To simulate a natural lightning strike, the breakdown should occur within 1–3 μs in an aircraft lightning zoning test. However, the breakdown voltage–time characteristic of the combined gap is hard to derive because of the complex breakdown process. In this paper, a series of positive lightning impulse discharge tests in a rod electrode–floating aircraft model–plate combined gap were conducted. The leader propagation processes in both sub-gaps were observed using a high-speed camera. The potential of the floating aircraft model was analyzed. We propose a potential calculation model for the floating aircraft during the leader propagation phase. After fitting test data, an empirical formula for the connected streamer plasma was given. Taking that model into account, the calculation result of the average breakdown time of the combined gap has an error of <5.7%.

The incidence of lightning strikes on commercial aircraft is about once a year on average.1 It is mandatory for aircraft to demonstrate adequate lightning protection design against direct lightning strike.2 The first step is to classify the surfaces depending on the lightning attachment points, which is known as lightning zoning, a premise for civil aircraft to pass the airworthiness certification.3 

The lightning striking points on the aircraft fuselage are determined during the last stepped leader propagation, also known as the final jump phase.4–6 Due to the last stepped leader being able to reach hundreds of meters, the impulse generators can hardly reproduce it in laboratory.7 Therefore, the work is fulfilled by a scale test constructed by a combined gap of rod–floating conductor (a conducting model aircraft)–ground, and the breakdown should occur in 1–3 μs.3 

The leader propagation model (LPM) is proposed to obtain the breakdown voltage–time characteristic of long air gaps, where the leader velocity is linked to the applied voltage and the leader length.8–13 However, the breakdown process of the combined gap is more complicated than that of the single air gap.4,5,14–19 The floating conductor potential will be influenced by the discharge plasmas, so it is hard to directly obtain the applied voltage across both sub-gaps by measurement or simulation during the final jump phase.19 

This paper presents aircraft model tests under positive lightning impulses, employing high-speed cameras to observe leader propagation processes. The floating conductor potential is analyzed during the final jump phase. Based on the leader propagation model, a calculation method for the breakdown voltage–time characteristics of the test gap is constructed. By fitting test data, the empirical formula for the resistance of connected streamer plasma is obtained. After taking the resistance into account, the calculation error is <5.7%.

The tests were carried out at the Anhui Provincial Laboratory of Aircraft Lightning Protection in Hefei, China, at an altitude of ∼31 m. The test setup is displayed in Fig. 1. A Phantom V2512 high-speed camera, with a frame rate of 660 000, a resolution of 128 × 64 pixel, and an exposure time of 1 μs, was applied to record the leader propagation processes in both sub-gaps.20 

FIG. 1.

Schematic diagram of the test platform. Reprinted with permission from Sun et al., AIP Adv. 14(3), 035148 (2024). Author(s), licensed under a Creative Commons Attribution (CC BY) License.24 

FIG. 1.

Schematic diagram of the test platform. Reprinted with permission from Sun et al., AIP Adv. 14(3), 035148 (2024). Author(s), licensed under a Creative Commons Attribution (CC BY) License.24 

Close modal

The combined gap consists of a rod electrode, a floating aircraft model, and a ground plate. The rod electrode had a 2 cm diameter hemisphere head. The metal aircraft model measured 162 cm in wingspan, 160 cm in length, and 45 cm in height. In this study, the length of the electrode–aircraft model gap is called sub-gap 1 of H of 143–200 cm, while the length of the aircraft model–ground gap is called sub-gap 2 of G of 100 cm.

The positive lightning impulse, with parameters (1.1/45) μs indicating a rise time of 1.1 μs and a decay time to half value of 45 μs, was generated using a 4800 kV Marx generator. The typical voltage waveform is shown in Fig. 2. The voltage applied to the rod electrode follows the lightning impulse used in tests and is expressed as21 
Ut=kUmeAtcosωt+BsinωteCt.
(1)
FIG. 2.

Applied lightning impulse and recorded voltage waveform.

FIG. 2.

Applied lightning impulse and recorded voltage waveform.

Close modal

Here, k is 0.846, Um is the voltage amplitude, A is 22 000, ω is 2 250 000, B is 0.5, and C is 1 080 000.

The tests underwent 20 repetitions, with a 10-min interval between each breakdown. Table I summarizes the gap length and the corresponding average peak voltage in tests.

TABLE I.

Average peak voltage and the corresponding standard deviation of each gap length.

H (cm)G (cm)Averaged peak voltage (kV)
143 100 2359 
150 100 2390 
160 100 2390 
170 100 2459 
175 100 2579 
180 100 2583 
190 100 2590 
200 100 2600 
H (cm)G (cm)Averaged peak voltage (kV)
143 100 2359 
150 100 2390 
160 100 2390 
170 100 2459 
175 100 2579 
180 100 2583 
190 100 2590 
200 100 2600 

For long air gaps or insulators, the leader propagation model (LPM) is often used to calculate the breakdown voltage–time characteristics under fast waveforms like lightning impulses.10–13 

The model considers an equivalent leader across the gap with a propagation time the same as that of the actual leader.

The equivalent leader length can be obtained by solving a set of differential equations. The formula recommended by CIGRE [International Conference on Large High Voltage Electric System (CIGRE)] has been widely used:22,
vt=kvugaptugaptdlE0,
(2)
where vt is the leader propagation velocity, kv is a constant of 0.8 × 10−6 m2 V−2 s−1 for positive leader propagation, ugapt is the voltage waveform across the gap, d is the gap length, l is the equivalent leader length, and E0 is the critical field strength for positive leader initiation, which is 600 kV/m.22 

The applied voltage across gap ugap is the necessary original input for LPM. The applied voltage across the combined gap is obtained from Sec. II.

The initial leader length in the gap was l0 = 0 in t = 0, and it would keep zero until the ugap was high enough in t0:
ugapt0dlE0>0.
(3)
After that, the leader can initiate and propagate. The time step Δt is 1 ns here. During the Δt, the leader velocity is assumed as the result of formula (2):
vt0=kvugapt0ugapt0d0E0.
(4)
And the leader length should be updated as
lt0=l0+vt0*Δt0.
(5)
After another Δt, the instantaneous velocity of the leader should be
vt0+Δt=kvugapt0+Δtugapt0+Δtdlt0E0,
(6)
and the length of the leader should be
lt0+Δt=lt0+vt0+Δt*Δt0.
(7)

The calculation will repeat to obtain instantaneous velocity and length of leader during the breakdown process, until the leader length exceeds the gap length or the ugap smaller than the [E0∗(dl)]. The former means the breakdown occurs, and the latter means the breakdown will not occur.

The classic application scenarios for the LPM method are rod–rod gaps, rod–plate gaps, or similar long insulators. The actual time for the actual leader across the gap is represented by that for the equivalent leader across the gap. It is clear that, under the same conditions (such as gap length, applied voltage, and air environment), the breakdown time of the rod–rod gap is always shorter than that of the rod–plate gap. Accordingly, the number of equivalent leader(s) can be different for different gap kinds.

Due to the complicated geometric features of the aircraft model, it is unreliable to assume the number of equivalent leader(s) in both sub-gaps, but experimental observation is a more reliable method. There are typical observations to illustrate the equivalent leader number in the two sub-gaps.

Apparently, there was only a positive downward leader on the rod and no upward leader on the floating model surface in sub-gap 1, as shown in Fig. 3. So, the corresponding equivalent leader number is one of the sub-gap 1. And after the leader bridged sub-gap 1, the downward leader was half as long as sub-gap 2, and there was no visible upward leader on the grounding plate, as shown in Fig. 4. So, the corresponding equivalent leader number is also one of the sub-gap 2.

FIG. 3.

Typical leader propagation process in sub-gap 1 under lightning impulse (H = 200 cm): (a) the downward leader initiation; (b) only the downward leader propagation; (3) the arc channel bridged the rod–floating model gap.

FIG. 3.

Typical leader propagation process in sub-gap 1 under lightning impulse (H = 200 cm): (a) the downward leader initiation; (b) only the downward leader propagation; (3) the arc channel bridged the rod–floating model gap.

Close modal
FIG. 4.

Typical comparison of leader propagation in both sub-gaps (H = 200 cm): (a) the downward leader was about to bridge sub-gap 1; (b) the downward leader propagated in sub-gap 2.

FIG. 4.

Typical comparison of leader propagation in both sub-gaps (H = 200 cm): (a) the downward leader was about to bridge sub-gap 1; (b) the downward leader propagated in sub-gap 2.

Close modal
Before the sub-gaps are bridged by streamer plasmas, the floating aircraft model potential (Uf) is mainly the geometric potential induced by the electric field.9 In Rizk’s study, the averaged electric potential gradient of the positive streamer zone is 400 kV/m. Therefore, it is considerate that the sub-gaps get bridged by streamer zones when the average electric field strength in sub-gaps reaches Ep = 400 kV/m. Before that, the kg represents the ratio of the geometric potential to the applied voltage [u(t)], as illustrated in Fig. 5:
uf=kgut.
(8)
FIG. 5.

Floating aircraft model potential is 170 V when the electrode is 1000 V (H = 200 cm).

FIG. 5.

Floating aircraft model potential is 170 V when the electrode is 1000 V (H = 200 cm).

Close modal

After a sub-gap was bridged by streamer plasma, once the average electric field strength reached E = 600 kV/m,22 the leader would initiate and propagate very soon. The high-speed camera was employed to observe the leader propagation in both sub-gaps.

As the downward leader initiated from the rod electrode head in Fig. 6(a), there were blurry streamer regions that bridged the electrode–aircraft model. After the streamer plasma connected the electrode and the aircraft model, the leader would propagate inside the streamer region along a random path, as shown in Fig. 6(b). That means the discharge had entered the final jump phase, and the streamer plasma path could be seen as an electric connection across the leader head and the aircraft model.

FIG. 6.

Typical leader propagation in the electrode–aircraft model gap (H = 200 cm): (a) the propagating downward leader; (b) the downward leader was about to bridge the sub-gap 1, while the short upward leader could be ignored.

FIG. 6.

Typical leader propagation in the electrode–aircraft model gap (H = 200 cm): (a) the propagating downward leader; (b) the downward leader was about to bridge the sub-gap 1, while the short upward leader could be ignored.

Close modal

The connected plasma channel consists of a leader and a streamer. The average potential gradient inside the leader channel is reported as 30–100 kV/m.15 A constant EL = 50 kV/m was chosen in this paper.9 And the average potential gradient inside the streamer region is much higher, which is at least 400 kV/m.

Therefore, the resistance of streamer plasma is the main body used to determine the equivalent current flowing through the connected plasma channel, which can be expressed as23 
Is1=utuftEL*l1/Rs1,
(9)
Is2=uftEL*l2/Rs2.
(10)

The Is1 and Is2 are equivalent current in two sub-gaps, uft is the instantaneous floating potential on the aircraft model, l1 and l2 are the leader length in two sub-gaps, and Rs1 and Rs2 are the resistance of streamer plasma in two sub-gaps.

The initial resistance of the connected streamer plasma and the gap length are positively correlated.23 And the shorter streamer plasma path causes a shrinking resistance value. Therefore, an empirical formula is proposed as
Rs=Rddl,
(11)
where R is a constant, whose value is obtained by fitting test data in Chap. 7.2, d is the gap length, and l is the leader length.
Those currents will change the net charge on the floating conductor surface, which affects the floating potential uft:
uft=kgut+(Is1Is2)dt/Cf,
(12)
where Cf is the stray capacitance of the floating aircraft model, which can be derived by simulations of different H, and a typical simulation result is shown in Fig. 7.
FIG. 7.

The floating potential is 18.9 kV with a 1 μC charge on the floating aircraft model (H = 200 cm).

FIG. 7.

The floating potential is 18.9 kV with a 1 μC charge on the floating aircraft model (H = 200 cm).

Close modal

The leaders propagate inside both sub-gaps, and the gap will be bridged by the leader channel. Figure 4 demonstrates the leader breakdown processes in both sub-gaps.

As shown in Fig. 4, when the downward leader almost bridged the electrode–aircraft model gap, the downward leader was already about 50 cm in the aircraft model–ground plate gap. Before the leader bridged the electrode–aircraft model gap, the downward leader had initiated and propagated for several ceilometers in the aircraft model–grounding plate gap. It can be concluded that the downward leader propagation in the two sub-gaps was not sequential. The calculation of leader propagation in both sub-gaps should also not be sequential but parallel. The central core of the leader channel is considered to be highly conductive. When the leader channel bridges the electrode–aircraft model gap, the floating aircraft model potential can be obtained by
uft=utELd1.
(13)

The lengths of sub-gaps are denoted as d1 of H and d2 of G. l1 and l2 are the leader lengths in two sub-gaps, u(t) is the applied lightning impulse voltage, and v(t) is the velocity of the developing leader. Rs and Is are the resistance of the streamer plasma channel and the current. The time step Δt is 1 ns, and the maximum simulation time tmax is 20 μs. The calculation stops when the time exceeds 20 μs or the leaders bridge the combined gap. The calculation flowchart is illustrated in Fig. 8.

FIG. 8.

Flowchart of the numerical calculation for the test gap.

FIG. 8.

Flowchart of the numerical calculation for the test gap.

Close modal

The calculation is implemented using a for loop structure. As the calculation begins, the first judgment statement is to determine whether the leader length exceeds the sub-gap length. If both leader lengths exceed the lengths of two sub-gaps, then the calculation stops. If both leader lengths do not exceed the lengths of two sub-gaps, then the calculation is about the discharge processed in both sub-gaps. If only one leader length exceeds the length of the corresponding sub-gap, then the calculation is about the leader breakdown in sub-gap 2.

If both leader lengths do not exceed the lengths of two sub-gaps, then a judgment statement is to determine whether the voltages across sub-gaps meet the threshold value for streamer zones that bridge the sub-gaps. Before the streamer zones bridge any sub-gap, the floating potential uf is calculated by formula (8). Once a sub-gap is bridged by a streamer, the floating potential uf shall be calculated by formula (12), which means the charge transferred inside the streamer zones starts to influence the floating potential uf.

After a sub-gap gets bridged by a streamer or the leader has initiated and propagated for a while, a judgment statement is made to determine whether the average electric field strength exceeds the threshold value for positive leader initiation, where E0 = 600 kV/m in formula (2). If so, the floating potential uf shall be calculated by formula (12). After that, the instantaneous velocity and length of the leader can be calculated by formulas (4) and (5).

Once the leader breakdown occurs in sub-gap 1, the floating potential uf is calculated by formula (13). After that, the instantaneous velocity and length of the leader in sub-gap 2 can be calculated by formulas (4) and (5). At last, but not least, the calculation should stop when the instantaneous voltage amplitude is hard to support leader propagation in this manuscript or the calculated breakdown time is far from the test results. Therefore, the maximum computation time is set at 20 μs. However, the time threshold value can be adjusted for different waveforms.

Figure 9 presents the breakdown time of the combined gap discharges with different H in this paper. Many of the data points overlap on the graph. And the average breakdown time under different H is depicted in Table II.

FIG. 9.

Breakdown time data of the lightning tests in this paper.

FIG. 9.

Breakdown time data of the lightning tests in this paper.

Close modal
TABLE II.

Average breakdown time of tests under different H.

H (cm) 143 150 160 170 175 180 190 200 
Average breakdown time (μs) 1.81 1.97 2.08 2.19 1.96 2.03 2.24 2.40 
H (cm) 143 150 160 170 175 180 190 200 
Average breakdown time (μs) 1.81 1.97 2.08 2.19 1.96 2.03 2.24 2.40 

In general, the breakdown time of the combined gap was positively correlated with H and negatively correlated with the applied voltage. The H is the leader propagation distance to bridge the gap, and the higher voltage leads to higher electric field intensity and higher leader velocity.

The constant R in Chap. 4.2 represents the conductivity of the connected streamer plasma channel. The initial value of R is set as 0.1 and then increases to 100 at a fixed step of 0.1. For each R, the sum of squared errors between the calculation result and the test result is counted. When R = 12.0 kΩ m−2, there is a minimum sum of squared error.

With R = 12.0 kΩ m−2, the calculation results and the test results are exhibited in Figs. 10 and 11. Sim 1 is the calculation result considering the current through the connected plasma channels. To verify the availability of the proposed method, the numerical calculation without considering the current through the connected plasma channels is conducted, and the result is Sim 2. Both calculation results are compared with the average test result.

FIG. 10.

Comparison of the calculated results and test results for H <170 cm.

FIG. 10.

Comparison of the calculated results and test results for H <170 cm.

Close modal
FIG. 11.

Comparison of the calculated results and test results for H >175 cm.

FIG. 11.

Comparison of the calculated results and test results for H >175 cm.

Close modal

As shown in Figs. 10 and 11, the maximum relative error between Sim 1 and the test result is about 5.7%. However, Sim 2 seems to be about at least 0.3 μs lower than the test result for each H. And the relative error between Sim 2 and the test result ranges from 14.9% to 21.0%. This illustrates that the proposed method is far more consistent with the test results in this paper.

The leader velocity shows strong nonlinear characteristics in Figs. 12 and 13. In the beginning of discharge, most of the applied voltage was across the sub-gap 1, which allowed the leader in sub-gap 1 to propagate much faster than the leader in sub-gap 2. When the leader breakdown occurred in sub-gap 1, the voltage gradient of the leader channel was about 100 kV/m. Therefore, there was a floating potential leap, leading to a leader velocity leap in sub-gap 2 too. After that, the leader velocity in sub-gap 2 increased nonlinearly, and the breakdown occurred very soon.

FIG. 12.

The calculated leader velocity in both sub-gaps (H = 200 cm).

FIG. 12.

The calculated leader velocity in both sub-gaps (H = 200 cm).

Close modal
FIG. 13.

The calculated leader length in both sub-gaps (H = 200 cm).

FIG. 13.

The calculated leader length in both sub-gaps (H = 200 cm).

Close modal

The leader length in Fig. 14 illustrates that the leader in sub-gap 2 propagated little before the leader bridged sub-gap 1, which is consistent with the trend of the experimental observation. And the leader breakdown in sub-gap 1 occurred earlier than that in sub-gap 2. Although the equivalent leader velocity is not the actual leader velocity, the leader breakdown sequence between two sub-gaps can be obtained by simulation.

FIG. 14.

Comparison of calculated floating potentials and the applied voltage (H = 200 cm).

FIG. 14.

Comparison of calculated floating potentials and the applied voltage (H = 200 cm).

Close modal

Most of the former studies about discharge in a combined gap focused on the disruptive critical voltage under a typical switching impulse, not the breakdown time under overvoltage. Therefore, the change in floating potential during the leader propagation phase is ignored because the leader propagation time is too short for the applied voltage to change significantly.

However, most of the lightning strikes and corresponding high voltage tests are overvoltage discharges. And the breakdown time is restricted to 1–3 μs in the aircraft lightning zoning test here. That indicates the leader propagation time is non-negligible, for the applied voltage could vary hundreds of kilovolts in 1 μs. It is fatal to simulate the leader breakdown process in both sub-gaps precisely, which is based on a refined analysis of the floating potential.

After the streamer bridges one sub-gap, the streamer plasma can be seen as a resistor for the low conductivity. As the transient floating potential is hard to measure, the formula for connected streamer plasma is derived by fitting test data. And the comparison indicates that the calculation error decreased from more than 14.9% to <5.7%.

It can be concluded that the assumption that the current may flow through the connected streamer plasma channel plays an important role in establishing an available calculation method for the breakdown voltage–time characteristics of the combined gap under lightning impulses.

In this study, we established an aircraft lightning zoning test platform and conducted positive lightning impulse discharges under different electrode–aircraft model gap lengths. The optical observations reveal that the leader propagation progress in two sub-gaps is not sequential but parallel, and the floating aircraft model potential has a great impact on leader propagation. After bridging the gap, the connected streamer plasma can be seen as a resistor element, whose instantaneous value is related to the gap length and leader length. By fitting the average breakdown time of tests, the numerical calculation has a minimum sum of squared error at constant R = 12 kΩ m−2. After taking the assumption into account, the error between calculation and test is <5.7% in this paper, while the error of the calculation ignoring the assumption is between 14.9% and 21.0%. A refined analysis of floating potential is essential to derive the available breakdown voltage–time characteristics of the combined gap under lightning impulses.

This work was supported by the Ministry of Industry and Information Technology of the People’s Republic of China (Grant No. MJZ5-2N22).

The authors have no conflicts to disclose.

Our work did not use any humans or animals.

Guoqing Sun: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Software (lead); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). Zemin Duan: Conceptualization (equal); Funding acquisition (lead); Investigation (supporting); Methodology (equal); Project administration (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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