The Coin-Cell Beacon’s Duty-Cycle Budget

The Coin-Cell Beacon’s Duty-Cycle Budget

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

The Coin-Cell Beacon's Duty-Cycle Budget

A BLE temperature sensor advertises every 1,000 ms, drawing 15 mA for 2 ms and just 2 µA asleep from a 220 mAh CR2032 coin cell — a 0.2% duty cycle and a 0.032 mA average. That yields 6,875 hours of life, which the chapter rounds up toward “approximately 1 year.” This audit re-derives the cascade and asks whether those inputs honestly reach a year, or land closer to 286 days.

Companion to the chapter Quizzes and Practice Problems — every number here comes from that chapter.

Try

A BLE temperature sensor advertises every 1,000 ms , drawing 15 mA for 2 ms and just 2 µA asleep from a 220 mAh CR2032 coin cell — a 0.2% duty cycle and a 0.032 mA average. Calculate this case.

Observe

That yields 6,875 hours of life, which the chapter rounds up toward “approximately 1 year.” This audit re-derives the cascade and asks whether those inputs honestly reach a year, or land closer to 286 days . Check shows this.

Explain

So 6,875 hours is about 286 days -- roughly 0.78 of a year. The chapter's follow-on "286-365 days ... approximately 1 year" is optimistic: real-world self-discharge (~15%/year) only subtracts from the 286-day figure, it cannot lift it toward 365. The honest, reproducible result from these inputs is 286 days. The design meaning is that the 0.2% duty cycle is what buys that286-day estimate; self-discharge can only shorten it. Check confirms it.

See the relationship before changing it

The figure reads from left to right. The blue input is advertising period. The middle card names the page’s rule. The green output is ideal battery life. The arrow matters: change the input, apply the rule once, then read the result with its unit.

Walk the arrow. A longer period dilutes the active pulse, until sleep current sets the floor.

Derive the baseline in four named moves

1. 1

   Name the input. The chapter baseline is 1000 ms.

2. 2

   Name the relationship. life = 220 mAh / (15 x 2/period + 0.002 x (1 - 2/period)) / 24

3. 3

   Substitute with units. period 1000 ms gives 286.5 days

4. 4

   Read the result. Keep the unit beside the value, then use the result only inside the technical boundary below.

Predict, then change advertising period

Try Predict how ideal battery life responds when advertising period moves. Calculate advertising period; compare ideal battery life with that prediction.

Advertising period

1000 ms

Chapter baseline

Ideal battery life

Observe Return to 1000 ms. Recheck ideal battery life with advertising period at its chapter value.

Explain A longer period dilutes the active pulse, until sleep current sets the floor.

Check yourself

What should you do before trusting a moved-slider result?

Answer: Predict its direction, apply the displayed relationship, keep the units, and compare the reset value with the chapter’s worked baseline.

What does this small model leave out?

Answer: Only advertising period moves here. The ideal battery life calculation excludes field effects listed below.

Honesty note. The ideal battery life result holds advertising period as the only changing input. It is not a field prediction.

Technical boundaries

Excluded from the “The Coin-Cell Beacon's Duty-Cycle Budget” calculation are battery self-discharge, temperature, advertising collisions, retransmissions, regulator losses, or current transients outside the stated duty cycle; “The Coin-Cell Beacon's Duty-Cycle Budget” therefore reports only its named fixtures.

Ada: The calculation problem builds a battery life from an advertising duty cycle. The duty-cycle and average-current steps are exactly right; the day count needs a careful look, so let me carry it through.

Starting from a 1,000 ms advertising period, 15 mA for 2 ms active, 2 µA (0.002 mA) asleep, and a 220 mAh cell:

• Duty cycle: 2 ms / 1,000 ms = 0.002 = 0.2%.
• Average current: (15 mA x 0.002) + (0.002 mA x 0.998) = 0.030000 + 0.001996 = 0.031996 mA, which rounds to the stated 0.032 mA.
• Battery life: 220 mAh / 0.032 mA = 6,875 hours, and 6,875 / 24 = 286.46 days.

So 6,875 hours is about 286 days – roughly 0.78 of a year. The chapter’s follow-on “286-365 days … approximately 1 year” is optimistic: real-world self-discharge (~15%/year) only subtracts from the 286-day figure, it cannot lift it toward 365. The honest, reproducible result from these inputs is 286 days. The design meaning is that the 0.2% duty cycle is what buys that life: the beacon is awake only 1/500 of the time, so the 15 mA active draw is diluted about 500-fold to the 0.032 mA average a coin cell can sustain for the better part of a year.

Every number above is taken from the chapter’s own material and re-derived step by step.