Math Bridge: Zigbee Router Battery Physics

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Math BridgeZigbee, Thread and MatterStruggle-friendly runway

Why can one Zigbee role drain a coin cell in 9.17 hours?

Turn active time, cycle time, active current, sleep current, derating, and self-discharge into an honest runtime comparison.

Radio Remi, the guideRadio Remi guides
The one targetConnect Zigbee device role to average current and runtime.
The chapter case24 mA awake, 1 µA asleep, 5 ms every 8 s, 220 mAh.
What it buys youReject a battery router before field failure makes the lesson expensive.

A field team has a real problem to settle: Why can one Zigbee role drain a coin cell in 9.17 hours? They must decide what happens before they change sleepy-device poll interval on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is sleepy-device poll interval. The middle card uses this page's rule. The green card is active duty. 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.

Sleepy-device poll interval changes active duty An input card leads through the page rule to the active duty result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Lengthening the poll interval shrinks the active fraction. The router cannot take that saving because listening is its network job.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for sleepy-device poll interval is 8.

  2. 2

    Name the rule. Iavg=(tactive/Tcycle)Iactive+[1-(tactive/Tcycle)]Isleep

  3. 3

    Put in the chapter value. Set sleepy-device poll interval to 8. The page rule gives active duty as 0.063%.

  4. 4

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

Predict, then change sleepy-device poll interval

Try Predict what happens to active duty. Move one control, calculate, then check your idea.

8
Chapter baseline
Active duty

Observe Lengthening the poll interval shrinks the active fraction. The router cannot take that saving because listening is its network job. Reset to 8 and compare active duty.

Explain Only sleepy-device poll interval 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 sleepy-device poll interval moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only sleepy-device poll interval. It replays the chapter baseline; it is not a field prediction.

1. Start with what the radio does

A router must listen almost continuously so it can relay arriving frames. A sleepy end device wakes for a short poll, then returns to microamp sleep. The role changes the fraction of time spent at the 24 mA current.

Radio Remi: A battery label gives charge capacity. It does not decide how quickly the device spends that charge.

2. Name every algebra move

1

Make a fractionduty=tactive/Tcycle.

2

Weight each currentIavg=duty Iactive+(1−duty)Isleep.

3

Discount the cellCusable=C0(1−k)t(1−δ).

4

Divide charge by currentruntime=Cusable/Iavg.

3. Keep units compatible

5 ms / 8 s = 0.005 s / 8 s = 0.000625

Convert 1 µA to 0.001 mA before adding it to 24 mA. Since mAh divided by mA gives hours, 172 mAh / 0.0160 mA produces about 10,800 hours.

4. Try one controlled change

Iavg=(tactive/Tcycle)Iactive+[1−(tactive/Tcycle)]Isleep

TryChange only the sleepy device's poll interval. The radio currents, 5 ms active window, cell, derating, and self-discharge stay fixed.

2.4 GHz wavelength
Active duty
Average current
Usable cell charge
Sleepy runtime
Sleepy runtime
Router runtime
Role runtime ratio

ObserveAt 8 s, duty is 0.0625%, average current is 0.0160 mA, usable charge is 172 mAh, and the sleepy estimate is about 10,800 hours or 1.23 years. The raw router limit is only 9.17 hours.

ExplainLengthening the poll interval shrinks the active fraction. The router cannot take that saving because listening is its network job.

Technical boundaries.

This is a first-order charge budget, not a battery qualification.

Radio and cell
24 mA and 220 mAh are catalog-typical teaching values
Pulse delivery
Internal resistance and brownout are not modelled here
Network traffic
Retries, sensing, processing, and reporting add charge

Measure the full current trace and validate the cell across temperature and age.

5. Reproduce the chapter numbers

duty=0.005/8=0.000625. Iavg=0.000625(24)+0.999375(0.001)=0.0160 mA. Cusable=220(0.99)²(0.80)=172 mAh. Runtime=172/0.0160=10,800 h=1.23 years. The un-derated always-on comparison is 220/24=9.17 h, making the displayed role comparison about 1,180×.

6. Carry the evidence forward

Record role, poll interval, wake duration, measured active and sleep current, payload rate, retry rate, cell chemistry, derating, temperature range, brownout threshold, and the retest trigger for a firmware or route change.

7. Check yourself

Why must 1 µA become 0.001 mA?
Answer: Both weighted-current terms must use the same unit before they are added.
Why is 1.23 years not a promise?
Answer: The model omits retries, sensor and processor work, temperature, ageing, pulse sag, and cell-rate effects.
Why can the router not simply sleep for 8 seconds?
Answer: A router must be available to receive and relay frames; sleeping would remove the behavior that makes it a router.
Honesty boundary.

The calculation proves that an always-listening role is incompatible with this coin-cell example. It does not predict a deployed lifetime.

9.17 hours
Raw 220 mAh / 24 mA limiting comparison
1.23 years
Sleepy charge-only estimate after stated allowances
1,180×
Comparison using the chapter's differently bounded worked cases

Approve the role only after current-trace, pulse, temperature, and network-behavior tests.

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