Math Bridge: Spend Antenna Gain

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Math BridgeWSNStruggle-friendly runway

Should the gateway spend 7 dB on range or battery?

Follow the same directional gain through two different engineering choices.

Packet Pete, the guidePacket Pete guides
The one targetConnect antenna gain to leaf-node power and runtime.
The chapter case2.15 to 9.15 dBi; 10 to 3 dBm.
What it buys youA visible choice between reach and energy.

A field team has a real problem to settle: Should the gateway spend 7 dB on range or battery? They must decide what happens before they change directional gain on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is directional gain. The middle card uses this page's rule. The green card is gain available. 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.

Directional gain changes gain available An input card leads through the page rule to the gain available result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Range uses the square-root dB relation. The battery path first converts dBm to power, then power to PA current, then duty-weights current.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for directional gain is 9.15.

  2. 2

    Name the rule. ΔG=9.15-2.15=7.00 dB; range=2.24x 10.0-7.00=3.00 dBm: 10.0 mW becomes 2.00 mW IPA=8.66 mA becomes 1.73 mA 1% TX ledger: 455 days becomes 5.13 years, a 4.11x extension

  3. 3

    Put in the chapter value. Set directional gain to 9.15. The page rule gives gain available as 7.00 dB.

  4. 4

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

Predict, then change directional gain

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

9.15
Chapter baseline
Gain available

Observe Range uses the square-root dB relation. The battery path first converts dBm to power, then power to PA current, then duty-weights current. Reset to 9.15 and compare gain available.

Explain Only directional 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 directional gain moves. Field effects named in the page limits stay fixed.

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

1. One gain can fund two budgets

A directional gateway can keep leaf transmit power fixed and increase ideal range, or keep the link budget fixed and let the leaf reduce transmit power.

Packet Pete: Do not claim both benefits in full from the same 7 dB.

2. Name every algebra move

1

Subtract antenna gainsΔG=Gdir−Gomni.

2

Spend on rangerange ratio=10^(ΔG/20).

3

Or spend on powerPnew,dBm=Pold,dBm−ΔG.

4

Convert dBm to milliwatts and currentPmW=10^(PdBm/10); I=P/(ηV).

5

Weight transmit and sleep, then divideIavg=DITX+(1−D)Isleep; t=C/Iavg.

3. Reproduce the chapter trade

ΔG=9.15−2.15=7.00 dB; range=2.24×
10.0−7.00=3.00 dBm: 10.0 mW becomes 2.00 mW
IPA=8.66 mA becomes 1.73 mA
1% TX ledger: 455 days becomes 5.13 years, a 4.11× extension

The battery comparison holds coverage and link margin fixed by spending the gain on lower transmit power.

4. Try the gateway gain

TryMove directional gain and watch both possible purchases change.

Directional gain
Gain available
Ideal range purchase
Reduced TX setting
Base RF power
Reduced RF power
Base PA current
Reduced PA current
Base average current
Reduced average current
Base life
Reduced-power life
Runtime ratio

ObserveAt 9.15 dBi, 7.00 dB can buy 2.24× ideal range or reduce the leaf setting to 3.00 dBm and extend the bounded ledger by 4.11×.

ExplainRange uses the square-root dB relation. The battery path first converts dBm to power, then power to PA current, then duty-weights current.

Technical boundaries.

This trade holds link budget, efficiency, duty cycle, sleep current, and capacity fixed.

Radio
Real PA current is not a single constant-efficiency curve
Traffic
Receive, processing, startup, retries, and acknowledgements are omitted
Antenna
Pattern, orientation, polarization, feedline, and field loss can spend the gain

Measure current states and link margin at the chosen transmit setting.

5. State which benefit you took

A design can divide gain between more range, lower transmit power, and more fade reserve. Record that allocation rather than quoting every maximum simultaneously.

6. Build the trade record

Record antenna patterns and mounting, leaf TX setting, PA current curve, duty states, sleep current, capacity and cutoff, measured margin, required coverage, allocation decision, owner, and retest trigger.

7. Check yourself

Why does 7.00 dB give 2.24× ideal range?
Answer: Range ratio is 10^(7/20)=2.24 under free-space spreading.
Why can TX fall from 10 to 3 dBm?
Answer: The extra 7 dB gateway gain can replace 7 dB of conducted leaf power if the rest of the link is fixed.
Does 5.13 years predict a real leaf node?
Answer: No. It is a PA-plus-sleep teaching ledger that omits the rest of the device and battery behaviour.
Honesty boundary.

All antenna, radio, efficiency, duty, sleep, and capacity values are the chapter's explicit catalog-typical example.

2.24×
Ideal range alternative
5.13 years
Bounded lower-power ledger
4.11×
Comparison within the same simplified model

Correct, not complete: field measurements decide how the gain can be spent.

Return to WSN Sensor Nodes with the maths unpacked →