16.4 Effective Radiated Power & Antenna Bandwidth

16.4 Effective Radiated Power & Antenna Bandwidth

Quick Answer: ERP = transmitter power adjusted by antenna gain and line/duplexer losses using dB: (P_{\mathrm{ERP}} = P_{\mathrm{TX}} \times 10^{(\mathrm{net\ dB})/10}). EIRP uses an isotropic reference (dBi); classic ERP often uses a dipole reference. Antenna bandwidth = frequency range of good performance. Series L lowers resonance (electrically longer); series C raises it; loading coils that shorten a whip also narrow bandwidth.

Key topics 3-J-065 (Frequency and Bandwidth) and 3-J-067 (Effective Radiated Power) finish Topic 3-J. One half is resonance craft; the other is the ERP arithmetic that shows up repeatedly on Element 3—and on real repeater and base-station paperwork.

Effective radiated power (ERP)

ERP answers: “How strong is the station in free space compared with a reference antenna fed with a known power?” You start with transmitter power output (TPO), subtract real losses, and add antenna gain.

dB method (preferred)

1. Convert every gain and loss to dB (antenna gain positive; feedline, duplexers, circulators, filters as negative losses).
2. Sum the net:

[ \mathrm{net\ dB} = G_{\mathrm{ant(dB)}} - L_{\mathrm{feed(dB)}} - L_{\mathrm{duplex/circ(dB)}} - \cdots ]

1. Apply to TPO:

[ P_{\mathrm{ERP}} = P_{\mathrm{TX}} \times 10^{(\mathrm{net\ dB})/10} ]

Linear method (same math)

Convert each dB figure to a power ratio (10^{dB/10}), multiply gains, divide by loss ratios:

[ P_{\mathrm{ERP}} = P_{\mathrm{TX}} \times g_{\mathrm{ant}} \times \ell_{\mathrm{feed}} \times \ell_{\mathrm{duplex}} ]

where each (\ell < 1) for losses.

Worked pool examples (memorize the pattern)

Example A: 50 W TX, 4 dB feedline loss, 3 dB duplexer/circulator loss, 6 dB antenna gain:

[ \mathrm{net} = 6 - 4 - 3 = -1,\mathrm{dB} ]

[ P_{\mathrm{ERP}} = 50 \times 10^{-0.1} \approx 50 \times 0.794 = \mathbf{39.7\ W} ]

Example B: 75 W, −4 dB line, −3 dB duplex, +10 dB antenna → net +3 dB → (75 \times 2 \approx \mathbf{150\ W}).

Example C: 75 W, −5, −4, +6 → net −3 dB → (75 \times 0.5 = \mathbf{37.5\ ≈ 37.6\ W}).

Example D: 100 W, −4, −3, +7 → net 0 dB → 100 W ERP.

Example E: 100 W, −5, −4, +10 → net +1 dB → (100 \times 1.26 \approx \mathbf{126\ W}).

Example F: 50 W, −5, −4, +7 → net −2 dB → (50 \times 0.631 \approx \mathbf{31.5\ W}).

ERP vs EIRP

Because a dipole has about 2.15 dBi gain:

[ \mathrm{EIRP_{dBW}} \approx \mathrm{ERP_{dBW}} + 2.15 ]

or in linear power, EIRP ≈ 1.64 × ERP when both describe the same physical field and the ERP reference is a dipole. Always read the regulation or datasheet: mixing ERP and EIRP without the 2.15 dB bridge is a classic paperwork error. Element 3’s 3-J-067 items are labeled effective radiated power and use the dB sum method above—drill those six numerical patterns until they are automatic.

Where the power path sits

[ \textbf{TX} \xrightarrow{L_{\mathrm{jumper}}} \textbf{duplexer/filter} \xrightarrow{L_{\mathrm{feed}}} \textbf{antenna} \xrightarrow{G_{\mathrm{ant}}} \textbf{free space (ERP)} ]

Section 16.3’s 6 W at the end of a lossy run is exactly the (P_{\mathrm{TX}}\times\ell_{\mathrm{feed}}) factor before antenna gain multiplies what is left. High antenna gain cannot fully rescue terrible feedline loss—and a high-gain antenna with a failed jumper still yields poor ERP.

Antenna bandwidth and Q

Antenna bandwidth (pool definition): the frequency range over which an antenna can be expected to perform well. “Perform well” usually means SWR under a limit (e.g., 2:1), gain and pattern still useful, and efficiency acceptable.

Bandwidth is not:

• Beamwidth (an angle on the pattern),
• Element length divided by number of elements,
• The geometric angle between element tips.

Resonance shifts with L and C

A Hertz (half-wave) antenna’s resonant frequency can be lowered by placing an inductance in series with the antenna—series L adds electrical length. Conversely, to lengthen an antenna electrically, add a coil (loading inductor).

To increase the resonant frequency of a λ/4 antenna, add a capacitor in series—series C makes the antenna electrically shorter, raising fr. (Physically cutting the element also raises resonance; the pool’s component answer is series C for the “increase fr” item.)

Loading coils and bandwidth

Mobile and marine whips are often physically shorter than λ/4 with a loading coil restoring resonance. What happens to bandwidth as the antenna is shortened through loading coils? It is decreased. Electrically shortened, high-Q antennas are “peaky”: SWR rises quickly as you move off the design frequency. That is why a short loaded HF whip may need a coupler for every channel, while a full-size dipole covers more of a band without retuning.

Rough intuition:

[ Q \approx \frac{f_0}{BW} \quad \Rightarrow \quad BW \approx \frac{f_0}{Q} ]

Loading that raises Q (more reactance, less radiation resistance relative to stored energy) narrows BW.

Pattern reminder tied to frequency use

A vertical λ/4 still receives equally from all horizontal directions across its usable band—bandwidth limits match and efficiency, not the basic omnidirectional idea. Stay inside the antenna’s rated frequency range; far outside resonance you get high SWR, low efficiency, and possibly transmitter foldback long before the pattern “changes modes.”

Service scenarios that combine ERP and bandwidth

1. VHF marine base: 25 W radio, 1.5 dB jumper+feed loss, 3 dBi whip → net ≈ +1.5 dB → ERP ≈ 35 W class—verify after installation with a wattmeter at the antenna if possible.
2. Repeater: Large duplexers add 2–4 dB loss each way; antenna gain must overcome that or ERP falls below TPO (Examples A and F).
3. HF loaded whip: Coil makes the 24 ft stick resonate on 2–3 MHz, but bandwidth is narrow—retune coupler on every frequency change (Element 1 + Element 3 together).
4. Harmonic check: After ERP is set, confirm harmonic attenuation (Section 16.2) so legal and clean radiation accompanies the intended ERP.

Exam-day ERP & bandwidth checklist (3-J-065, 3-J-067)

1. ERP: net dB = Gant − Lfeed − Lduplex…; (P \times 10^{\mathrm{net}/10}).
2. Drill anchors: 39.7 W, 150 W, 37.6 W, 100 W, 126 W, 31.5 W pool family.
3. EIRP ↔ isotropic; ERP often dipole-referenced; ~2.15 dB bridge.
4. Bandwidth = frequency range of good performance.
5. Series L → lower fr / longer electrically; series C → higher fr.
6. Loading coils on shortened antennas → decreased bandwidth.
7. λ/4 vertical → omnidirectional horizontally across its operating band.
8. Excited λ/2 → both electric and magnetic field components in the radiated wave.

Topic 3-J is complete when you can describe the radiator, sketch V/I on a half-wave, budget the feed line, compute ERP, and predict how coils and capacitors move resonance and bandwidth. Those skills transfer directly to aircraft antennas, radar arrays, and marine VHF plants in later Element 3 topics.