**1. Introduction**

Analysis of the propagation behavior of lightning surges on overhead lines is essential for lightning overvoltage protection and insulation coordination of electric power apparatus. It is well known that lightning surges undergo distortion and attenuation when they propagate along overhead lines. Surge corona plays a significant part in the distortion and attenuation phenomena. The design of lightning overvoltage protection depends strongly on accurate knowledge of the amplitude and wavefront steepness of lightning surges [1–6]. In the calculation of lightning overvoltages, surge corona is taken into account by its charge-voltage characteristic, namely the *q*–*<sup>u</sup>* curve. Measurements of the *q*–*<sup>u</sup>* curves have been taken in the corona cages and on the actual overhead lines [7–10]. Nevertheless, the measured data were obtained mainly under double exponential surges. The distortion and attenuation were always calculated according to the *q*–*<sup>u</sup>* curves under double exponential surges, no matter what the actual surge waveshapes are [3,11–13]. As a matter of fact, the grea<sup>t</sup> majority of lightning surges intruding into substations are damped oscillation surges due to the refraction and reflection of surge waves [14,15]. There is a practical need for taking into account the corona effect on the distortion and attenuation of damped oscillation surges. Some work suited to this need was reported in literature [16–19]; however, the systematic research results were still not reported in the previous work. In view of this situation, an attempt is made in this paper to comprehensively investigate the corona effect on the distortion and attenuation of damped oscillation surges. An experimental measurement is made in a corona cage. The basic difference is found between the *q*–*<sup>u</sup>* curves under double exponential and damped oscillation surges. The trajectory feature of the minor loops is also observed under positive and negative damped oscillation surges. Owing to the fact that the traditional approach is only suitable to modeling of the monotonic *q*–*<sup>u</sup>* curves under double exponential surges [20], an extended approach is

**Citation:** Zhang, X.; Huang, K. Lightning Surge Analysis for Overhead Lines Considering Corona Effect. *Appl. Sci.* **2021**, *11*, 8942. https://doi.org/10.3390/ app11198942

Academic Editors: Massimo Brignone and Daniele Mestriner

Received: 31 August 2021 Accepted: 22 September 2021 Published: 25 September 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

presented for modeling of the hysteresis-like *q*–*<sup>u</sup>* curves under damped oscillation surges. With the extended approach implemented into the transient analysis, an efficient method is proposed to calculate the lightning transients on overhead lines. The proposed method can effectively predict the distortion and attenuation of damped oscillation surges on overhead lines with corona. The calculated results are compared with the field test results to check the validity of the proposed method. Then, we further discuss the calculated results in the presence and absence of the minor loops to inquire into the influence of the minor loops on the distortion and attenuation of damped oscillation surges under positive and negative polarities.

### **2. Experimental Investigation on Corona Characteristics**

An experimental arrangemen<sup>t</sup> was built in a high voltage laboratory for measuring the corona *q*–*<sup>u</sup>* curves, as shown in Figure 1. Its schematic diagram is illustrated in Figure 2. Double exponential and damped oscillation surge voltages can be generated by putting the wavefront resistor *R*W and inductor *L*W into operation in the impulse generator IG, respectively. The corona cage consists of an inner electrode and three sections of outer electrodes. These electrodes are coaxially assembled with the longer outer electrode EL (1 m) in the middle and the two shorter outer electrodes ES (0.52 m) at both sides to shield the end effect of the electric field. When a surge voltage is applied to the inner electrode Ei, corona discharge is produced in the corona cage. The charge signal *q* and voltage signal *u* are taken from the integral capacitor *C*e and voltage divider VD, respectively. The two signals are recorded by a digital oscilloscope DS, so that the *q*–*<sup>u</sup>* curves can be obtained from the data processing for the signals *q* and *u*. For the sake of comparison, the wavefront time and amplitude are held to be approximately equal for double exponential and damped oscillation surges. Figure 3 shows a group of measured *q*–*<sup>u</sup>* curves under negative and positive polarities. It can be noticed that the *q*–*<sup>u</sup>* curves under damped oscillation surges roughly coincide with those under double exponential surges until their respective charges reach the maximum values. The difference appears between the curve parts subsequent to the maximum charge points. In these parts, the *q*–*<sup>u</sup>* curves under double exponential surges descend monotonically; however, those under damped oscillation surges follow a hysteresis-like trajectory. The possible forming mechanism of the hysteresis-like trajectory might be attributed to the occurrence of the opposite polar corona [21] and is illustrated by Figure 4. With the voltage *u* decreasing to the first wave trough, the slope of the section FG becomes larger than the geometrical capacitance C0, which causes the section FG to deviate from the section EF. In fact, as the voltage *u* decreases to a certain extent, the electric field near the surface of the inner electrode could be reversed. Once the reversed field strength exceeds a critical value, the opposite polar corona discharge may occur, which produces the opposite polar space charge near the inner electrode. Thence a reduction in the net space charge leads the total charge *q* on the section FG to decrease more rapidly than that on the section EF. A similar interpretation can be reached for the section HP as the voltage *u* decreases to the second wave trough. For this reason, the minor loops is formed after the first oscillation cycle. The minor loops enclosed by the hysteresis-like trajectory are significantly larger under negative polarity than under positive polarity. The area of the minor loops represents the energy loss and can distort and attenuate damped oscillation surges in the subsequent oscillation cycles.

**Figure 1.** Experimental arrangemen<sup>t</sup> in high voltage laboratory.

**Figure 2.** Diagram of experimental system (Ei—inner electrode; EL—longer outer electrode; ES—shorter outer electrode; DS—digital oscilloscope; VD—voltage divider; IG—impulse generator).

**Figure 3.** Measured *q*–*<sup>u</sup>* curves. (**a**) Under negative polarity, (**b**) under positive polarity.

### **3. Modeling of Corona** *q***–***<sup>u</sup>* **Curves**

The traditional approach is only suitable for modeling of the *q*–*<sup>u</sup>* curves under double exponential surges [20]. It introduced the corona charge *q*C by subtracting the induced charge *C*0*<sup>u</sup>* from the total charge *q* [20,22,23], as illustrated in Figure 5. The corona current is described by: − − −− −−

$$i\_{\mathbb{C}} = \begin{cases} 0 & q\_{\mathbb{C}} > (\mathbb{C}\_2 - \mathbb{C}\_0)(\mu - \mathcal{U}\_2) \\\ f(\mu, q\_{\mathbb{C}}) & (\mathbb{C}\_1 - \mathbb{C}\_0)(\mu - \mathcal{U}\_1) < q\_{\mathbb{C}} < (\mathbb{C}\_2 - \mathbb{C}\_0)(\mu - \mathcal{U}\_2) \\\ h(\mu, q\_{\mathbb{C}}) & 0 < q\_{\mathbb{C}} < (\mathbb{C}\_1 - \mathbb{C}\_0)(\mu - \mathcal{U}\_1) \end{cases} \tag{1}$$

where,

$$\begin{cases} f(u, q\_{\mathbb{C}}) = a[(\mathbb{C}2 - \mathbb{C}0)(u - l\mathbb{L}2) - q\_{\mathbb{C}}] \\ h(u, q\_{\mathbb{C}}) = f(u, q\_{\mathbb{C}}) + \beta[(\mathbb{C}\_1 - \mathbb{C}\_0)(u - l\mathbb{L}\_1) - q\_{\mathbb{C}}] \end{cases} \tag{2}$$

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where *C*1 and *C*2 are the dynamic capacitances (*C*2 > *C*1 > *C*0); *U*1 and *U*2 are the critical voltages (*U*1 > *U*2); *α* and *β* are the coefficients (*α* > 0, *β* > 0), as shown in Figure 6.

**Figure 5.** Diagram of corona charge.

**Figure 6.** Model parameters.

In view of the trajectory complexity of the *q*–*<sup>u</sup>* curves under damped oscillation surges, an extended improvement is made on the traditional approach [20,23]. For such a *q*–*<sup>u</sup>* curve, it is divided into different curve segments, as shown in Figure 7. The first curve segmen<sup>t</sup> AB is still described by Equation (1). For modeling of the second curve segmen<sup>t</sup> BB-, the origin of coordinate system is translated to point B, where *u*- = *u* − *U*B and *q*- = *q* − *Q*B. In this way, the curve segmen<sup>t</sup> BB- is described by:

$$i\_{\mathbb{C}} = \begin{cases} 0 & q\_{\mathbb{C}}' > (\mathbb{C}\_2' - \mathbb{C}\_0)(\boldsymbol{u}' - \mathbb{U}\_2') \\\ f(\boldsymbol{u}', q\_{\mathbb{C}}') & (\mathbb{C}\_1' - \mathbb{C}\_0)(\boldsymbol{u}' - \mathbb{U}\_1') < q\_{\mathbb{C}}' < (\mathbb{C}\_2' - \mathbb{C}\_0)(\boldsymbol{u}' - \mathbb{U}\_2') \\\ h(\boldsymbol{u}', q\_{\mathbb{C}}') & 0 < q\_{\mathbb{C}}' < (\mathbb{C}\_1' - \mathbb{C}\_0)(\boldsymbol{u}' - \mathbb{U}\_1') \end{cases} \tag{3}$$

where,

$$\begin{array}{l} f(u', q'\_C) = \alpha' \left[ (\mathbb{C}'\_2 - \mathbb{C}\_0)(u' - \mathbb{U}\_2) - q'\_C \right] \\ h(u', q'\_C) = f(u', q'\_C) + \beta' \left[ (\mathbb{C}'\_1 - \mathbb{C}\_0)(u' - \mathbb{U}'\_1) - q'\_C \right] \end{array} \tag{4}$$

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− − where *C*-2 > *C*-1 > *C*0, *U*-1 > *U*-2, *α*- > 0 and *β*- > 0. In a similar manner, the third curve segmen<sup>t</sup> B-B--, as shown in Figure 8, is described by:

$$i\_{\mathbb{C}} = \begin{cases} 0 & q\_{\mathbb{C}}'' > (\mathbb{C}\_2'' - \mathbb{C}\_0)(\mu'' - \mathbb{U}\_2'') \\\ f(\mu'', q\_{\mathbb{C}}'') & (\mathbb{C}\_1'' - \mathbb{C}\_0)(\mu'' - \mathbb{U}\_1'') < q\_{\mathbb{C}}'' < (\mathbb{C}\_2'' - \mathbb{C}\_0)(\mu'' - \mathbb{U}\_2'') \\\ h(\mu'', q\_{\mathbb{C}}'') & 0 < q\_{\mathbb{C}}'' < (\mathbb{C}\_1'' - \mathbb{C}\_0)(\mu'' - \mathbb{U}\_1'') \end{cases} \tag{5}$$

where,

$$\begin{array}{l} f(\boldsymbol{\mu}^{\boldsymbol{\nu}}, \boldsymbol{q}\_{\boldsymbol{\zeta}}^{\boldsymbol{\nu}}) = \boldsymbol{\mu}^{\boldsymbol{\nu}} \left[ (\mathbf{C}\_{2}^{\boldsymbol{\nu}} - \mathbf{C}\_{0})(\boldsymbol{\mu}^{\boldsymbol{\nu}} - \boldsymbol{\mathcal{U}}\_{2}^{\boldsymbol{\nu}}) - \boldsymbol{q}\_{\boldsymbol{\zeta}}^{\boldsymbol{\nu}} \right] \\\ h(\boldsymbol{\mu}^{\boldsymbol{\nu}}, \boldsymbol{q}\_{\boldsymbol{\zeta}}^{\boldsymbol{\nu}}) = f(\boldsymbol{\mu}^{\boldsymbol{\nu}}, \boldsymbol{q}\_{\boldsymbol{\zeta}}^{\boldsymbol{\nu}}) + \boldsymbol{\beta}^{\boldsymbol{\nu}} \left[ (\mathbf{C}\_{1}^{\boldsymbol{\nu}} - \mathbf{C}\_{0})(\boldsymbol{\mu}^{\boldsymbol{\nu}} - \boldsymbol{\mathcal{U}}\_{1}^{\boldsymbol{\nu}}) - \boldsymbol{q}\_{\boldsymbol{\zeta}}^{\boldsymbol{\nu}} \right] \end{array} \tag{6}$$

where *C*--2 > *C*--1 > *C*0, *U*--1 > *U*--2 , *α*-- > 0 and *β*-- > 0.

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**Figure 7.** Modeling of the second curve segment.

**Figure 8.** Modeling of the third curve segment.

Description of the subsequent curve can be made by analogy with that of either the curve segmen<sup>t</sup> BB- or the curve segmen<sup>t</sup> B-B--. In Equations (1)–(6), the parametric values of *<sup>α</sup>*~*α*--, *β*~*β*--, *C*1~*C*2--, *U*1~*U*1-- and *U*2~*U*2-- can be determined by fitting the given *q*–*<sup>u</sup>* curve. The integral of the corona current *i*C from *t* − Δ*t* to *t* gives the corona charge

$$q\_{\mathbb{C}}(t) = \frac{\Delta t}{2} i\_{\mathbb{C}}(t) + q\_{\mathbb{C}P}(t - \Delta t) \tag{7}$$

where Δ*t* is the time step and *q*CP(*<sup>t</sup>* − Δ*t*) is corona charge at the preceding time step:

$$q\_{CP}(t - \Delta t) = \frac{\Delta t}{2} i\_{\mathbb{C}}(t - \Delta t) + q\_{\mathbb{C}}(t - \Delta t) \tag{8}$$

According to Figure 3, the total charge *q* on the *q*–*<sup>u</sup>* curve is evaluated by:

$$
\eta(t) = q\_{\mathbb{C}}(t) + \mathbb{C}\_0 u(t) \tag{9}
$$

### **4. Transient Calculation Considering Corona Effect**

Figure 9a shows an overhead line with corona. It is subdivided into *M* line segments (see Figure 9b). On each line segment, the corona sheath is approximately considered to be uniform and replaced as a lumped non-linear branch carrying a corona current [24,25], as shown in Figure 9c. After separating the corona sheath from each line segment, the remainder is free of corona and has the linear circuit parameters per unit length, i.e., *R*1, *L*1 and *C*1. Its equivalent circuit is depicted in Figure 10 [26], where Δ *R* = Δ*lR*1, Δ*L* = Δ*lL*1, Δ *C* = Δ*l*C1, *Z* = ( Δ*L*/Δ *C*)1/2 and *τ* = Δ*l*/*v* (*v* is the wave velocity). The expressions of the historical current sources *Ij*−<sup>1</sup> (*t* − *τ*) and *Ij* (*t* − *τ*) were also given in [26]. Considering Figures 9c and 10, the overhead line with corona is converted into a complete equivalent circuit, as shown in in Figure 11. This is a cascade circuit containing *M* non-linear branches carrying corona current. Topologically each unit including the nonlinear branch is disconnected from other. The *j*th (*j* = 1, 2,···, *M*) circuit unit in Figure 11 can be represented as a one–port circuit, as shown in Figure 12. With the non-linear branch removed from the port (see Figure 13), the corresponding open-circuit voltage *uOCj* and input impedance *Zthj* is found by using the Thevenin's equivalent technique. The non-linear branch can be solved according to the Thevenin's equivalent circuit shown in Figure 14: −

$$u\_{\rm OCj} - Z\_{\rm thj} i\_{\rm cj} = u\_{\rm Cj}(j = 1, 2, \cdot, \cdot, M) \tag{10}$$

where the corona current *iCj* is described by Equations (1), (3) and (5). The fundamental studies made on an equivalent circuit model may refer to [27–29]. For the first curve section on the *q*–*<sup>u</sup>* curve, as described by Equation (1), if the corona charge satisfies (*C*1 − *<sup>C</sup>*0)(*uj* − *U*1) < *qcj* < ( *C*2 − *<sup>C</sup>*0)(*uj* − *U*2), the corona current and branch voltage are given by:

$$\begin{array}{l} i\_{\mathbb{C}\bar{\jmath}} = \frac{\not\mathbb{P}}{\mathbb{1} + \mathbb{E}\mathsf{E}} [(\mathsf{C}\mathsf{2} - \mathsf{C}\mathsf{o})(\mathsf{u}\_{\mathbb{C}\bar{\jmath}} - \mathsf{U}\mathsf{L}) - q\_{\mathbb{C}P}(t - \Delta t)] \\\ u\_{\mathbb{C}\bar{\jmath}} = \frac{\not\mathbb{P}}{\mathbb{1} + \mathbb{T}} [u\_{\mathbb{C}\mathsf{e}} + \Gamma \mathsf{U}\mathsf{L} - \frac{\mathbb{P}}{\mathsf{C}\mathsf{2} - \mathsf{C}\mathsf{e}\_{0}} q\_{\mathbb{C}P}(t - \Delta t)] \end{array} \tag{11}$$

where *ξ* = Δ*t*/2 and Γ is,

$$
\Gamma = \frac{\beta Z\_{thj}(\mathbf{C}\_2 - \mathbf{C}\_0)}{1 + \beta \xi} \tag{12}
$$

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**Figure 9.** Segmentation of an overhead line with corona. (**a**) An overhead line with corona, (**b**) line segmen<sup>t</sup> with corona, (**c**) non-linear branch carrying corona current.

**Figure 10.** Equivalent circuit of a line segmen<sup>t</sup> free of corona.

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**Figure 11.** Complete equivalent circuit of an overhead line with corona.

**Figure 12.** One-port circuit containing nonlinear branch.

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**Figure 13.** Thevenin's equivalent procedure.

**Figure 14.** Solution to non-linear branch.

If the corona charge satisfies 0 < *qcj* < (*C*1 − *<sup>C</sup>*0)(*uj* − *U*1), the corona current and branch voltage are given by:

$$\begin{array}{l}\mathrm{i}\_{\mathrm{C}\dot{\jmath}} = \frac{1}{\Lambda\_{1}} [\mathrm{a}(\mathrm{C}\_{1}-\mathrm{C}\_{0})(\mathrm{u}\_{\mathrm{C}\dot{\jmath}}-\mathrm{l}\mathrm{l}\_{1}) + \beta(\mathrm{C}\_{2}-\mathrm{C}\_{0})(\mathrm{u}\_{\mathrm{C}\dot{\jmath}}-\mathrm{l}\mathrm{l}\_{2}) - (\mathrm{a}+\beta)q\_{\mathrm{CP}}(\mathrm{t}-\Delta\mathrm{t})] \\\ u\_{\mathrm{C}\dot{\jmath}} = \frac{1}{\Lambda\_{1}+\Lambda\_{2}+\Lambda\_{3}}[\Lambda\_{1}u\_{\mathrm{o}\dot{\jmath}}+\Lambda\_{2}\mathrm{l}\mathrm{l}\_{1}+\Lambda\_{3}\mathrm{l}\mathrm{l}\_{2}+\Lambda\_{4}q\_{\mathrm{CP}}(\mathrm{t}-\Delta\mathrm{t})] \end{array} \tag{13}$$

where,

$$\begin{array}{l} \Lambda\_1 = 1 + \xi(\alpha + \beta) \\ \Lambda\_2 = Z\_{tlhj}\alpha(\mathcal{C}\_1 - \mathcal{C}\_0) \\ \Lambda\_3 = Z\_{tlhj}\beta(\mathcal{C}\_2 - \mathcal{C}\_0) \\ \Lambda\_4 = Z\_{tlhj}(\alpha + \beta) \end{array} \tag{14}$$

For the second, third and subsequent curve segments on the *q*–*<sup>u</sup>* curve, the corresponding solutions of the corona current and branch voltage can be obtained in a manner similar to Equations (12) and (14). After determining *iCj* and *uCj* (*j* = 1, 2,···, *M*), the nonlinear branch is replaced by a voltage source equal to *uCj*. Figure 12 is converted into a purely linear circuit, as shown in Figure 15, and thereby the transient responses in the remaining linear part can be calculated by performing the nodal voltage analysis. The calculation procedure has been stated in detail in [26,30,31] and is not repeated here due to the limitation of space.

**Figure 15.** Solution to remaining linear circuit.

### **5. Calculated Results and Discussions**

A test overhead line is considered here [17]. Its line conductor is a single-core copper wire with a cross-section of 16 mm2. The average line height is 10 m above the ground and total length *l* is 3084 m. The geometrical capacitance *C*0 is 6.41 pF/m and the soil resistivity is about 50 Ω·m. The corona onset voltages under positive and negative polarities are taken as 70 kV and 80 kV, respectively. The damped oscillation surge voltages with different amplitudes and polarities are applied to the sending end of the line. Two capacitive voltage dividers are installed at the sending end and at 1674 m from the sending end, respectively, to measure the voltage waveshapes. The parameter values for describing the *q*–*<sup>u</sup>* curves are given in Table 1, where the signs (**+**) and (−) denote positive and negative polarities. Using the method proposed above, the voltage waveshapes are calculated at a distance of 1674 m from the sending end, as shown in Figures 16 and 17. The calculated voltage values at the first two wave crests and first wave trough are listed in Tables 2 and 3. In Figures 16 and 17 and Tables 2 and 3, the field test results [17] and those calculated in the presence and absence of the minor loops are given together for comparison, which shows the calculated results can agree well with the field test results. This confirms the validity of the proposed method.

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**Table 1.** Parameter values of modeling *q*–*<sup>u</sup>* curves.


**Table 2.** Voltage values under positive polarity (kV).

**Table 3.** Voltage values under negative polarity (kV).


(**a**) **Figure 16.** *Cont*.

**Figure 16.** Calculated and field test voltage waveshapes under positive polarity (*U*m is the voltage amplitude at the sending end; sending end; field test; calculated). (**a**) In the presence of minor loops, (**b**) in the absence of minor loops.

**Figure 17.** Calculated and field test voltage waveshapes under negative polarity (*U*m is the voltage amplitude at the sending end; sending end; field test; calculated). (**a**) In the presence of minor loops, (**b**) in the absence of minor loops.

As can be seen from Figures 16 and 17 and Tables 2 and 3, the distortion and attenuation of the surge voltages in the first oscillation cycle are considerably more severe under positive polarity than under negative polarity. This is due to the fact that the area of the main loop formed by the curve in the first oscillation cycle is much larger under positive polarity than under negative polarity according to the measured *q*–*<sup>u</sup>* curves shown in Figure 3. Hence the corona discharge in the first oscillation cycle can cause much higher energy loss under positive polarity than under negative polarity. The attenuation decrement of the first wave crest is 44.0~56.4% under positive polarity, whereas that is only 24.1~26.7% under negative polarity. In addition, the minor loops also have different influences on calculation of the distortion and attenuation of the surge voltages. Under positive polarity, as shown in Figure 16 and Table 2, the calculated errors (relative to the field test results) at the first wave trough and second wave crest are 1.7~6.1% and 1.1~4.7% in the presence of the minor loops and 2.8~6.7% and 1.5~4.5% in the absence of the minor loops, respectively. It is thus clear that the calculated errors in the presence and absence of the minor loops are close to each other. The reason is that the area of the minor loops is relatively small on the *q*–*<sup>u</sup>* curves under positive polarity in terms of the measured results given above, and incapable of producing considerable energy loss for the distortion and attenuation after the first oscillation cycle. However, the case under negative polarity is different from that under positive polarity. As shown in Figure 17 and Table 3, the calculated errors at the first wave trough and second crest reach 5.2~10.2% and 5.1~10.6% in the absence of the minor loops, whereas those are only 1.5~4.8% and 0.94~5.4% in the presence of the minor loops, respectively. It follows that there is an appreciable difference in the calculation precision in the presence and absence of the minor loops. Moreover, the calculated results in the presence of the minor loops more closely approximate the field test results. This can be interpreted as the area of minor loops being relatively large on the *q*–*<sup>u</sup>* curves under negative polarity in the light of the aforementioned experimental observation and causing considerable energy loss to distort and attenuate the surge voltages in the subsequent oscillation cycles. Therefore, neglect of the minor loops in the transient calculation may give rise to non-negligible error for predicting the distortion and attenuation of negative damped oscillation surges.

For a further verification of the proposed method, the calculated voltage waveshapes are also compared with those obtained from the FDTD method [32], as shown in Figure 18. On the whole, the former agrees with the latter and both are close to the field test voltage waveshapes.

**Figure 18.** A comparison with FDTD method ( *U* m is the voltage amplitude at the sending end; sending end; field test; proposed method; FDTD).
