10 MHz ground constants at eleven locations in Great Britain. At each location the soil was sampled at several depths to 10′. Linear interpolation can appear curved on a log scale. 10−8 ESU = 11.1 mS/m.
This writeup compares ground constants from short ground probes with those derived from a stratified ground model. I model eight locations in Great Britain at one frequency and one location in Oregon at eight frequencies.
To reduce sampling artifacts, I oversample the British data by a factor of 10. I use linear interpolation between the original samples. Imagine the curves with tiny stairstep layers, each with uniform ground constants. Layer boundaries are halfway between samples, with the last layer infinitely deep. I calculate the Fresnel reflection coefficient for a vertically incident plane wave. Then I find ground constants for uniform soil that yield the same coefficient. The two ground models should be equivalent whenever the reflection coefficient is valid. I calculate the response of a 12″ ground probe by averaging ground permittivity and resistivity at the surface and 1′ below. For a 24″ probe I use a 1:2:1 weighted average of 0′, 1′, and 2′.
Location Geology Peq Ceq P12 C12 P24 C24 ■ Rugby 1 Lower lias 22 7.3 2.22 0.99 2.14 1.16 ■ Baldock Chalk 20 0.9 1.17 1.32 1.15 1.05 ■ Tatsfield Upper greensand 44 3.7 0.78 1.15 0.61 1.04 ■ Brookmans Park London clay 16 1.6 1.56 0.96 1.54 0.95 ■ Washford Cross Red marls 12 2.6 2.68 0.91 2.95 1.00 ■ Moorside Edge Millstone grit 45 3.1 0.85 1.13 0.81 0.99 ■ Westerglen Boulder clay 16 1.2 1.28 1.03 1.30 1.14 ■ Teddington London clay 24 0.8 0.57 1.78 0.55 1.56
Peq and Ceq are equivalent ground constants. P12, C12, P24, and C24 are ratios of the ground probe response to the equivalent. No equivalent exists for ■ Rugby 2, ■ Daventry, or ■ Brendon Hills.
C12 is within 32% of 1 except for one outlier. C24 is within 16% with the outlier reduced. P12 is within 28% for only half the locations and has big outliers. Overall, P24 is not quite as good as P12.
Fresnel reflection coefficients are valid for plane waves, but fields near ground may not be planar for low antennas. If Sommerfeld-Norton and reflection coefficient ground differ little for a NEC antenna model, equivalent ground constants should be valid for the reflection coefficient method. Use values from the calculator to model antenna currents, impedance, and overhead gain. Equivalence is unlikely to be valid at lower elevation angles since it is derived from RF incident at 90°.
Several of the Peq values seem far from an eyeball average of the permittivity curve. But stratified ground does not average. It acts like a cascade of lossy transmission lines, each of which can transform its source impedance to something other than its characteristic impedance. Permittivity seems to transform more than conductivity.
A stratified layer may resonate. It is an artifact of sparse sampling unless the layer actually is thick. Oversampling inhibits artificial resonance. Oversampling by a factor of 10, 100, or 1000 yields virtually identical results. Oversampling seems to remove resonance artifacts without introducing any of its own. Use the calculator to explore transformation and resonance.
The notion of equivalent ground constants is not entirely coherent. Consider an example with two soil layers. Reduce the ground constants of the upper layer until they become those of air. This increases antenna height by the layer depth. But no set of equivalent ground constants can do this. In general, stratified ground requires a model more complex than NEC provides.
Within the limitations described above, I believe the stratified ground model is accurate. Reflection coefficients agree with published results. But I have no way to verify the equivalence aspect of the model.
The British data are for 10 MHz. Subsurface soil effects are more likely at lower frequencies where skin depth increases. Ground probe accuracy may degrade since the probe samples a smaller percentage of soil with current. To assess the sampling limitation, use this program to calculate skin depth.
8″ of concrete over pastoral ground Water table 200′ below desert
This Windows program calculates equivalent ground constants for stratified ground with two layers. The documentation includes typical permittivity and conductivity values for snow, ice, asphalt, concrete, and rock. Generic values for soil and water are here. Results have not been verified.
Rudy Severns, N6LF, used a 12″ ground probe and NanoVNA-H4 to measure ground constants in a newly dug trench in Oregon bottomland. He inserted the probe horizontally in the trench wall. The soil was loam down to the bottom at 6′ where it became gravel. He inserted the probe vertically at 6′ and at the surface near the trench.
The graph shows the values I used for the stratified ground analysis. I derived ground constants for the 3″ surface layer from the surface measurement and the measurements at 0.5′ and 1′ it overlapped. For the deepest measurement I used the mean probe depth of 6.5′. I oversampled the data by a factor of 10, but it had little effect.
MHz Peq Ceq P12 C12 P24 C24 28.5 13 42 1.78 0.67 2.25 0.83 21.2 15 40 1.62 0.63 2.05 0.78 14.2 20 35 1.40 0.62 1.78 0.77 10.1 23 33 1.34 0.59 1.70 0.74 7.1 32 33 1.09 0.55 1.37 0.69 3.7 74 26 0.58 0.58 0.71 0.74 1.8 124 17 0.43 0.78 0.52 1.00 1.0 158 13 0.40 0.93 0.48 1.21
P12 is close to 1 only at 7.1 MHz and C12 only at 1 MHz. I simulated a 24″ probe by combining measurements at the surface, 1′, and 2′. P24 is worse than P12 above 3.7 MHz while C24 is somewhat better than C12 above 1 MHz.
Ground probe results do not compare well with those from the stratified ground models. But probes should work fine for efficiency analysis of antennas with ground radials, where ground current concentrates near the surface.
Ground current should drop quickly near a buried wire as fields spread radially. As they expand, the fields gradually become more planar and current decay approaches exponential. For symmetrical radials and soil, currents cancel directly below. Away from center, they partially cancel at depth. Since it samples the soil with highest current (by far, I believe), a ground probe should provide suitable ground constants for antennas with buried radials. This should be true for radials on the ground as well. However, subsurface soil can affect ground reflection coefficients. This may limit accuracy for far-field results since ground probes ignore subsurface soil. But accuracy should be good for studying the effect of radial number, length, and depth on antenna efficiency.
Smith-Rose, R. L., "Electrical Measurements on Soil with Alternating Currents," Proc. IEE, Vol. 75, pp. 221-237, 1934. I skipped the three locations with indefinite depths or no surface measurement.
88–108 MHz