Portable …. ?
For a possible vacation in a relatively rare DXCC country there may be opportunities to bring along equipment.
Because I consider the lower HF-bands more challenging from a technical point of view, my first thoughts were
about 160 or 80m. But.., it will be summer there and the amount and material/weight exceeds that of an average tourist.

Considering relative DXCC rareness, a ‘more serious’ 40m-antenna is obligatory to get the most out of the situation,
but also to ‘manage expectations’ when I (hope to) appear on the air.

As I don’t know what material might be available there, some boundary conditions apply:

1. Light weight (travelling by plane as a normal tourist, not DHL or FedEx!)
2. One 12m-pole (luggage!), and several supporting poles (to be ‘retrieved there’, e.g. tree branches).
Hopefully there is a flag pole or high tree nearby a shack.
3. Simple but relatively rigid construction
4. Material compatible with deltaloop setup (‘plan B’)

Disclaimer: the following is not intended to seduce the Nobel Prize Committee for a nomination, nor do I claim
owner- and/or inventorship of the used knowledge. Look at it on how I interpret public available knowledge about
four square (rocket) science. There is no list of references, but reference is made to respective call signs when cited.
Material of them can be easily found by search engines. One reference will be made in advance: ON4UN : -)

…. 40m four square array?

Very likely >95% of the potential contacts will be ‘DX’ (at least ~6000km away), requiring an antenna with as
much gain and a low take off angle. Yagi’s are good candidates but have some practical difficulties.

Another approach is a four square array. These vertical arrays have been discussed in the literature extensively.

I made my own NEC2-model and performed modelling studies on a four square array with 4NEC2.
The array has to be realised around one (non conductive) 12m high central pole and bent verticals.
The pole is guyed with four 20m ropes. Element spacing of 0.25λ is not mandatory, but improves the F/B significantly (ca. +8dB).
With an element spacing of 9m (0.21λ) 75Ω vf=0.66 ¼λ-coax cables (e.g. RG-59) may be used.
The array uses only four (4) sloping resonant elevated radials. The contrapsion is a nice goniometric exercise : -)

Various setups were modelled to study radiation patterns. It appeared only minor improvements
were visible, mainly for the F/B (ca. +2 dB). Foward gain was almost not (ca. +0.1dB) affected. With the K3LC setup
the take off angle was 2′ lower. The contrapsion was also modelled with four ‘conventional’ elevated radials a la DF6QV,
i.e. at ground level and then sloping upwards: no difference in calculated performance appeared.

See picture below for some (not all) calculated contrapsions (click on image to enlarge, opens in new tab).

Sloping top ends, and only one resonant radial per element introduce a horizontal component and skewed radiation patterns
for each element, resulting in a more spherical vertical radiation pattern of the array. I.e. it lacks the ‘characteristic rear lobe dip’.

As I will not be in Europe and very likely will not receive in band strong nearby signals ‘from above’, it may be a pity, but it’s fine.
Using more radials compresses the ‘radiation sphere’ somewhat, but according to my calculations this does not influence the
forward gain and doesn’t have impact on the take off angle (which remains 24′ above the modelled ground used).

Having modelled different contrapsions extensively AND considering materials used and efforts, four resonant radials are the way to go.
In total ~86m 2mm diam wire (‘two deltaloops’ ; -) is needed.

Calculated gain is 5.86 dBi above agricultural ground (clay), F/B is ~31 dB. RDF = ca. 10 dB. Above equal ground one can squeeze
+1dB gain and +3dB F/B out this four square design, but the additional complexity of the phasing unit and alignment procedures
involved are disproportionally high.

Note: all gains are relative, actual gains are influenced by ground properties and may be more, due to the NEC2 ground model.
NEC4 might give more accurate results, but I don’t have this engine. Besides, the antenna has to be tweaked in the field anyway.

An interesting article on how to fire a 4-square into eight (8) directions by K3LC led me to simulate his setup and compare this with
my bent vertical approach. The whole thing was x2 scaled towards 80m, subsequently optimised and compared with his article.
The picture below shows the comparison. (click on image to enlarge, opens in new tab)

When you have a big tree of around 24m high and ca. 160m wire somewhere, it may be worth a try on 80m ; -)

Below the 3D-visualisation of the 40m 4-square, modelled with 4NEC2, is depicted. Thank you Arie, for bringing 4NEC2 to the public domain!

The whole looks like a tropical  insect ; -)  (click on image to enlarge, opens in new tab)


The setup will be temporary, and has to be as simple as possible (as I can hardly explain non-HAMs why I want to build this contrapsion ; -)

Besides element spacing, one can fiddle with phase relationships of elements to achieve a little more gain and ‘cleaner’ radiation pattern.
Below a comparison between 0.25λ (blue) and 0.21λ (red) element spacing with 4 elevated radials, 0.25λ with 8 radials (green), and
0.25λ in conjunction with -so called- WA3FET-feeding (-112′, -224′, -112′, and 0′) with 4 radials (purple).

Gain differences between my calculated contrapsion and ‘super duper’ state-of-the-art knowledge on (feeding) 4-square arrays
differ less than 1dB above equal ground. F/B of 8 radials and WA3FET-feeding is better, and the latter feed method may result
in a 2′ lower take off angle. As already stated, I take that for granted as it’s not worth the complexity

Click on the image below to enlarge (opens in new tab).


Why and how does this work?
The easiest way to fire a 4-square is feeding each element with equal currents (aka ‘current forcing’), but with different
phase angles (φ) between voltage (real) and current (imag). Relative voltage (‘real’) and current (‘imag’) fractions of sine waves
are determined by cosφ and sinφ respectively. If φ=0′ then real is cos(0′)=1, and imag is sin(0′)=0.
When current is lagging 90′ (i.e. -90′), then real is cos(-90′)=0, and imag is sin(-90′)=-1.
By connecting these phase shifted signals in a smart manner they are added in one direction, and substracted in another direction.
This way of feeding is known as ‘quadrature feeding’ (k1 = k2 = 1.0). Anyway, net result is gain and directivity. Presto!

Let’s consider the NE-direction. When elements 1, 2, 3, and 4 are fed with relative phases (φ) of -90′, -180′, -90′ and 0′ respectively, the
total mathematical sum will be 4 (i.e. 6dB) in this direction. Yagi-Uda antennas are based on the same principle but then elements are mostly in line.

It should be noted that, although the array elements are electrically fed with ‘defined’ phase differences, the geometrical contrapsion
does not heed these phasing relationships of the wave front due to geometrical issues. E.g. elements 2 and 4 are fed with 180′ phase difference,
but are spaced 2*√2*0.25λ = 0.7λ, and not 0.5λ. This is even more valid for the upper sloping parts of the array, but these are current minima.

Chapter 11 of ON4UN’s book on Low Band DXing elaborates on how to feed each element properly and thoroughly describes how the
impedances of each element vary/are influenced by the presence of other nearby (resonant)  elements. It may be a gullible assumption
but laziness induced me to rely on the mathematics in the model while optimising the overall impedance of the array to Z = 50 + j0 Ω. ; -)

Looking at a lot of radiation pattern plots in the internet, lots of people seem to use 4-squares above perfect ground,
or at least put up their antennas above warm seawater (DXpeditions on remote/rare islands). My situation will not be that ideal, but… for fun:
a simulation above 20′ seawater is depicted below. Above warm seawater gain increases with 4dB and the elevation angle
amounts 10′, and F/B = ~40dB  (!) Unfortunately seawater is ~800 kms away… Click on image below to enlarge (opens in new tab).

Alignment procedure.

First make one bent vertical and connect it to an antenna analyser with the ability to show R, and X ( VSWR only is useless! Sorry…).
The bent vertical is made resonant (Z = 50 +j0 Ω ) by trimming the top length (ca. 528 cm -> R = 50Ω) first, and subsequently the
radial length (ca. 919 cm -> X = 0). Copy the resulting contrapsion three times as accurately as possible and measure the impedance
of each element leaving the other elements open. Individual element impedances should be as equal as possible,
but that is not really necessary (due to ‘current forcing’).

Guy ropes (sloping angle is arctan(12/16) = 37′) support the upper sloping parts of each bent vertical.
Make a knot in the guy ropes at the desired length (Pythagoras is your friend) and guide the vertical wire through it.
The end side is fixed upwards (with a knot or cable tie).
Vertical feed points might have been lower (increasing the ‘non bent’ vertical part, to increase radiation resistances -and band widths-
of the verticals), but 1m seems a practical compromise considering the elevated nature of the radials relative to
the operating frequency (7.1 MHz), and obtaining R = 50Ω.

The central pole also supports the switch box with phasing circuitry at 1m above ground with vf=0.66 coax.
If RG6-/H125-coax (foam dielectric, vf ~0.8) is used, the box can lie on the ground. Feed points of each vertical are fed through
common mode chokes (about 40cm coax is left when using vf=0.66 coax), and are supported with 1m poles to make the contrapsion more rigid.

The reactance (Xl) of the chokes has to be ‘significant’. Like with transformers/UNUN’s: around 10x the connected impedance will do:
Xl = ~500Ω, to be realised by winding the feed side of the stubs into suitable toroids.
DF6VQ uses the same value for Xl, and his approach has proven to be successful.

I had an idea to deliberately use the ¼λ-lines as (elevated) radials and put a choke at the phasing box side, but simulations didn’t show
improvements for the radiation pattern. Moreover, this may introduce additional surprises ‘in the field’, which has to be prevented.

The array is optimised for 7.1 MHz and VSWR at the band limits is calculated to be <1.6 on both sides. The calculated resistive part of the impedance
only changes around +1.5 Ω (7200 kHz) and -1.5Ω (7000 kHz) while the imaginary part changes ca. 15x more (around +/- 22Ω)!
See picture below:

Another strategy could be trimming the contrapsion resonant (Z = 50 +j0 Ω) for the CW portion of the band (7000 – 7050 kHz)  and shorten the
radials for the high portion of the band 7150 – 7200 (SSB DX). DF6QV did this for his 80m-4-squares.
Calculations result in around 45cm less radial length for 7200 vs 7000 kHz. When operating in the lower part of the band, the radials are lengthened with
45cm wires with a connector at their hi-Z  ends. Delta VSWR across the band then is neglectable (calculated VSWR ‘worst case’ is 53/50 = 1.06).
The hi-Z side of the radials are pegged on the ground with extension ropes.