The Duty Cycle Where Sleep Takes Over

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

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.

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.

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