The result is for stretched width antennas in free space, medium or minimum 2 x λ over ground with sinusoidal current and l>>d. For the resonances frequency and wire diameter have to be specified.
Lenghts are related to free space. The resonance-calculation delivers the lenght for free space. To get the geometric lenght in medium you have to watch lenght in medium.
For experts I have transfered the 2kl in the auxilliary calculation. For arguments greater than 36 the calculation of Si(x) and Ci(x) is not exact.
Continuing calculations [here].
See also: Antenna Theory - Analysis and Design, BALANIS, Arizona State University, Tempe, AZ

BALANIS-impedance of antenna / dipol

antenna

integral-sine and -cosine

trend of impedance over l/λ, λ=1 m, d=1 mm, vacuum:

Calculation of the influence of ideal ground for the impedance of dipols

For horizontal and vertical dipols BALANIS calculation. The input impedances will be inserted by the programs above, You can use befor this calculation.
Also see: BALANIS, Arizona State University, Tempe, AZ

Influence of ideal ground for dipol impedance BALANIS

For the resonances frequency and wire diameter have to be specified. Lenghts are related to free space. The resonance-calculation delivers the lenght for free space. To get the geometric lenght in medium you have to watch
lenght in medium. For experts I have transfered the 2kl in the auxilliary calculation. For arguments greater than 36 the calculation of Si(x) and Ci(x) is not exact.
Continuing calculations [here].
See also: Antenna Theory - Analysis and Design, BALANIS, Arizona State University, Tempe, AZ

BALANIS-impedance of antenna / vertical

antenna

integral-sine and -cosine

trend of impedance over l/λ, λ=1 m, d=1 mm, vacuum: