Choosing Length & Number of Shallow-Buried Radials, Version 2

Author: R.J.Edwards G4FGQ © 5th August 2004

A buried radial is a lossy single-wire transmission line. It has four 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. All these parameters can be estimated from line dimensions of length, wire diameter and burial depth in conjunction with soil resistivity and permittivity.

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 14 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 when Zin = Zo serves no practical purpose insofar as transmitter performance 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 a number which will provide acceptable efficiency commensurate with the considerable expense and labour of installation.

Accuracy of Estimation
The accuracy of a mathematical model of a radial system is uncertain. 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, there is no great handicap in using this program. In a dearth of numerical data even a small amount of logical ball-park quantitative information is valuable.

Behaviour of a System of N Radials
Interactions occur. The following relationships are the most important: Radial input impedance is calculated from an open-circuit at the far end. Overall radial attenuation increases in direct proportion to radial length. As line attenuation increases input impedance converges towards Zo.

Attenuation per 1/4 wavelength of line 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 lower, line attenuation per 1/4-wave is high enough to damp down peaks and troughs in input impedance. The first peak at 1/4-wave is almost nonexistent. The best way to see these two effects is to vary length while observing the small maximum in efficiency at approximately 1/4-wavelength of line where the line's input resistance versus line length passes through a minimum.

Except at VLF, Zo of a radial wire has a positive angle. This is due to normal wire inductance in conjunction with the large shunt conductance of the soil.

Zo increases with frequency which is not good for efficiency. This is due to skin effect in the wire and the increase in wire resistance versus frequency.

The program includes a simplified model of a vertical or an inverted-L antenna of the same length. This helps to show how a radial system affects radiating efficiencies of antennas of different lengths.

The antenna base-loading coil or capacitor is a reminder that ground reactance is tuned-out by a tuner simultaneously with the antenna input reactance.

The program user must 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 will be taken into account. The user may have reason to bias his judgment above or below a typical value of 14 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 very high ground resistivities the loss in the ground 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 antenna type.

Remember that a 1/4-wavelength along a buried radial is determined by its own propagation velocity. This depends on soil properties including permittivity. The free-space velocity value must be disregarded.

Frequency may be varied between 1 KHz and 30 MHz. Zo and VF are small at VLF. Attenuation per 1/4-wavelength is large. The latter determines whether peaks and troughs will occur in the input-impedance versus frequency curve. None occur at LF and VLF. Zin smoothly converges onto Zo.

Table of Ground Characteristics
A guide to the type of soil to be found in the locality of an antenna. Selection is non-critical. +/- 33 percent is OK for R.

R = Resistivity
K = Permittivity (Depends 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 Radials2 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 3 of this program.