Simple answer, yes, you can use a hairpin.
To find out, I modeled a dipole alone at 20' in the air over real ground and got this:
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Now to add the parasitic element. For this, I chose to go with a director instead of a reflector, because if you only have two elements total, you only get a tiny improvement in gain with the reflector over a director, but using the director, the spacing is much smaller. So, here we have a driven with a single director (no reflector).
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The feed point impedance dropped as expected (as it would have with the reflector too). This is what happens if we do our best at matching with just a hairpin. In this case, the hairpin must have an inductance of 620nH
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but if we also use a series capacitor, you can land it right on 50Ω and the values become 430nH and 560pF
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The bottom line is that none of these numbers really matter, as you will probably use a different height, have different ground conductivity etc. This is where a nanoVNA or another antenna analyzer is worth its weight in gold. You can end all matching problems just by putting in your known feed point impedance and correct for that perfectly.
Hairpins are simply shorted stubs, and just like the coax equivalent, they are inductive under 1/4λ. The first step is to find the characteristic impedance of the open wire transmission line the hairpin is made of Zo = 276 * log(2s/d) where s is wire spacing center to center and d is diameter of wire (in same units). Then, to know the inductive reactance of the stub, you just multiply that by the tangent of its length in degrees.
Here is an example. Lets rearrange that and solve for the 430nH hairpin.
Find inductive reactance of 430nH:
Xl = 2*pi*27200000*.000000430 = 73.5Ω
Next, we need to decide on some dimensions for the hairpin, lets say 1.9" spacing with a wire diameter of .15". Lets figure out the characteristic impedance of a line of those dimensions:
Xo = 276 * log (2*1.9*.15) = 387.4Ω
The final question is at what length will the stub exhibit the necessary 106Ω?
invtan(73.5Ω / 387.4Ω) = 10.7°
The VF in air is 299705000m/s so 1λ at 27.2MHz is 433.8"
433.8" * 10.7°/360° = 12.9" is the hairpin length
With that out of the way, you can probably see why it is beneficial to try to adjust the antenna so the complex impedance lands on that .02s constant conductance curve, because if it were there, the hairpin would get you to 50Ω without other components like that capacitor in pic 4
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I know this reply is way too late as the OP got it working, but I figured details on the hairpin calculations might be useful info to someone..
