Choosing Length & Number of Shallow-Buried Radials, Version 3

Author: R.J.Edwards G4FGQ © 18th July 2006

A buried radial is a lossy single-wire transmission line. It has 4 primary parameters, R,L,C & G. It has secondary parameters: Alpha the attenuation constant and Beta the phase constant closely related to the velocity factor. The characteristic impedance is complex with components Ro and Xo. These parameters can be estimated using classical formulae from radial length, wire diameter, depth of burial, in conjunction with soil resistivity and permittivity. There is also input reactance which is of little consequence. It is automatically tuned out. The most important 'measurements' are the input impedance of all the radials in parallel at the system's focal point, and antenna input impedance. The final figure of interest is antenna radiating efficiency given by Rr/(Rr+Rs), where Rr is antenna radiation resistance and Rs is the radial system's input resistance.

Loss along a radial wire is proportional to length. As length increases input impedance converges towards its characteristic impedance Zo. At roughly 20 dB convergence is practically complete, input impedance ceases to change and radiating efficiency reaches a maximum. At that distance from the input end little current flows in a radial. There is nothing to be gained in efficiency by extending it any further. To lengthen a radial beyond the point where Zin = Zo serves no practical purpose insofar as radiating efficiency is concerned.

The Number of Radials
Once radial input impedance ceases changing with an increase in length, further increase in efficiency can occur only by increasing the number of radials. The relationship is not a linear one. But it is easy to use this program to help decide on the number of radials which will provide acceptable efficiency commensurate with the labour and expense of installation.

Accuracy of Estimation
The accuracy of a mathematical model of a radial system is not high. But in view of the insensitivity of efficiency to radial input resistance and the fact that soil resistivity and permittivity at HF are at best only crudely known, this is no handicap. This program is the only one of its type in existence. The general behaviour of a set of N radials is adequately represented.

Behaviour of a System of N Radials plus Test Antenna
Interaction occurs between inputs. The most important output data are:

• Input impedance of N wires - calculated with the far ends open circuited.
• Overall attenuation of 1 wire - increases in direct proportion to length. As attenuation increases up to about 20 dB, input impedance converges on Zo. dB per 1/4 wavelength indicates how resistance loss damps-down resonance.
• Input impedance of a simple vertical, sloping or inverted-L testing antenna. Radiating efficiency of the whole system summarises overall performance.

Attenuation per 1/4 wavelength of wire decreases at the higher frequencies. Consequentially, at 30 MHz attenuation is small enough to see peaks and troughs in input impedance as length is increased up to 3/4-wavelengths and beyond.

At 2 MHz and below, wire attenuation per 1/4-wave is high enough to damp down peaks and troughs in input impedance. The first trough at 1/4-wave is almost non-existent. The best way to see these effects is to vary length while observing the small increase in efficiency at approximately 1/4-wavelength of wire where the wire's input resistance versus wire length passes through a minimum.

Except at ELF, Zo of a radial wire has a positive angle. This is due to normal wire inductance in conjunction with the low shunt resistance of the soil. This is opposite to a normal transmission line. Zo increases with frequency. This is due to skin effect in the wire which reduces efficiency. So thicker wire may be better at the higher frequencies but the improvement is small even at 30 MHz

The program includes a simplified model of a vertical or a sloping antenna of the same length. This shows how a radial system affects radiating efficiencies of antennas of different heights.

The base-loading coil or capacitor shows that radial input reactance is tuned out by a tuning component simultaneously with antenna reactance.

The program user can judge the number of decibels attenuation at which the line's input impedance can be assumed to have converged on Zo. The behaviour of the antenna efficiency calculation can be taken into account. The user may have reason to bias his judgment above or below the typical value of 20 dB.

If a user has confidence in the accuracy of his knowledge about ground resistivity he may decide to shorten radials to 1/4-wavelength at a particular frequency to take advantage of the small improvement in efficiency which occurs at that length and frequency even though the Zin = Zo limit may not be met.

At high ground resistivities and at 30 MHz, attenuation is small, large standing waves occur on the radials and they behave similar to elevated radials. It becomes necessary to cut them to be 1/4-wave resonant. Or change to a dipole.

Remember that a 1/4-wavelength along a buried radial is determined by its own propagation velocity. At 10 MHz and above this strongly depends on permittivity. But at VLF and below, changing permittivity has little or no effect.

Frequency may be varied between 1 KHz and 30 MHz. Zo and VF are small at VLF. Attenuation per 1/4-wavelength is large. This determines whether peaks and troughs will occur in the input-impedance versus frequency curve. None occur at MF and below and Zin smoothly converges onto Zo without any ripples.

Table of Ground Characteristics
A guide to the type of soil to be found in the locality of an antenna. Selection is non-critical. +/- 40 percent is OK for R.

• R = Resistivity, Ohm-metres
• K = Relative Permittivity, Air=1

(R and K both depend on soil salts and moisture content.)

Nature of Area	R	K
Salt seas, oceans, remote from large freshwater river estuaries	0.22	81
Agricultural plains, warm, moist, dark, highly fertile soil	25	25
Temperate climate, warm rainfall, steppes, pampas, prairies	40	23
Pastoral, undulating, damp, fertile soil, streams, trees	60	20
Rural undulating farmlands, woods, fields, grasslands, cattle	100	17
Flat, cool, marshy, slow streams, grasses, weeds, bushes, birds	150	15
Undulating, drier but some streams, woods, medium fertility soil	200	14
Hilly, some woods or forests, grasses, weeds, poor dry clay soil	300	13
Fairly dry climate, grasses, weeds, poor sandy or stony soil	500	12
Dry climate, hilly, poor soil, small rural towns and villages	700	11
Suburban, low-rise housing, roads, back gardens, parks, pools	1000	10
Hilly, rocky, semi-desert, small rainfall, weeds, cactus	1500	8
City blocks, roads, streets, river bridges, industrial areas	2000	6
High-rise city blocks, spaghetti road systems, railways, bridges	4000	5
Mountainous regions, bare rock, vegetation only in valleys	7000	4
Arid sand deserts, minimal plant, insect, animal and bird life	15000	3
Unpolluted, deep, fresh water lakes, water weeds, fish, birds	1000	80

Run this Program from the Web or Download and Run it from Your Computer
This program is self-contained and ready to use. It does not require installation. Click this link Radial_3 then click Open to run from the web or Save to save the program to your hard drive. If you save it to your hard drive, double-click the file name from Windows Explorer (Right-click Start then left-click Explore to start Windows Explorer) and it will run.

Also see Version 2 of this program.