To get an equivalent circuit we have to move the RF source toward the top end of the counterpoise. Since a good balun is transparent for differential signals, we can do this. Then, assuming a common mode inductance of the balun is high enough, we can cut the counterpoise at the balun. The result is on the figure above, right side.
What we have after all this transformation is just an asymmetric dipole. The easiest way to analyse this dipole is to put it into an antenna simulation program like EZNEC or MMANA. Fortunately we can take the MMANA program for free, with many examples of antenna models. Thanks to Makoto Mori JE3HHT, Igor (Gary) Gontcharenko DL2KQ/EU1TT, and Nob Oba JA7UDE. You can download it from http://www.qsl.net/mmhamsoft/mmana/index.htm . Although our dipole can be represented by just two "wires" of different length and diameter, we can try using a wire mesh model for a relatively fat radiator. This experience will help us in future EH analysis. Again, we can take a ready for use model made by DL2KQ with little modifications: DL7PE_1.maa . A picture below shows this grid model of the DL7PE radiator. It has approximately 2 cm diameter and 30 cm length. The coaxial feeder is modelled by a single 6 mm wire (3 mm radius).
For 14.05 MHz frequency, aluminium wires, inductor Q factor of 200, real ground and 1.4m elevation (the bottom end of counterpoise) it needs L=15.35 uH and has Rin=27 Ohm input impedance. Its gain is Ga=-2 dBi at 23 degrees elevation. Probably the minus 10 dBi figure reported by DL7PE himself was measured for a counterpoise lying on the concrete floor. If we put the counterpoise horizontally: DL7PE_1H.maa , while keeping the radiator vertical, then it behaves like a dipole. For horizontal polarization its gain Ga=+2.7 dBi at 41 degrees elevation angle, it needs L=14.63 uH and has Rin=22 Ohm.
Replacing a wire grid radiator model for a solid stick: DL7PE_2.MAA , we get for a vertical counterpoise a slightly different result: L=22.75 uH, Rin=38 Ohm. However the gain and the elevation angle are almost the same: Ga=-2.6 dBi at 23 degrees. This allows us to have a choice for the EH antenna modelling. Since it has some wires inside its bottom cylinder, we have to model it by a wire mesh, while we can use a solid wire model for its top cylinder.
Now we can test some more arrangements of the counterpoise. First, take a half wavelength vertical counterpoise: DL7PE_3.MAA . We have to increase its elevation because our Z=0 origin is fixed at the top of the counterpoise. Now Ga=-0.75 dBi at 20 degrees elevation. It needs L=28.5 uH inductor for an active Rin=904 Ohm input impedance. This value is typical for a parallel resonance of the counterpoise.
Second, bend this half wavelength vertical counterpoise in the middle: DL7PE_3a.MAA . Now it looks like on the figure below. Red and blue lines are current distribution curves.
This result is even more interesting. We have L=36.5 uH, Rin=2154 Ohm, Ga=1.6 dBi at 33 degrees elevation, and even F/B=4.2 dB for vertical polarization. The horizontal polarization is just -6 dB below the vertical one for the same elevation angle. The directivity pattern is presented below.
I guess a relatively strong (-1 S-unit) sensitivity of such an "antenna" (recall, it's just a feeder) to a horizontal polarization explains why it can outperform a pure vertical antenna of the same height.
Our third (and the last) exercise with DL7PE microvert is 1/4 wavelength
grounded counterpoise: DL7PE_2a.MAA . A picture
of its current distribution is the same as for 1/4+1/4 counterpoise above
with its horisontal part removed. Now we have L=36.2 uH, Rin=1957 Ohm and
Ga=-2 dBi at 29 degrees elevation. If we make our counterpoise a bit shorter
(5.0 m instead of 5.4 m): DL7PE_2a1.MAA , then
L=11 uH, Rin=2705 Ohm, Ga=-1.84 dBi at 29 degrees elevation. An interested
reader can try constructing a famous 5/8 wavelength vertical using this
"top feed" technique.