Math Bridge: Domain Radio Requirement Ledger

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Math BridgeApplicationsStruggle-friendly runway

How can the weaker radio keep more margin and run for years?

Separate the decibel comparison from the reporting-duty calculation.

Radio Remi, the radio guideRadio Remi guides
The one targetTurn a radio requirement into margin and current budgets.
The chapter case150 mW at 2.4 GHz versus 35 mW at 915 MHz.
What it buys youAn honest link advantage and service-life screen.

A technician must decide whether radio duty cycle is safe before changing report interval on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is report interval. The middle card applies this page's rule. The green card is radio duty cycle. 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 model keeps those stated values fixed and changes only report interval, so the numeric fixture does not switch without explanation.

Report interval changes radio duty cycle An input card leads through the rule duty = 0.2 s x 100 / (interval x 60 s/min) to the radio duty cycle result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Longer report intervals reduce the share of time spent active.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 10 min.

  2. 2

    Name the relationship. duty = 0.2 s x 100 / (interval x 60 s/min)

  3. 3

    Substitute with units. 0.333333 / 10 = 0.0333%

  4. 4

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

Predict, then change report interval

Try Predict the direction of duty = 0.2 s x 100 / (interval x 60 s/min). Test another report interval, then compare radio duty cycle.

10 min
Chapter baseline
Radio duty cycle

Observe Longer report intervals reduce the share of time spent active. Reset report interval to 10 and compare radio duty cycle.

Explain Longer report intervals reduce the share of time spent active.

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 report interval moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only report interval. It is not a field prediction.

1. Start with the physical story

Lower transmit power does not automatically mean shorter range. The lower carrier loses less energy at the same distance, while sparse reports spend the high radio current for only a tiny fraction of time.

Radio Remi: Keep decibels and duty fraction in separate ledgers before joining the decision.

2. Name every algebra move

1

Log each powerConvert milliwatts to dBm with 10 log10(P).

2

Compare bandsUse 20 log10(2400/915) for the same-distance path-loss advantage.

3

Net the termsSubtract the transmit-power penalty from the frequency advantage.

4

Find dutyDivide 0.2 seconds by the selected report interval.

5

Average currentWeight active and sleep currents by their time shares.

6

Derate chargeCompound retention, reserve 20%, then divide charge by current.

3. Reproduce the chapter case

PWiFi=10log10(150)=21.76 dBm; PLoRa=10log10(35)=15.44 dBm
ΔL=20log10(2400/915)=8.38 dB
ΔM=8.38−6.32=2.06 dB
D=0.2/(10×60)=0.000333=0.0333%
Iavg=D×32.4 mA+(1−D)×0.0015 mA=12.3 µA
Qusable=1200×0.993⁸×0.8=907 mAh; life=8.42 years

The lower band recovers more link margin than the weaker transmitter gives up, and the real duty fraction—not a friendly 1%—makes multi-year service plausible.

4. Try one real input

TryMove the report interval. Predict which values stay fixed and which service-life terms move.

Report interval
Wi-Fi power
LoRa power
Band advantage
Power penalty
Net margin gain
Active duty
Active current
Average current
Usable charge
Ideal life
Derated life

ObserveChanging interval moves duty and lifetime but cannot change the 2.06 dB radio comparison.

ExplainFrequency and power set link margin; firmware scheduling sets average current. One technology choice contains both requirements.

Technical boundaries.

This is a same-distance free-space comparison and fixed-state current screen.

Channel
Terrain, foliage, installation, antennas, interference, receiver sensitivity, and required fade reserve are omitted.
Traffic
Join, acknowledgements, retries, receive windows, sensing, processing, and exceptional reports add current.
Cell
Temperature, pulse resistance, cutoff voltage, ageing, and measured retention determine usable charge.

Correct, not complete: this ledger does not prove LoRaWAN coverage or an eight-year field deployment.

5. Use the result in the design

Write separate acceptance tests for installed link reserve, measured report current, reporting policy, and worst-case cell behavior.

6. Record the evidence state

Keep carrier, conducted power, antenna, site loss, receiver sensitivity, payload, interval, retry count, current trace, firmware, cell lot, temperature, and cutoff.

7. Check yourself

Why can 35 mW beat 150 mW here?
Answer: The 915 MHz band saves 8.38 dB while the power cut costs 6.32 dB, leaving 2.06 dB net.
Why is one report every ten minutes not 1% active?
Answer: 0.2 seconds divided by 600 seconds is 0.000333, or 0.0333%.
Does 8.42 years prove field life?
Answer: No. It omits installed traffic, temperature, pulse delivery, ageing scatter, and maintenance conditions.
Honesty boundary.

The arithmetic reproduces the chapter's 150 mW, 35 mW, 2.4 GHz, 915 MHz, and ten-minute screening case.

Channel
Terrain, foliage, installation, antennas, interference, receiver sensitivity, and required fade reserve are omitted.
Traffic
Join, acknowledgements, retries, receive windows, sensing, processing, and exceptional reports add current.
Cell
Temperature, pulse resistance, cutoff voltage, ageing, and measured retention determine usable charge.

Correct, not complete: this ledger does not prove LoRaWAN coverage or an eight-year field deployment.

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