Surge behavior at the discontinuity of a vertical line over the ground
Introduction
Lightning surges over a conductor system are an important issue in the design of electric power systems and communication systems. In case of parallel conductors or the conductors running in parallel with the ground surface, the lightning surges are analyzed using transmission line theory [1]. At the discontinuity of a line, transmission and reflection of a surge can be described with characteristic impedance of the line. For a single conductor or a vertical conductor above the ground, the associated electromagnetic fields are not in TEM mode. The classic transmission line theory is then inapplicable. A modified transmission line theory would be necessary in order to address lightning surges on vertical conductors over the ground.
Surge propagation on a vertical line over the ground has been increasingly of concern in the recent years. It is noted in [2] that the electromagnetic field is a spherical TEM when an unattenuated current propagates along a vertical line with the speed of light. In case of nonzero radius for the line the unattenuated current propagation could not be supported [3]. This current attenuation is primarily caused by the presence of “scattering” current arising from the boundary condition on the conductor surface, or can be interpreted with the internal “reflection” of a nonuniform transmission line.
Surge impedance of a vertical conductor has been addressed significantly in the past decade. The impedance at the time when the reflected surge travels back to the top of the conductor has been extensively discussed. A number of theoretical formulas have been derived [4], using either basic circuit theory or transmission line theory. More rigorous analysis of surge impedance has been made recently using numerical electromagnetic approaches, such as PEEC [5], FDTD, MOM (NEC2 and TWTD) and others [6], [7], [8]. Note that surge impedance of a vertical conductor is time-dependent. It increases with time even if the conductor is perfectly conducting [9]. It is also found in [10] that the current waveform has a significant influence on surge impedance. Single-value impedance would be insufficient in traveling wave analysis. When a surge encounters a discontinuity, reflection and transmission of the surge are observed. These transmitted and reflected waves can be determined by characteristic impedance of the line in classic transmission line theory. Specially, the surge reflection at the ground surface is discussed in [11], [12]. The authors tried to interpret the current attenuation of the reflected surge in the ‘scatter theory’ and improved the lightning model. However, the reflection and transmission of a surge on the vertical line has not been well addressed.
In this paper, the surge behavior at the discontinuity of a vertical line over the ground is discussed. The line conductor has a nonzero radius, and may or may not be perfectly conducting. Similar to the classic transmission line theory, the reflection and transmission at the discontinuity are determined by surge impedances of the conductor. However, these are not the single-value impedances. They are defined against the nature of the waves on the line. In Section 2, three surge impedances are respectively introduced for incident wave, transmitted wave and reflected wave. Modified transmission coefficients at the discontinuity are then derived. These surge impedances are further discussed in Section 3. Finally, numerical verification using the Partial Element Equivalent Circuit (PEEC) method is presented.
Section snippets
Modified transmission equations at a discontinuity
When a vertical line is struck by lightning, a current surge is injected into the line and a voltage surge is generated on this line [10]. Surge impedance is then defined as the ratio of the voltage over the current on the line. Unlike the impedance defined for a TEM transmission line, this surge impedance is not constant, and varies with time and position on the line.
It is noted in [10] that the surge impedance varies with the surge waveform. At the discontinuity the waveforms of transmitted
Surge impedances of vertical conductors
Transmission coefficients are determined by incident, reflected and transmitted wave impedances. These impedances are totally different from characteristic impedance of a traditional transmission line. It is necessary to know how these impedances are calculated. Generally speaking, these impedances can be obtained analytically or numerically. In this paper, the numerical method using the retarded partial element electrical circuit (PEEC) approach is employed. Note that surge impedance is a
Numerical simulations
A numerical example is presented in this section to illustrate three distinct impedances of a vertical line at a discontinuity, and to validate the formulas of transmission coefficients Fig. 1 shows the configuration of a vertical transmission line excited by a current source on Wire 1. Wire 1 is made of a conductor with radius of 0.05 m and length of 50 m, and Wire 2 with radius of 0.005 m and length of 100 m. Both Wire 1 and Wire 2 join together at Point A. The source current Is has a waveform of
Effect of the lossy conductor on the transmission coefficient
The discussion above has revealed the surge behavior on a perfectly conducting vertical line in detail, and several important conclusions have been made. However, the conductors in fact are made of non-perfectly conducting material, and the impact of a lossy conductor needs to be discussed further. For the convenience of theoretical discussion, the current attenuation is neglected here.
For a lossy conductor, the basic equations are:
Conclusions
This paper investigated the transient surge impedances of a vertical line over the ground and transmission of a surge over a line discontinuity. Surge impedances of the vertical line were defined, and the methods for impedance evaluation were presented. The transmission coefficients for voltage and current at the discontinuity of the vertical line was derived, and expressed by using the distinct surge impedances. A numerical example using the retarded PEEC method was presented to demonstrate
Acknowledgements
The work leading to this paper was supported by grants from the Research Committee of the Hong Kong Polytechnic University, the Research Grants Council, University Grants Committee, Hong Kong (Project No. 514912E), and under a joint project of the Hong Kong Polytechnic University and Shanghai Lightning Protection Center.
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