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The electromagnetic field radiated from a lightning channel is the excitation for analyzing the field-to-transmission line coupling problem. The purpose of this paper is to propose a novel efficient approach to evaluate the horizontal electric field of the lightning channel expressed by the generalized Sommerfeld integral.

The asymptotic integral is extracted from the original one, which actually makes the Sommerfeld integral tail reach its convergence very quickly. To handle the sharp variance around k0, a closed-form integral, which is obtained by replacing the original kernel with an approximated function, is presented in detail. The numerical examples validated the proposed approach in the both frequency and time domain.

The approach proposed in this paper has been validated by the comparison with results in other papers. The agreement among these results reaches very well, and the approach proposed in this paper is more efficient and easy to implement, especially for the calculation of the tail integral part.

In accordance with the numerical experiments, the proposed approach can be served as a qualified candidate in terms of computational efficiency to evaluate the electromagnetic field generated by the lightning channel.

The earth-return mutual impedances between underground and overhead conductors can be expressed by Pollaczek integrals. Many efforts have been exerted to calculating this kind of integrals. However, most of numerical methods turn out to be time-consuming as integrands become highly oscillatory and strongly singular. Therefore, efficient algorithms should be devised. The paper aims to discuss these issues.

The paper separates the singularity from the whole integral and couple with the singularity and oscillation, respectively. A sinh transformation is applied for the finite part and complex integration method is used to calculate the tail.

Numerical experiments show that the given method shares the property that the stronger the singularity and the higher the oscillation, the better the accuracy of the calculation.

The sinh transformation is first proposed to calculate Pollaczek integrals. This efficient algorithm can be used to evaluate mutual impedances between conductors. Also, it provides a new aspect of the research on fast calculation of Pollaczek integrals and Sommerfeld integrals.

Solutions for the earth return mutual impedance play an important role in analyzing couplings of multi-conductor systems. Generally, the mutual impedance is approximated by Pollaczek integrals. The purpose of this paper is devising fast algorithms for calculation of this kind of improper integrals and its applications.

According to singular points, the Pollaczek integral is divided into two parts: the finite integral and the infinite integral. The finite part is computed by combining an efficient Levin method, which is implemented with a Chebyshev differential matrix. By transforming the integration path, the tail integral is calculated with help of a transformed Clenshaw–Curtis quadrature rule.

Numerical tests show that this new method is robust to high oscillation and nearly singularities. Thus, it is suitable for evaluating Pollaczek integrals. Furthermore, compared with existing method, the presented algorithm gives high-order approaches for the earth return mutual impedance between conductors over a multilayered soil with wide ranges of parameters.

An efficient truncation strategy is proposed to accelerate numerical calculation of Pollaczek integral. Compared with existing algorithms, this method is easier to be applied to computation of similar improper integrals, such as Sommerfeld integral.

The paper aims to estimate the instantaneous breakdown voltage of the lightning discharge from simulated figures in an energized rod-plane configuration protected by a lightning rod. The same configuration of electrodes has been the subject of experimental investigations for the measurement of the instantaneous breakdown voltage using oscillographic monitoring. This study validates the simulation model by making a comparison with experimentation and involves the role of the inception field of the upward discharge in the propagation of this last one.

The research methodology is based on the development of a fractal lightning protection model based on real physical conditions of the discharge propagation, such as the downward discharge and the upward one emanating from protection. The voltage drop and the randomness character of the lightning discharge are also taken into account. The electrical field is an important parameter in discharge development; it is considered in this work at each step of the discharge propagation by the finite element method. The instantaneous breakdown voltage is measured and estimated by both empirical equations and simulated figures of lightning discharge

The established model that allows estimating the instantaneous breakdown voltage from simulated discharges and empirical equations gives results in a good agreement with experimentation. The involvement of the upward discharge inception field emanating from the lightning rod in the evolution of electrical discharge is illustrated.

The work presented in this paper aims to develop a new fractal lightning protection model taking into consideration physical phenomena intervening in the development of the lightning discharge.

The originality of this work consists of the combination between fractals modelling of the electrical discharge and the protection against lightning, in addition, to use one of the characteristics of the electrical discharge, which is the instantaneous breakdown voltage, to prove the importance of the inception field emanating from the upward discharge in the propagation criterion of this last one.

The per-unit-length earth return mutual impedance of the overhead conductors plays an important role for analyzing electromagnetic transients or couplings of multi-conductor systems. It is impossible to have a closed-form expression to evaluate this kind of impedance. The purpose of this paper is to propose an efficient numerical approach to evaluate the earth return mutual impedance of the overhead conductors above horizontally multi-layered soils.

The expression of the earth return mutual impedance, which contains a complex highly oscillatory semi-infinite integral, is divided into two parts intentionally, i.e. the definite and the tail integral, respectively. The definite integral is calculated using the proposed moment functions after fitting the integrand into the piecewise cubic spline functions, and the tail integral is replaced by exponential integrals with newly developed asymptotic integrands.

The numerical examples show the proposed approach has a satisfactory accuracy for different parameter combinations. Compared to the direct quadrature approach, the computational time of the proposed approach is very competitive, especially, for the large horizontal distance and the low height of the conductors.

The advantage of the proposed approach is that the calculation of the highly oscillatory integral is completely avoided due to the fact that the moment function can be evaluated analytically. The contribution of the tail integral is well included by means of the exponential integral, though in an asymptotic way. The proposed approach is completely general, and can be applied to calculate the earth return mutual impedance of overhead conductors above a soil structure with an arbitrary number of horizontal layers.

A fast algorithm is proposed to calculate the lightning electromagnetic field over a perfectly conducting earth surface.

The channel base current is approximated by a number of sub‐domain quadratic functions using the proposed adaptive sampling technique, and the derivative and integral of the channel base current with respect to time can be analytically expressed. With the help of these approximations, the ideal electromagnetic field of the lightning channel can be evaluated along the lightning channel with respect to the height.

The computational time can be greatly reduced using the proposed approach to evaluate the electromagnetic field of a lightning channel in the time domain.

The adaptive sampling technique is a general‐purposed approach, which can be potentially used in other applications to fit a function with the minimal number of intervals.

Since the end of the Second World War, many spectacular advances have been made in aeronautics, thanks chiefly to the development of more powerful and economical jet engines. As to the parasitic drag of manned aircraft, progress has been confined to reducing unfavourable compressibility effects (area rule, Whitcombe bodies); methods to suppress separation have been developed but no new methods to reduce the drag resulting from turbulent boundary layers developing over the exposed surfaces have as yet found practical application.