Math Bridge: Sector Antenna Directivity

A field team faces an unresolved physical question: Where does a macro sector's 19.6 dBi come from? They must answer it before changing sector horizontal beamwidth on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is sector horizontal beamwidth. The middle card applies this page's relationship. The green card is beam-area proxy. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Walk the arrows. Narrowing the horizontal beam reduces the angular area and raises the on-axis ratio. The gain is useful only where the engineered sector points; the device stays broadly oriented so it can be installed or moved.

Derive the baseline in four named moves

1
Name the input. The chapter baseline for sector horizontal beamwidth is 65.
2
Name the relationship. D≈41253/(θHθV); GdBi=10log10(D)
3
Substitute the chapter fixture. Set sector horizontal beamwidth to 65. The page ledger gives beam-area proxy as 455 degrees squared.
4
Read the result. Keep degrees squared beside the value. Use it only inside the technical boundary on this page.

Predict, then change sector horizontal beamwidth

Try Predict the direction of beam-area proxy. Move one control, calculate, then check your prediction.

Observe Narrowing the horizontal beam reduces the angular area and raises the on-axis ratio. The gain is useful only where the engineered sector points; the device stays broadly oriented so it can be installed or moved. Reset the control to 65 and compare beam-area proxy.

Explain Only sector horizontal beamwidth moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only sector horizontal beamwidth moves. Field effects named in the page's technical boundary stay fixed.

1. Conserve the radiated power

An isotropic reference spreads power in all directions. A sector aims much of that power into a narrow region. Power on the beam axis rises. Other directions lose power.

Radio Remi: Gain redirects power; it does not create power.

2. Name the algebra moves

Approximate beam areaAbeam=θHθV in square degrees.

Form a direction ratioD≈41253/Abeam.

Convert ratio to dBGdBi=10log10(D).

Keep the device separateEIRPUE=23 dBm+0 dBi≈23 dBm.

3. Read the angular trade

65×7=455 deg²; 41253/455=90.7; 10log10(90.7)=19.6 dBi

The 455 deg² rectangle is about 1.10% of 41,253 deg². This teaching estimate links a narrow serving wedge with a large on-axis ratio.

4. Try one controlled change

D≈41253/(θHθV); GdBi=10log10(D)

TryChange only horizontal half-power beamwidth. The 7° vertical beam, Kraus constant, and 23 dBm near-isotropic user equipment stay fixed.

ObserveAt 65° by 7°, the proxy beam area is 455 deg², 1.10% of all directions. The ratio is 90.7× and the Kraus estimate is 19.6 dBi. A 23 dBm, 0 dBi UE is about 200 mW EIRP.

ExplainNarrowing the horizontal beam reduces the angular area and raises the on-axis ratio. The gain is useful only where the engineered sector points; the device stays broadly oriented so it can be installed or moved.

Technical boundaries.

The Kraus beamwidth formula is a rough directivity estimate, not an antenna pattern or operator specification.

Beam rectangle: Real main lobes are not rectangular and have sidelobes and nulls
Gain: Efficiency, mismatch, cable loss, downtilt, polarization, and pattern shape alter realized gain
Service: Serving sector, terrain, clutter, load, interference, handover, and network policy remain

Use the actual antenna pattern, installation loss, and measured service evidence.

5. Reproduce the chapter values

For 65°×7°, Abeam=455 deg². D≈41253/455=90.7 and G=10log10(90.7)=19.6 dBi. The direction fraction is 455/41253=1.10%. On the UE side, 23 dBm+0 dBi=23 dBm and 10^(23/10)=200 mW.

6. Carry the evidence forward

Record device conducted power and installed gain, base-station antenna model, horizontal and vertical patterns, downtilt, feeder loss, serving sector identity, band, RSRP/SINR, handover, enhancement mode, and installation orientation.

7. Check yourself

Does 19.6 dBi mean the antenna creates 90.7 times more power?

Answer: No. It redirects available power into a narrower angular region, increasing on-axis density.

Why is a 0 dBi device antenna still useful?

Answer: Broad coverage supports arbitrary device orientation; a narrow high-gain beam could point away from the serving cell.

Is 455/41253 an exact coverage fraction?

Answer: No. It is a rectangular beam-area proxy used by the approximate formula, not a measured 3D pattern.

Honesty boundary.

The page connects beamwidth and directivity without pretending that two beamwidth numbers describe a deployed sector.

19.6 dBi: Kraus estimate from illustrative 65° and 7° values
1.10%: Angular rectangle proxy, not service coverage
200 mW: 23 dBm conducted-power conversion before device losses

Coverage approval requires actual patterns, losses, serving-cell data, radio measurements, and network behavior.