LF: FS formula

["Dick Rollema" <d.w.rollema@freeler.nl>]

To All from PA0SE

William Oorschot, PA0WFO, has sent me a copy of a three part article by Karl H. Hille, DL1VU, in the German 2002 magazine FUNK: "Die Bergantenne am Herzogstand - Vor 75 Jahren eine technische Grosstat" (The mountain antenna at the Herzogstand - For 75 years a technical amazing feat).  This concerns the following.

The large VLF stations at Nauen and Eilvese in Germany that were on the air around 1920 were built by Telefunken. The German firm Carl Lorenz AG would like to supply VLF stations as well but were unable to do so. They could supply arc-transmitters of sufficient power but not the enormous antenna masts they had to offer with the transmitters in order to compete with Telefunken. Steel masts were unacceptably expensive around 1920 due to the unbelievable strong inflation that started at the same time Lorenz began planning the station. So they looked for an alternative. There was an example. For the VLF link between the Netherlands and the Netherlands East Indies (now Indonesia) a large arc-transmitter hand been constructed at Malabar on the island Java. The antenna hung over a valley between two mountains, obviating the use of high masts.
Lorenz decided to do the same. They strung an antenna between two mountain tops in the South of Germany, the mountains were the Herzogstand at 1735m and the Stein at 940m. The horizontal distance between the mountain tops was 2700m.  About one third from the low end of the sloping wire an insulator was inserted and a vertical wire connected the part to the Herzogstand to the transmitter in the valley. So an L-antenna resulted, fed against an extensive earth system, using multiple electrodes in marsh land around the station.. The efficiency of this configuration was found to be higher than for a T-antenna, in which the whole sloping wire between the mountain tops was used.
Using a 500W valve transmitter strength measurements were performed over the whole of Germany and the efficiency of the antenna system was found to be comparable to the one of Nauen, that consisted of two 250m masts, seven 210m masts, two 180m masts and four200m masts.
In 1926 the station was ready but it did not become operational as by that time it  had been found that on short waves a few kilowatt and a small antenna could provide long distance communication equal to or even better than on VLF.
The reason I mention all this that the article contains an interesting formula for field strength at the surface of the earth around  a vertical antenna. The formula comes from a 1926 publication by  M. Bäumler.
(also as attachment).

E = field strength in V/m
I = current in the current maximum of the antenna.
hw = effective height in m.
lambda = wave length in m.
d = distance in m.
j = operator for 90 degrees phase shift.

The first part gives the far field; the second part the electric component of the near field. The far field diminishes with 1/d, the near field with 1/d squared. At 2 wavelength distance the near field is 8% of the far field, at 4 wavelength 4% and at 16 wavelength 1% of the far field.
It is clear that the formula is only correct over perfect earth. But at the distances where field strength measurements are usually performed, and certainly at VLF, the influence of the real earth on the field strength is negligible.
The term effective height is now mostly used in connection with antennas for reception.

Now an example:
A vertical antenna of 20m at 136kHz has an effective height of 10m and a radiation resistance of 0.028 ohm.
To radiate 1kW the current must be 189A. The wavelength is 2205m. Entering these figures into the formula we find the far field at 1km distance to be 0.323V/m, which compares well with the 300mV/m that is given by the CCIR curves for ground wave propagation.

73, Dick, PA0SE
JN22GD