Coupling between over-head transmission lines and lightning electromagnetic fields

Abstract

In terms of the generation of the over-voltage, caused by the coupling between overhead transmission lines and lightning electromagnetic fields, an experimental model was built to simulate the coupling, based on the theoretical analyses. It is concluded that when the overhead transmission lines couple with the lightning electromagnetic fields, there is less low frequency component but more high frequency one in its spectrum than the source, thus the coupled waveform is steeper, and the time is shorter than the excitation waveform, about 2 $\mu$ s; Meanwhile, the coupled voltage linearly increases with the addition of the electric resistance while the resistance of overhead transmission lines is below 510 $\Omega$ , however, if the resistance is above 300 k $\Omega$ , the circuit approximates an open one; The peak value of the coupled voltage has a positive correlation with the increase of the height and the length, while the height ranging in 1–3 m and the length in 10–30 m. There is an accordance between the experiments and the theoretical analyses, which is significant to the lightning protection on transmission lines.

Keywords

Overhead transmission line lightning electromagnetic field coupling spectrum analysis

1. Introduction

The lightning electromagnetic pulse has a wide frequency range, high energy density, strong interference and damage. The hazard of lightning to transmission mainly concentrates on the power supply system and transmission system [1, 2, 3].

The coupling of the circuit includes the magnetic induction coupling and the electric induction. In fact, it is difficult to distinguish the intertwined radiation disturbance and conduction disturbance [4, 5]. It would cause transient high voltage and current, which lead to the equipment malfunction, lower precision, even the equipment damage [6]. Therefore, it is necessary to research the process and characteristics of coupling between the transmission cables and the lightning electromagnetic fields and has references to the arrangement of power systems, the electromagnetic compatibility engineering for information systems and the subsequent lightning protection.

Since the mid-twentieth century, amount of analyses have been carried out on the interference coupling of cables in western developed countries [7]. There are three international acknowledged models to describe the coupling between the electromagnetic fields and the transmission line, derived from the Maxwell equation respectively: Taylor model [8], Agrawal model [9] and Rachidi model [10]. If the application is proper, diverse methods on transmission line coupling could acquire equal response results [11]. It is worth to mention that the transmission line theory is not a complete solution but an approximate when the transmission line excited by the incident electromagnetic field [12].

Baba et al. calculated the induced over-voltage on the internal transmission lines of the tall buildings stricken by lightning, using the FDTD method [13].

Xie et al. compared the results of the elevated ground cable coupling of the transient electromagnetic fields which were calculated by Agrawal and Taylor model respectively, and were equal [14].

Wen et al. used numerical calculation to study the over-voltage of the lightning electromagnetic fields in the overhead power distribution lines, obtained that the distribution and the related characteristics of the distribution line coupling over-voltage [15]. Ren et al. adopted FDTD to calculate the near field distribution of the lightning current return stroke, and the transient over-voltage of the overhead power transmission line terminal was calculated with the transmission line equation [16].

Most of the above researchers calculated the pattern in the coupling of transmission lines through simulation models, however, it lacks of experimental results. On the physical conditions, there are differences between the cable coupling on lightning electromagnetic fields and the theoretical model. Especially in the near field, the cable coupling would not be the standard plane wave excitation source for the most of time. In this paper, the coupled characteristics of the lightning electromagnetic fields on overhead cable by scale model discussed, we obtain that when the overhead transmission lines are coupling with lightning electromagnetic fields, the coupled wave head is steeper than the excitation waveform; The coupled voltage linear increases with the augment of the electric resistance while it is below 510 $\Omega$ ; The peak value of the coupled voltage has a positive correlation with the augment of the height and length.

2. The theoretical analyses on the overhead transmission line

The external incident field $E_{in}$ of the transmission line can be a uniform plane wave generated by the long distance lightning, or an inhomogeneous wave generated by the near distance lightning. The influence of lightning electromagnetic field can be used as the source of distribution along the transmission line in the transmission line equation. Assuming a uniform conductor transmission line in parallel with the earth and is placed at a distance of h from the earth (the length of the transmission line is much longer than h). Parallel to the earth direction, defined as the Z axis direction. That is, the transmission line is in (X, Z) surface. Between the wire and the earth, make a square area S with a boundary of L. The region satisfies the Maxwell equation and Stokes’s theorem.

$\displaystyle\int\limits_{S}{\nabla\times\bm{E}_{1}}\cdot\bm{u}_{n}dS=\int% \limits_{L}{\bm{E}_{1}}\cdot\bm{u}_{l}dl=-\frac{\partial}{\partial t}\int% \limits_{S}{\bm{B}_{1}}\cdot\bm{u}_{n}dS.$ (1)

In the Eq. (1), U ${}_{\text{l}}$ is the unit tangent vector along the counterclockwise Loop L. U ${}_{\text{b}}$ is the unit vector pointing in the paper. The above Eq. (1) can be written as:

$\displaystyle\int\limits_{ab}{\bm{E}_{1}}\cdot\bm{u}_{l}dl+\int\limits_{bc}{% \bm{E}_{1}}\cdot\bm{u}_{l}dl+\int\limits_{cd}{\bm{E}_{1}}\cdot\bm{u}_{l}dl=-% \frac{\partial}{\partial t}\int\limits_{S}{\bm{B}_{1}}\cdot\bm{u}_{b}dS.$ (2)

Figure 1.

Theoretical model of transmission line.

The total induced electric field can be decomposed into the sum of the incident field and the scattered field:

$\displaystyle\bm{E}_{1}=\bm{E}_{in}+\bm{E}_{s},\bm{B}_{1}=\bm{B}_{in}+\bm{B}_{% s}.$

Load response model of ideal overhead single transmission on the ground can be seen as a kind of load response model extending of two-wire transmission lines. Its characteristic impedance is calculated as half of the two-wire transmission lines while the reflection coefficient is unchanged, still $\rho$ . Since the propagation constant is $\gamma$ , so the inductance coefficient $L^{\prime}$ of the overhead single wire transmission line on the ground is half of the two-wire transmission lines and the capacitance $C^{\prime}$ is doubled. Thus, the only difference is that the current amplitude requires to be multiplied by a factor 2.

According to the literature [17] can be obtained the load voltage and current at the terminal overhead transmission lines, which can be written as compact matrix forms, namely, BLT equations (Baum-Liu-Tesche).

$\displaystyle\begin{array}[]{l}\left[{{\begin{array}[]{*{20}c}{V(0)}\hfill\\ {V(L)}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{1+\rho_{1}}\hfill\\ 0\hfill\\ \end{array}}{\begin{array}[]{*{20}c}0\hfill\\ {1+\rho 2}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{-\rho_{1}}\hfill\\ {e^{\gamma L}}\hfill\\ \end{array}}{\begin{array}[]{*{20}c}{e^{\gamma L}}\hfill\\ {-\rho 2}\hfill\\ \end{array}}}\right]^{-1}\left[{{\begin{array}[]{*{20}c}{e^{\gamma x_{s}}(V_{0% }+Z_{c}I_{0})/2}\hfill\\ {-e^{\gamma(L-x_{s})}(V_{0}-Z_{c}I_{0})/2}\hfill\\ \end{array}}}\right]\\ \left[{{\begin{array}[]{*{20}c}{I(0)}\hfill\\ {I(L)}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{1-\rho_{1}}\hfill\\ 0\hfill\\ \end{array}}{\begin{array}[]{*{20}c}0\hfill\\ {1-\rho 2}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{-\rho_{1}}\hfill\\ {e^{\gamma L}}\hfill\\ \end{array}}{\begin{array}[]{*{20}c}{e^{\gamma L}}\hfill\\ {-\rho 2}\hfill\\ \end{array}}}\right]^{-1}\left[{{\begin{array}[]{*{20}c}{e^{\gamma x_{s}}(V_{0% }+Z_{c}I_{0})/2}\hfill\\ {-e^{\gamma(L-x_{s})}(V_{0}-Z_{c}I_{0})/2}\hfill\\ \end{array}}}\right]\\ \end{array}$ (3)

Local current and local voltage are represented as the integral form of the transmission lines. For example, the following calculation equation of BLT can be obtained by integrating $x_{s}$ coordinate and distributed field source [18].

$\displaystyle\begin{array}[]{l}\left[{{\begin{array}[]{*{20}c}{V(0)}\hfill\\ {V(L)}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{1+\rho_{1}}\hfill\\ 0\hfill\\ \end{array}}{\begin{array}[]{*{20}c}0\hfill\\ {1+\rho 2}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{-\rho_{1}}\hfill\\ {e^{\gamma L}}\hfill\\ \end{array}}{\begin{array}[]{*{20}c}{e^{\gamma L}}\hfill\\ {-\rho 2}\hfill\\ \end{array}}}\right]^{-1}\left[{{\begin{array}[]{*{20}c}{S_{1}}\hfill\\ {S_{2}}\hfill\\ \end{array}}}\right]\\ \left[{{\begin{array}[]{*{20}c}{I(0)}\hfill\\ {I(L)}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{1-\rho_{1}}\hfill\\ 0\hfill\\ \end{array}}{\begin{array}[]{*{20}c}0\hfill\\ {1-\rho 2}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{-\rho_{1}}\hfill\\ {e^{\gamma L}}\hfill\\ \end{array}}{\begin{array}[]{*{20}c}{e^{\gamma L}}\hfill\\ {-\rho 2}\hfill\\ \end{array}}}\right]^{-1}\left[{{\begin{array}[]{*{20}c}{S_{1}}\hfill\\ {S_{2}}\hfill\\ \end{array}}}\right]\\ \end{array}$ (4)

vector sources as follows [19]:

$\displaystyle\left[{{\begin{array}[]{*{20}c}{S_{1}}\hfill\\ {S_{2}}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{\frac{1}{2}\int_{0}^{L}{e% ^{\gamma x_{s}}[{V}^{\prime}_{S1}(x_{s})+Z_{c}{I}^{\prime}_{S1}(x_{s})]dx_{s}}% }\hfill\\ {-\frac{1}{2}\int_{0}^{L}{e^{\gamma(L-x_{s})}[{V}^{\prime}_{S1}(x_{s})-Z_{c}{I% }^{\prime}_{S1}(x_{s})]dx_{s}}}\hfill\\ \end{array}}}\right]$ (5)

$V^{\prime}_{s1}$ and $I^{\prime}_{s1}$ are distributed voltage and current sources respectively.

If using the reflected voltage (Agrawal) formula, the full load current and voltage on the load can be given, but the excitation sources are different, which should be as follows:

$\displaystyle\left[{{\begin{array}[]{*{20}c}{S_{1}}\hfill\\ {S_{2}}\hfill\\ \end{array}}}\right]=\left[{{\begin{array}[]{*{20}c}{\frac{1}{2}\int_{0}^{L}{e% ^{\gamma x_{s}}{V}^{\prime}_{S2}(x_{s})dx_{s}-\frac{V_{1}}{2}+\frac{V_{2}}{2}e% ^{\gamma L}}}\hfill\\ {-\frac{1}{2}\int_{0}^{L}{e^{\gamma(L-x_{s})}{V}^{\prime}_{S2}(x_{s})dx_{s}+% \frac{V_{1}}{2}e^{\gamma L}-\frac{V_{2}}{2}}}\hfill\\ \end{array}}}\right]$ (6)

Either Eq. (5) or Eq. (6) of the BLT source expressions, can be used to calculate the terminal response. As long as you use the correct components, you can get the same results.

As shown in Fig. 2a below, considering the case of plane wave excitation. The Agrawal transmission telegraph equations of a single overhead transmission line on the ground are derived from the double transmission lines equations. To make it easier to analyze ground reflected field, the incident field is still divided in to a vertically polarized component and a horizontally polarized components. The incident electromagnetic wave in Fig. 2b, the angle $\alpha$ is used to describe the relationship between the direction of the electric field vector and the direction of the perpendicular incident plane. $\psi$ and $\Phi$ are respectively defined as the incident angle of elevation and angle of incidence.

Figure 2.

The incident plane wave excitation system.

The distribution source $V^{\prime}_{s2}(x)$ in Eq. (7) is given by the literature.

$\displaystyle{V}^{\prime}_{s2}(x)=E_{x}^{ex}(x)=E_{0}(\cos\alpha\sin\psi\cos% \phi+\sin\alpha\sin\phi)\times(e^{jkh\sin\psi}-e^{-jkh\sin\psi})e^{-jkx\cos% \psi\cos\phi}$ (7)

As requested, the height of the overhead wire is $h<<\lambda$ from the ground. Thus, needing to use the approximation of frequency domain to simply the exponential in Eq. (7), just as Eq. (8):

$\displaystyle{V}^{\prime}_{s2}(x)\approx jk\sin\psi 2hE_{0}(\cos\alpha\sin\psi% \cos\phi+\sin\alpha\sin\phi)e^{-jkx\cos\psi\cos\phi}$ (8)

The entire vertical excitation field at $Z$ height distance from the ideal ground contains only the vertically polarized component, expressed as:

$\displaystyle E_{x}^{ex}(x,z)=E_{0}\cos\alpha\cos\phi(e^{jkz\sin\psi}+e^{-jkz% \sin\psi})e^{-jkx\cos\psi\cos\phi}$ (9)

The lumped voltage source at the end of the wire, that is $x=$ 0, can be expressed as:

$\displaystyle V_{1}=-\int_{0}^{H}{E_{z}^{ex}(0,z)dz}=-E_{0}\cos\alpha\cos\psi% \int_{0}^{h}{(e^{jkz\sin\psi}+e^{-jkz\sin\psi})}dz=-E_{0}\cos\alpha\cos\psi% \frac{e^{jkh\sin\psi}-e^{-jkh\sin\psi}}{jk\sin\psi}$ (10)

In the low frequencies, the expression is:

$\displaystyle V_{1}\approx-2E_{0}h\cos\alpha\cos\psi$ (11)

As the vertical field from in Eq. (9), the lumped voltage source at $x=LV_{2}$ is obtained by $V_{1}$ multiplied by a phase shift factor.

$\displaystyle V_{2}=V_{1}e^{-jkL\cos\psi\cos\phi}$ (12)

The source integral can be calculated by $V_{1}$ and $V_{2}$ , which are defined in the Eqs (10), (11) and (12). Results are as follows:

$\displaystyle\left({{\begin{array}[]{*{20}c}{S_{1}}\hfill\\ {S_{2}}\hfill\\ \end{array}}}\right)=\left({{\begin{array}[]{l}-\left({\frac{E_{0}(\cos\alpha% \sin\psi\cos\phi+\sin\alpha\sin\phi)jkd\sin\psi}{\gamma-jk\cos\psi\cos\phi}-E_% {0}d\cos\psi\cos\alpha}\right)\\ \quad(1-e^{(\gamma-jk\cos\psi\cos\phi)L})\hfill\\ -e^{\gamma L}\left({\frac{E_{0}(\cos\alpha\sin\psi\cos\phi+\sin\alpha\sin\phi)% jkd\sin\psi}{\gamma+jk\cos\psi\cos\phi}+E_{0}d\cos\psi\cos\alpha}\right)\\ \quad(1-e^{(\gamma-jk\cos\psi\cos\phi)L})\hfill\\ \end{array}}}\right)$ (13)

The approximate solution of overhead lines coupled the plane wave in the frequency domain can be obtained by making Eq. (13) into Eq. (4). A wide band spectrum can be transformed into time domain by Fourier transform. Then, the transient load response current and voltage of overhead transmission lines in the domain can be gained.

3. Experimental model and data analyses

3.1 Establishment of experimental model

As shown in Fig. 3, a simulation model for the ratio of the lightning current channel is established [22]. At the outdoor, we set up a insulating rod with length of 30 m and diameter of 2 cm (d $=$ 2 cm), which is twined uniformly with an insulation copper cable length of 150 m and diameter of 7 mm (r $=$ 7 mm). At the bottom end of the channel, the impulse current is injected and the upper end is connected with the bottom line to form a loop. In this way, the impulse current of the lightning is simulated and the speed of the impulse current to the vertical ground upward is c/3 (1/3 light speed), which is similar to the return stroke development speed of the natural lightning [23]. In the experiment, the simulation of the lightning current channel is used as the simulation of the electromagnetic wave radiation source.

Figure 3.

The experimental model.

According to the requirements of the International Electrical Commission, the waveform of standard lightning impulse are shown in Fig. 4. In the experiment, the 8/20 $\mu$ s generator was used to generate the impulse current, which was injected into the vertical cable to simulate the lightning channel. Measured lightning channel impulse current is shown in Fig. 5. LEMP (lightning electromagnetic pulse) is generated by the strong electromagnetic field radiation of transient current from lightning current discharge. By simulating the lightning channel, the transient electromagnetic radiation field around the lightning can be simulated well.

Figure 4.

Analytical lightning current waveform.

Figure 5.

Measured simulated lightning current waveform.

8/20 $\mu$ s generator used in the experiment generates the voltage amplitude of 5–40 kV and the current amplitude of 3–24 kA impulse current. Voltage step size is 5 kV.

The overhead line used a copper core insulated wire with a diameter of 9.0 mm. Core cross-sectional area of 47.7 mm ${}^{2}$ . Both ends of the wire were connected with analog load R ${}_{1}$ , R ${}_{2}$ then to the buried grounding wires, which is depth of about 0.3 m. As shown as Fig. 3a, the horizontal and vertical distances from the lightning channel were set to be 1.5 m and 8 m. In the experiment, the voltage of R ${}_{1}$ was collected by a digital oscilloscope, and the transient voltage waveform of the overhead wire coupling the lightning electromagnetic fields was recorded. The current of the lightning channel was collected by the Roche coil in the simulation of the lightning channel loop which was a comparison with the later coupling waveform.

The day for testing was sunny, humidity 48%. Before and after the experiment, we have used 61557 electrical test equipment to test the soil resistivity on the test site (below the overhead line). The soil resistivity are 33 $\Omega\cdot$ m, 31 $\Omega\cdot$ m respectively. So it is considered that the soil resistivity is constant in the experiment, which will not produce error in the results.

In the test, diverse conditions could be changed to analyze the characteristics of the lightning electromagnetic wave, such as the height, the length and the value of analog load.

3.2 Results and analyses

3.2.1 Typical waveform analyses

According to the experimental data, a graph was plotted on time domain voltage waveform of overhead transmission line coupling lightning electromagnetic fields. As presented in Fig. 3a, the length of the line was 30 m, height being 3 m, the analog load R ${}_{1}=$ R ${}_{2}$ being 680 k $\Omega$ . The impulse voltage was 30 kV (Fig. 6a) and 20 kV (Fig. 6b) respectively.

Figure 6.

Typical waveform.

As demonstrated as Fig. 6a, there was a high frequency oscillation, lasting about 3 $\mu$ s, after the trigger. The first peak of the oscillation waveform was about 370 V, which might be due to the mismatch caused by the self excitation oscillation. In the subsequent 3–6 $\mu$ s, the second peak value reached 220 V, the peak value of the coupled lightning electromagnetic fields. Compared to the impact waveform, the rising speed of the coupled waveform was faster (6 $\mu$ s), and the waveform was more steep. The entire period was 44–50 $\mu$ s, with the relatively low frequency recoil oscillation.

Figure 7.

Waveform comparison in time domain.

As shown in the Fig. 7, the coupling voltage wave (shown in Fig. 6a) and the the lightning channel excitation source voltage waves simulated are compared in time domain. By comparison, it can be known that the time of the voltage waveform of the overhead transmission line is much shorter than that of the excitation source. At the wave head of the coupling wave has a more obvious high-order mode. And at the rising edge and the peak of the wave, there are obvious oscillations, which is caused by the impedance mismatch of the overhead cable and the impedance mutation.

Figure 8.

Waveform comparison in frequency domain.

The amplitude frequency curves were obtained by the Fourier transform of the current waveform which were converted by the coupled voltage waveform of Fig. 3a. Similarly, making Fourier transform to the simulation of lightning channel current collected at the impact platform. Two results were normalized and exhibited as the Fig. 4. It is presented that the simulated lightning current spectrum has the highest point in the extremely low frequency, 0–10 kHz, while a rapid decline, down to 100 kHz. Then the trend slows down, and gradually tends to 0. However, the coupled spectrum curve shows a rapid rise in the 0–5 kHz part. Normalized current A ${}_{\text{n}}$ increases from 0.5 to 1, then declines rapidly, and the speed of decline is far faster than lightning current spectrum. When the frequency is 100 kHz, the coupled frequency spectrum is coincident with the lightning current frequency spectrum, and tends to be 0. Nevertheless, the A ${}_{\text{n}}$ still has a vibration in the range of 0–0.1. The results present that the lightning current energy is mainly concentrated upon the low frequency parts below the 50 kHz. In 50–100 kHz, the amplitude frequency curve of the coupling waveform declines much faster than the lightning current source waveform, thus the coupling component in the 50–100 kHz frequency band is also far less than that in the lightning current source. From 100 kHz to infinite frequency band, the amplitude frequency curve of the current source tends to 0. And the coupling waveform has a higher A ${}_{\text{n}}$ value at various frequency points. So, above 110 kHz, the coupling waveform contains more.

3.2.2 The influence of the coupled overhead wire from impulse lightning current source

The other test conditions were invariant. The amplitude of the simulated lightning impulse current aggrandized linearly from 5 kV to 40 kV, and the step size is 5 kV.

Figure 9.

Coupled waveform with 2.5 m high and 510 $\Omega$ analog load.

As demonstrated in Fig. 9, the four figures are the overhead wires coupling the lightning electromagnetic voltage waveform whose length were 30 m, both ends of the load impedance being 510 $\Omega$ . In Fig. 9a, b, c and d, the load impedance R ${}_{1}$ was measured at the overhead wires to get the coupled voltage waveform, while the simulated lightning voltages were 10 kV, 20 kV, 30 kV and 40 kV respectively. In Fig. 9a, the coupled voltage value reached the first peak in less than 1 $\mu$ s, which was the maximum value of the whole coupling wave, about 36 V. Then, the coupled voltage waveform appeared a damped oscillation, which was due to the overhead wire impedance mismatching caused by the mismatching of the connecting dots of wires and simulation loads, overhead wires grounding, simulated lightning current incentive loop and so on. Within the subsequent 20 $\mu$ s, the oscillation was obviously weakened and connected with the coupling simulated lightning waveform head at 20 $\mu$ s, where reached a second peak value about 8 V. The first oscillation waveform of the coupling waveform in Fig. 9b, c and d were resemble to that in Fig. 9a where the impulse voltage was 10 kV, reaching the maximum value within 1 $\mu$ s. However, the oscillation duration became shorter compared to the 10 kV condition, and it took only 10 $\mu$ s to reach the peak. As demonstrated as Fig. 9, the voltage waveform of the coupled overhead cable had a steeper second peak and a faster acceleration when impulse voltage was over 10 kV.

Figure 10.

The coupled voltage change trend with difference impulse voltages.

As present as Fig. 10, the overhead height was set as 3 m, and the experiment indicate being 510 $\Omega$ and 680 k $\Omega$ of simulated load. With the increase of impulse period, the coupling voltage were a linear trend. In comparison, the 680 k $\Omega$ g had a greater increasing rate than the 510 $\Omega$ . The fitting formulas of two sets are listed as follows:

$\displaystyle 510\,\Omega:\text{U}=28.543\text{U}_{\text{c}}-24.243;\text{R}^{% 2}=0.96251$ (14) $\displaystyle 680\,\text{k}\Omega:\text{U}=73.23\text{U}_{\text{c}}-52;\text{R% }^{2}=0.9761$ (15)

3.2.3 The influence of the analog load on the coupling overhead wire

The peak values of the coupled voltage data were shown in Table 1.

Table 1
The coupled peak voltage in different analog load

Coupling peak voltage with different analog load/V
High	3 m				1.5 m
Analog load	510 $\Omega$	300 k $\Omega$	680 k $\Omega$	R ${}_{1}$ open	510 $\Omega$	300 k $\Omega$	680 k $\Omega$	R ${}_{1}$ open
Impulse voltage				circuit				circuit
5 kV	19	54	56	40.6	20	42	51	28
10 kV	38	102	104	71	42	84	88	51
15 kV	56	154	160	113	60	128	128	76
20 kV	80	206	208	154	82	168	180	102
25 kV	104	264	280	196	112	232	220	126
30 kV	134	360	376	266	144	280	264	166
35 kV	174	464	464	343	188	360	344	220
40 kV	228	568	576	455	224	408	408	260

Figure 11.

The coupled voltage change trend with different analog loads.

As presented in Fig. 11, the overhead cable coupled voltage increases with the augment of impulse voltage from excitation source in a linear pattern under different load conditions. On Fig. 11a and b, the load was respectively 300 k $\Omega$ and 680 k $\Omega$ . Meanwhile, their coupled voltage lines nearly coincided, which means that the circuit was a open loop because of the huge impedance, and the coupled value reached maximum. It would not affect the coupled voltage seriously if the R ${}_{1}$ and R ${}_{2}$ increased continuously. In addition, the peak of coupled voltage was not in a linear relationship with the enlarging impedance.

3.2.4 The influence of coupling overhead wire from height and length

Figure 12.

The coupled voltage change trend with different heights.

As demonstrated as Fig. 12a, altering the height of transmission line when load resistance was set as 300 k $\Omega$ , the peak value transformation was observed. When the impulse voltage raised, the peak of the coupled voltage rose as well. On the uniform condition, the higher overhead line owned a higher coupled value. In addition, the coupled voltage increasing by 210 V, the overhead height raised from 1 m to 1.5 m, when the impulse voltage reached 40 kV. However, the coupled voltage only augmented 160 V while the increment was 1.5 m in height, within the range of 1.5–3 m. Therefore, the transform had a greater amplitude of variation in the height of 1–1.5 m.

According to Fig. 12b, higher overhead lines brought higher coupled voltage peak values. The coupled voltage of 2 m, 3 m high overhead lines had greater recoil waveform than the 1 m, 1.5 m ones. The smallest coupled amplitude value appeared at the overhead height of 1 m. 1.5 m, the second smallest value. Although the 2 m and 3 m high had alike waveform that had coincide elements, the peak in 3 m height was significantly greater.

The overhead line length was altered in the pursuant series of experiments. The overhead height was maintained as 3 m, the 40 m wire used. There were four sets of length: 10 m, 15 m, 20 m and 30 m. The extra length of wire was horizontally placed on the ground instead of cutting off when the length decreased in the experiment, so it maintained the resistance. As presented as Fig. 13, the augment of the overhead length made a greater coupled voltage peak in the impulse voltage range of 5–40 kV. In Fig. 13a, when the lightning impulse voltage was 40 kV, the wire with a load of 300 k $\Omega$ had a coupled voltage peak value, 570 V, the height being 3 m. When the wire shorten to 15 m, the coupled voltage peak diminished to 320 V (56% of 30 m in length). In Fig. 13b, as the load being 510 $\Omega$ , the coupled peak had a voltage of 230 V while the length was set as 30 m. By shortening the length to 15 m, the peak lessened to 118 V (51% of 30m in length). That is to say, the overhead length had a proportional relation with the coupled voltage, which was significant in Fig. 13b.

Figure 13.

The coupled voltage change trend with different length.

In conclusion, when the other conditions remained invariant, it was presented that the expansion of proportion, adding the height and the length of the overhead cables, would increase the coupled voltage extremely.There is an accordance between the experiments and the theoretical analyses.

3.2.5 The influence of different distance on the coupling of overhead line

When the length of the overhead transmission line is 3 m, and the load impedance is 300 k $\Omega$ , we study the coupling characteristic only by changing the distance between the overhead transmission line and the simulating lightning channel. As shown in Fig. 14, when the excitation source voltage is 30 kV, the coupling voltage waveform have been measured under three different distance. From the figure, we can see that the maximum peak value of the coupling is 364 V when the distance from the source is 8 m. When the distance is increased to 16 m, the coupling peak value is 200 V. When the distance is 24 m, the peak value of the measured voltage is 89 V. According to the test data, it is known that when the distance between the overhead transmission line and the simulated lightning channel is changed, the amplitude of the coupling voltage is influenced, and there is no effect on the coupling voltage waveform. The coupling voltage of the overhead transmission line decreases with the increase of the distance. Some test data are shown in Table 2.

Table 2
The coupled peak voltage in different different distances

Coupling peak voltage with different distance/V
Analog load	510 $\Omega$			300 k $\Omega$
[height=0.cm,width=2.4cm]ImpulseDistance	8 m	16 m	24 m	8 m	16 m	24 m
5 kV	19	7	5	54	30	12
10 kV	38	13	6	102	66	31
15 kV	56	21	6	154	92	43
20 kV	80	27	11	206	132	55
25 kV	104	39	18	264	171	70
30 kV	134	51	24	360	220	89
35 kV	174	68	33	464	252	132
40 kV	228	77	43	568	308	151

Figure 14.

Coupled waveform in different distances.

4. Conclusion

In terms of the generation of the over voltage, caused by the coupling between over-head transmission lines and lightning electromagnetic fields, we have built the model, and tested on multiple conditions, gaining following results:

On the experimental conditions, the coupled waveform of overhead transmission line has smaller coupled energy within low frequency bands (0–100 kHz), compared to lightning current waveform. However, the coupled energy is higher than the source waveform while the frequency reaches 100 kHz. It is presented in the time domain as following characteristics: the faster accelerated waveform; the steeper waveform; the high-frequency oscillation in waveform.

The coupled voltage peak values of overhead lines raise with the augment of the lightning impulse voltages.

The coupled voltage has a positive correlation with the augment of electric resistance while the resistance of overhead transmission line is below 510 $\Omega$ , however, it tends to be placid, implying that the determination of the alternative resistance has been diminutive, when the resistance being 510 $\Omega$ . If the resistance is above 300 k $\Omega$ , the circuit approximates an open one.

The coupled voltage linearly increases with the add of the height whose range is 1–3 m, so does the length, ranging 10–30 m, since the coupled area of the lightning radiation fields enlarges while the magnetic flux amplifies. The experiment results are conformed to the theories.

Footnotes

Acknowledgments

The authors are grateful for the support from the Chinese Natural Science Foundation (Grant No. 41175003) and a project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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