The Duty Cycle Where Sleep Takes Over

The Duty Cycle Where Sleep Takes Over

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

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Ada ADA · CALCULATION AUDIT

The Duty Cycle Where Sleep Takes Over

The chapter claims a crossover duty cycle below which trimming the active burst stops helping. With an active current of 20 mA and a sleep current of 0.005 mA, that crossover lands at 0.025%; a 1%-duty node instead runs about 1.1 years on a 2000 mAh cell. This audit derives the duty cycle where sleep takes over, flipping the audit target from the transmit routine to leakage.

Companion to the chapter WSN Duty Cycle Worked Examples — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is active duty cycle. The middle card applies the page rule. The green card is ideal battery life. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Active duty cycle changes ideal battery life An input card leads through the rule life = 2000 mAh / (D x 20 mA + (1 - D) x 0.005 mA) / 8760 to the ideal battery life result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Lower duty helps until the sleep current becomes the floor.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 1 %.

  2. 2

    Name the relationship. life = 2000 mAh / (D x 20 mA + (1 - D) x 0.005 mA) / 8760

  3. 3

    Substitute with units. 2000 / (0.01 x 20 + 0.99 x 0.005) / 8760 = 1.11 years

  4. 4

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

Predict, then change active duty cycle

Try Predict the direction of life = 2000 mAh / (D x 20 mA + (1 - D) x 0.005 mA) / 8760. Test another active duty cycle, then compare ideal battery life.

1 %
Chapter baseline
Ideal battery life

Observe Lower duty helps until the sleep current becomes the floor. Reset active duty cycle to 1 and compare ideal battery life.

Explain Lower duty helps until the sleep current becomes the floor.

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 active duty cycle moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only active duty cycle. It is not a field prediction.

TryThe chapter claims a crossover duty cycle below which trimming the active burst stops helping. Use Check derivation.
ObserveThe displayed ledger resolves 20 mA, 0.005 mA, 0.025%, 1%, 1.1 years at full precision. This audit derives the duty cycle where sleep takes over, flipping the audit target from the transmit routine to leakage. Check derivation shows this.
ExplainThe design meaning is where to spend effort: above 0.025% you win by waking less often, but below it the microamps you cannot switch off own the battery life, so the audit target flips from the transmit routine to leakage and sleep-mode current. Check derivation confirms it.

Ada: This section claims there is a crossover duty cycle below which trimming the active burst stops helping. That is a checkable statement, so let me derive it from the same I_avg = D x I_active + (1 - D) x I_sleep used all through the chapter, with I_active = 20 mA and I_sleep = 0.005 mA.

The active and sleep contributions are equal when D x I_active = (1 - D) x I_sleep. Solving:

  • D x 20 = (1 - D) x 0.005 gives 20D + 0.005D = 0.005, so D = 0.005 / 20.005 = 0.00024994, i.e. 0.025% – the chapter’s crossover.

Now test the section’s 0.01% example against it (D = 0.0001, which is below the crossover):

  • Active contribution: 0.0001 x 20 = 0.002 mA.
  • Sleep contribution: (1 - 0.0001) x 0.005 = 0.0049995 mA, about 0.005 mA.

So at 0.01% duty the active term (0.002 mA) really has fallen below the sleep term (0.005 mA) – sleep now sets the floor, exactly as claimed. For continuity, the practitioner node at 1% duty gives I_avg = 0.01 x 20 + 0.99 x 0.005 = 0.20495 mA, and 2000 mAh / 0.20495 mA = 9758 h, about 1.1 years. The design meaning is where to spend effort: above 0.025% you win by waking less often, but below it the microamps you cannot switch off own the battery life, so the audit target flips from the transmit routine to leakage and sleep-mode current.

Technical boundaries
The crossover calculation deliberately does not simulate wake-transition energy, regulator efficiency, cell voltage sag, self-discharge, or temperature; it compares only the stated active and sleep current contributions at a fixed cycle.

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.