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