Radiated Power From MF Verticals

Rudy Severns N6LF

July 2014

Unlike the higher bands where the maximum transmitting power limit is stated in terms of the transmitter output power, on 630m the maximum allowable power is stated in terms of the effective isotropic radiated power (EIRP) from the antenna. Which raises the question "how do we determine the radiated power (Pr)?" The standard broadcast approach has been to measure the E-field intensity at a point some distance from the antenna and calculate Pr from that. That's fine for the pros but most amateurs won't be able to use that approach. There are other ways we might go. If we can measure the current at the feedpoint (Io) and if we know the radiation resistance (Rr) we can find Pr = Io2 Rr. An alternative would be to measure the feedpoint resistance (Ri) and the input power (Pi), Pr=(Rr/Ri)Pi. We can measure quantities like Io, Pi and Ri but there is no way to measure Rr directly.

The traditional assumption has been that Rr for a vertical over real ground is equal to that for the same vertical over perfect ground. The value we measure for Ri is assumed to be the sum of the perfect ground Rr and additional loss terms due to the soil close to the antenna and other loss elements. For many simple cases and especially for short antennas (<λ/8), we can determine the ideal value of Rr either graphically or with a bit of algebra. Examples are given in appendix D. More complicated antennas can be easily modeled with software. A program like 4NEC2 (which is free[1]!) is well suited for this purpose. Finding the ideal value for Rr is straightforward.

I've used the idea that Rr in a real antenna equals the ideal value for a long time but over the years I've become a bit skeptical after seeing experimental and modeling results that didn't seem to fit. To keep down the clutter in the main text I've placed examples of this earlier work in appendix D. I've also placed almost all the mathematics and many supporting technical details in other appendices. This makes life much easier for the casual reader but provides the gory details for those who want them.

The key question for me has become: is the perfect ground value for Rr close enough to the lossy ground value to be used for calculating Pr? I think the question is worth some thought and that's what the following discussion is about.

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Equivalent circuit model

Figure 1 shows the traditional circuit model for the resistive part of an antenna's feedpoint impedance where Rr is the radiation resistance, Rg represents loss in the soil around the antenna and RL represents the sum of other losses such as conductor loss, matching network loss, etc. The input resistance at the feedpoint (Ri) is assumed to be the sum of these resistances, i.e. Ri = Rr + Rg + RL. Assuming the feedpoint rms current is Io then Pi = Io2 Ri, Pr = Io2 Rr, Pg= Io2 Rg, etc. For most of this discussion and the associated modeling I have assumed Io = 1 A rms so that Pi=Ri, Pr=Rr, etc.

Figure 1 - Typical model for the feedpoint resistance.

Determining PL is reasonably straightforward but Pg is trickier. In the following discussion I will be ignoring RL. This not because these losses are unimportant but the interest here is in determining Pr and the design of effective ground systems to reduce Pg. PL is a subject for another day.

What we're after is Pr:

Pr=R rPiRi

=Rr PiRr+Rg

=I o2R r

Rr for a lossless antenna

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We will need to be careful with our use of the term "radiation resistance". A definition for the Rr associated with a lossless antenna in free space, can be found in almost any book on antennas. A typical example is given in Terman[2]:

"The radiation resistance referred to a certain point in an antenna system is the resistance which, inserted at that point with the assumed current Io flowing, would dissipate the same energy as is actually radiated from the antenna system. Thus

Radiation resistance=radiated powerI o2

Although this radiation resistance is a purely fictitious quantity, the antenna acts as though such a resistance were present, because the loss of energy by radiation is equivalent to a like amount of energy dissipated in a resistance. It is necessary in defining radiation resistance to refer it to some particular point in the antenna system, since the resistance must be such that the square of the current times radiation resistance will equal the radiated power, and the current will be different at different points in the antenna. This point of reference is ordinarily taken as a current loop, although in the case of a vertical antenna with the lower end grounded, the grounded end is often used as a reference point."

An exposition of the lossless case for Rr is common but I have not see discussion of Rr associated with the ground loss close to the antenna where only the effect of the near-field losses are considered. Kraus[3] does tease us with a comment:

"The radiation resistance Rr is not associated with any resistance in the antenna proper but is a resistance coupled from the antenna and its environment to the antenna terminals."

The underline is my addition! The implication that the environment around the antenna plays a role is important but unfortunately Kraus does not seem to explore this point further.

Calculation of Rr and Rg

As pointed out earlier if you know Io and Pr you can calculate Rr. A standard way to calculate the total radiated power is to sum (integrate) the power density over a hypothetical closed surface surrounding the antenna. For lossless free space calculations the surface can be anywhere from right at the surface of the antenna to an enclosing sphere with a very large radius (large in terms of wavelengths). For a Pr calculation using E and H-field field intensities in equation form, a large radius has the advantage of reducing the equations to their far-field form which usually greatly simplifies the math. This is fine for lossless free space or over

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perfect ground where, as long as the surface is closed around the antenna, the distance doesn't matter. Near-field or far-field values will give the same answer. However, when we add a lossy ground surface in close proximity to the antenna then things get more complicated.

Take for example a vertical dipole with the bottom a short distance above lossy soil. You could create a closed surface which surrounds the antenna but does not intersect ground and then calculate the net power flow through that surface. When you do this you find that the Ri provided by NEC will be the same as the Rr calculated from the power passing through the surface. Technically this is Rr by the usual definition since the antenna is lossless as is the space within the enclosing surface, but that's not how we usually think of Rr. The conventional point of view is that the near-field of the antenna induces losses in the soil which we designate as Rg which are separate from Rr as indicated in figure 1. The power absorbed in the soil near the antenna is not considered to be "radiated" power although clearly it is being supplied from the antenna. When we run a model on NEC or make a direct measurement of the feedpoint impedance of the actual antenna, we get a value for Ri= Rr + Rg.

Can we separate Rr from Rg and if so, how? Assuming we're going to use NEC modeling, we could simply use the average gain calculation (Ga). The problem with Ga is that it includes all the ground losses, near and far-field, ground wave, reflections, etc. For verticals Ga gives a realistic if somewhat depressing estimate of the power radiated for skywave communications but the far-field loss is not normally included in Rg. Typically Rg represents only the losses due to the reactive near-field interaction with the soil. In the case of a λ/4 ground based vertical for example, that would be the ground losses out to ≈λ/2 (see appendix C). Instead of using Ga we can have NEC give us the E and H field intensities on the surface of a cylinder which protrudes into the soil a couple of skin depths as indicated in figure 2.

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Figure 2 - cylindrical surface enclosing a ground mounted vertical.

The power density is integrated over the surface of the cylinder and over the surface of the disc which forms the top of the cylinder. When you reach a depth in the soil of two skin depths the fields become very small and it is not necessary to integrate over a disc at the bottom of cylinder, although you could if you wished. Using the field intensities (amplitude and phase!) provided by NEC we can calculate the net power passing out of the cylinder which is Pr and we can find Rr from Rr = Pr/Io2. NEC also gives us Pi, so we can calculate Pg = Pi - Pr. From Pi, Pg and Io you can determine both Rr and Rg. Of course this is much more complicated that simply using Ga! But it turns out, if you're moderately clever in your choice of surface and field components, to be quite practical using a spreadsheet like EXCEL. The mathematical details are in appendix A. We could also use a hemisphere for the power integration surface but NEC gives the field values in Cartesian coordinates so the cylinder is a little simpler although you could convert the Cartesian coordinates to spherical if you wished.

Another alternative would be to calculate Pg directly by using a surface in the form of a disc centered on the antenna a small distance above the air/soil interface. In many cases this approach is simpler than using a cylinder and gives similar results. A few of the examples in the text were done this way.

Because the boundaries between the field regions are not sharply defined, the choice for the cylinder or disc radius is somewhat arbitrary. Details of the reactive near-field, Fresnel and Fraunhofer regions and the choice of integration surface radius are discussed in appendix C.

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Rr and Rg for a λ/2 vertical dipole

We'll start with the example of a resonant vertical λ/2 dipole like that shown in figure 3. The analysis was done at three frequencies, 475 kHz, 1.8 MHz and 7.2 MHz, over several different soils. Note that the three frequencies are a factor of ≈4X apart.

Figure 3 - Vertical dipole.

The bottom of the antenna was at either 1m (7.2 MHz) or 4m (1.8 MHz) or 16m (475 kHz) above ground. Table 1 shows the values for Ri from NEC over several different grounds.

Table 1 - Ri for resonant λ/2 dipoles.

soil Ri from NEC [Ω]7.2 MHz

Ri from NEC [Ω]1.8 MHz

Ri from NEC [Ω]475 kHz

Free space 72.29 72.29 72.290.001/5 85.74 90.10 92.84

0.005/13 90.46 92.95 93.690.03/20 93.31 93.83 93.89perfect 94.11 94.11 94.11

As we would expect, in free space Rr≈72Ω and over perfect ground Rr≈94Ω for all these antennas. Over real ground however, Ri differs with both soil characteristics and frequency.

One point is obvious:

Ri is not a combination of Rr over perfect ground and some Rg.

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The values we see for Ri over real soils are all lower than the perfect ground case and the values change as either frequency or soil characteristics are changed. As ground conductivity is increased Ri converges on the ideal ground case. Also, as we go down in frequency Ri converges on the perfect ground case. What's going on?

Figure 4 - example of an antenna image (from Kraus [3]).

As illustrated in figure 4, one way to analyze a vertical over ground is to use a hypothetical image. If the ground is perfect then the image antenna will be a duplicate of the actual antenna. For a dipole a short distance above ground the image is another dipole the same distance below ground. We now have a system of two coupled dipoles and it's no surprise that Ri of the upper dipole is no longer ≈72Ω but, for this particular case, Ri≈94Ω. What's happening is that the upper vertical (the real one) has a self resistance of ≈72Ω but added to that is a mutual resistance (Rm) coupled from the image antenna of ≈+22Ω. This mutual impedance will be a function of the separation distance between the two antennas (i.e. 2X the height above ground of the bottom of upper vertical).

However, if the ground is not perfect then the image antenna will not be a exact replica of the real antenna and we should not be surprised if Ri does not have the same value as either the free space or perfect ground cases. Information on the behavior of real soils with frequency is given in appendix B. Viewing Ri as a combination of the free space value and some mutual ±Rm due to the soil is perfectly valid and this is Wait's approach[4] where he calculates the ±ΔRi

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as the soil and/or radial fan is changed. But ±ΔRi is a combination of radiation and ground losses. Rr and Rg are not separated!

We can attempt to separate Rr from Ri by integrating the power lost in the soil near the antenna. The results from integrating the power lost in the soil out to a given radius are shown if figure 5. This integration gives us Rg which is then subtracted from Ri to yield Rr.

Figure 5 - Rr for λ/2 dipoles over average ground as a function of integration radius.

Figure 5 illustrates a problem: what radius do we use for the power integration? As we move out from the base of the vertical there will be more ground loss which is subtracted from Pi to yield progressively lower values for Rr. Which value do we use? No matter what radius is chosen it is clear that Rr is not the value over perfect ground! There is no sharp distinction between the reactive near-field and Fresnel regions leading to the ambiguity we see in figure 5. These boundary problems and the choice of integration radius are discussed in detail in appendix C and also in the next section on λ/4 verticals with radial ground systems.

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Rr and Rg for a λ/4 vertical at 7.2 MHz

The λ/4 vertical with a buried radial screen shown in figure 6 is more representative of typical amateur antennas for 40m and 160m. Amateurs are not likely to use a full λ/4 vertical on 630m which would be ≈500' high! We'll look at a more typical 630m antenna in a later section. I calculated data points for 16, 32 and 64 radials, with lengths of 2, 5, 10 and 16m over poor (0.001/5), average (0.005/13) and very good (0.03/20) soils.

Figure 6 - λ/4 vertical with a buried radial screen.

The first issue we have to settle is the radius of the cylinder over which we are integrating the power.

Figure 7 - Rr as a function of integration surface radius.

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Figure 7 shows the values we get for Rr as we vary the integration radius. The larger we make the radius the more power is lost in the soil within the cylinder so the value for Rr decreases with radius. Obviously the choice of radius is critical to determining Rr. Where is the boundary between the reactive near-field and the radiating Fresnel zones? One way to make a judgment is to look at the variation of the fields (Ez and Hy) near the antenna as the radius is varied. An example is given in figure 8.

Figure 8 - E and H field intensities as a function of distance.

In the far-field both Ez and Hy will decrease proportional to 1/radius. For Hy the decrease is 1/r everywhere but for Ez there is a clear break between 10 and 20m. 20m is ≈λ/2.

We can also look at the ratio Ez/Hy. For a plane wave in free space Ez/Hy = 376.8Ω. Figure 9 shows Ez/Hy for this example.

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Figure 9 - Ez/Hy as a function of distance.

We see that between 10 and 20m Ez/Hy starts converging on the free space value. We also know from experience with ground systems that radial lengths approaching λ/2 are very much in a region of vanishing returns. You gain very little additional efficiency from using radials longer than ≈0.4λ. From this observation and the information in figure 8 and 9 I have chosen to use a radius of 20m (0.48λ) for the 7.2 MHz vertical and 80m for the 1.8 MHz vertical Rr calculations.

Figure 10 is a typical graph showing the behavior of Ri, Rr and Rg as a function of radial length when 64 radials are employed over average ground.

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Figure 10 - Ri, Rr and Rg as a function of radial length.

There is a dashed line labeled "36Ω" which represents the value of Rr for a resonant λ/4 vertical over infinite perfect ground.

The curve for Ri replicates actual measurements taken on real antennas. The fact that Ri does not decrease or even flatten out for radial lengths >λ/4 but instead starts to increase has been predicted analytically (for example Wait[4]), my earlier NEC modeling (see appendix D) and seen in practice. The decrease in Rg with longer radials is also typical. What's interesting is that Rr36Ω. Rr starts out well below the value for an infinite perfect ground-plane but as the radial length is extended it does approach 36Ω. Increasing the radial number and/or extending the radial lengths also moves Rr closer to 36Ω.

Figure 10 represents only one case: 64 radials over average ground. Figure 11 gives a broader view of the behavior of Rr for different soils and radial numbers as radial length is varied.

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Figure 11 - Rr as a function of soil, radial number and radial length.

It is abundantly clear that Rr36Ω but as we improve the soil conductivity and/or increase the number and/or length of the radials Rr approaches the value for an infinite perfect ground.

We can also graph the values for Rg as shown in figure 12. This nicely illustrates how more numerous and longer radials reduce ground losses.

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Figure 12 - Rg as a function of radial length and number.

Figure 13 - Efficiency as a function of radial length and number.

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For a given model NEC will give us Ri, Ga and the field data from which we can determine Rr. With this information we can have some fun! Rr/Ri is the radiation efficiency including only the ground losses in the near-field. Ga is the radiation efficiency including all the losses, near and far-field. The ratio Ga/(Rr/Ri) will give us the loss in the far-field separate from the near-field losses. Figure 13 graphs all three, Ga, Rr/Ri and Ga/(Rr/Ri) for a 40m λ/4 vertical with various numbers of radials over average ground. Note that the far-field loss is almost independent of the radial number or radial lengths, which is what you would expect because we haven't changed anything in the far-field as we modified the radials. In fact any bumps or anomalies in that graph would indicate a screw-up in the calculations! It serves as a much needed cross check on the calculations.

Rr and Rg for λ/4 vertical at 1.8 MHz

By repeating the calculations for a λ/4 vertical at 1.8 MHz we can compare the results to expose the effect of frequency on Ri, Rr and Rg. An example is given in figure 14. The solid lines are for 1.8 MHz and the dashed lines 7.2 MHz.

Figure 11 - Ri, Rr and Rg for a λ/4 vertical with 64 radials at 1.8 and 7.2 MHz.

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What we see is that even though both antennas are λ/4, with the same length radials (in λ) and the same soil characteristic, the values for Ri, Rr and Rg are substantially different. At 1.8 MHz Rr is much closer to 36Ω. Using λ/4 radials at 7.2 MHz and integrating the radiated power, Rg≈8Ω. However, if you subtracted the Ri value given by NEC from 36Ω you would think Rg was essentially zero! At 1.8 MHz Rg=36-Ri≈2Ω, which seems like a reasonable value. However, the power integration for the 1.8 MHz vertical gives Rg≈6Ω which means the efficiency is lower than we thought. The soil characteristics which give rise behavior differences with frequency are discussed in appendix B.

As the soil conductivity (σ) increases, the values for Rr move closer to 36Ω and if we lower the frequency to towards the lower broadcast region (say 600 kHz) for a λ/4 vertical and add 120 0.4λ radials, Rr moves very close to 36Ω. This is a frequency range where a great deal of profession work has been done which might explain why the discrepancy between estimated and actual Rg and Rr went unnoticed. The difference was small, easily within the range of experimental error, at least for older instrumentation! This would be even more likely for the good soils typically chosen for commercial installations.

A check!

It's a good idea to check results using a different approach, i.e. calculating Rg directly without NEC, to see if we get the same answers. The values for Rr and Rg derived from integrating NEC near-field data look reasonable but are they? We can calculate Rg directly using the ground loss equations from Abbott[5], Wait[4] or Watt[6]. The equations are bit complicated but I happened to have Watt's equations already set up in an Excel spreadsheet so I could quickly run a calculation. Table 2 shows the result for f=7.2 MHz.

Table 2 - NEC-equation comparison

Radius of ground system Rg from Watt Rg from NEC, 64 radials5 [m] 18.1 Ω 16.1 Ω

10 [m] 9.3 Ω 8.4 Ω16 [m] 3.5 Ω 5.2 Ω

The calculation assumed an infinite number of radials and NEC used 64 radials. As we might expect for radial lengths out to 10m (0.25λ) the two are similar (within 10%) because the radials are quite dense. When we go to radial lengths of 16m (≈0.38λ) the ends of the 64

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radials are further apart so we would expect Rg to be higher. The correlation isn't perfect but it's reasonable.

A small 630m vertical

On 630m (472-479 kHz) any practical amateur antenna is very likely to be small in terms of wavelength. Figure 12 shows an example of a short top-loaded vertical for 630m.

Figure 12 - 630m antenna example.

The vertical is 15.24m high (50', 0.024λ) with 7.62m (25') radial arms in the hat. The usual practice for very short verticals is to have a dense ground system which extends some distance beyond the edge of the top-hat and/or radials a bit longer than the height of the vertical. Two cases were run: 64 and 128 radials, 18m long.

Figure 13 -Ez as a function of distance.

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Again we have the question of an appropriate radius for the power integration cylinder. Figure 13 is a graph of Ez as a function of radius. Close to the antenna the field intensity is very high and varies slowly with radius (≈1/r) but at about 20m the intensity starts falling off very rapidly (≈1/r3). This continues out to ≈100m where the rate of change goes back to 1/r corresponding to the far-field rate. We can also look at the change in Hy with radius as shown in figure 14.

Figure 14- Hz as a function of distance.

Again we see an increase in the rate of decrease (to≈1/r2) at ≈20m which returns to 1/r above 100m. For this antenna 100m appears to be approximately the boundary between reactive near-field and the radiating Fresnel zone and so I've chosen r=100m (0.16λ) for the power integration radius. The results obtained from the NEC field data are given in table 3.

Table 3A - 630m vertical 64 radials, integration radius =100m.

soil Ri [Ω] Rr [Ω] Rg [Ω] Rr/Ri 0.69/Ri Ga0.001/5 5.50 1.01 4.49 0.18 0.13 0.060

0.005/13 2.01 0.844 1.17 0.42 0.34 0.2320.03/20 1.09 0.76 0.32 0.70 0.63 0.533

perfect 0.69 0.69 0 1.00 1.00 1

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Table 3B - 630m vertical 128 radials, integration radius =100m.

soil Ri [Ω] Rr [Ω] Rg [Ω] Rr/Ri 0.69/Ri Ga0.001/5 4.90 1.009 3.895 0.21 0.14 0.067

0.005/13 1.883 0.843 1.04 0.45 0.37 0.2470.03/20 1.033 0.78 0.253 0.76 0.67 0.561perfect 0.69 0.69 0 1.00 1.00 1.00

For this antenna with real soils Rr is somewhat higher than the perfect ground case and converges on the perfect ground case as the soil conductivity improves. In this example using the perfect ground value for Rr yields an efficiency somewhat lower than actual as shown in the 0.69/Ri column, but the difference is not very large.

Summary

For a lossless antenna in a lossless environment, the calculation of radiation resistance Rr is very straight forward: integrate the power density over a closed surface enclosing the antenna. The net power outflow divided by the square of the rms current at the feedpoint gives Rr. The radius of the surface doesn't matter, you get the same answer at all radii. However, when the antenna is in a lossy environment, such as a vertical close to ground, the radius of the surface matters a great deal. You will get different values for Rr at different radii and the choice of radius will always be a compromise because the transitions from one zone to another are not sharply defined. In general as you go to larger radii you accumulate more loss (larger Rg!) and a lower value for Rr. Initially the losses are dominated by the reactive near-field but as you increase the radius the losses include ground reflections and attenuation of the ground wave. How you define Rr depends on your definition of what's loss and what's useful radiation.

The traditional view is that only the loss in the soil due to the reactive near-field close to the base of the antenna is considered to be ground loss Rg. Any power propagating beyond the near-field is classified as radiation. But even this definition does not give a unique value for Rr because the transition from the reactive near-field to the Fresnel zone is not sharply defined. If we accept that caveat we can still derive useful information on the efficiency of a vertical close to ground.

At HF, values for Rr over perfect ground appear to be significantly higher than the values for the same antennas over real soils, at least in the case of λ/4 verticals! However, for short

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verticals at MF, the real ground Rr appears to be close to the perfect ground value depending on the details of the soil and the ground system. It's my opinion that calculating Pr and efficiency using the perfect ground value for Rr is probably a good approximation for the very short verticals likely to be used by amateurs at 630m and down in frequency.

If you make a careful measurement of the ground characteristics at your site as suggested in my QEX article[7] and then, using these values, model the antenna and calculate Rr by integrating the near-field data, it may be possible to get a better estimate of efficiency. Realistically however, very few people will want to take the trouble.

The work reported here has been a delightful and informative exercise but I'm not deluded into thinking it's something many people will want to do.

References

[1] Ari Voors, www.qsl.net/4nec2

[2] Frederick Terman, Radio Engineers’ Handbook, McGraw-Hill, 1943

[3] John Kraus, Antennas, McGraw-Hill, 1988, second edition

[4] R. Collin and F. Zucker, Antenna Theory, Chap 23 by J. Wait, Inter-University Electronics Series (New York: McGraw-Hill, 1969), Vol 7, pp 414-424.

[5 ] F. R. Abbott, "Design of Optimum Buried-Conductor RF Ground System", IRE Proceedings, July 1952, pp. 846-852

[6] Arthur Watt, VLF Radio Engineering, Pergamon Press, 1967, see section 2.4

[7] Rudy Severns, N6LF, Measurement of Soil Electrical Parameters at HF, QEX magazine, Nov/Dec 2006, pp. 3-9

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