The reference antennas
What is an isotropic radiator?
A hypothetical, lossless antenna having equal radiation intensity in all directions used as a reference for antenna gain
Hypothetical — it cannot exist, which is exactly why it is useful as a reference. Every direction equal, no losses, a perfect sphere of radiation.
Two references, and the 2.15 dB between them:
How much gain does an antenna have compared to a half-wavelength dipole if it has 6 dB gain over an isotropic radiator?
3.85 dB.
6 − 2.15 = 3.85 dBd. The dipole itself has 2.15 dBi, so referencing to it always reads 2.15 dB lower.
Where 2.15 comes from — one integral, and 73 ohms falls out of it too
The 2.15 is not a convention or a measurement. It is the directivity of a half-wave dipole, and it is computable in closed form.
A half-wave dipole carries a standing current . Integrating that current distribution against the free-space Green’s function gives a far-field pattern
Directivity is peak radiation intensity over the average, so with normalised to 1 at :
= \frac{2}{\displaystyle\int_0^{\pi} F^2(\theta)\,\sin\theta\,d\theta}$$ That integral has a name. It is $\tfrac{1}{2}\mathrm{Cin}(2\pi)$, where $\mathrm{Cin}$ is the entire cosine integral, and it evaluates to $$\int_0^{\pi} F^2 \sin\theta \; d\theta = 1.21883 \quad\Longrightarrow\quad D = \frac{2}{1.21883} = 1.6409$$ $$G_{dBi} = 10\log_{10}(1.6409) = \mathbf{2.15\ dBi}$$ **And the same integral gives the feed impedance.** Radiation resistance is defined by $P_{rad} = \tfrac{1}{2}I_0^2 R_r$, and working it through: $$R_r = \frac{\eta_0}{2\pi}\int_0^{\pi} F^2\sin\theta\,d\theta = \frac{376.73}{2\pi}(1.21883) = \mathbf{73.1\ \Omega}$$ So the two numbers every antenna discussion opens with — *2.15 dB of gain* and *73 ohms* — are the same integral read two ways. Neither is a rule of thumb. Two things worth taking from this. First, the 2.15 dB is **small**: a dipole is barely directive, and its "gain" is almost entirely the absence of radiation along the wire. Second, 73 Ω is why 50 Ω cable is a *compromise* rather than a match — the mismatch is 73/50 = 1.46:1, which is why a dipole fed with 50 Ω coax reads about 1.5:1 at resonance and that is **correct**, not a fault. *Both figures, and the closed form, are recomputed in `scripts/tests/test_rf_numbers.py` on every test run.* </details> ## ERP and EIRP > **What term describing total radiated power takes into account all gains and > losses?** > **Effective radiated power.** $$P_{ERP} = P_{tx} \times 10^{(G - L)/10}$$ Sum the losses in decibels, subtract from the gain, apply the net as a power ratio. **ERP uses dBd; EIRP uses dBi** — and the question tells you which by naming the gain's reference. **Three worked answers, exactly as the pool asks them:** **150 W, 2 dB feed line, 2.2 dB duplexer, 7 dBd antenna.** Losses 4.2 dB; net 7 − 4.2 = **+2.8 dB**. 150 × 10^0.28 = 150 × 1.905 = **286 watts ERP**. **200 W, 4 dB feed line, 3.2 dB duplexer, 0.8 dB circulator, 10 dBd antenna.** Losses 8.0 dB; net **+2 dB**. 200 × 10^0.2 = 200 × 1.585 = **317 watts ERP**. **200 W, 2 dB feed line, 2.8 dB duplexer, 1.2 dB circulator, 7 dBi antenna.** Losses 6.0 dB; net **+1 dB**. 200 × 10^0.1 = 200 × 1.259 = **252 watts EIRP**. The method never changes. Add up losses, subtract from gain, one power ratio. The only thing to watch is that **every** loss goes in — repeater questions deliberately list three. ## Efficiency > **What is antenna efficiency?** > **Radiation resistance divided by total resistance.** $$\eta = \frac{R_{radiation}}{R_{total}}$$ An antenna's feed point resistance has two parts: **radiation resistance**, which represents power that leaves as radio waves, and **loss resistance**, which represents power that becomes heat. Both look identical to the transmitter. This is the number that explains the loaded mobile antenna. A short whip may have 2 ohms of radiation resistance and 8 ohms of loss — 20% efficient, and 80% of your power warming a coil, while the SWR meter reads a contented 1:1. > **Which of the following improves the efficiency of a ground-mounted quarter-wave > vertical antenna?** > **Installing a ground radial system.** > **Which of the following determines ground losses for a ground-mounted vertical > antenna operating on HF?** > **Soil conductivity** A ground-mounted vertical returns its current through the earth, and earth is a poor conductor. Radials give that current a metal path instead, cutting the loss resistance and raising efficiency. **Soil conductivity** is why the same antenna performs differently in salt marsh and in dry sand — and it is not something you can change, which is why you add radials instead. ## Ground gain > **What does the term "ground gain" mean?** > **An increase in signal strength from ground reflections in the environment of the > antenna** The ground reflects, and at low elevation angles the reflected wave can arrive in phase with the direct one — adding up to 6 dB over free space in that direction. It is genuinely free gain, and it is why antenna **height** is the single most effective HF investment: height sets the elevation angle at which the reflection reinforces, and low angles are where DX is. > **Which of the following factors affect the feed point impedance of an antenna?** > **Antenna height.** The same coupling to ground that produces ground gain also changes feed point impedance — which is the General-level observation that a dipole's impedance falls as you lower it, now with the mechanism. ## Two more terms > **What is the difference in radiated power between a lossless antenna with gain and > an isotropic radiator driven by the same power?** > **They are the same** A lossless antenna radiates **all** the power it is given, exactly as an isotropic radiator would. Gain redistributes that power; it does not create any. This is the "gain is not amplification" point from the Technician track, stated as a total-power identity. > **Which frequency band has the smallest first Fresnel zone?** > **5.8 GHz** — of the choices offered. The Fresnel zone is the ellipsoidal region around a line-of-sight path that must be kept clear of obstructions. Its radius shrinks with wavelength, so the highest frequency has the smallest zone — which is why microwave links tolerate a narrower cleared path than VHF ones. ## Check yourself 1. A repeater: 100 W out, 3 dB feed line loss, 1 dB duplexer loss, 6 dBd antenna. ERP? 2. Your vertical has 3 ohms radiation resistance and 12 ohms of loss. Efficiency? 3. An antenna is quoted at 9 dBi. What is that in dBd, and which figure would a manufacturer prefer? <details> <summary>Answers</summary> 1. Losses 4 dB, net +2 dB. 100 × 10^0.2 = **158 watts ERP**. 2. 3 / (3 + 12) = **20%**. Four-fifths of your power becomes heat. 3. 9 − 2.15 = **6.85 dBd**. The manufacturer prefers **dBi**, because it is the bigger number for the same antenna. </details>