Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 150756, 9 pages

http://dx.doi.org/10.1155/2015/150756

## On the Lightning Electromagnetic Fields due to Channel with Variable Return Stroke Velocity

^{1}Centre for Electromagnetic and Lightning Protection Research (CELP), Faculty of Engineering, University Putra Malaysia, (UPM), 43400 Serdang, Selangor, Malaysia^{2}Department of Electrical Engineering, Islamic Azad University, Firoozkooh Branch, 3981838381 Firoozkooh, Iran

Received 1 October 2014; Accepted 30 January 2015

Academic Editor: Xiao-Qiao He

Copyright © 2015 M. Izadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Numerical field expressions are proposed to evaluate the electromagnetic fields due to the lightning channel with variable values of return stroke velocity. Previous calculation methods generally use an average value for the return stroke velocity along a lightning channel. The proposed method can support different velocity profiles along a lightning channel in addition to the widely used channel-base current functions and also the general form of the engineering current models directly in the time domain without the need to apply any extra conversions. Moreover, a sample of the measured lightning current is used to validate the proposed method while the velocity profile is simulated by the general velocity function. The simulated fields based on constant and variable values of velocity are compared to the corresponding measured fields. The results show that the simulated fields based on the proposed method are in good agreement with the corresponding measured fields.

#### 1. Introduction

Lightning is an important natural phenomenon that can affect power lines. Induced voltages are a major effect of lightning on distribution lines that can be created by coupling between the electromagnetic fields of the lightning and the power line [1–5]. Therefore, the evaluation of electromagnetic fields associated with lightning is an important objective when considering lightning induced voltages and setting an appropriate protection level for power systems. Several studies have been undertaken to estimate the electromagnetic fields due to a lightning channel [6–11]. Such studies depend on the lightning channel parameters, the geometrical parameters, channel shape, channel condition, and the ground conductivity parameters. Among the different channel parameters, the return stroke velocity is an important variable for the evaluation of lightning electromagnetic fields [10, 12, 13], and this is usually entered into field calculations as an average value of velocities at different heights along a lightning channel, with a typical value between and ( is speed of light in free space) [14].

On the other hand, some experimental work has been carried out to measure the return stroke velocity at different heights along a channel where the velocity is measured as a height-dependent variable [14–17]. Therefore, the variation of velocity along a lightning channel can have an effect on the values of the lightning electromagnetic fields. In this study, numerical field expressions are proposed to evaluate lightning electromagnetic fields based on a channel with variable values of return stroke velocity directly in the time domain. Likewise, the proposed method is applied to a typical current sample whereby the corresponding return stroke velocity function is used for considering the velocity profile along a lightning channel. Further, the simulated fields are compared to the corresponding fields based on a constant value of velocity and the results are discussed accordingly. The proposed method can support a wide range of velocity profiles along the channel, along with various current functions and current models directly in the time domain without the need to apply any extra conversions. The basic assumptions in this study are expressed as follows.(1)The lightning channel is assumed to be vertical without any branches.(2)The ground conductivity is assumed to be infinite.

#### 2. Return Stroke Current

The return stroke currents at the channel base (ground surface) and at different heights along a lightning channel can be simulated using current functions and current models, respectively. In this study, the sum of two Heidler current functions [22, 23] is used to simulate the channel-base current as expressed by the following equation: where is the current amplitude of first/second Heidler function in (1), is the front time constant of first/second Heidler function in (1), is the decay-time constant in first/second Heidler function in (1), and are the exponents (2~10) The general form of the engineering current models is considered in this study to cover a wide range of current models as given by (3). Table 1 shows the constant factors of some common engineering current models, where is the cloud height, is the decay factor, and is the attenuation factor of the peak [9, 20, 21]. It should be mentioned that the MTLE (Modified Transmission Line current model with Exponential decay factor) current model was used for the simulation of electromagnetic fields; however, the proposed method can support a wide range of current models based on (3) [20, 21, 24] where is temporary charge height along lightning channel, is current distribution along lightning channel at any height and any time , is the channel-base current, is attenuation height dependent factor, is the current-wave propagation velocity, is upward propagating front velocity, and is Heaviside function defined as