The QHA is relatively easy to construct. The difficult part is to get the dimensions exactly right. A self-phasing QHA consists of two loops with slightly different overall lengths. At the operating frequency the impedance of the larger loop must be inductive and the reactive part must be equal to the resistive part to produce a 45° phase shift between the voltage and current of the loop. The smaller loop must be capacitive and again the reactive part and the resistive part must be equal to get a -45° phase shift. With the loops connected in parallel to the same voltage source, the currents in the loops are 90° out of phase. This is essential for the performance of the QHA.

The problem is that for relatively thin conductors, like the outer conductor of RG58 coax that was used, the antenna is rather critical. Almost always fine-tuning is needed after construction to achieve the required 90° phase difference. And this is by no means a simple task.

That is why I opted for a somewhat different approach to make the antenna easier to tune and a different measurement method to get a more accurate result. Please scroll down for further details.

But first let's look at the basic principle of the QHA.
Let's consider a straight rectangular full wavelength loop with an height / width ratio equal to that of the QHA. (0.44)

The radiation pattern shows a healthy gain in the horizontal direction perpendicular to the plane of the loop and a deep dip at 90° from the main direction.
If we want to make an omni-directional antenna, we could use a second loop perpendicular to the first one to fill up the dip. The sum of both loops would be omni-directional in the horizontal plane. This is only true, however, if the current in the second loop is 90° out of phase with the first one.  Simply adding the fields of two loops without any phase difference would result in the same pattern as the one of a single loop only rotated over 45° in the horizontal plane.
The radiation pattern of two perpendicular loops carrying equal currents with a 90° phase difference is shown on the left. It is almost an ideal (isotropic) omni-directional pattern.

However, if the currents are not equal, or the phase difference is not 90°, the pattern will be distorted resulting in extra gain in some directions and loss of gain in other directions.

Now let's look again at a single loop, only this time bend into a half turn double helix. (Like a DNA molecule)
The radiation patterns below are for Left Handed Circular Polarization (LHCP) on the left and Right Handed Circular Polarization (RHCP) on the right.
Note that the LHCP radiation is at least 15 dB down compared to the RHCP radiation. So even a single helix shaped loop produces almost perfect circular polarization in almost all directions. The radiation pattern of just a single loop, however, is not very useful. Adding a second helix shaped loop in the same manner as with the straight loops gives an omni-directional pattern in de horizontal plane and a good coverage of the upper hemisphere. This is ideal for satellite reception with a fixed antenna. The resulting radiation patterns are shown below.
Based on experience with other antennas, most people would associate feeding orthogonal elements with 90° phase difference with circular polarization. But in the case of a QHA, it is the helix shape that is responsible for the circular polarisation and not the 90° phase difference. So if we fail to properly tune the QHA and end up with a phase difference other than 90° the radiation pattern will suffer, but the polarization will remain almost perfectly circular.

LHCP

RHCP
Tuning the antenna...... the hard way.

First we need to measure the resonance frequency of each loop. Then from the SWR at resonance we need to calculate the impedance (radiation resistance). This should be somewhere around 20 to 30 Ohm. Now we have to calculate the SWR for a loop impedance of R+jR or R-jR. For example if the radiation resistance is 25 Ohm the SWR at resonance would be 2. The SWR for 25+j25 Ohm is 2.6. So we have to find the frequency where the SWR has increased from 2 to 2.6. This gives us the required frequency offset between the operation frequency of 137.5 MHz and the resonance frequency of the loop. It will be somewhere around 3 MHz. Note that this should be done for each loop separately (without the other one connected). Cutting back the length in small increments until we reach the desired resonance frequency will get us there eventually. But it is a tedious job. Moreover, this method has a more serious problem. There is always some mutual coupling between the loops. So tuning one loop will slightly detune the other one. This could be a problem in the more critical designs.

Tuning the antenna...... the easy way.

So I decided to try a different approach. Initially the resonance frequency of both loops was far too low. The reason for this is that most designs are based on bare conductors. The insulating outer sheet of the coax however decreases the velocity factor by about 5 to 10 %. So the loops have to be about 5 to 10 % shorter than designed. I intentionally kept them a bit longer and inserted a variable capacitor in series with each loop. Now I could tune the loops by just trimming the capacitors. With both loops connected I wanted to directly monitor the loop currents.

So I made a sensing loop with an oscilloscope probe by simply connecting the earth strap to the tip of the probe. Using two probes, one coupled to the large loop and one to the small loop you can easily monitor both the magnitude of the loop currents and the phase difference between them.  See the illustrations below.

The final result: equal currents and 90° phase difference at 137.5 MHz
Fine tuning just took a few minutes.
A final check revealed that the nonuniformity in the horizontal direction was about 1dB.  That corresponds well with results from antenna modelling.