Theory

 

Appendix C - Mutual Impedance Between Wire Elements

The mutual impedance in a general case between two wires arbitrarily positioned above ground (Figure 40) can be calculated with the use of the following equations, presented here without derivation [9].

Care must be taken both for the direct interaction and the indirect interaction due to the presence of the ground.

Figure 40 Elements of a thin-wire structure above ground

The following expression gives the mutual impedance between two wire elements of length D ln and D lm of any orientation. The elements is positioned above an imperfectly conducting ground and the currents flowing through the elements are and . Linear triangular functions are used for approximation. The imperfectly conducting ground is modeled by the use of a complex permittivity in the form .

Equation 192

Three new functions are now introduced, these are the Sommerfeld integrals for air to air interface, having a semi-infinite integration path and thereby they are the most demanding in computational power for this calculation.

Equation 193

Equation 194

Equation 195

where

Equation 196

Equation 197

Equation 198

, Re{u0} > 0

Equation 199

, Re{u1} > 0

Equation 200

The calculations also make use of Green functions in free space associated with the source and its image:

Equation 201

Equation 202

where

Equation 203

Equation 204

In these equations z is the height above ground for the point of observation and z’ is the height above ground for the source point.

 

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EMC of Telecommunication Lines
A Master Thesis from the Fieldbusters © 1997
Joachim Johansson and Urban Lundgren