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 line-charge +q at (s,0) and its mirror image.

According to Gauss's law, the electric field E created by a line-charge 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 line-charges +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 line-current +I at (s,0) and its mirror image.

According to Ampere's law, the magnetic field H created by a line-current I is:

Therefore, the total magnetic field at point (x,y) created by the two line-currents 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 line-current I, passing between the wires, is:

Then the total magnetic flux created by the line-current +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: