Characteristic impedance of vertical wire segment (inf. short) over PEC ground?

[dd2023]

Hi coleagues,

for the purpose of TL transformer simulation I need solution for segment characteristic impedance of vertical wire, with radius a, at the infinitesimally short segment - point, at the height h above perfect ground.

For horizontal wire over (PEC) ground, we have analytical solution: Zo' = (Zfs/Pi) * acosh(h/a)
That solution, derived from characteristic inductance and capacitance of the wire against its mirror image, works well even for slightly sloped wire (if we make very short segments).
I am not sure, does that equation applies any more for angles approaching vertical and finally vertical wire over PEC ground? Most of theoretical framework is based on mirror image.
Despite best efforts to find solution for the vertical wire, I couldn't find any.

Does it exists? My math level is far below needed to solve it by hand.

 

[dd2023]

Let me explain, in full, what is my intention:

Helical antenna has that vertical feed extension, that sticks above reflector to the actual start of helix winding. That vertical wire acts as transformer, with sweep of its own impedances, which interferes with attempts to design and simulate inline transformer. With known impedances, along that vertical feed, we can de-embed that part (backpropagating of helix impedance results, conjugated).

 

[PlanarMetamaterials]

Welcome dd2023,

"Characteristic" impedances refer to properties of propagation modes, which by definition only exist in/on longitudinally invariant (or other similar forms, such as infinitely periodic) structures.

I think you are looking for input impedance of such a wire segment. I've never seen such an equation -- I would recommend simulation.

 

[dd2023]

Welcome dd2023,

"Characteristic" impedances refer to properties of propagation modes, which by definition only exist in/on longitudinally invariant (or other similar forms, such as infinitely periodic) structures.

I think you are looking for input impedance of such a wire segment. I've never seen such an equation -- I would recommend simulation.

No, I am looking for solution (impedances) along the vertical wire. Or L' and C' so Zo' = sqrt (L'/Z')
If I can get impedances along the sloped wire, with degradation of solution as the wire slope approaches vertical, I don't see any obstacle, except math skills to get L' and C' along any wire. It already exist for horizontal wire (what works for smaller slopes), but that does not work for vertical wire, because of different distribution of charges and currents.