## Mutual Z in a square array

The "four-square" array of vertical antennas has been around a long while in various forms. The idea is to put 4 vertical antennas in a square with sides that are 1/4 wavelength, 3/8 wavelength, etc.. The array is symmetrical, and a fairly simple feed scheme (feed 2 in phase, and 2 lagging by 90 or 135 degrees) and some relays can be used to form a beam that can be rotated to 4 (or 8) positions. Given that the beamwidth is on the order of 90 degrees, this works fairly well. The moderately large aperture also means that not too much power is being radiated at high angles.

This page gives some mutual impedances calculated for an array in free space using NEC2.

### Historical notes

It seems to be quite popular for 160m, where the reasonable aperture with small radiators, and the ability to point away from noise is probably the selling point. The ARRL handbook had a 40 meter version using a Wilkinson power divider for a number of years. Unfortunately, that design didn't allow for the mutual Z of the elements, and didn't perform particularly well, a topic discussed at some length in the ARRL Antenna Book. Subsequently, the Antenna Compendium has had a number of versions, including some with center loaded sleeve dipoles, various schemes using sloping dipoles off a tower, and so forth.

I've often thought about building a small monopole array for use at 2m and 70cm by putting 4 antennas on the roof of the car, and using a suitable rotary switch to move the beam. The extra 5-6 dB or so from the array gain would help hit those distant repeaters. On the other hand, a 40 Watt amplifier might work just as well, and is omnidirectional.

### Modelling results

Some NEC studies were done to determine the mutual Z for a square array. Two cases were run with the square sides 1/4 wavelength and 3/8 wavelength. (At 1/2 wavelength, you don't form a unidirectional beam). In both cases, the length of the dipoles were adjusted to make the driven element "resonant", or at least, with a reactive impedance less than an ohm.

The 4 elements are numbered as follows:

 2 4 1 3

There is a lot of symmetry: Z12=Z13, Z14 = Z23, etc..

The model was run at 29.98 MHz with copper wires 0.00127 in diameter (approximately #10 AWG). The wavelength of 10.0 meters made spacing the elements easy, which is why it was chosen. Element 1 was driven, and the other elements were loaded with a 10K resistor. The voltages and currents from the NEC output were then used to calculate the mutual Z.

In the 1/4 wave case, the elements wound up being 4.83 meters long. In the 3/8 wavelength spacing case, the elements were slightly shorter: 4.82 meters. The one cm difference manifested itself as a change in the reactive impedance of about 1 ohm. While the impedances in the following table are reported to 0.01 ohm, I suspect that the real accuracy is somwhat poorer.

 Spacing (wavelength) Z11 Z12 Z13 Z14 1/4 69.77-j0.87 -36.16+j29.76 -36.16+j29.76 -11.35+j37.99 3/8 67.64+j0.82 -9.98+j35.16 -9.98+j35.16 17.41+j22.45

Note that the "resonant" impedance is slightly different from the nominal 72 ohms you would expect for an ideal thin dipole. This is no doubt due to several factors: the finite size of the wires, the resistive loading, and the effect of the "parasitic" elements, even though they are effectively open circuited.

For 1/4 monopoles above a ground plane, divide these numbers by two.

The impedances will vary somewhat when operating above a real ground.

zsquare.htm - 25 Nov 2000 - Jim Lux
Phased Arrays