Math Bridge: Wi-Fi Gain, Beamwidth, and Range

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Math BridgeWi-FiStruggle-friendly runway

Why does antenna gain buy range by spending coverage angle?

Follow gain through EIRP and free-space distance, then expose the narrower beam that paid for it.

Eddie, the electronics guideEddie guides
The one targetConnect dBi to both idealised range and the beamwidth trade.
The chapter case20 dBm, 3 dBi versus 8 dBi, 2400 MHz, and a −67 dBm target.
What it buys youA reason to measure installed coverage instead of treating gain as free power.

A field team has a real problem to settle: Why does antenna gain buy range by spending coverage angle? They must decide what happens before they change gain on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is gain. The middle card uses this page's rule. The green card is eirp. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

Gain changes eirp An input card leads through the page rule to the eirp result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Gain concentrates the fixed transmit power, so the favoured direction gains link margin while other angles lose coverage.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for gain is 8.

  2. 2

    Name the rule. 20 + 8 = 28 dBm EIRP 28 - (-67) = 95 dB allowable loss d = 10^((95 - 20log10(2400) - 32.44)/20) = 0.559 km 10^((8-3)/20) = 1.778 range ratio sqrt(41,253 / 10^(8/10)) = 80.9°

  3. 3

    Put in the chapter value. Set gain to 8. The page rule gives eirp as 28.0 dBm.

  4. 4

    Read the result. Keep dBm next to the value. Use it only within the limits on this page.

Predict, then change gain

Try Predict what happens to eirp. Move one control, calculate, then check your idea.

8
Chapter baseline
EIRP

Observe Gain concentrates the fixed transmit power, so the favoured direction gains link margin while other angles lose coverage. Reset to 8 and compare eirp.

Explain Only gain moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?
Answer: Predict its direction, use the shown rule, 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 limits stay fixed.

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

1. Start with the physical story

An antenna does not create energy. It redirects the same input power, increasing power density in one direction by reducing it elsewhere. Range can grow inside the main beam while angular coverage shrinks.

Eddie: Read gain twice: once in the link budget and once in the beam shape.

2. Name every algebra move

1

Add decibelsEIRP in dBm equals transmit power plus antenna gain in dBi.

2

Find allowable lossSubtract the negative receive target from EIRP.

3

Undo path lossRearrange FSPL and raise ten to the resulting exponent to find kilometres.

4

Compare rangesA gain change of ΔG dB changes free-space range by 10^(ΔG/20).

5

Expose the tradeFor a symmetric idealised beam, beamwidth is the square root of 41,253 divided by linear gain.

3. Reproduce the chapter case

20 + 8 = 28 dBm EIRP
28 − (−67) = 95 dB allowable loss
d = 10^((95 − 20log10(2400) − 32.44)/20) = 0.559 km
10^((8−3)/20) = 1.778 range ratio
sqrt(41,253 / 10^(8/10)) = 80.9°

The idealised 8 dBi sector reaches 1.78 times farther than the 3 dBi reference, but its symmetric beam estimate narrows from about 144° to 81°.

4. Try one real input

TryMove antenna gain above and below 8 dBi. Watch range and beamwidth move in opposite design directions.

Gain
EIRP
Allowable path loss
Ideal distance
Linear gain
Range versus 3 dBi
Symmetric beamwidth
Beamwidth versus 3 dBi

ObserveAt 8 dBi, the idealised distance is 559 m and the symmetric beam is 80.9°. More gain raises the first value and lowers the second.

ExplainGain concentrates the fixed transmit power, so the favoured direction gains link margin while other angles lose coverage.

Technical boundaries.

This is an idealised free-space and symmetric-beam comparison.

Environment
Walls, floors, people, multipath, noise, and regulatory EIRP limits are omitted.
Antenna
The beamwidth-product approximation does not reproduce a real pattern, sidelobes, or efficiency.
Capacity
Meeting −67 dBm says nothing about contention, airtime demand, or client density.

Correct, not complete: this ledger does not design an installed WLAN.

5. Use the result in the design

Use the calculation to form a survey hypothesis. Then measure both the intended beam and the areas that lose coverage, while checking EIRP limits, client capacity, interference, and roaming.

6. Record the evidence state

Record radio power, antenna model and orientation, gain pattern, cable loss, EIRP, frequency, receive target, floor plan, measured RSSI/SNR, client load, and the retest trigger.

7. Check yourself

Why does a 5 dB gain increase produce only 1.78 times the range?
Answer: Received power falls with distance squared, so range uses 10^(ΔG/20), not 10^(ΔG/10).
What paid for the higher signal in the favoured direction?
Answer: Reduced radiation into other angles; the antenna redirected input power rather than creating it.
Does 559 m predict an indoor installed range?
Answer: No. It is a free-space comparison before walls, bodies, multipath, noise, regulations, and capacity.
Honesty boundary.

This is an idealised free-space and symmetric-beam comparison.

Environment
Walls, floors, people, multipath, noise, and regulatory EIRP limits are omitted.
Antenna
The beamwidth-product approximation does not reproduce a real pattern, sidelobes, or efficiency.
Capacity
Meeting −67 dBm says nothing about contention, airtime demand, or client density.

Correct, not complete: this ledger does not design an installed WLAN.

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