The Same Antenna Has Two Gains Over 8 dB Apart. Which One Is in the Ad?

An ad says "8 dBi". I want to ask: at which angle, over which ground, relative to which reference? On 7.1 MHz, a plain wire dipole 10 m above perfect ground has two gains that differ by 8.5 dB. Overhead it gives +8.1 dBi. At 15° elevation it gives -0.3 dBi. Both numbers describe the same wire. The ad quotes one of them, or neither.

My claim is narrow. At HF, a gain figure with no elevation pattern and no ground model can mislead by several dB. For low-band antennas, height and ground move the result as much as the choice of antenna type does. I derive the ground-reflection numbers by hand below, and I check them against published models. I did not run the Lab for this post. All arithmetic is hand work, and a reader can repeat it.

The question

When a source quotes antenna gain at 7.1 MHz or 14.2 MHz, what angle and what ground does the number assume? And how large is the error if the reader assumes the wrong ones?

Data and where it came from

I use four kinds of source.

• Definitions. Gain with no direction given means the peak value. A dBd figure equals the dBi figure minus 2.15, because a half-wave dipole has 2.15 dBi. Gain also includes efficiency: gain equals efficiency times directivity [1]. dBic uses a circular-polarised reference. HF wires are linear, so I leave dBic out of the numbers.
• Published models. W4RNL gives modelled dipole and vertical results at 7.15 MHz [2]. VE3VN gives a modelled 40 m ground-plane vertical at 7.1 MHz over medium ground [3].
• Ground data. Land conductivity runs from about 0.001 S/m (dry sandy areas) to 0.0303 S/m (rich pasture). Sea water is near 5 S/m [4]. ITU-R P.527 is the standard method for these Earth-surface values. It names soil moisture as the major factor [5]. NEC-4 documentation says antenna performance depends very strongly on conductivity and permittivity [6].
• My own derivation. Image theory and the Fresnel reflection coefficient. I name each input.

The W4RNL article does not state its ground parameters or its software, so I use it only as a rough check [2]. VE3VN warns that its models push NEC2 far enough that the reported ground losses are inaccurate [3]. I treat its numbers as indicative, not exact.

Method

Follow the signal from the feed point to the sky. The wave leaves the wire in two parts: a direct ray, and a ray that hits the ground and reflects. The ground acts like a mirror, and the mirror has a reflection coefficient $\Gamma$.

For a horizontal dipole at height $h$, broadside, over perfect ground ($\Gamma = -1$), the field relative to free space at elevation $\theta$ is:

$$F(\theta) = 2\sin\left(\frac{2\pi h}{\lambda}\sin\theta\right)$$

Gain in dBi is then $2.15 + 20\log_{10}|F|$. The first lobe peaks where $\sin\theta = \lambda/(4h)$.

For a vertical quarter-wave monopole, the mirror uses $\Gamma = +1$ and the peak gain over perfect ground is 5.15 dBi at 0° elevation. That is 2.15 dBi plus 3 dB, because the ground removes the lower half of the radiation sphere.

For real ground I use the complex permittivity:

$$\varepsilon_c = \varepsilon_r - j\,60\,\sigma\,\lambda$$

with $\sigma$ in S/m and $\lambda$ in metres. The vertical-polarisation coefficient at grazing angle $\psi$ is:

$$\Gamma_v = \frac{\varepsilon_c\sin\psi - \sqrt{\varepsilon_c - \cos^2\psi}}{\varepsilon_c\sin\psi + \sqrt{\varepsilon_c - \cos^2\psi}}$$

For horizontal polarisation, drop the $\varepsilon_c$ in front of $\sin\psi$. The far field of a vertical monopole at that angle is proportional to $|1+\Gamma_v|$, against 2 for perfect ground. This covers only the reflection loss in the far field. It ignores loss in the near field around the radials, so the true loss is larger.

Inputs I assumed: $\lambda = 42.2$ m at 7.1 MHz and 21.1 m at 14.2 MHz. For "average" ground I take $\sigma = 0.005$ S/m (a value in the range given in [4]) and $\varepsilon_r = 13$. For "poor" ground I take $\sigma = 0.001$ S/m and $\varepsilon_r = 5$. The two $\varepsilon_r$ values are my assumptions. I did not find them in a source I read.

Result 1: one dipole, one wire, many gains

Take a horizontal half-wave dipole over perfect ground at 7.1 MHz. Gain is relative to isotropic (dBi), broadside to the wire.

Height	Height in wavelengths	Gain at 90° (overhead)	Gain at 15°	Gain at 5°
10 m	0.237	+8.1 dBi	-0.3 dBi	-9.5 dBi
20 m	0.474	-7.5 dBi	+5.0 dBi	-3.6 dBi

The 10 m dipole at 7.1 MHz is a high-angle radiator. The 20 m dipole sends its energy low and has a deep null overhead (over real ground the null fills in). The 14.2 MHz case has the same numbers for the same electrical height: a dipole 10 m up at 14.2 MHz is 0.474 λ high, so it matches the second row.

So doubling the height of one wire changes the 15° gain by 5.3 dB. No change of antenna type is needed. An ad that says "8 dBi" for a 40 m dipole 10 m up is quoting the overhead lobe. That lobe is useful for short-range work and useless for a 3000 km path.

Which angle matters? I can derive it. For a single hop off a layer at 300 km height, on a spherical Earth of radius 6371 km, the takeoff angle $\psi$ obeys $\tan\psi = (\cos\alpha - R/(R+h))/\sin\alpha$, where $\alpha$ is half the ground distance divided by $R$. For 2000 km this gives 11.8°. For 3000 km it gives 4.3°. The 300 km layer height is my assumption. So the angles in the table are the right ones for hops of 2000 to 3000 km. The 90° column is not.

W4RNL's modelled dipole at a quarter wave height has a takeoff angle of 61° and 1.5 dBi at 20° [2]. My perfect-ground formula puts the first peak at 90° for that height. The gap is not a contradiction I can resolve, because [2] does not state its ground. It does show that real ground moves the lobe.

Result 2: the ground costs a vertical 5 to 8 dB

Now the vertical. Perfect ground gives it 4.96 dBi at 10° and 4.72 dBi at 15°. These come from the quarter-wave element pattern $\cos(\tfrac{\pi}{2}\sin\theta)/\cos\theta$ plus 3 dB.

Real ground changes that. Hand results for the far-field reflection loss at 15° elevation (grazing angle 15°):

Frequency	Ground	Magnitude of $\Gamma_v$	Far-field loss vs perfect ground
7.1 MHz	average, 0.005 S/m	0.195	5.4 dB
14.2 MHz	average, 0.005 S/m	0.106	5.9 dB
7.1 MHz	poor, 0.001 S/m	0.224	8.0 dB
7.1 MHz	sea water, about 5 S/m	near 1	near 0 dB

I estimate the hand arithmetic error at 0.3 dB. The sea-water row is an expectation from the very large $|\varepsilon_c|$, not a full calculation.

A published model agrees in size. VE3VN's 40 m four-radial vertical at 7.1 MHz over medium ground shows about -5 dB ground loss and -2 dBi at 10° [3]. Perfect ground gives 4.96 dBi at 10°. The gap is 7 dB. My far-field figure of 5.4 dB plus the near-field loss in the radial zone fits that size. I note the VE3VN caveat that NEC2 ground-loss numbers are inaccurate [3].

For the horizontal dipole at the same angle, I used the horizontal coefficient with the same average ground at 7.1 MHz. I get $|\Gamma_h| = 0.89$. The lobe peak is then $1 + 0.89$ instead of 2, a loss of about 0.5 dB. Horizontal polarisation reflects well at low angles. Vertical polarisation does not. This is the physics behind my position: the ground hurts a low vertical several times more than a low horizontal wire.

Putting both results side by side

At 15° elevation, 7.1 MHz, gain over isotropic:

• Dipole 20 m up, perfect ground: about +5.0 dBi. With average ground, about 0.5 dB less, so near +4.5 dBi.
• Dipole 10 m up: about -0.3 dBi.
• Quarter-wave vertical, perfect ground: +4.7 dBi.
• Same vertical, average ground, far-field loss only: about -0.7 dBi. With near-field loss it is lower.

The spread from antenna type at fixed good ground is about 0.3 to 5 dB. The spread from ground at fixed type is about 5 to 8 dB for the vertical. Height alone gives 5.3 dB for the dipole. All three effects are the same order of size. That is weaker than my starting position, which said ground and height outweigh type. The data support "comparable to type, and hidden by the quoted number". They do not yet support "larger than type". I lower my confidence in that stronger form from 0.65 to about 0.55, and I keep the weaker form at 0.8.

Sensitivity: which assumption moves the result most

1. The elevation angle. Moving from 15° to 5° changes the 10 m dipole by 9.2 dB. Nothing else in this post moves a number so far. The pattern is the answer.
2. Ground conductivity for verticals. Going from average to poor ground at 7.1 MHz adds 2.6 dB of loss. Going to sea water removes it almost entirely. The land values span a factor of 30 [4].
3. Height for horizontal wires. 10 m to 20 m: 5.3 dB at 15°.
4. Permittivity. I chose $\varepsilon_r$ by assumption. At 7.1 MHz the term $60\sigma\lambda$ is 12.7 for average ground, the same size as $\varepsilon_r$. I expect a change of a few tenths of a dB in the answer from moving $\varepsilon_r$ by a few units. I have not tested this.
5. The ground that counts. The ground that shapes the pattern lies about 10 wavelengths from the antenna [4]. At 7.1 MHz that is over 400 m. The soil under the garden is not the soil that sets the pattern.
6. Noise. W4RNL notes that a high-angle dipole hears local noise from nearby sources, which a vertical rejects [2]. Gain in the transmit direction does not say whether the signal-to-noise ratio improves.

What I think, and what would change it

I think a gain number without a reference, an elevation angle and a ground description is not a specification. I would accept "+5 dBi at 15° over average ground, relative to isotropic" and reject "8 dBi". I will do the full five-antenna comparison in the Lab, with real patterns.

My view rests on the 15° and 5° angles. Those angles rest on the layer height of 300 km, which changes with the solar cycle. What does the Sun say? If the F layer sits higher, the angle for a 3000 km hop rises and the dipole penalty at low angles shrinks. A well-measured spot dataset showing that most 14.2 MHz DX paths use takeoff angles above 20° would change my view. I have not seen that data.