Electromagnetic modelling of the first stroke of a lightningflash

Abstract

Lightning is one of the most frequently encountered and spectacular natural phenomena which induces disturbances and sometimes damages to electrical and communication systems. Aircrafts, ships, and explosive storage units represent other systems which are also frequently affected, sometimes disastrously, by lightning. Therefore, protection against lightning has been a serious concern in the design of such systems. However, due to the nature of lightning and its macroscopicity, design calculations are generally performed by employing simplified models. The assumptions underlying the simplifications often remain without any serious experimental or theoretical justification. Also, the extensive lightning research carried out in various parts of the globe has resulted in a vast amount of field data on natural lightning. Both the above aspects call for a detailed electromagnetic model for natural lightning. This would not only serve to better analyze the measured data but would also serve as a benchmark for evaluating the simplified models. The present work aims at developing such an electromagnetic model for the first complete stroke of natural lightning. It comprises models for the stepped leader and the return stroke. For simplifying the computations, it is assumed that the lightning channel is vertically straight without branchings. Most of the related quantities like voltages, currents, and velocities are all derived from the model.

Modelling of the stepped leader involves computations of the transient electric field associated with the leader and modelling of the air breakdown process. The transient fields associated with the stepped leader are unconventional in that the governing equations remain Laplacian/Poissonian, which are time-independent. The transients arise only because of the time-dependent material boundary conditions. In classical electromagnetic literature, transients arising out of time-dependent governing equations are well studied. However, transients of the above type are scarcely discussed. Therefore, a rather detailed analysis of this type of fields has been carried out, giving specific attention to the problem of development of the stepped leader wherever necessary.

Following the circuit theoretical convention, these types of transient fields are identified as Resistive-Capacitive (RC) and Resistive-Inductive-Capacitive (RLC) transient fields. The RC type of transient fields exist in conducting media where the associated field changes are slow enough to neglect the contribution from the changing magnetic field to the electric field. The governing equations reduce to Laplacian/Poissonian. However, at the material interface, the electric field must satisfy the relevant material boundary conditions. Electric field must be continuous across the interface to ensure the potential continuity. Further, the normal component of the electric flux density must be continuous, and the flow of current must satisfy the charge conservation condition. This general form of boundary condition (to be satisfied by the normal component of the current density vector) is identified as the root cause for the transients in the field distribution. Transients in nonlinear and inhomogeneous materials are also discussed briefly. A general algorithm has been developed to solve this type of transient fields with the help of any method for solving Laplace’s/Poisson’s equation. One illustrative and two practical high-voltage problems have been solved to demonstrate the technique. The performance and stability of the algorithm are qualitatively discussed, and a filtering algorithm (from fluid dynamics) has been used to improve its performance.

Next, the concept of generalized inductance is revisited. This definition does not demand the circuit (path) to be closed for defining the inductance. Also, it is shown that this definition reduces to the conventional flux-linkage concept if the circuit is closed. Using this definition and the RC transient theory, RLC transient fields are addressed for electrically thin wires or conductors. To demonstrate the validity of the new approach, the resonant frequencies of an air-core coil were computed, which compared favorably with the measured values reported in literature.

For the numerical solution of the field of the leader, CSM was employed in the initial stages. Later, leader charges were directly employed for computing the field and potentials. An attempt has been made to consider the earth more realistically by developing a generalized image theory applicable for linear lossy dielectrics. It is shown to reduce to the conventional images in case of an ideal dielectric/conductor. Also, for the steady-state sinusoidal case, the generalized images are shown to be the same as the complex images employed in the literature.

The present work considers the stepped leader as a nonlinearly conducting channel surrounded by a corona sheath carrying all the charges. Whenever the gradient at the tip exceeds the local critical value, a new step is assumed to be formed. Air breakdown has been simulated by a semiphysical model including the effects of ambient pressure and temperature. The step length is governed by the electrical gradient profile along the axis and the streamer propagation gradient. With the formation of a step, charge redistribution takes place with current flowing through the nonlinear leader channel. This current flow and the charge accumulation are governed by the RC & RLC transient fields discussed earlier. This cycle repeats if the gradient ahead of the new step exceeds the critical gradient. Detailed attention has been given to numerical implementation and circumventing the associated problems. A method is suggested and implemented for overcoming the large computational burden. First, a sensitivity analysis has been carried out for checking the dependence of the proposed model on its parameters. Both spherical and cylindrical cloud-charge models are considered. The results of the simulation are as follows. The leader always propagates in steps. Step length depends on the streamer propagation gradient and the leader radius. For negative leaders, due to the higher streamer gradients compared with positive leaders, computed step lengths are smaller. Step duration depends on the strength of the field produced by the cloud charges and the value of the critical gradient. Importantly, both step length and duration lie well within the measured values reported in the literature. The ground electric field also follows the measured pattern. The computed leader voltage and current profiles give a good insight into the phenomena. Leader currents are reported in the literature to be of the order of a few hundred amperes all along and to be of the order of kiloamperes at least at the tip portion. These are very clearly portrayed by the present model. Also, as observed in practice, it was found from the simulation that the longer the step length, the longer the step duration. Thus, a satisfying comparison of the simulation results of the model proposed with the measured data gives credibility to it.

Next, the modelling of the return stroke is considered. Many researchers feel that a transmission line model for the return stroke intrinsically assumes a transverse electromagnetic mode (TEM) of wave propagation. However, in reality, a significant component of the field exists along the direction of propagation. Therefore, a more reliable and accurate method would be to employ an axisymmetric field solution of Maxwell’s wave equation which obviates the TEM assumption and models the earth more realistically. In addition, the cloud charges which contribute significantly to the field can be accurately considered. The present work attempts a fully field-oriented approach to the problem on an axisymmetric system and in the time domain. Due to the complexity and nonlinearity of the problem, an analytical solution seems impractical and hence a numerical method is adopted. Among the various well-known methods for numerical solution for solving Maxwell’s wave equations, the Finite Difference Time Domain (FDTD) method is selected because of its appropriateness and simplicity. Yee’s scheme has been the most used FDTD algorithm. It is an explicit leap-frog time marching scheme, and nonlinear materials have been modeled using it. The present work has employed this technique. However, the implementation of the method is not straightforward due to the following:

(a) Most of the FDTD analyses are carried out using a transient Gaussian pulse and not a long-duration excitation. The FDTD method needs a radiating or absorbing boundary, and this boundary is generally kept far from the source location. Therefore, with the above conditions, the source boundary interactions and the stability of the boundary conditions employed do not pose any problem. However, with lightning return stroke computations, neither of the above two could be realized. First of all, the total electric field (including that due to the charges in the cloud) extends well beyond the domain of dominance of the field produced by the charges on the channel. Secondly, the outer radiating/absorbing boundary cannot be kept far away as it would result in an impossible computational burden.

(b) Importantly, the geometry of the lightning problem has a very bad aspect ratio. The diameter of the main conducting channel is less than 1-2 cm and is surrounded by a corona sheath of diameter of a few meters. The length of the channel is in kilometers, and the charge centers influencing the phenomena span several kilometers. If an attempt is made to directly model the channel, it would impose an unmanageable number of FDTD cells and computation time steps. The sub-cell approach was found to be very suitable for modelling the channel and the surrounding corona sheath. At the same time, it should be noted that the channel possesses highly nonlinear conductivity, and the initial conductivity could be very low at the newly formed portions. Therefore, conditions for retaining the sub-cell approach at the latter conditions were analyzed, and the minimum FDTD cell width needed is assessed. For further minimizing the computational burden, a combined sub-cell sub-grid approach was visualized. For the verification of the method and the codings, the impedance of a monopole antenna was computed. Firstly, the impedance of a coated monopole over a conducting earth plane was computed and compared with the published results. Instead of the commonly used technique of applying a Gaussian pulse and making the calculations using FFT, sinusoidal excitation was given, and impedances are all computed at steady-state. This validated the sub-cell approach and the stability of the outer radiating boundary over a long time scale. For verifying the method for the adverse aspect ratio encountered in the lightning problem, a monopole of dimensions similar to the lightning channel was considered. The computed impedances are compared with the formula given by Schelkunoff. Next, for different conductivities of the conductor forming the monopole, pulse propagation characteristics are studied. A lossy coating was also included later. These exercises have clearly shown that the velocity of propagation of the pulse does not get significantly affected by either the conductivity of the central conductor or the coating. Finally, the initial conditions are taken from the results of the stepped leader model, and return stroke computations are carried out for perfectly conducting earth and for various earth conductivities. The charge held by the corona sheath is found to cause a significant reduction of return stroke velocity. The earth connection not only affects the return stroke velocity but also influences the magnitude of the return stroke current. Any increase in arc time constant is found to decrease the magnitude and rate of rise of return stroke current. Computations were carried out only till the return stroke reaches the cloud end. With a few examples, the method developed is shown as a tool for the lightning return stroke computations.

In conclusion, the following may be summarized. An attempt has been successfully made to present a model based on the electromagnetic approach for the first complete stroke of a natural lightning flash. The stepped leader and the return strokes are modelled separately. For the field computations associated with the leader, RC & RLC transient fields are analyzed with their solution methodology. A generalized image theory has been developed for modeling the earth more realistically. The concept of generalized inductance is revisited. The model for the stepped leader gives a step-wise propagating leader whose step length and step duration agree very well with the field data. The effect of the variation in the values of the different parameters used in the modeling of the air breakdown has been analyzed. For the return stroke modeling, a time-domain solution of Maxwell’s equations on an axisymmetric geometry is considered. The FDTD method on a nonorthogonal grid with PML absorbing outer boundary is employed for the numerical solution. The sub-cell-sub-grid approach has been adopted for modeling the channel with its corona sheath. Computed current magnitudes and waveshapes compare favorably with the field data. In developing these numerical simulation methods, minimization of the computational time has been given high priority. Thus, an electromagnetic model for the first complete stroke of a lightning flash has been developed which can be used for digital simulation.