Math Bridge: Antenna gain and harvested energy

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Math BridgePrototypingStruggle-friendly runway

How can an antenna halve a harvester's radio recovery time?

Keep target EIRP fixed and follow antenna gain through conducted power, one transmit burst, and the energy a small harvester must replace.

Voltage Vera, the prototyping guideVoltage Vera guides
The one targetJoin the wireless and energy benches with one ledger.
The chapter case14 dBm EIRP, 60.0 ms airtime, and a 0.100 mW harvester.
What it buys youSee antenna choice as part of the harvested-energy cadence.

A field team faces an unresolved physical question: How can an antenna halve a harvester's radio recovery time? They must answer it before changing antenna 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 antenna gain. The middle card applies this page's relationship. The green card is radio setting. 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.

Antenna gain changes radio setting An input card leads through the page relationship to the radio setting result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. A passive RF choice changes the energy cadence because the harvester replaces joules, not link-budget labels.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for antenna gain is 3.

  2. 2

    Name the relationship. All-radio case: 14.0 dBm=25.1 mW With 3.00 dBi gain: Ptx=14.0-3.00=11.0 dBm=12.6 mW Eall=25.1x0.0600=1.51 mJ Egain=12.6x0.0600=0.755 mJ Recharge=15.1 s versus 7.55 s at 0.100 mW

  3. 3

    Substitute the chapter fixture. Set antenna gain to 3. The page ledger gives radio setting as 11.00 dBm.

  4. 4

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

Predict, then change antenna gain

Try Predict the direction of radio setting. Move one control, calculate, then check your prediction.

3
Chapter baseline
Radio setting

Observe A passive RF choice changes the energy cadence because the harvester replaces joules, not link-budget labels. Reset the control to 3 and compare radio setting.

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

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

1. Hold the link target still

EIRP combines the radio setting with antenna gain. If the target EIRP stays fixed, more antenna gain lets the radio conduct less power. The harvester pays for conducted power over time, not for dBm as a label.

Voltage Vera: Subtract gain in decibels first; convert the remaining radio setting to milliwatts second.

2. Name every algebra move

1

Keep EIRP fixedPtx(dBm)=EIRP−G.

2

Undo dBmPtx(mW)=10^(Ptx/10).

3

Pay for airtimeEtx=PtxTtx; mW×s=mJ.

4

Replace the bursttrecharge=Etx/Pharvest.

5

Compare honestlyUse the same target EIRP and airtime in both cases.

3. Reproduce the 3 dBi case

All-radio case: 14.0 dBm=25.1 mW
With 3.00 dBi gain: Ptx=14.0−3.00=11.0 dBm=12.6 mW
Eall=25.1×0.0600=1.51 mJ
Egain=12.6×0.0600=0.755 mJ
Recharge=15.1 s versus 7.55 s at 0.100 mW

The gain case uses 49.9% less ideal conducted RF power at the same target EIRP, so its transmit-only energy and recovery time are also about half.

4. Try the antenna gain

TryChange antenna gain while target EIRP, airtime, and harvested power stay fixed.

Antenna gain
Radio setting
Conducted power
0 dBi power
Power saving
Burst energy
0 dBi burst
Recharge time
0 dBi recharge

ObserveEach 3 dB of ideal antenna gain roughly halves conducted power and transmit-only recharge time.

ExplainA passive RF choice changes the energy cadence because the harvester replaces joules, not link-budget labels.

Technical boundaries.

This is a transmit-RF energy ledger, not a complete radio or harvester model.

Radio
DC input includes PA efficiency, synthesizer, processor, startup, receive, and sleep energy
Antenna
Gain helps only in its pattern direction and can be lost to mismatch, mounting, or cable
Harvester
Input power and conversion efficiency vary with light, source voltage, storage state, and temperature

Measure the complete current waveform, installed radiation behavior, and harvested input across the intended environment.

5. Test the closed energy cycle

Measure the radio's full current trace for both antenna configurations at equal delivered-link performance. Then measure storage voltage and harvested input until the node returns to the same starting energy state.

6. Record the evidence state

Store target EIRP, radio setting, antenna gain and pattern, mismatch and cable loss, payload, data rate, airtime, current trace, harvester input and efficiency, storage voltage, environment, and recovery criterion.

7. Check yourself

Why does 3 dBi reduce a 14 dBm setting to 11 dBm?
Answer: At fixed EIRP, Ptx=EIRP−G=14−3=11 dBm.
Why is 11 dBm about 12.6 mW?
Answer: Undo dBm with 10^(11/10)=12.59 mW.
Does 7.55 s predict the complete node recovery time?
Answer: No. It replaces ideal transmit RF energy only and omits radio inefficiency and all other loads.
Honesty boundary.

The arithmetic reproduces the chapter's illustrative EIRP, gain, 60.0 ms burst, and 0.100 mW harvest cases.

49.9%
Ideal conducted RF saving, not complete node-current saving
0.755 mJ
RF power times airtime, before PA and platform overhead
7.55 s
Ideal replacement time at constant harvested power

Correct, not complete: this ledger does not prove a harvested node's energy neutrality or link coverage.

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