Characteristic impedance of a wire over a ground plane
The characteristic impedance of a wire over a ground plane:

We will calculate the characteristic impedance, using an analytically solvable example: a wire over a ground plane.
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First, we will calculate the capacitance per unit length between the wire and plane. It is convenient to replace the wire and plane with a linecharge +q at (s,0) and its mirror image.
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 (x=0) be zero, the potential at a position (x,y) created by the linecharges +q and q is:
Then the equipotential line is a circle:
We determine the parameter s and K to fit the equipotential line in the outline of the wires.
These two solutions correspond to the potential of the wire 1 and 2 respectively:
We obtain the capacitance between the wire and ground plane.
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Next, we will calculate the inductance per unit length of the wire. It is convenient to replace the wire and plane with a linecurrent +I at (s,0) and its mirror image.
According to Ampere's law, the magnetic field H created by a linecurrent I is:
Therefore, the total magnetic field at point (x,y) created by the two linecurrents is:
The magnetic field H must be perpendicular to the radius of the wire.
Then the line of magnetic force is a circle:
We determine the parameter s to fit the line of magnetic force in the outline of the wire.
The magnetic flux created by the linecurrent I, passing between the wires, is:
Then the total magnetic flux created by the linecurrent +I and I is:
The magnetic flux between the wire and the ground plane is half of that between two wires. So we obtain the inductance of the wire over the ground plane:
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Using the capacitance and inductance above, we obtain the characteristic impedance:
Similarly, the propagation speed is:
Note that the propagation speed does not depend on the form of conductors.
Assuming the radius a is much smaller than the height h, the form factor is simplified:
The permittivity and permeability of vacuum are
,and those of a material are
where ε_{r} is the relative permittivity of the material.
Material 
ε_{r} 
Glassreinforced epoxy (FR4) 
4.04.8 
Polyethylene 
2.3 
Air 
1.0 
Then the characteristic impedance and the propagation speed are computed as: