Math Bridge: Z-Wave Repair Energy

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Math BridgeZ-Wave simulationEnergy boundary

What does a route repair cost outside the simulator?

Turn retry and explorer counters into radio energy, usable cell budget, and a bounded repair-only lifetime.

Eddie, the electronics guideEddie guides
The one targetConvert simulated repair steps into a reproducible battery-budget screen.
The chapter caseEight 40 ms radio steps on a 3 V, 220 mAh coin-cell model.
What it buys youA hardware-validation trigger for repair-heavy topology.

A field team has a real problem to settle: What does a route repair cost outside the simulator? They must decide what happens before they change repairs per day on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is repairs per day. The middle card uses this page's rule. The green card is steps per repair. Follow the arrows: 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.

Repairs per day changes steps per repair An input card leads through the page rule to the steps per repair result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Frequency multiplies the fixed physical event cost, so the bounded lifetime falls in direct inverse proportion.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for repairs per day is 24.

  2. 2

    Name the rule. E_step = 0.030 x 3.0 x 0.040 = 3.60 mJ E_repair = (1 + 2 + 5) x 3.60 = 28.8 mJ = 0.008 mWh C_usable = 220 x 0.99^5 x 0.80 = 167.4 mAh

  3. 3

    Put in the chapter value. Set repairs per day to 24. The page rule gives steps per repair as 8.

  4. 4

    Read the result. Keep the stated output unit next to the value. Use it only within the limits on this page.

Predict, then change repairs per day

Try Predict what happens to steps per repair. Move one control, calculate, then check your idea.

24
Chapter baseline
Steps per repair

Observe Frequency multiplies the fixed physical event cost, so the bounded lifetime falls in direct inverse proportion. Reset to 24 and compare steps per repair.

Explain Only repairs per day moves here. The other chapter values stay fixed.

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 repairs per day moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only repairs per day. It replays the chapter baseline; it is not a field prediction.

1. Start with the physical story

A graph treats a retransmission as a counter. A battery must supply current at voltage for the whole radio interval, so every counter has an energy cost.

Eddie: The repair can be logically successful and still be energetically expensive when it repeats often.

2. Name every algebra move

1

Count stepsAdd normal, retry, and explorer transmissions.

2

Price one stepMultiply current, voltage, and time: E = IVt.

3

Derate the cellApply calendar self-discharge and the reserved margin.

4

Scale frequencyMultiply repair energy by repair events per day.

5

Screen lifetimeDivide usable cell energy by daily repair energy.

3. Reproduce the chapter case

E_step = 0.030 × 3.0 × 0.040 = 3.60 mJ
E_repair = (1 + 2 + 5) × 3.60 = 28.8 mJ = 0.008 mWh
C_usable = 220 × 0.99^5 × 0.80 = 167.4 mAh

One repair is tiny. At one event each hour, repair traffic alone consumes about 0.192 mWh per day and the bounded repair-only lifetime is about 7.2 years.

4. Try one real input

TryMove the repair frequency. The event cost stays physical while daily drain and the repair-only lifetime recompute together.

Repairs per day
Steps per repair
Energy per step
Energy per repair
Repair in mWh
Usable charge
Usable energy
One-event budget
Daily repair energy
Daily budget share
Repair-only lifetime
Repair-only years

ObserveAt 24 repairs per day, repair traffic uses 0.192 mWh per day and the repair-only screen is about 2,615 days.

ExplainFrequency multiplies the fixed physical event cost, so the bounded lifetime falls in direct inverse proportion.

Technical boundaries.

This is a repair-traffic screen, not a battery-life prediction.

Cell
Voltage, capacity, self-discharge, resistance, temperature, and cutoff vary by cell and load.
Radio
Current and 40 ms step time are explicit model inputs, not a Z-Wave guarantee.
Workload
Sleep current, sensing, ordinary reports, inclusion, and battery conversion losses are omitted.

Correct, not complete: this ledger does not predict device lifetime or validate RF behavior.

5. Use the result in validation

Carry repair frequency and step energy into the hardware plan. Measure current pulses and route recovery on the real controller and endpoint.

6. Record the evidence state

Keep firmware, route, retry and explorer counts, radio timing, cell state, temperature, current trace, and the trigger for repeating the test.

7. Check yourself

Why is mAh alone not the event energy?
Answer: mAh is charge. Voltage is required to convert charge to energy, while IVt prices one timed radio event.
Why does doubling repair frequency halve this lifetime screen?
Answer: Daily repair energy doubles while the usable energy assumption stays fixed.
Does a seven-year screen prove a seven-year sensor?
Answer: No. It omits every load except repair traffic and uses illustrative cell and radio inputs.
Honesty boundary.

The arithmetic prices only the named simulated repair traffic.

Reproducible
Step count, IVt energy, derating, and frequency scaling are explicit.
Measured
Real current pulses, sag, topology, and retries belong in hardware evidence.
Excluded
Sleep, sensing, reporting, leakage, temperature, and conversion losses are outside the ledger.

Correct, not complete: use this as a test trigger, not a battery claim.

Return to the Z-Wave simulation lab with the energy boundary unpacked →