The MicroVert by Jürgen Schäfer, DL7PE, based on foreign sources was described briefly in different places, e.g. [1].
But his version could not inspire me, since the necessary lengths of the 22 mm chunky pipes in the lower amateur radio bands become very fast unmanageable.
After the in [2] published formula

the following dimensions results for the low-frequency bands, whereby h is the length of the used 22 mm pipe and f the frequency in MHz:

f = 7.03 MHz h = 669 mm ≈ 67 cm

f = 3.56 MHz h = 1320 mm ≈ 1.3 m

f = 1.85 MHz h = 2541 mm ≈ 2.5 m

Such long things are usable only rarely by radio amateurs with antenna restrictions without to exite attention.

**Solution in sight**

Since 2003 an advancement is spooking by the numbers of the radio amateurs, whom avoids these disadvantages.
Arthur Wenzel, DL7AHW, was concerned about the use of different materials and forms, which do not possess in addition the proportions of a pipe.
And I can say that he make a find!

My building references here use to a large extent the descriptions on Arthur's antenna webpage and come of to expanded discussions with him.
Arthur offers a MS-DOS program for the free download and webpages for the calculation of different antenna forms, whereby one saves the arithmetic exercise.
As alternative I publish here the calculation for a cylindric antenna.
The © copyright for the formulas used thereby is with Arthur Wenzel, DL7AHW.

In order to be able to sawy the formulas used in this script well, I would like to describe their emergence step by step.
Whom less this interests, who can read on according to the last formula.

**Calculate the capacity C**

With me stood about already longer an empty tin can with a height of h = 150 mm and a diameter of D = 200 mm.
In order to be able to use the cylinder as radiators, first the surface A of the coat

is to calculate. I ascertained approximately 94248 mm².
The next step is the calculation of a fictitious height of h', if a pipe with a diameter of d' by 22 mm would be used, but the same surface is demanded.
The picture on the right shows the size comparison of both antennas.

If one merged both formulas together, it becomes still simpler.

With this fictitious height of h' (with me 1364 mm) the capacity C in pF of the can can be calculated.
An adapted formula from the AntenneX file [2] serves as basis.

After a few simplifications and with the real dimensions of the radiator the following equation implies.

Unfortunately it's not possible due to the logarithm in the denominator to calculate the capacities of small cylinders, because it must be

But in the meantime Arthur found a solution for it:
He uses now linear equations, which replace the logarithm step by step and simplify the formula at the same time.
Smart!

The can I used by me got approximately 16 pF.

**Calculate the inductance L**

In order to transform the capacity of the radiator (can) into a circuit with the desired resonance frequency f, still another inductance L is necessary.
The calculation is possible with the oscillation formula by Thomson.

After transposing the equation and some simplifications it is better usable.

If one enters the capacity in pF and the frequency f in MHz, one receives the necessary inductance in µH.
The can I used by me requested an inductance of approximately 129 µH for 3.5 MHz.

The best construction method for the coil is to wind it as air coil from a wire as thick as possible on a plastic pipe, which has the same diameter as the can - but other variants function also.
The calculation of the mechanical dimensions is comfortably done with the mini Ring Core Calculator by Wilfried Burmeister, DL5SWB.

Otherwise one also can use the long-known formulas for air coils, which are however often only usable for certain diameter/length conditions.
Arthur also was concerned about this problem and he developed based on the well-known formulas a new one, which is usable for arbitrary conditions.
The formulas above uses his new equation, which approximat the logarithm as straight lines.

**Counterweight K and decoupling coil X**

Since this form of the antenna is a monopole, it needs a counterweight K.
For it direct at the end of the coil is attached a coaxial cable with an electrically length of a quarter wavelength.

with K in meters and f in MHz.
VF is here the velocity factor of the coaxial cable, for RG58 i.e. 0.66.
By courtesy of the decoupling coil X the radiating antenna is uncoupled from the following feed line.
The coil can be realized in the simplest case by the fact that one rolled up the feed line some turns and stuck it with insulating tape.
In addition ring cores can be used: Thus e.g. a FT140-43, on which one applies with the RG58 coaxial cable 10 until 12 turns, results in good results according to Arthur.
In addition variants with the smaller FT114-43 and twisted wire are suitable according to his datas.
Afterwards any long feed cable can follow.

**Assembly**

The assembly is done nearly faster than the whole calcutations.
Solder or screw the coaxial cable with the decoupling coil to the tin can.
So that a fine tuning is later possible for frequency change of the finished thing, you can attache a small telescope antenna at the radiator to change the capacity and/or you can change the mechanical dimension of the coil by pushing it together or pulling apart.
An antenna analyzer facilitates the alignment very much, but it's however not a condition.
The photo on the right is the antenna version by Peter, DL2FI, which is named "Cudgel of Berlin".

**Result**

The antenna is a good variant for all, who cannot errect uncurtailed antennas.
Their operational area is not limited to the short wave bands.
Variations for the 136 kHz band were already sighted, which uses as radiators empty oil barrels - always more manageably than the otherwise necessary enormous antenna lengths.

Arthur, DL7AHW, develops on basis of the formulas specified above also angular antennas and he is busy with the development for spherical radiators.
I recommend to you to visit also his webpage already mentioned above.

**Sources**

[1] Prof. Friedrich Landstorfer, Prof. Hans Heinrich Meinke: Ein neues Ersatzschaltbild für die Impedanz kurzer Strahler. Nachrichtentechnische Gesellschaft im VDE, Nachrichtentechnische Zeitschrift, 26 (1973), issue. 11, pp. 490-495

[2] Jürgen Schäfer, DL7PE: Progess in design of extremely short antennas (PDF), antenneX, April 2001