Math Bridge: LoRaWAN Gateway Gain and Coverage

← Back to LoRaWAN Network Architecture
Math BridgeLoRaWANGateway pattern

Why does a gateway trade hearing directions for antenna gain?

Start with dBi and carry it through solid angle, EIRP, and an honest range screen.

Eddie, the electronics guideEddie guides
The one targetExplain what a 3-to-12 dBi antenna swap changes.
The chapter case13.95 dBm conducted power, a 3 dBi omni, and a 12 dBi directional comparison.
What it buys youA gateway plan that separates boresight gain from campus-wide hearing.

A field team faces an unresolved physical question: Why does a gateway trade hearing directions for antenna gain? They must answer it before changing gain 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 gain. The middle card applies this page's relationship. The green card is linear gain. 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.

Gain changes linear gain An input card leads through the page relationship to the linear gain result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The same gain that strengthens one direction removes hearing elsewhere; three campus gateways address geometry that one pointed antenna cannot.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for gain is 12.

  2. 2

    Name the relationship. Glinear = 10^(GdBi/10) ohm = 4π/Glinear EIRP = 13.95 dBm + GdBi range ratio = 10^((GdBi - 3 dBi)/20)

  3. 3

    Substitute the chapter fixture. Set gain to 12. The page ledger gives linear gain as 15.85 times.

  4. 4

    Read the result. Keep times beside the value. Use it only inside the technical boundary on this page.

Predict, then change gain

Try Predict the direction of linear gain. Move one control, calculate, then check your prediction.

12
Chapter baseline
Linear gain

Observe The same gain that strengthens one direction removes hearing elsewhere; three campus gateways address geometry that one pointed antenna cannot. Reset the control to 12 and compare linear gain.

Explain Only gain 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 gain moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only gain. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

An antenna does not create radio power. It reshapes a fixed response: more power density in favoured directions means less response elsewhere. A gateway that must hear scattered devices therefore values coverage shape as much as headline gain.

Eddie: Think of squeezing the same blanket over a smaller patch. The covered patch gets thicker, but the blanket does not grow.

2. Name every algebra move

1

Undo dBTurn dBi into linear gain with 10^(G/10).

2

Find the favoured angleDivide the sphere, 4π steradians, by linear gain.

3

Close EIRPAdd antenna dBi to conducted dBm.

4

Compare antennasUse the gain difference for coverage and ideal range ratios.

3. Reproduce the chapter case

Glinear = 10^(GdBi/10)
Ω = 4π/Glinear
EIRP = 13.95 dBm + GdBi
range ratio = 10^((GdBi − 3 dBi)/20)

At 3 dBi, gain is 2.00× and the favoured solid angle is 6.30 sr. At 12 dBi, gain is 15.85× and the angle is 0.79 sr: 7.94 times narrower than the omni screen.

4. Try one real input

TryMove directional gain while conducted power and the 3 dBi omni reference stay fixed.

Gain
Linear gain
EIRP
Favoured solid angle
Sphere share
Gain over omni
Coverage shrink
Ideal range ratio

ObserveAt 12 dBi the boresight screen gains 9 dB and 2.82× ideal range, while its favoured solid angle is only one eighth of the 3 dBi screen.

ExplainThe same gain that strengthens one direction removes hearing elsewhere; three campus gateways address geometry that one pointed antenna cannot.

Technical boundaries.

This is an ideal pattern ledger, not a gateway approval.

Pattern
Real antennas have elevation shape, nulls, sidelobes, mounting loss, and polarization effects.
EIRP
Use regional limits and the installed conducted-power and cable-loss record.
Hearing
Gateway diversity needs measured uplink records from the actual device group.

Correct, not complete: use the measured state named above before release.

5. Use the result in the architecture review

Keep the omni when the gateway must hear devices around it. Consider directionality only when the target sector, mounting, overlap, and loss of other directions are explicit.

6. Record the evidence state

Record conducted power, antenna model and pattern, gain, cable loss, height, polarization, azimuth, gateway overlap, representative device positions, RSSI, SNR, and deduplication evidence.

7. Check yourself

Does 12 dBi create extra transmitter power?
Answer: No. It concentrates the existing response into fewer directions.
Why is the coverage shrink 7.94× from 3 to 12 dBi?
Answer: Nine decibels is a 7.94× linear-gain ratio, and solid angle is inversely proportional to gain.
Does the 2.82× ratio prove campus range?
Answer: No. It is a boresight free-space screen; installed patterns and field hearing still decide.
Honesty boundary.

The bridge keeps ideal pattern arithmetic separate from installed gateway evidence.

Computed
Linear gain, EIRP, solid angle, sphere share, coverage shrink, and ideal range ratio.
Specified
Conducted power, reference gain, candidate gain, mounting, and regulatory profile.
Observed
Installed pattern, hearing diversity, RSSI, SNR, packet delivery, and blind directions.

Correct, not complete: this page does not approve an antenna, gateway layout, coverage, regulation, or deployment.

Return to LoRaWAN Network Architecture with gain and hearing shape kept together →