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Can anyone tell me if the equations below for the impedance at the center feed points of a half wave dipole include terms for self inductance and velocity factor of the elements, and if so provide an explanation ?

A dipole half wave in length has + 45 ohms of reactance in the feed point impedance, i'm trying to figure out exactly what causes this reactance.

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If the equations assume no self inductance and a velocity factor of 1, what causes the reactance ?

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The impedance formula you quoted assumes that the radiator is made of an ideal cylindrical conductor, ie., completely lossless idealized material that is represented by the boundary conditions: $n\cdot H=0$ and $n\times E=0$. Whatever low frequency inductance and/or capacitance it may have as a pair of metal pieces is already built in the formula.

It has a reactance because the pair of cylinders do not have a constant wave impedance as a pair of transmission lines (imagine you are pushing the far ends of parallel wires apart while keeping their feed points together) and also because the vacuum impedance ($377\Omega$) is not a proper termination at the far ends.

If by including the "velocity factor" you mean whether the formula would work for an antenna embedded in a homogeneous and lossless linear dielectric medium or some such then the answer is yes, $k=2\pi\sqrt {\mu_r \epsilon_r}/\lambda_0$, and $\lambda_0$ is the free-space wavelength.

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  • $\begingroup$ Thanks for that answer, you are one of the very few if not only person who has been able to comment intelligently on this subject ! Your answers raise two more questions, i will post these as new questions rather than asking in a string of comments to see if there is the possibility that these can be answered as well. $\endgroup$
    – Andrew
    Commented Aug 28, 2021 at 22:37

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