Math Bridge: Antenna Gain on a Remote Link

← Back to Infrastructure-Denied IoT Connectivity
Math BridgeWSNStruggle-friendly runway

Can focused gain rescue the 20 km link?

Follow one remote path from wavelength to loss, received power, and honest reserve.

Packet Pete, the guidePacket Pete guides
The one targetTurn antenna gain into a bounded link-margin decision.
The chapter case868 MHz over 20 km, compared with 2.4 GHz.
What it buys youA visible reserve before field losses.

A field team has a real problem to settle: Can focused gain rescue the 20 km link? They must decide what happens before they change gain on the device. Predict the direction first.

See the relationship first

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.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for gain is 6.02.

  2. 2

    Name the rule. λ=3.00x10⁸/(868x10⁶)=0.346 m L868=32.44+20log10(868)+20log10(20)≈117 dB L2400≈126 dB; penalty=8.83 dB 10^(6.02/20)=2.00x range; 10^(6.02/10)=4.00x concentration

  3. 3

    Put in the chapter value. Set gain to 6.02. The page ledger gives margin after reserve as 7.79 dB.

  4. 4

    Read the result. Keep dB beside the value. Use it only within the limits on this page.

Predict, then change gain

Try Predict what happens to margin after reserve. Move one control, calculate, then check your idea.

6.02
Chapter baseline
Margin after reserve

Observe Predict an increase in margin after reserve as the input increases over this range. Reset to the chapter baseline and compare.

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

Source: content/wsn/wsn-infrastructure-denied-radio-options.qmd — Phoebe's Field Notes: Why Antenna Gain Buys Range by Spending Coverage. Use the chapter equation and fixed ledger above; only the named input varies.

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. Gain moves power; it does not create it

Antenna gain focuses the same transmitted power into fewer directions. That can raise received power toward a fixed gateway, but aiming error and blocked coverage become part of the design.

Packet Pete: Keep frequency loss, antenna focus, and fade reserve in separate rows.

2. Name every algebra move

1

Convert frequency to wavelengthλ=c/f.

2

Add the free-space termsLfs=32.44+20log10(fMHz)+20log10(dkm).

3

Balance the dB ledgerPrx=Ptx+Gtx+Grx−Lfs−Lcable−Lmisc.

4

Compare with sensitivitymargin=Prx−Srx−reserve.

5

Translate gain into ideal geometryrange ratio=10^(G/20); concentration=10^(G/10).

3. Reproduce the chapter case

λ=3.00×10⁸/(868×10⁶)=0.346 m
L868=32.44+20log10(868)+20log10(20)≈117 dB
L2400≈126 dB; penalty=8.83 dB
10^(6.02/20)=2.00× range; 10^(6.02/10)=4.00× concentration

The ideal beam shrinks from 4π to about π steradians. That is a geometry result, not a field acceptance.

4. Try the directional gain

TryMove the gain and watch both the dB ledger and ideal geometry change.

Gain
868 MHz wavelength
868 MHz path loss
2.4 GHz path loss
Frequency penalty
Received power
Raw margin
Margin after reserve
Ideal range ratio
Power concentration
Ideal beam solid angle

ObserveAt 6.02 dB, ideal range doubles and power concentration is fourfold, while the frequency penalty stays 8.83 dB.

ExplainGain changes the antenna term. It cannot erase the path's frequency, distance, cable, terrain, or installation terms.

Technical boundaries.

This is a free-space teaching ledger with fixed catalog-style radio inputs.

Propagation
Terrain, foliage, Fresnel blockage, diffraction, weather, and multipath are omitted
Antenna
Efficiency, pattern, polarization, height, feedline, and aiming require measurement
Receiver
Sensitivity depends on bandwidth, coding, data rate, noise, and implementation

Use a site survey and measured fade distribution before accepting the link.

5. Read reserve as a requirement

A positive arithmetic margin is only the amount left after the terms entered. Add a named reserve for seasonal and installation uncertainty, then test whether field evidence spends it.

6. Build the field record

Record frequency, distance, conducted power, antenna patterns and mounting, cable loss, receiver mode, measured RSSI/SNR, weather, obstructions, packet delivery, reserve, owner, and retest trigger.

7. Check yourself

Why is 2.4 GHz about 8.83 dB harder at the same distance?
Answer: Free-space loss contains 20log10(f); 20log10(2400/868)=8.83 dB.
Why does 6.02 dB double ideal range?
Answer: Range follows 10^(G/20), so 10^(6.02/20)=2.00.
Does a positive calculated margin prove the remote link?
Answer: No. The simplified ledger omits the field losses and variability that must be measured.
Honesty boundary.

The frequencies, distance, wavelength, and free-space comparisons come from the chapter; radio budget inputs are explicit teaching assumptions.

117/126 dB
Rounded free-space losses
2.00×
Ideal gain-for-range exchange
Margin
Not a deployment acceptance without measured losses

Go deeper in the chapter's link-budget and field-validation sections.

Return to Infrastructure-Denied IoT Connectivity with the maths unpacked →