Math Bridge: Antenna Gain and Beamwidth

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Math BridgeOne learning thread

Why antenna gain narrows coverage

Follow a fixed transmit-power budget from a sphere to the chapter’s 6 dBi rack-aisle patch, including EIRP, on-axis density, and beamwidth.

Phoebe guides this bridge
One targetConnect gain to its coverage cost.
Chapter caseTwenty sensors, 17 dBm radio, 20 dBm cap.
What it buys youTest both battery and coverage gates.

See the relationship before changing it

Set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only antenna gain, so the numeric fixture does not switch without explanation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 6 dBi.

  2. 2

    Name the relationship. linear gain = 10^(gain in dBi / 10)

  3. 3

    Substitute with units. 10^(6 / 10) = 3.98 times

  4. 4

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

Predict, then change antenna gain

Try Predict the direction of linear gain = 10^(gain in dBi / 10). Test another antenna gain, then compare linear antenna gain.

6 dBi
Chapter baseline
Linear antenna gain

Observe More directional gain concentrates ideal radiated power into less solid angle. Reset antenna gain to 6 and compare linear antenna gain.

Explain Decibels encode multiplicative directional power ratios, so increasing antenna gain by equal decibel steps multiplies linear gain rather than adding equal power increments.

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 antenna gain moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only antenna gain. It is not a field prediction.

1. Begin with power spread over area

An isotropic reference spreads transmit power Pt equally over a sphere. At distance d, that sphere has area 4πd².

Siso = Pt / (4πd²)

Phoebe: A directional antenna does not create energy. It moves energy away from some directions and into others.

2. Turn concentration into gain

A full sphere contains 4π steradians. If the useful lobe occupies solid angle Ω, the concentration ratio is:

G = 4π/Ω    and    Sdir = G Siso
1

Compare areasFull-sphere angle ÷ lobe angle gives the dimensionless gain G.

2

Multiply density by gainInside the lobe, the same Pt produces G times the isotropic density.

3

Convert ratio to decibelsG(dBi) = 10log10(G).

3. Respect the 20 dBm EIRP cap

EIRP(dBm) = Pt(dBm) + G(dBi)
1

Baseline17 dBm + 0 dBi = 17 dBm EIRP, leaving 3 dB below the chapter’s cap.

2

Make conducted power the subjectPt = EIRP − G.

3

Substitute the patch gainPt = 20 − 6 = 14 dBm.

4

Find the power ratio10−3/10 = 0.501× the conducted power.

4. Find the on-axis payoff

1

Find the EIRP change20 − 17 = 3 dB.

2

Convert decibels to ratio103/10 = 2.00×.

The on-axis power density doubles even though conducted power is roughly halved. The chapter calls this a 2 dB range win; the exact range change still depends on propagation and receiver thresholds.

5. Calculate the beamwidth bill

The chapter uses the Kraus estimate G ≈ 41253/(θEθH) with angles in degrees.

1

Convert 6 dBi to linear gainG = 106/10 = 100.6 = 3.98.

2

Cross-multiplyθEθH = 41253/G.

3

Substitute and divide41253/3.98 = 10362 deg².

4

Assume a symmetric beamθ² = 10362.

5

Take the square rootθ = √10362 ≈ 102°, or about ±51° from boresight.

6. Check yourself

1. With 6 dBi gain and a 20 dBm cap, what conducted power is allowed?

14 dBm, because 20 − 6 = 14.

2. What does a 3 dB on-axis rise mean as a ratio?

10^(3/10) = 2.00× power density.

3. What coverage warning follows from the 102° estimate?

Sensors more than about 51° off boresight may sit beyond the half-power lobe, so placement must be measured.

7. Honesty boundary

The gain conversion uses an ideal antenna that redistributes power without loss; the beamwidth estimate assumes a simple main lobe. Sidelobes, efficiency, polarisation, reflections, cable loss, mounting, regional rules, and whole-device battery current need separate evidence.

The isotropic radiator is a reference
Ideal equal-direction radiation reference
not a physical antenna
Physical-antenna limitation of isotropic reference
The Kraus relation is an estimate for a simple main lobe
Main-lobe approximation used for beamwidth
sidelobes
Unmodelled radiation outside main lobe
efficiency
Unmodelled antenna efficiency
polarization
Unmodelled polarisation alignment
reflections
Unmodelled reflected propagation
cable loss
Unmodelled feed-cable attenuation
mounting
Unmodelled installation effects
the exact regulatory domain matter
Region-specific radio limits require checking
Conducted RF power is not automatically proportional to whole-device battery current
Radio output power does not determine total current

Go deeper in Compare Finalists and Validate, then use a site survey and production antenna measurements.