On the Issue of Attenuation and Distortion of High-Voltage Aperiodic Pulses in a Coaxial Transmission System
Abstract Generated abstract
This paper examines attenuation and waveform distortion of high-voltage aperiodic pulses transmitted through coaxial cable systems over long distances. It first analyzes the proposed condition for distortionless transmission, deriving current-density distributions in the conductors for exponential pulse fronts with allowance for skin effect, and argues that a constant effective resistance is not a satisfactory assumption over practical front durations. The authors then calculate cable transient response using conductor-loss theory and the Duhamel integral, and compare these results with experiments on an RK-103 cable. The measurements show substantial pulse-front lengthening after 530 m, from about 7 ns to 200 to 300 ns, while calculated and observed waveforms agree fairly well up to 20 kV, suggesting that corona has little influence under the studied conditions.
Full Text
ON THE ATTENUATION AND DISTORTION OF HIGH-VOLTAGE APERIODIC PULSES IN A COAXIAL TRANSMISSION SYSTEM
A. A. Dulzon, I. I. Kalyatskii
(Submitted by the Scientific Seminar of the Research Institute
of High Voltages)
When electrical energy is channeled by means of high-voltage aperiodic pulses with amplitudes up to 200–300 kV over distances of hundreds and thousands of meters, it becomes necessary to estimate the attenuation and distortion of the pulses and to develop measures for reducing them.
On the calculation of distortionless transmission of aperiodic pulses through a coaxial system
In the works of T. Yu. Mogilevskaya [1, 2], the idea was proposed of the possibility of creating a distortionless line for transmitting high-voltage pulses. It was shown that if, taking account of the skin effect, a certain effective value of the resistance \(R_0=\mathrm{const}\) is adopted, then, by selecting the dimensions of the coaxial system and the conductor material, one can achieve fulfillment of the condition
\[ \frac{R_0}{L_0}=\frac{G_0}{C_0}, \tag{1} \]
where \(R_0\), \(L_0\), \(G_0\), and \(C_0\) are, respectively, the active resistance, inductance, active conductance, and capacitance of the coaxial system per unit length.
When equality (1) is satisfied (the condition for a distortionless line), the coaxial system introduces no phase distortions, i.e., the length of the pulse front remains constant. The amplitude of the pulse decreases as it propagates along the system according to an exponential law with attenuation coefficient
\[ \beta=\frac{R_0}{L_0}\sqrt{L_0 C_0}. \tag{2} \]
We have attempted to estimate how valid the above assumption \(R_0=\mathrm{const}\) is for an exponential pulse shape. Let, in the general case,
\[ U=U_0\left(e^{-p_1 t}-e^{-p_2 t}\right)=k(t)\cdot U_0. \tag{3} \]
Let us try to obtain an expression for the current density in the central conductor of the coaxial system. As is known [3], the equa-
Maxwell’s equations for a conducting medium in a cylindrical coordinate system have the form:
\[ \left\{ \begin{aligned} \frac{\partial E_\varphi}{\partial r}+\frac{E_\varphi}{r} -\frac{1}{r}\frac{\partial E_r}{\partial \varphi} &=-\mu_a\frac{\partial H_z}{\partial t},\\ \frac{1}{r}\frac{\partial E_z}{\partial \varphi} -\frac{\partial E_\varphi}{\partial z} &=-\mu_a\cdot\frac{\partial H_r}{\partial t},\\ \frac{\partial E_r}{\partial z} -\frac{\partial E_z}{\partial r} &=-\mu_a\frac{\partial H_\varphi}{\partial t}; \end{aligned} \right. \tag{4} \]
\[ \left\{ \begin{aligned} \frac{\partial H_\varphi}{\partial r}+\frac{H_\varphi}{r} -\frac{1}{r}\frac{\partial H_r}{\partial \varphi} &=\frac{E_z}{\rho},\\ \frac{\partial H_z}{r\cdot\partial \varphi} -\frac{\partial H_\varphi}{\partial z} &=\frac{E_r}{\rho},\\ \frac{\partial H_r}{\partial z} -\frac{\partial H_z}{\partial r} &=\frac{E_\varphi}{\rho}. \end{aligned} \right. \tag{5} \]
Taking (3) into account, we may, in the first approximation for the given cross section, set
\[ \overline{H}=H_0\left(e^{-p_1t}-e^{-p_2t}\right)=k(t)\overline{H}_0, \tag{6} \]
\[ \frac{\partial H}{\partial t}-\overline{H}_0\left(p_2e^{-p_2t}-p_1e^{-p_1t}\right)=k_1(t)\cdot\overline{H}_0, \tag{7} \]
\[ \overline{E}=\overline{E}_0\cdot k(t). \tag{8} \]
Let us make the \(z\) axis coincide with the axis of the conductor. Since inside the conductor the currents flow only in the direction \(z\), it is necessary to take into account only three components \(E_z\), \(H_r\), and \(H_\varphi\).
Then, taking (6), (7), and (8) into account, equations (4) and (5) take the form:
\[ \left\{ \begin{aligned} \frac{\partial E_{0z}}{\partial r} &=\mu_a\cdot\frac{k_1(t)}{k(t)}\cdot H_{0\varphi},\\ \frac{\partial H_{0\varphi}}{\partial r} +\frac{H_{0\varphi}}{r} &=\frac{E_{0z}}{\rho}. \end{aligned} \right. \tag{9} \]
From (9)
\[ H_{0\varphi} = \frac{k(t)}{\mu_a\cdot k_1(t)} \cdot \frac{\partial E_{0z}}{\partial r}. \tag{10} \]
Then
\[ \frac{\partial H_{0\varphi}}{\partial r} = \frac{k(t)}{\mu_a\cdot k_1(t)} \cdot \frac{\partial^2 E_{0z}}{\partial r^2}. \tag{11} \]
Substituting (10) and (11) into (9), we obtain
\[ \frac{k(t)}{\mu_a\cdot k_1(t)} \cdot \frac{\partial^2 E_{0z}}{\partial r^2} + \frac{k(t)}{\mu_a\cdot k_1(t)} \cdot \frac{1}{r} \cdot \frac{\partial E_{0z}}{\partial r} - \frac{E_{0z}}{\rho} =0. \tag{12} \]
or
\[ \frac{\partial^2 E_{0z}}{\partial r^2} + \frac{1}{r} \frac{\partial E_{0z}}{\partial r} - \frac{k_1(t)}{k(t)} \cdot \frac{\mu_a}{\rho} E_{0z} =0. \tag{13} \]
Denoting
\[ a(t)=-\frac{k_1(t)}{k(t)}\cdot\frac{\mu_a}{\rho}, \tag{14} \]
we obtain
\[ \frac{\partial^2 E_{0z}}{\partial r^2}+\frac{1}{r}\frac{\partial E_{0z}}{\partial r}+a(t)\cdot E_{0z}=0 . \tag{15} \]
As is known [4], equation (15) is readily reduced to Bessel’s equation. The general integral of equation (15) is
\[ E_{0z}=A_1\cdot I_0\!\left(\sqrt{a(t)}\cdot r\right)+A_2\cdot K_0\!\left(\sqrt{a(t)}\cdot r\right), \tag{16} \]
where \(I_0\!\left(\sqrt{a(t)}\cdot r\right)\) and \(K_0\!\left(\sqrt{a(t)}\cdot r\right)\) are, respectively, Bessel functions of the first and second kind of zero order.
Let us find the constants of integration.
As \(r\to 0\), \(K_0\!\left(\sqrt{a(t)}\cdot r\right)\to \infty\); therefore, for physical reasons (\(E_{0z}\) must always be finite), it is necessary to take \(A_2=0\). Then
\[ E_{0z}=A_1\cdot I_0\!\left(\sqrt{a(t)}\cdot r\right). \tag{17} \]
Using equation (10) and taking into account that \(I_0'(x)=-I_1(x)\), we obtain
\[ H_{0\varphi}=-\frac{k(t)}{\mu_a\cdot k_1(t)}\cdot A_1\cdot \sqrt{a(t)}\cdot I_1\!\left(\sqrt{a(t)}\cdot r\right), \tag{18} \]
where \(I_1\!\left(\sqrt{a(t)}\cdot r\right)\) is the Bessel function of the first kind of first order.
To determine the constant of integration \(A_1\), we use the total-current law
\[ \oint \overline{H}\, d\overline{l}=I. \tag{19} \]
For a line of force of radius \(r_1\),
\[ H_{0\varphi}=\frac{I_0}{2\pi r_1}, \tag{20} \]
where \(r_1\) is the radius of the conductor.
Equating (18) and (20), we find
\[ A_1=-\frac{\mu_a I_0 k_1(t)}{2\pi r_1\cdot k(t)\sqrt{a(t)}}\cdot \frac{1}{I_1\!\left(\sqrt{a(t)}\cdot r_1\right)}. \tag{21} \]
Substituting \(A_1\) into (17), we obtain
\[ E_{0z}=-\frac{\mu_a I_0 k_1(t)}{2\pi r_1\cdot k(t)\sqrt{a(t)}}\cdot \frac{I_0\!\left(\sqrt{a(t)}\cdot r\right)} {I_1\!\left(\sqrt{a(t)}\cdot r\right)}. \tag{22} \]
Differentiating with respect to \(r\) and substituting into (10), we obtain
\[ H_{0\varphi}=\frac{I_0}{2\pi r_1}\cdot \frac{I_1\!\left(\sqrt{a(t)}\cdot r\right)} {I_1\!\left(\sqrt{a(t)}\cdot r_1\right)}. \tag{23} \]
Taking into account that, within the pulse front, \(a(t)\) is a negative quantity and that
\[ I_n\!\left(i\sqrt{|a(t)|}\cdot r\right)=i^n\cdot I_n\!\left(r\sqrt{|a(t)|}\right), \]
we obtain
\[ E_{0z}=\frac{\mu_a I_0}{2\pi r_1}\cdot \frac{k_1(t)}{k(t)\cdot \sqrt{|a(t)|}}\cdot \frac{I_0\!\left(r\sqrt{|a(t)|}\right)} {I_1\!\left(r_1\sqrt{|a(t)|}\right)}, \tag{24} \]
\[ H_{0\varphi}=\frac{I_0}{2\pi r_1}\cdot \frac{I_1\!\left(r\sqrt{|a(t)|}\right)} {I_1\!\left(r_1\sqrt{|a(t)|}\right)}, \tag{25} \]
where
\(I_0\!\left(r\sqrt{|a(t)|}\right)\) and \(I_1\!\left(r_1\sqrt{|a(t)|}\right)\) are modified Bessel functions, respectively—
respectively, of zero and first order. Taking (8) into account, we finally obtain
\[ E_z=\frac{\mu_a I_0}{2\pi r_1}\cdot \frac{k_1(t)}{\sqrt{|a(t)|}}\cdot \frac{I_0\!\left(r\sqrt{|a(t)|}\right)} {I_1\!\left(r_1\sqrt{|a(t)|}\right)} . \tag{26} \]
Dividing \(E_z\) by the resistivity of the conductor material \(\rho\), we obtain the current density in the conductor
\[ j_z=\frac{\mu_a I_0}{2\pi r_1\rho}\cdot \frac{k_1(t)}{\sqrt{|a(t)|}}\cdot \frac{I_0\!\left(r\sqrt{|a(t)|}\right)} {I_1\!\left(r_1\sqrt{|a(t)|}\right)} . \tag{27} \]
Analogously, one can obtain expressions for \(E_z\), \(H_\varphi\), and \(j_z\) in the outer conductor of the coaxial system as well.
In Fig. 1 the curves of the current-density distribution in the core of the RK-103 cable are presented for a number of points on the front of an exponential pulse, where it is assumed that
\[ p_1=0,\qquad p_2=\frac{\tau_{\phi}}{3}. \]
Fig. 1. Distribution of current density over the cross section of the inner conductor of a coaxial system at \(\tau_{\phi}=5\) ns.
a) pulse shape, b) cable cross section, c) current-density distribution curves:
1 — \(t=5\) ns, 2 — \(t=8\) ns, 3 — \(t=10\) ns, 4 — \(t=20\) ns, 5 — \(t=50\) ns and greater.
In Fig. 2 the curves of the current-density distribution in the core of the RK-103 cable are presented for the maximum pulse-front duration used by us, \(\tau_{\phi}=0.3\) μs. The calculation was carried out with the aid of tables [5, 6].
From Figs. 1 and 2 it is seen that it is hardly possible to adopt some constant value for the resistance of a coaxial system, since within the limits of the pulse front it will vary over wide ranges (for all practically possible pulse-front durations). It is probably more correct to speak of establishing certain optimum parameters of the transmitting coaxial system. As shown in [3], the optimum diameter ratios for obtaining minimum attenuation and minimum field strength do not coincide. The choice of optimum dimensions should therefore be made with allowance for the specific conditions.
Fig. 2. Same as in Fig. 1, but for \(\tau_{\phi}=0.3\) μs.
1 — \(t=0.05\) μs, 2 — \(t=0.1\) μs, 3 — \(t=0.2\) μs, 4 — \(t=0.3\) μs, 5 — \(t=0.4\) μs, 6 — \(t=0.637\) μs.
Toward the Calculation of the Attenuation and Distortion of Pulses in Transmission over a Coaxial Cable
As is known [7], at frequencies up to hundreds of MHz, the attenuation in the metal for polyethylene cables is many times greater than the attenuation in the dielectric. For our conditions the maximum frequency can be found if it is assumed that the wave front corresponds to one quarter of the period of a sinusoid. Then, for \(\tau_{\phi\min}=5\) nsec, we obtain \(f=50\) MHz. Naturally, at such a frequency it is not necessary to take into account the attenuation in the dielectric.
In the article by L. A. Zhekulin [8] it was shown that the propagation of a “unit voltage step,” when only the losses in the conductors are taken into account, is described by the following expression:
\[ U_1(t_1,l)=1-\Phi\left(\frac{b_1 l}{2\sqrt{t_1}}\right), \tag{28} \]
where \(\Phi\left(\dfrac{b_1 l}{2\sqrt{t_1}}\right)\) is the probability integral, or Kramp’s function;
\[ b_1=\frac{1}{4\pi}\sqrt{\frac{C_0}{L_0}\sqrt{\mu\rho}\left(\frac{1}{r_1}+\frac{1}{r_2}\right)}, \tag{29} \]
\[ t_1=t-t_z=t-l\sqrt{L_0C_0}, \]
\(L_0, C_0\) are the inductance and capacitance of the cable per unit length,
\(r_1, r_2\) are the radii of the inner and outer conductors of the cable,
\(\mu,\rho\) are, respectively, the magnetic permeability and the resistivity of the conductor material,
\(l\) is the cable length,
\(t_z\) is the cable delay time.
Fig. 3. Transient characteristics of cable RK-103.
1 — \(l=70\) m, 2 — \(l=250\) m, 3 — \(l=400\) m, 4 — \(l=530\) m.
Expression (28) is called the transient characteristic of the cable and is usually denoted by \(h(l,t_1)\). The transient characteristics of cable RK-103, calculated with the aid of tables [9], are shown in Fig. 3.
Knowing \(h(t_1)\), one can, by means of the Duhamel integral, find the pulse shape at the end of the cable, \(U_2(t)\), from the known pulse shape at the beginning of the cable, \(U_1(t)\):
\[ U_2(t)=\int_0^t U_1'(\tau)h(t-\tau)\,d\tau, \tag{30} \]
where \(\tau\) is the running variable,
\(U_1'(\tau)\) is the derivative of the voltage at the beginning of the cable.
Since integral (30) is not expressible in elementary functions, a graphical-analytical method was used to compute it [10]. The calculation results are given in Fig. 8 together with the experimental curves.
Experimental evaluation of attenuation and distortion of high-voltage pulses in an RK-103 cable
A. Experimental procedure
A schematic diagram of the experimental setup is shown in Fig. 4. From a high dc-voltage source (up to \(\pm 50\) kV), through a resistance \(R_1 = 680\) megohms, an IK-2 type cable 91 m long was charged. The beginning and end of the cable were connected in parallel in order to reduce the equivalent wave resistance.
Fig. 4. Schematic diagram of the experimental setup.
\(P\) — discharger, \(C\) — solenoid, \(\Pi\) — spring, \(D_1, D_2\) — voltage dividers, \(O_1, O_2\) — high-voltage oscillographs OK-19M-2, \(K_1\) — cable under test, \(K_2\) — generating cable, \(LZ\)-1, \(LZ\)-2 — artificial delay lines.
In the diagram: “Start \(O_1\),” “Start \(O_2\).”
The discharger is of coaxial design and is filled with nitrogen at a pressure of 14 atm. One electrode of the discharger is made movable and is controlled by a solenoid.
An RK-103 type cable 630 m long was taken as the test cable. The connections of the individual cable sections were made with couplings filled with epoxy resin. At a distance of 37 m from the beginning of the cable the first divider was placed, and at a distance of 530 m from it, the second divider. The end of the cable is grounded. The dividers are made according to the Fletcher-divider type, but in order to prevent sparking at high voltages they are cast from Wood’s alloy. This ensures reliable contact between the cable sheath and the divider body. The second electrode is installed on the cable insulation and filled with epoxy resin. By means of a simple structure, the dividers are fastened directly to the cathode-ray tubes of the oscillographs. The design of the dividers and of the discharger was developed by Engineer B. M. Kovalchuk.
For recording the pulses, two high-voltage oscillographs of the OK-19M2 type were used. The oscillographs were triggered from two separate voltage dividers (Fig. 4). To synchronize the phenomenon and the sweep, we used specially made artificial delay lines \(L-C\).
Calibration of both dividers was performed with S-96 and S-100 kilovoltmeters according to the circuit of Fig. 5. In doing so, it was assumed that the pulse at the distance \(l_1 = 4\) m is not attenuated and that the pulse amplitude is equal to the voltage across the capacitance \(C_1\).
Fig. 5. Circuit for calibrating the voltage dividers.
\(P\) — controlled spark gap,
\(C_1 = 0.25\) μF, 30 kV; \(C_2 = 1000\) pF, 60 kV; \(C_3 = 0.25\) μF, 500 V;
\(R_1 = 220\) MΩ, \(D\) — calibrated voltage divider.
Typical oscillograms of the pulses at the beginning and at the end of the cable are shown in Fig. 6. The experiments were carried out for both pulse polarities. To eliminate gross errors, a series of experiments was also performed in which the pulse was applied from the end of the cable.
Fig. 6. Typical oscillograms of pulses at the beginning (a) and at the end (b) of the cable. \(U_0 = 10\) kV.
a) calibration frequency 105 MHz, b) — 11.8 MHz.
B. Results Obtained
The experimental results are shown in Fig. 7. Each point represents an rms value obtained from 10—
30 oscillograms. In Fig. 7 the values of $t$ are plotted for a point on the pulse front at the end of the cable, taken at the level $0.3U_0$, where $U_0$ is the amplitude of the initial pulse. The values of $t$ for the level $0.75U_0$ are also given there.
It follows from Fig. 8 that the pulse has a kink on its front. The coordinates of the kink point are given in Fig. 7.
Fig. 7. ○—$t$ at the level $0.3U_0$ for positive polarity;
●—the same, for negative polarity;
△—coordinates of the kink point for positive polarity;
▲—the same, for negative polarity;
$x$—$t$ at the level $0.75U_0$ for positive polarity;
□—the same for negative polarity.
It should be noted that the length of the pulse front at the beginning of the cable remained practically constant for voltages from 3.7 to 22 kV and was equal to 7–7.5 nsec.
Fig. 8. Comparison of the calculated and experimental pulse shapes.
1 — oscillogram of the pulse at the beginning of the cable; 2 — the same, after a run of 530 m; 3 — calculated pulse shape at the end of the cable for initial pulse 1; 4 — calculated pulse shape at the end of the cable for $\tau_{\phi 1}=0$.
In Fig. 8 the pulse shape at the beginning of the cable (curve 1) and at the end of the cable after a run of 530 m (curve 2) is presented. In addition, the calculated pulse shape at the end of the cable (curve 3) is also given here,
obtained with the aid of the transient characteristic of the cable and the Duhamel integral for the initial pulse. Curve 4 is the calculated pulse shape after traveling 530 m under the condition \(\tau_{\phi1}=0\).
As is seen from Fig. 7, \(t\) at the levels \(0.3\cdot U_0\) and \(0.75\cdot U_0\), in the voltage range from 4 to 20 kV, remains practically constant. This permits one to say that corona has little effect on the distortion of the pulse in the cable.
As is seen from a comparison of the curves in Fig. 8, the calculated and experimental curves are fairly close in shape, which also permits one to say that corona has a comparatively small effect on the transient processes in the cable in the voltage range studied.
Conclusions
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It is hardly meaningful to speak of an undistorting line for the transmission of high-voltage aperiodic pulses. The question should be considered of the optimum ratios of the conductor diameters of the coaxial system from the point of view of minimum attenuation and maximum electric strength.
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As the experiments showed, the pulse front, after traveling 530 m, is lengthened from \(\tau_{\phi1}=7\) nsec to \(\tau_{\phi2}=200\text{--}300\) nsec. At voltages up to 20 kV \((E_{\max}\simeq160\ \text{kV/cm},\ E_{\min}\simeq23\ \text{kV/cm})\), the pulse shape can be calculated approximately by the known method without taking pulse corona into account.
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