Differential impedance of two wires over a ground plane
The even and odd mode impedances of two wires over a ground plane are:
where Z_{0} is the characteristic impedance of each line:
and ƒÌ is a parameter that indicates the strength of coupling:

We will calculate the even and odd mode impedances, using an analytically solvable example: two wires over a ground plane. For convenience, we assume the radius of wire is negligibly small.
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It is convenient to replace the ground plane with the mirror image of line1 and 2.
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First, we will calculate the selfcapacitance of each line1 and 2 and the mutual capacitance between line1 and 2.
According to Gauss's law, the electric field E created by a linecharge q is:
Therefore, the potential at distance r from the line is:
Let the potential at the ground plane be zero, the potential of the line1 created by the charge of line1,2,3 and 4 is:
Similarly, the potential of the line2 created by the charge of line1,2,3 and 4 is:
Expressing the line distances in terms of s and h,
we obtain the total potential of the line1 and 2 created by line charges:
where F and F_{m} are the form factors:
Conversely, the linecharges are:
The linecharges can also be expressed in terms of the capacitances Cs,Cm:
Comparing the coefficients of these equations, we obtain the selfcapacitance of each line1 and 2 and the mutual capacitance between line1 and 2:
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Next, we will calculate the selfinductance of each line1 and 2 and the mutualinductance between line1 and 2.
According to Ampere's law, the magnetic field H created by a linecurrent I is:
The magnetic flux created by the current of line1, passing between line1 and 3, is:
Therefore, the total magnetic flux created by the current of line1 and 3 is:
The magnetic flux between line1 and the plane is half of that between line1 and line3. So we obtain the self inductance of line1:
The mutual inductance between line1 and 2 can be similarly calculated.
The magnetic flux created by the current of line1, passing between line2 and 4, is:
Then the total magnetic flux created by the current of line1 and 3 is:
The magnetic flux between line2 and the plane is half of that between line2 and 4. So we obtain the mutual inductance between line1 and 2:
Expressing the line distances in terms of s and h,
we obtain the self inductance and mutual inductance per unit length:
where F and F_{m} are the same form factors as those of the capacitances.
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Using the capacitances and inductances above, we can calculate the propagation speeds in even and odd mode:
The form factors vanished, so we have confirmed that the propagation speed does not depend on the crosssection of conductors. Furthermore the propagation speeds in even and odd mode are just equal to the light speed in the media.
Similarly, the characteristic impedances in even and odd mode can be calculated:
Reminding that the characteristic impedance of a single wire over a ground plane is
,we can express the even and odd mode impedances in terms of Z_{0} and a parameter ƒÌ:
This parameter ƒÌ increases as the line distance becomes closer.