Two Budgets a Coin Cell Must Pass

Two Budgets a Coin Cell Must Pass

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

foundations
math-foundations
calculation-audit
energy-power
Ada ADA · CALCULATION AUDIT

Two Budgets a Coin Cell Must Pass

The chapter’s sharpest claim is that a CR2032’s real limit is its resistance, not its 225 mAh: a 10 mA radio pulse sags 0.1 V on a fresh 10-15 ohm cell but 0.4 V on a 40-ohm aged one, near brownout with most charge still inside. The cell easily passes a 2.6 uA energy check yet can still reset on every transmission. This audit runs both budgets a coin cell must pass, energy and pulse-sag, together.

Companion to the chapter Battery-Life Target Contracts — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is cell resistance. The middle card applies the page rule. The green card is pulse sag. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Cell resistance changes pulse sag An input card leads through the rule sag = 0.010 A x resistance to the pulse sag result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. A cell can keep charge yet fail a radio burst because resistance, not capacity, sets sag.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 40 ohm.

  2. 2

    Name the relationship. sag = 0.010 A x resistance

  3. 3

    Substitute with units. 0.010 x 40 = 0.40 V

  4. 4

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

Predict, then change cell resistance

Try Predict the direction of sag = 0.010 A x resistance. Test another cell resistance, then compare pulse sag.

40 ohm
Chapter baseline
Pulse sag

Observe A cell can keep charge yet fail a radio burst because resistance, not capacity, sets sag. Reset cell resistance to 40 and compare pulse sag.

Explain A cell can keep charge yet fail a radio burst because resistance, not capacity, sets sag.

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 cell resistance moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only cell resistance. It is not a field prediction.

TryThe chapter’s sharpest claim is that a CR2032’s real limit is its resistance, not its 225 mAh : a 10 mA radio pulse sags 0.1 V on a fresh 10-15 ohm cell but 0.4 V on a 40-ohm aged one, near brownout with most charge still inside. Use Check derivation.
ObserveThe displayed ledger resolves 225 mAh, 10 mA, 0.1 V, 10-15 ohm, 0.4 V at full precision. This audit runs both budgets a coin cell must pass, energy and pulse-sag, together. Check derivation shows this.
ExplainCapacity governs average-energy life, but pulse voltage follows current times internal resistance; an aged 40-ohm coin cell therefore sags 0.4 V and can brown out while substantial charge remains. Check derivation confirms it.

Ada: This chapter’s sharpest claim is that a CR2032’s real limit is its resistance, not its milliamp-hours. The number that proves it is a 0.4 V pulse sag on an aged cell. Let me run both budgets the chapter insists must pass together — the energy check and the pulse-sag check — using its stated values (225 mAh label, 10 mA / 5 ms radio pulse, 10-15 ohm fresh rising to 40 ohm aged).

  • Energy check: a ten-year average allowance is 225 mAh / (10 x 8,760 h) = 0.00257 mA = 2.6 uA — easily met by a mostly-sleeping beacon.
  • Pulse sag, fresh: V = I x R = 0.010 A x 10 ohm = 0.1 V, so a 3.0 V terminal holds 2.9 V under the burst.
  • Pulse sag, aged: 0.010 A x 40 ohm = 0.4 V, dropping a 2.7 V open-circuit cell to 2.3 V — near a 2.0-2.2 V brownout with most of the charge still inside.
  • Brownout headroom (2.6 V cell, 10 mA pulse): a 2.4 V threshold allows only (2.6 - 2.4) / 0.010 = 20 ohm, while a 2.0 V threshold allows (2.6 - 2.0) / 0.010 = 60 ohm of internal resistance before reset.
  • Reservoir fix: for a 0.2 V droop, C = I x t / dV = 0.010 A x 0.005 s / 0.2 V = 250 uF — but its leakage counts too: 1 uA x 10 yr x 8,760 h/yr = 87.6 mAh, about 87.6 / 225 = 39% of the cell.

The audit confirms the whole message: the cell clears the energy check by a wide margin (2.6 uA against a tiny average) yet can still reset on every transmission once resistance climbs, because 0.4 V of sag is set by Ohm’s law, not by remaining charge. Usable capacity is therefore a design variable — a lower brownout threshold triples the resistance headroom (60 ohm versus 20 ohm) and unlocks charge that a higher threshold would strand.

Technical boundaries
The coin-cell checks deliberately do not simulate electrochemical recovery, pulse-shape variation, temperature, regulator dropout, contact resistance, or aging beyond the stated internal-resistance points; energy and instantaneous sag are tested separately.

Work the audit first, then check the displayed derivation.

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