EH at close look.

Now the EH itself is just another exercise. We can take a model prepared by DL2KQ  W0KPH_1.MAA , and first test it as a fat dipole in free space:

Each radiator is 6 inches long, they are spaced by 2 inches ( approximately equal to their diameter) gap. With no inductor added and lossless wires the dipole shows a radiation impedance of 17 mOhm - j 1.7 kOhm, or a Q factor of 100000. It has Ga=1.76 dBi. When its capacitance is compensated by a real coil (L=19.7 uH, Q=200), all its input resistance Rin=9 Ohm is mostly coil's equivalent series resistance. Its gain drops to Ga=-25 dBi. Increasing the Q factor twice (Q=400) we can get Rin=4.5 Ohm and 3 dB increase in gain: Ga=-22 dBi.
Replacing the top radiator by a single thick wire (W0KPH_1a.MAA) only slightly changes the result, but greatly reduces the calculation time. Now L=20.15 uH, Rin=9.22 Ohm, Ga=-25.24 dBi (for Q=200). This is what one may expect for a "no feeder" EH test.

Now we can connect a counterpoise. This is a bit simpler for a non-symmetric L+T network, so we try it first, leaving only a single series coil connected to the top radiator. First, we add only 1 metre of coax, which should model an "in shack" adjustment setup: W0KPH_2.MAA . This is shown on the pictire below.

With no inductor we have a real part of the radiation impedance equal to 0.168 Ohm, which is a tenfold increase. With L=16.9 uH (Q=200) we have Rin=7.8 Ohm and Ga=-14.6 dBi, which is also a 10 dB increase comparing to Ga=-25 dBi. If we take a 3 meters counterpoise and a real ground 5 m below its bottom end, we get something like a "car battery test", assuming our 3 m wire is a sum of short coax between the antenna and a transceiver, and a power cord from the transceiver to the car battery: W0KPH_2a.MAA . This variant has not too bad Ga=-7.4 dBi at 20 degrees elevation.
A quarter wavelength grounded counterpoise W0KPH_2b.MAA gives us Rin=146 Ohm (L=19.05 uH) and Ga=-2.15 dBi at 30 degrees elevation angle. This is already good. Longer counterpoise gives better result, but takes more CPU time.