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In the design of most of the modern systems, lightning threat needs to be considered at the design phase itself. This demands a suitable model and owing to associated complexity, only simplified modeling have been attempted. As a consequence, it does not provide adequate insight into the phenomena. Considering these, a more realistic time-¬ domain electromagnetic model for the return stroke current evolution has been developed by incorporating the following underlying physical processes: (i) excitation formed by the electric field due to charge distribution along the channel, cloud and that induced on ground, (ii) the transient enhancement of series conductance at the bridging regime, which initiates the return stroke,( iii) the non¬linear variation of channel conductance along with (iv) the associated dynamic Electromagnetic Fields(EMFs) that supports the current evolution. The intended modeling begins from the instant of bridging and the pre-¬return stroke charge distribution along the channel is calculated using Charge Simulation Method(CSM). For the calculation of dynamic EMFs, the thin wire Time Domain Electric Field Integral Equation(TD¬EFIE) is employed The transient enhancement of conductance at the bridging/streamer region is emulated using Toepler’s spark law while that along the matured section of the channel is described by first order arc model. The macroscopic physical model developed depicts most of the salient features of current evolution and resulting remote electromagnetic fields in a self¬ consistent manner. The work is not limited by the simplifications adopted for the channel geometry. The strength of the model was exploited for investigating a couple of practically important questions, one of which had divided opinion in the literature. Firstly, analysis showed that the "secondary" current waves generated by successive reflection within struck TGO and that fed by branches do not get reflected at the main wave front. It is shown that the dynamic spatial resistance profile of the channel at the main wave front is primarily responsible for this behavior. Secondly, it is shown that the abrupt change in radii at TGO ¬channel junction is mainly responsible for reflection at the junction. In summary, a novel time-¬domain macroscopic physical model for the first return stroke of a downward cloud¬-to-¬ ground lightning has been successfully developed, which is capable of providing much deeper insight in to the complex phenomena.

Being one of the most spectacular events in nature, lightning is basically a transient high current electric discharge in the atmosphere, which extends up to kilometres. Cloud to ground discharge is the most hazardous one as far as ground based structures are considered. Among the different phases of a lightning flash, return stroke is considered to be the most energetic phase and is basically responsible for most of the damages. Hence, much emphasis has been given to return stroke modeling. A more realistic modeling of return stroke is very essential to accurately study the interaction of return stroke with the structures on ground. As return stroke is dominated by electromagnetic phenomenon, an electromagnetic model will be the most suitable one. It does not call for any assumption on the mode of wave propagation, as well as, electromagnetic coupling between the different channel portions. There are mainly two approaches adopted for electromagnetic models i.e. frequency domain and time domain approach. Time domain approach is more reliable as it can handle, in principle, the nonlinear processes in the lightning channel. It is also free of numerical frequency domain to time domain inversion problem, which are found to be quite severe. However, most of the previous works on time domain electromagnetic models suffered from following two serious limitations - (i) the initial charge on the channel, which forms the true excitation for the problem, is not considered and (ii) instead of the non-linearly rising conductivity of the channel, a constant resistance is employed. For a realistic simulation of the interaction between the channel and any intercepting system, a time domain model with the above two major aspects being fully represented is very essential. In an earlier work, all these aspects have been fully considered but a domain based numerical modelling was employed. Consequently, it was difficult to consider the down conductor and further the number of unknowns was considerably large. In view of this, the present work is taken up and its scope is defined as to develop a boundary based numerical time domain electromagnetic model in which the initial charge on the channel and the non-linearly evolving channel conductance are fully considered. For the electrical engineering applications, electromagnetic aspects of the lightning phenomena is more important than the other associated physical processes and hence, importance is given only to the electromagnetic aspects. In other words, the light emission, thunder, chemical reactions at the channel etc. are not considered. Also, for most of the electrical engineering applications, the critical portion of current would be the region up to and around the peak and hence, modeling for this regime will be given prime importance. Owing to the complexity of the problem, some simplifying assumptions would be very essential. The literature indicates that these assumptions do not affect the adequate representation of the phenomena. Lightning channel is considered to be vertically straight without any branches. Earth is considered to be perfectly conducting. Explicit reference to dynamically varying channel radius, temperature and the air density is not made. However, it is assumed that the arc equation employed to describe the temporal changes in conductivity would adequately take care of these parameters. Lightning channel is represented by a highly conducting small core, which is surrounded by a weakly conducting corona sheath. The initial charge on the channel is deduced by solving for electrostatic field, with leader portion set to possess an axial gradient of 60 V/cm and the streamer portion to 5 – 10 kV/cm. The radius of the corona sheath is set iteratively by enforcing a gradient of 24 kV/cm up to its radial boundary. As analytical solution for the problem is impractical, suitable numerical solution is sought. Since the spatial extension of this time marching problem is virtually unbound and that the significant conduction is rather solely confined to an extremely small cross section of the channel core, a boundary-based method is selected. Amongst the numerical methods, the present work employs the moment method for the solution of the fields associated with the return strokes. A numerical solution of the Electric Field Integral Equation (EFIE) for thin structures has been developed in the literature. The same approach has been employed in the present work, however, with suitable modifications to suit the lightning problem. The code was written in MATLAB and integrations involved in the EFIE were solved using MATLAB symbolic computation. Before introducing the channel dynamic conductance and the initial charge on the channel, the code developed is validated by comparing the results for a center fed dipole antenna with that given in the literature. Also, NEC (Numeric Electromagnetic Code) simulations for various cases of monopole and dipole antenna were performed. The results from the code developed are shown to have good matching with that obtained from NEC based time domain results. In an earlier work, the dynamic conductance of the return stroke channel core, which is a high current electric arc, was represented by a first order arc equation. The same approach is employed in the present work also. Similarly, the transition from streamer to leader was modeled by Braginskii’s spark law and the same has been considered in the present work. A value of 10-5 S/m was used for minimum value of streamer conductance. For numerical stability, upper (Gmax = 3 S/m) and lower bounds (Gmin = 0.0083 S/m) for the channel conductance are forced. Preliminary simulations were run with and without dynamic channel conductance. The initial charge distribution along the channel formed the excitation. Results clearly show that without the dynamically varying channel conductance, no streamer to leader transition and hence, no return stroke evolution can occur. In other words, the non-linearly evolving channel conductance is mainly responsible for the evolution of the return stroke. In order to consider the charge neutralization by the return stroke, the charge deposited by it is diffused into the corona sheath. A fixed value of the corona sheath conductance is employed and the diffusion process is modeled by an equation derived from the continuity equation. To study the effect of corona sheath, simulations were run with and without corona. From the simulation results it was observed that the corona sheath causes increase in peak value of the stroke current, as well as, time to front and a decrease in the velocity of propagation. For the validation of the model, the basic characteristics of the return stroke current like the current wave shape, temporal variation of stroke current at different heights, velocity of propagation and the vertical electric fields at various radial distances were compared with available field/experimental data. A good agreement was seen and based on this, it is concluded that the present work has successfully developed a boundary based time domain numerical model for the lightning return stroke. Natural lightning being a stochastic process, the values of the parameters associated with it would differ in every event. On other hand, any deterministic model like the one developed in the present work predicts a fixed pattern of the simulated quantities. Therefore, it was felt that some of the model parameters must be permitted to vary so that a range of results could be obtained rather than a single pattern of results. Incidentally, the model parameters like arc time constant, settling value of arc conductivity/gradient, bounds for channel conductivity, streamer gradient, radius of the core etc. are not precisely known for the natural lightning environment. Further, some of them are known to vary within an event. Considering these and that simplicity is very important in already complex model, the above-mentioned parameters are taken as tunable parameters (of course to be varied within the prescribed range) for deducing the return stroke currents with some desired characteristics. A study on the influence of these parameters is made and suggestions are provided. Simulations for the nominal range of stroke currents are made and results are presented. These simulations clearly show the role of cloud potential, which in turn dictates the length of final bridging streamer, on the return stroke currents. The spatio-temporal variation of the current, charge deposited by the return stroke and the channel conductivity are presented which, reveal the dynamic processes leading to the evolution of return stroke current. Subsequently, simulations for two cases of stroke to elevated strike object are attempted. The upward leader was modeled quite similar to the descending one. Many interesting findings are made. In summary, the present work has successfully developed a boundary-based time domain numerical electromagnetic model for the lightning return stroke, wherein, the initial charge deposited on the channel and the non-linearly rising channel conductance have been appropriately considered. Simulation using the model clearly depicts the dynamic evolution of the return stroke. The characteristics of the simulated return strokes are in good agreement with the field data. Some of the parameters of the model are suggested as tunable parameters, which permit simulation of strokes with different characteristics.

The indirect effects of a nearby lightning strike on an airborne vehicle with its long trailing conducting plume is not well understood. Since airborne vehicles and its payload are expensive, their loss as a result of either a direct strike or due to the induced current and voltage of a nearby lightning strike is not desirable. The electromagnetic field generated due to the induced current on the skin of the vehicle may get coupled with the internal circuitry through the apertures on the vehicle body. If the coupled electromagnetic energy is more than the damage threshold level of the sensitive devices of the control circuit, they may fail which may lead to aborting the mission or a possible degradation in the vehicle performance. It has been reported that lightning induced phenomena was the cause of malfunctioning as well as aborting of some of the lunar missions. So in the present work, the computation of induced current and voltage on the skin of the vehicle body in the presence of an ionized long trailing exhaust plume has been attempted. The lightning channel is assumed to be vertical to the ground plane and extends up to a height of 7.5 km. The radiated electric and magnetic fields from the lightning channel at different heights from 10 m to 10 km and for lateral distances varying from 20 m to 10 km from the lightning channel are computed and the field waveforms are presented. For the computation of the radiated electric and magnetic fields from the lightning channel, modified transmission line with exponential current decay (MTLE) model for representing the lightning channel and the Heidler’s expression for the lightning channel base current are used. The peak amplitude of the lightning current used is 12 kA with a maximum current derivative of 40 kA/µs. It is seen that the vertical electric field in general, is bipolar in nature and that the height at which the change in polarity reversal takes place increases with increase of lateral distance from the lightning channel. The vertical electric field just above the ground is unipolar for all lateral distances from the channel and this is because the contribution due to the image of the lightning channel dominates the vertical electric field. The horizontal electric field is always unipolar for all heights and all lateral distances from the lightning channel studied. The effect of variation in the rate of rise of lightning current (di/dt) and the velocity of lightning current on the radiated electric and magnetic fields for the above heights and distances have also been studied. It is seen that the variation in maximum current derivative does not have a significant influence on the electric field when ground is assumed as a perfect conductor but it influences significantly the horizontal electric field when ground has finite conductivity. The velocity of propagation of lightning current on the other hand has a significant influence for both perfectly as well as finitely conducting ground conditions. For the computation of the induced current and voltage on the body of the airborne vehicle due to the coupling of the above mentioned electromagnetic fields radiated from a near by lightning discharge, the vehicle and its exhaust plume have been modeled as a transmission line and Finite Difference Time Domain (FDTD) numerical technique has been used for the computation. Regardless of the vehicle size, the structure can be modeled as a nonuniform transmission line consisting of a series of sections consisting of capacitive and inductive components. These components of the vehicle and its exhaust plume are computed using the Method of Moment (MoM) technique. The interaction of the electromagnetic wave with the plume depends on the electrical conductivity as well as the gas dynamic characteristics of the plume. Hence, in this research work an attempt has also been made to study the electrical conductivity (σe) and permittivity (εe) as well as the gas dynamic properties of the exhaust plume taking into consideration its turbulent nature. In general, the airborne vehicle can be considered as perfectly conducting (conductivity 3x107 S/m) where as the plume has finite conductivity. The electrical properties of an airborne vehicle exhaust plume such as electrical conductivity and the permittivity and their distribution along axial and radial directions depend on several factors. They are (i) propellant composition, (ii) impurity content in the propellants which generate ionic charge particles in the exhaust and (iii) the characteristics of the exhaust plume intensive parameters such as temperature, pressure, velocity and the presence of shock waves. These properties of the exhaust plume are computed in the two separate regions of interest as discussed next. The first region is inside the combustion chamber and up to the nozzle throat of the vehicle and the second region is from the throat to the exterior i.e., the ambient atmosphere or the downstream of the plume. In the first region where chemical reaction kinetics have to be considered, NASA Chemical Equilibrium with Application (CEA) software package has been used to compute the intensive parameters of the fluid at the throat of the nozzle. The pressure in the combustion chamber is taken as 4410 kPa and the back pressure at the exit plane is taken as 101.325 kPa. In the second region, FLUENT software package have been used for the fluid dynamic study of the exhaust plume from the vehicle nozzle throat to the exterior domain. The data obtained from the first region using CEA provides the parameters at the nozzle throat that are used as input parameters for the second region. In the study, a conical nozzle configuration of throat radius (rt) of 0.0185 m (nozzle exit plane radius is 0.05 m), half cone angle of 18º and nozzle expansion ratio (Ae/At) of 7.011 are used. The contour plot of the intensive parameters of the exhaust plume and the mass fraction of the charged particles are presented. The vehicle exhaust flow passes through different types of expansion and compression waves. In the present work, simulation is done for a slightly under expanded nozzle i.e. nozzle exit static pressure is slightly more than the ambient static pressure. Since the exit pressure is more than the ambient pressure, the exhaust gases expand to reach the ambient pressure. As the expansion waves reach the contact discontinuity (i.e. the boundary where the outer edge of the gas flow meets the free stream air), they again reflect back inward to create compression waves. These compression waves force the flow to turn back inward and increase its pressure. If the compression waves are strong enough, they will merge into an oblique shock wave. In the present work, more than eight such barrel shocks are captured. When the shock waves are generated, Mach number reduces sharply and static temperature and static pressure increases where as the total temperature of the exhaust remains constant in the shock wave formations. The characteristics of the plume such as pressure, temperature, velocity and concentration of the charged particles (i.e., e¯, Na+ and Cl¯) and neutral species such as CO, CO2 , Cl, H, HCl, H2O, H2 , N2, Na, NaCl, O, OH and O2 along axial and radial directions in the external domain have been studied. The above parameters are used to compute the collision frequencies and plasma frequencies of the charged particles as well as the number density of the species along axial and radial directions of the exhaust plume. These parameters are used to compute the effective conductivity distribution in the axial and radial directions for an incident electromagnetic field of frequency 1 MHz. The peak value of the conductivity computed is 0.12 S/m near the exit plane and it reduces to 0.02 S/m at an axial distance of 7.5 m from the exit plane which is well within the range suggested in the published literature. It has been observed that the oscillation in the conductivity along axial direction is a reflection of the shock wave formation in the exhaust plume. The electrical conductivity and the relative permittivity of the exhaust plume have been computed for three different radii of the nozzle at the exit plane i.e., 0.025 m, 0.05 m and 0.075 m. It is seen that the distribution of the conductivity and relative permittivity along the axial direction of the exhaust are independent of the nozzle exit plane radius. To study the coupling of lightning electromagnetic field with the vehicle and its exhaust plume two cases have been considered. These are (i) when the vehicle and its exhaust plume are at certain height above the ground and (ii) when the exhaust plume is touching the ground. The dimensions of the vehicle used in the study are as follows: length of the vehicle is 20 m and the length of its exhaust plume is 75 m. The radius of the vehicle is taken as 0.5 m. The vehicle and its exhaust plume are assumed to be at a lateral distance of 250 m from the lightning channel. In case one, when the vehicle and its inhomogeneous exhaust plume tip is at a height of 10 m above the ground, both the ends are open. So the reflection coefficients of the current wave and voltage wave at the end points are -1 and +1 respectively irrespective of the characteristic impedances of the vehicle and its exhaust plume. So when the reflected current propagates it will tend to reduce along the length of the object. Hence, the induced current at the end points are zero and the currents in the end segments are less than those in the intermediate segments. The spatial distribution of the peak magnitude of the time varying induced current, |Imax|, in each segment along the length of the vehicle without and with the exhaust plume are presented. In case of vehicle without plume, the maximum value of the induced current is at the middle segment of the vehicle and its value is 4.8 A. The presence of the inhomogeneous plume enhances the maximum value of the induced current to 33 A and its position is shifted to the exhaust plume side. When the voltage wave propagates, it will enhance the induced voltage in the vehicle body. The time varying potential difference between the end points of the vehicle without plume and the vehicle with its exhaust plume which drives the induced current are computed and it is seen that the potential difference for the vehicle without plume is unipolar whereas it is bipolar for the vehicle with exhaust plume. The lightning induced current on the skin of the vehicle will generate an electromagnetic field which may couple with the internal electronic devices and circuits through the apertures. The amount of electromagnetic energy that will be transmitted through an aperture on the vehicle skin and coupled with the internal electronic equipments depends on the characteristics of the induced current on the skin of the vehicle, the electrical size, shape, orientation and location of the aperture and the location of the internal electronic devices with respect to the aperture. So the time varying induced current and its di/dt at three different locations on the vehicle body i.e., tail of the vehicle, middle of the vehicle and vehicle nose are computed. It is seen that the induced current on the vehicle and its di/dt in the absence of the plume are oscillating in nature but they are critically damped in the presence of the trailing inhomogeneous exhaust plume. It also shows that the enhancement of induced current and its di/dt at the tail are much more than at the middle or at the nose of the vehicle which is true for an electrically short vehicle i.e., lv/λmin ≈ 0.067 as cited in the literature. So the presence of an aperture on the skin of the vehicle near to tail will transmit maximum electromagnetic energy into the inside of the vehicle. Therefore during design of the electrically short airborne vehicles, any aperture should be avoided near the tail of the vehicle or internal electronic devices should be placed away from the tail of the vehicle. In case 2, when the plume is touching the ground, the transient induced current in the plume will propagate into the soil. The effective impedance for smaller currents will be quite high (the inductance and capacitance effect are not taken into consideration for calculating the impedance. So the impedance of the soil is dominated by only the resistance). However, as soon as the current exceeds a certain value, the resulting soil gradient can reach the breakdown gradient of the soil i.e., 200-500 kV/m as cited in literature resulting in soil ionization. This will effectively lower the soil impedance. These dynamic characteristics of the soil resistance with induced current are incorporated by considering the expression for the soil resistance. To study the effect of soil resistivity on the time varying induced current and the voltage, computations have been done for various resistivities of the soil i.e., 0 Ωm, 100 Ωm and 200 Ωm. For soil resistivity of 0 Ωm, the reflection coefficients at the ground and at the open ends for the current wave are +1 and -1 respectively. So at the ground end, the reflected current wave will enhance and at the open end it will diminish as it propagates along the length of the vehicle and its exhaust. As the resistivity of the soil increases, the reflection coefficient of the current at the ground end decreases from +1, so the peak magnitude of the current reduces along the length till the length is half of the total length of the plume and the vehicle. Therefore, the peak magnitude of the induced current in the ground segment is much more than the peak magnitude of the current in the segment at the open end. For a finitely conducting plume, the peak value of the potential difference between the two ends of the vehicle and its exhaust plume are 92 kV, 91 kV and 90 kV for soil resistivities of 0 Ωm 100 Ωm and 200 Ωm respectively. Therefore the influence of the soil resistivity on the induced current is found to be not much significant. The spatial distribution of the peak magnitude of the time varying induced current in each segment along the length of the vehicle with inhomogeneous exhaust plume for the above three different soil resistivities are presented at a lateral distance of 250 m from the lightning channel. It is seen that when the plume is touching the ground, the induced current on the vehicle at the tail, middle and nose sections are marginally more than when the vehicle and its exhaust are at a height of 10 m above the ground. The effects of different parameters such as peak value and maximum di/dt of lightning current, velocity of lightning current, lateral distance of the vehicle from lightning channel and the height of the tip of the exhaust plume above the ground on the induced current and voltage on the airborne vehicle have also been studied. The peak amplitude of the lightning current used are 30 kA and 100 kA in addition to 12 kA mentioned earlier for the field computation. Also maximum di/dt values of 40 kA/µs and 120 kA/µs for the lightning current have been used for the computation. It is observed that the induced current increases with increase of the peak value, maximum di/dt as well as the velocity of propagation of the lightning current where as the induced current will reduce with increase of lateral distance and height of the tip of the exhaust plume above the ground. As an offshoot of the present work, the axial and radial distribution of the parameter, σe/ωεe (loss tangent of the exhaust plume) for an incident electromagnetic wave (lightning electromagnetic field) frequency of 1 MHz have been computed to study the conducting properties of the exhaust plume. σe/ωεe of the exhaust plume at 1 MHz frequency varies from 2324 to 365. Since σe/ωεe >>1, the plume behaves as a good conductor and the displacement currents can be neglected. In addition to this, the variation of parameter σe/ωεe for frequency ranges of 0.1 MHz to 5 GHz are also studied where σe and εe are the maximum effective conductivity and permittivity of the exhaust plume at the chosen frequency of an incident EM wave. It shows that the parameter σe/ωεe is 1.8x104 at 0.1 MHz and reduces to 0.45 for 5 GHz and its value is 1 at a frequency of 2.285 GHz. Therefore at lower EM wave frequency, the exhaust plume behaves as a good conductor and that conductivity reduces with increase of the frequency. The exhaust plume in the present study behaves as a good conductor below or at the EM wave frequency of 2.285 GHz. The microwave attenuation of electromagnetic wave through the ionized plume (the angle of incidence of microwave is 90o and transmission of microwave is always transverse to the exhaust plume) has also been studied using the above electrical characteristics computed and it is seen that the attenuation follows the axial variation in the conductivity of each cross section of the plume. In the present work, a theoretical model has also been developed to compute the microwave attenuation through the vehicle exhaust plume using the electrical conductivity computed earlier for any angle of incidence of the microwave. The thesis also lists some additional topics for further studies.