Math Bridge: Two WSN Duty Cycles

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Math BridgeWSNStruggle-friendly runway

Why can two duty cycles give two different safety answers?

Use the chapter's S-MAC values to separate receiver scheduling, battery life, and legal transmit airtime.

Packet Pete, the guidePacket Pete guides
The one targetNever substitute listen duty for transmit duty.
The chapter case10%, 1.8 mA off, 20 mA active.
What it buys youSeparate energy and regulatory evidence.

A field team has a real problem to settle: Why can two duty cycles give two different safety answers? They must decide what happens before they change mac receiver listen duty cycle percentage on the device. Predict the direction first.

See the relationship first

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.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for mac receiver listen duty cycle percentage is 10.

  2. 2

    Name the rule. Iavg=(1-D)Ioff+DIactive; Vterm=Voc-IactiveR; trun=Qusable/Iavg; tair=Dreg3600

  3. 3

    Put in the chapter value. Set mac receiver listen duty cycle percentage to 10. The page ledger gives runtime days as 18.4 days.

  4. 4

    Read the result. Keep days beside the value. Use it only within the limits on this page.

Predict, then change mac receiver listen duty cycle percentage

Try Predict what happens to runtime days. Move one control, calculate, then check your idea.

10
Chapter baseline
Runtime days

Observe Predict a decrease in runtime days as the input increases over this range. Reset to the chapter baseline and compare.

Explain Only mac receiver listen duty cycle percentage moves here. The other chapter values stay fixed.

Source: content/wsn/wsn-adaptive-duty-cycle.qmd — Phoebe's Field Notes: Two Different Things Both Called "Duty Cycle". Use the chapter equation and fixed ledger above; only the named input varies.

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 mac receiver listen duty cycle percentage moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only mac receiver listen duty cycle percentage. It replays the chapter baseline; it is not a field prediction.

1. One name hides two clocks

MAC duty says how often the receiver is awake. Regulatory duty limits how long a transmitter occupies a shared band. The first is a schedule chosen by firmware. The second is a band rule. Neither alone gives battery life.

Packet Pete: Label every percentage with the state and time window it measures.

2. Name every algebra move

1

Turn percent into a fraction10% becomes 0.10.

2

Weight each stateIavg=(1−D)Ioff+DIactive.

3

Discount nameplate chargeQusable=fderateQnominal.

4

Divide charge by currentRuntime hours=Qusable/Iavg.

5

Use the regulatory clockAllowed transmit seconds=Dreg×3600.

3. Work the S-MAC numbers

Iavg=0.9(1.8)+0.1(20)=3.62 mA; 1600/3.62=442 h=18.4 days

The cell has 1600 mAh usable. The 20 mA active pulse across 3 Ω sags 0.0600 V, leaving 3.54 V. The illustrative cell holds 5.76 Wh after derating. A separate 1% transmit limit means 36 s/hour.

4. Try one controlled change

Iavg=(1−D)Ioff+DIactive; Vterm=Voc−IactiveR; trun=Qusable/Iavg; tair=Dreg3600

TryMove only MAC listen duty. Currents, cell, voltage, resistance, and the separate 1% regulatory limit stay fixed.

Average current
Active sag
Terminal voltage
Usable charge
Usable energy
Runtime
Runtime days
Regulatory airtime/hour

ObserveAt 10%, average current is 3.62 mA and runtime is 18.4 days. Changing MAC duty moves runtime, but active sag and the 36 s/hour transmit allowance do not move.

ExplainThe control changes how often the receiver uses each current state. It does not change pulse current, cell resistance, or the regulation's separate clock.

Technical boundaries.

This is a two-state constant-current model.

MAC
Wake-up, sensing, CPU, transmit, receive, retries, and clock drift need more states
Cell
Capacity and resistance vary with temperature, age, and pulse history
Regulation
Actual sub-band rules, exceptions, dwell limits, and conducted power must be checked

Use measured state traces and the rules for the installed region and band.

5. A low percentage can still drain quickly

The off current is 1.8 mA, not zero. It runs for 90% of the time at the chapter setting and contributes 1.62 mA. The active part contributes 2.00 mA. Both matter.

6. Keep two audit columns

For energy, record state currents, durations, measured average, cell condition, and runtime. For spectrum compliance, record region, band, channel, transmit airtime, power, and the exact rule. Do not copy a percentage between columns.

7. Check yourself

Why is average current not 2.00 mA?
Answer: The off state still draws 1.8 mA for 90% of the time, adding 1.62 mA.
What does 36 s/hour describe?
Answer: An illustrative 1% regulatory transmit airtime budget, not receiver listen time.
Does an 18.4-day estimate prove field life?
Answer: No. More current states, real cell behaviour, environment, failures, and retries remain.
Honesty boundary.

The duty and current values come from the chapter evidence record; the cell and 1% band example remain labelled typical.

3.62 mA
Two-state weighted average
18.4 days
Illustrative constant-condition life
36 s/hour
Separate example band limit

Go deeper in the chapter, then verify the measured schedule and applicable radio rules.

Return to WSN Duty Cycling with the maths unpacked →