Math Bridge: Deep-Indoor Battery Bound

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Math BridgeCellular IoTStruggle-friendly runway

How can 3 dB of installation loss shorten a meter's battery estimate?

Follow one ideal margin penalty through repetitions, radio-on time, average current, and service-life bounds.

Radio Remi, the guideRadio Remi guides
The one targetTurn an installation-loss assumption into an explicit battery bound.
The chapter case220 mA active, 5 µA sleep, 2400 mAh, one report per day.
What it buys youSee why an installed current trace outranks a spreadsheet promise.

A field team faces an unresolved physical question: How can 3 dB of installation loss shorten a meter's battery estimate? They must answer it before changing deep indoor installation loss 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 deep indoor installation loss. The middle card applies this page's relationship. The green card is radio-active time. 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.

Deep indoor installation loss changes radio-active time An input card leads through the page relationship to the radio-active time result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The battery effect comes from extra radio-on time, not from the dB number alone. The model makes that causal chain visible so the release test can measure every term.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for deep indoor installation loss is 3.

  2. 2

    Name the relationship. N=10^(ΔM/10); Iavg=Isleep(1-d)+Iactive d; tlife=Q/Iavg

  3. 3

    Substitute the chapter fixture. Set deep indoor installation loss to 3. The page ledger gives radio-active time as 4.99 s.

  4. 4

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

Predict, then change deep indoor installation loss

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

3
Chapter baseline
Radio-active time

Observe The battery effect comes from extra radio-on time, not from the dB number alone. The model makes that causal chain visible so the release test can measure every term. Reset the control to 3 and compare radio-active time.

Explain Only deep indoor installation loss 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 deep indoor installation loss moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only deep indoor installation loss. It replays the chapter fixture; it is not a field prediction.

1. Keep margin and energy linked

A weak installed link may need more radio work. In a simple ideal model, each 3 dB margin deficit doubles repetitions. Attach time grows, so average current grows. Battery life then falls.

Radio Remi: This is a bound for asking better test questions, not a modem forecast.

2. Name the algebra moves

1

Convert margin to repeatsN=10^(ΔM/10).

2

Find active timeTactive=N×Tattach+Ttransfer.

3

Find duty fractiond=Tactive/Treport.

4

Average and divideIavg=Isleep(1−d)+Iactive d; tlife=Q/Iavg.

3. Build the once-daily case

ΔM=3 dB ⇒ N=2.00; Tactive=2×2+1=5.00 s; Iavg=17.7 µA

With no deficit, the 2 s attach plus 1 s transfer gives 12.6 µA. A 2400 mAh ideal division gives 21.7 years. The 3 dB case gives about 15.5 years.

4. Try one controlled change

N=10^(ΔM/10); Iavg=Isleep(1−d)+Iactive d; tlife=Q/Iavg

TryChange only the ideal installation-loss penalty. The current levels, attach and transfer times, daily interval, and capacity stay fixed.

Ideal repetitions
Radio-active time
Duty fraction
Average current
Ideal life bound
Zero-loss life bound
Life reduction

ObserveAt 3.00 dB, ideal repeats are 2.00×, active time is 4.99 s, average current is 17.71 µA, and the ideal life bound is about 15.5 years. The zero-loss bound is about 21.7 years.

ExplainThe battery effect comes from extra radio-on time, not from the dB number alone. The model makes that causal chain visible so the release test can measure every term.

Technical boundaries.

Ideal coherent repetition is not a modem state machine or field-current trace.

Radio time
Search, synchronization, attach failures, retries, paging, and network timers can dominate
Cell
Ageing, pulse capability, self-discharge, temperature, cutoff, and reserve reduce usable capacity
Margin
Antenna loss does not map one-to-one to a supported repetition setting

Measure installed current across representative weak and strong sites.

5. Reproduce the chapter values

At zero loss, active time is 3 s/day, average current is 12.6 µA, and the ideal life is 21.7 years. At 3 dB, N=1.995, active time is 4.99 s/day, current is 17.7 µA, and life is 15.5 years: a 1.40× reduction.

6. Carry the evidence forward

Record enclosure and antenna, RSRP and SINR, attach-search time, retry count, transfer time, PSM current, pulse current, cell voltage and temperature, report interval, cutoff, reserve, and repeated current traces.

7. Check yourself

Why does 3 dB give about two ideal repetitions?
Answer: Because 10^(3/10) is about 2.
Does 2400 mAh divided by average current predict warranty life?
Answer: No. It is an ideal capacity bound before ageing, temperature, pulse delivery, reserve, and field behaviour.
What should the spreadsheet trigger?
Answer: An installed current-trace plan that measures the assumed states and timings.
Honesty boundary.

The page connects stated assumptions without claiming that ideal repetition predicts a deployed modem.

2.00×
Ideal energy-combining ratio for 3 dB
15.5 years
Capacity-division bound
17.7 µA
Average of two stated current states

Deployment approval requires installed RF and current evidence plus a cell model.

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