\sqrt{f}\left(\frac{\sqrt{\mu_{r1}\rho_1}}{r_1}+\frac{\sqrt{\mu_{r2}\rho_2}}{r_2}\right),\ \text{ohm/m}, \tag{2} \]
is suitable, where \(r_1\) and \(r_2\) are the radii of the conductors.
The internal inductance of the conductors of a coaxial system, when structural steels are used in the frequency range under consideration, does not exceed several percent of the external inductance [4]. Therefore [1]
\[ L \simeq 4.6 \cdot 10^{-7}\lg \frac{r_2}{r_1}, \qquad \text{H/m}. \tag{3} \]
The capacitance of the coaxial system is
\[ C=\frac{2\pi \varepsilon_r \varepsilon_0}{\ln \dfrac{r_2}{r_1}}, \qquad \text{F/m}. \tag{4} \]
The active conductance of the coaxial system is
\[ G=\frac{2\pi}{\rho_g \lg \dfrac{r_2}{r_1}}+\omega C \operatorname{tg}\delta, \qquad \text{mho/m}. \tag{5} \]
Already for \(f>10\) kHz, when insulating liquids are used, one may with high accuracy assume
\[ G=\omega C \operatorname{tg}\delta. \tag{6} \]
Calculation of the propagation coefficient
As is known [3], the propagation coefficient is
\[ \overline{\gamma}=\sqrt{(R+j\omega L)(G+j\omega C)}=\beta+j\alpha, \tag{7} \]
where
\[ \beta=\sqrt{\frac{1}{2}(RG-\omega^2LC)+\frac{1}{2}\sqrt{(R^2+\omega^2L^2)(G^2+\omega^2C^2)}}, \tag{8} \]
\[ \alpha=\sqrt{-\frac{1}{2}(RG-\omega^2LC)+\frac{1}{2}\sqrt{(R^2+\omega^2L^2)(G^2+\omega^2C^2)}}. \tag{9} \]
When calculating the attenuation coefficient \(\beta\), it becomes necessary to subtract two large numbers close in magnitude. To avoid this, we transform expression (8), using Newton’s binomial:
\[ \beta=\sqrt{\frac{1}{2}\omega^2LC\left(\frac{1}{2}\frac{R^2}{\omega^2L^2}-\frac{1}{8}\frac{R^4}{\omega^4L^4}\right)+\frac{1}{2}RG}. \tag{10} \]
Similarly, we can obtain
\[ \alpha=\omega\sqrt{LC}\left(1+\frac{R^2}{8\omega^2L^2}-\frac{3}{128}\cdot\frac{R^4}{\omega^4L^4}\right). \tag{11} \]
Calculation of the transient characteristics of the coaxial system
As is known [4], the transform of the solution (according to Carson) of the system of differential equations of a long line has the form:
\[ U(p)=U_1(p)\cdot \frac{Z_2(p)\cdot \operatorname{ch}\gamma(l-x)+Z(p)\cdot \operatorname{sh}\gamma(l-x)} {Z_2(p)\cdot \operatorname{ch}\gamma l+Z(p)\cdot \operatorname{sh}\gamma l}, \tag{12} \]
where \(U_1(p)\) is the transform of the pulse at the beginning of the line,
\(Z_2(p)\) is the transform of the load at the end of the line,
\(Z(p)\) is the transform of the characteristic impedance of the line.
For an infinitely long line we may put \(Z_2(p)=Z(p)\). Then for \(x=l\) we obtain:
\[ U(p)=U_1(p)\cdot \frac{1}{\operatorname{ch}\gamma l+\operatorname{sh}\gamma l}. \]
For a unit “step” \((U_1(p)=1)\) we find:
\[ U(p)=h(p)=\frac{1}{\operatorname{ch}\gamma l+\operatorname{sh}\gamma l} =\frac{1}{e^{\gamma l}}=e^{-\gamma(p)\cdot l}, \tag{13} \]
where \(h(p)\) is the transform of the transient characteristic of the coaxial system.
The original of expression (13) may be represented in the form [5]:
\[ h(p)=e^{-\gamma(p)\cdot l}=h(l,t) =\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty} e^{-\beta l}\frac{\sin(\omega t-\alpha l)}{\omega}\,d\omega . \tag{14} \]
Taking into account that the front of the pulse arrives at the point \(l\) of interest to us after the time \(t_3=l\sqrt{LC}\), it is expedient in formula (14) to substitute, instead of \(t\), \(t_1=t-l\sqrt{LC}\). Then, taking into account that for \(t_1<0\), \(h(l,t_1)=0\), one can obtain
\[ h(l,t_1)=\frac{2}{\pi}\int_{0}^{\infty} e^{-\beta l}\cos(\omega l\sqrt{LC}-\alpha l)\, \frac{\sin\omega t_1}{\omega}\,d\omega . \tag{15} \]
The quantity \(P(\omega)=e^{-\beta l}\cdot \cos(\omega l\sqrt{LC}-\alpha l)\), representing the real part of the transmission coefficient, is the real frequency characteristic of the coaxial system. Finally,
\[ h(l,t_1)=\frac{2}{\pi}\int_{0}^{\infty} P(\omega)\cdot \frac{\sin\omega t_1}{\omega}\,d\omega . \tag{16} \]
In the particular case when \(G\simeq 0\) and \(R\ll \omega L\), the transient characteristic can be expressed [5] in terms of Kramp’s function, or the probability integral:
\[ h(l,t_1)=1-\Phi\left(\frac{b_1l}{2\sqrt{t_1}}\right), \]
where
\[ b_1=\frac{1}{4\pi}\sqrt{\frac{C}{L}} \left( \frac{\sqrt{\mu_{r_1}\rho_1}}{r_1} + \frac{\sqrt{\mu_{r_2}\rho_2}}{r_2} \right). \]
\[ t_1=t-l\sqrt{LC}=t-t_3, \]
where \(t_3\) is the delay time of the line.
Kramp’s function
\[ \Phi(z)=\int_{0}^{z} e^{-x^2}\,dx \]
is not expressed in elementary functions; however, very detailed tables are available [6].
To compute the transient characteristics from expression (16), one can use the frequency-domain method for calculating transient processes [7].
The idea of the frequency-domain method is to represent the curve \(P(\omega)\) in the form of a sum of certain standard curves \(r_i(\omega)\)
\[ P(\omega)=\sum_{i=1}^{n} r_i(\omega) \]
so that, when computing expressions of the form
\[ h_{ri}(t)=\frac{2}{\pi}\int_{0}^{\infty} r_i(\omega)\frac{\sin \omega t}{\omega}\,d\omega \tag{18} \]
one could use tables and reduce the calculation of the transient process \(h(t)\) to summation of the tabulated functions \(h_{ri}(t)\)
\[ h(t)=\sum_{i=1}^{n} h_{ri}(t). \]
It is assumed that each of the functions \(r_i(\omega)\) has the form of a rectangular trapezoid (Fig. 1) with slope coefficient
\[ x=\frac{\omega_d}{\omega_0}. \]
Fig. 1. Function \(r_i(\omega)\).
For the calculation, tables \(h_x(t)\) are used, computed in accordance with expression (18) for unit trapezoids with \(r_0=1\) and \(\omega_0=1\). The transient functions \(h_{xi}(t)\) found from the tables (for the specified \(x\)) are recalculated to the actual values \(\omega_{0i}\) and \(r_{0i}\) in accordance with the similarity theorem. For this, the tabulated values \(h_x\) are multiplied by \(r_{0i}\), and the argument \(t\) is divided by \(\omega_{0i}\). Summing the obtained curves \(h_{xi}(t)\), we obtain the transient characteristic of the coaxial system.
Calculation of the pulse shape at the end of a coaxial system
The calculation of the pulse shape at the output of a coaxial system \(U_2(t)\), from the known pulse shape at the input of the system \(U_1(t)\), is carried out with the aid of the Duhamel integral [4]:
\[ U_2(t)=\int_{0}^{\infty} U_1'(\tau)\cdot h(t-\tau)\cdot d\tau, \tag{19} \]
where \(h(t-\tau)\) is the transient function,
\(U_1'(\tau)\) is the derivative of the voltage curve at the beginning of the line.
Since the computation of integral (19) is performed by a graphical-analytical method, we replace it by the sum [8]:
\[ U_2(t)\simeq \sum_{n=1}^{N} \Delta U_{1n}\cdot h(t-\tau_n), \tag{20} \]
where \(N\) is the number of intervals into which the region of existence of the initial pulse is divided;
\(\Delta \tau\) is the length of an interval;
\(\tau_n = n \cdot \Delta \tau\).
Results of calculations of attenuation and distortion of pulses in a coaxial system
In the present work, a calculation was carried out of the attenuation and distortion of pulses in a coaxial system made of steel tubes filled with transformer oil, intended for the transmission of pulses used for technological purposes.
The outer tube (with inner diameter \(2r_2 = 81\ \mathrm{mm}\)) is made of steel 40Kh, and the inner tube (with outer diameter \(2r_1 = 21.25\ \mathrm{mm}\)) is made of steel 3. The magnetic permeability of the material of the outer tube in the frequency range of interest to us is \(\mu_{r_2}=50\), and of the inner tube, \(\mu_{r_1}=75\); the specific resistance of the material of the outer tube is \(\rho_2 = 0.225\ \Omega \cdot \mathrm{m}/\mathrm{mm}^2\), and of the inner tube, \(\rho_1 = 0.157\ \Omega \cdot \mathrm{m}/\mathrm{mm}^2\). Measurements of the electrical properties of the transformer oil gave the following results: dielectric permittivity \(\varepsilon_r = 2.19\), volume resistivity \(\rho_v = 10^{12}\ \Omega \cdot \mathrm{cm}\), \(\tg \delta = 5 \cdot 10^{-4}\). The values of \(\varepsilon_r\) and \(\tg \delta\) remain practically constant in the frequency range \(2 \cdot 10^3 \div 10^8\ \mathrm{Hz}\).
The real frequency characteristics for system lengths of \(30\ \mathrm{m}\) and \(300\ \mathrm{m}\) are given in Fig. 2. We approximate curve 1 (Fig. 2) by rectangular trapezoids (Fig. 3). Using the tables [7], we calculate the transient functions for each trapezoid and, summing them, find the transient characteristic of the coaxial system for \(l = 300\ \mathrm{m}\) (curve 1, Fig. 4). There also (curve 3) is given the transient characteristic calculated by the simplified formula (17).
Fig. 2. Real part of the transfer coefficient of the coaxial system. \(1\)—\(l = 300\ \mathrm{m}\), \(2\)—\(l = 30\ \mathrm{m}\).
In Fig. 5 the pulse shape is shown at the beginning of the coaxial system and after a run of \(300\ \mathrm{m}\).
Fig. 3. For the calculation of the transient characteristic of the system. 1 — trapezoid 1; 2 — trapezoid 2; 3 — trapezoid 3; 4 — trapezoid 4; 5 — trapezoid 5.
Fig. 4. 1 — transient characteristic of the system for \(l = 300\ \mathrm{m}\) (calculation by the frequency method), 2 — the same for \(l = 30\ \mathrm{m}\), 3 — transient characteristic of the system for \(l = 300\ \mathrm{m}\), calculated with the aid of Kramp functions. 4–8 — transient functions \(h_{x_i}\), calculated for the individual trapezoids.
Fig. 5. Pulse shape at the beginning of the coaxial system (1) and after traveling 300 m (2).
Conclusions
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When transmitting powerful high-voltage pulses with a front duration of 0.1 μsec and more and an amplitude of up to 300 kV over distances of several tens of meters through a coaxial system using an insulating liquid, the attenuation and distortion of the pulses are negligibly small.
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When transmitting the indicated pulses through the coaxial system under consideration over distances of hundreds of meters, the pulse-front duration increases by a factor of 2–5.
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It has been shown that an approximate estimate of the transient response of a coaxial system can be obtained on the basis of Kramp’s function in accordance with (17).
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