Characteristic impedance of a wire between two ground planes

The characteristic impedance of a wire between two ground planes:

In the previous section, we calculated the characteristic impedance of a wire over a ground plane. Here we will calculate the characteristic impedance of a wire between two ground planes. For convenience, we assume the radius of wire is negligibly small.

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In the same manner as in the previous section, we will calculate the capacitance per unit length. Because of multiple mirroring, the mirror images are infinitely arranged in series.

Let the potential at the ground plane (x=0) be zero, the potential at a position x created by the pairs of line-charges +q and -q is:

Therefore, the potential of the wire is:

Noting the radius a is negligible compared with the height h, the infinite product results in Wallis' product:

We obtain the capacitance of the wire between ground planes:

The characteristic impedance is determined by the capacitance:

Sandwiching a wire with two ground planes is equivalent to moving a wire to a ground plane at the 2/π(=0.64) times distance.