Math Bridge: Why can a full drone pack still fail a climb?

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Why can a full drone pack still fail a climb?

Connect stored energy to loaded voltage so a mission protects both runtime and worst-segment headroom.

Motion Marley, the guideMotion Marley guides
The one targetSeparate watt-hour capacity from voltage delivery under load.
The chapter case22.2 V, 10.0 Ah, 40 mΩ, 25 A cruise, 45 A climb, and 20% reserve.
What it buys youA mission gate that checks the hardest segment.

A field team has a real problem to settle: Why can a full drone pack still fail a climb? They must decide what happens before they change climb current on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is climb current. The middle card uses this page's rule. The green card is nameplate energy. 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.

Climb current changes nameplate energy An input card leads through the page rule to the nameplate energy result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Watt-hours bound mission duration; IxR sets the instant voltage drop. The launch and segment gates need both checks.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for climb current is 45.

  2. 2

    Name the rule. E=22.2x10.0=222 Wh; Eusable=0.80x222=177.6 Wh Cruise: ΔV=25x0.040=1.00 V; climb: ΔV=45x0.040=1.80 V; Vterm=20.4 V

  3. 3

    Put in the chapter value. Set climb current to 45. The page rule gives nameplate energy as 222.00 Wh.

  4. 4

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

Predict, then change climb current

Try Predict what happens to nameplate energy. Move one control, calculate, then check your idea.

45
Chapter baseline
Nameplate energy

Observe Watt-hours bound mission duration; IxR sets the instant voltage drop. The launch and segment gates need both checks. Reset to 45 and compare nameplate energy.

Explain Only climb current 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 climb current moves. Field effects named in the page limits stay fixed.

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

1. Start with two battery questions

Energy asks how much work the pack can supply over a mission. Loaded voltage asks whether it can supply the current demanded right now. Passing one does not pass the other.

Motion Marley: A climb can hit the cutoff even while plenty of watt-hours remain.

2. Name every algebra move

1

Convert charge to energyEnameplate=V×Q.

2

Protect reserveEusable=k×Enameplate.

3

Calculate sagΔV=I×Rint.

4

Subtract the dropVterm=Voc−ΔV.

5

Age stored chargeQ(t)=Q0(1−r)^t.

3. Reproduce the chapter case

E=22.2×10.0=222 Wh; Eusable=0.80×222=177.6 Wh
Cruise: ΔV=25×0.040=1.00 V; climb: ΔV=45×0.040=1.80 V; Vterm=20.4 V

After half a month at 2% monthly self-discharge, the model retains 9,899 mAh and loses about 101 mAh.

4. Try the climb current

TryRaise the worst-segment current and watch voltage headroom shrink while stored energy stays unchanged.

Climb current
Nameplate energy
Usable energy
Cruise sag
Cruise terminal
Climb sag
Climb terminal
Climb sag share
Retained charge
Storage loss

ObserveAt 45.00 A, climb sag is 1.80 V or 8.11%, leaving 20.40 V. The 222.00 Wh nameplate and 177.60 Wh planned budget do not move with current.

ExplainWatt-hours bound mission duration; I×R sets the instant voltage drop. The launch and segment gates need both checks.

Technical boundaries.

This compact engine is a first-order pack model, not a flight certificate.

Voltage
22.2 V is treated as a fixed open-circuit teaching value rather than a discharge curve
Resistance
40 mΩ is held constant despite temperature, age, state of charge, and connector heating
Mission
Energy excludes wind, manoeuvres, payload variation, conversion loss, and return uncertainty

Use measured pack and flight logs, with an explicit cutoff and reserve policy.

5. Build the segment gate

For every transit, climb, dwell, collection, and return segment, estimate energy and maximum current. Reject or trim the plan if either reserve energy or loaded-voltage margin fails.

6. Keep the pack record

Record pack chemistry, voltage, measured capacity, resistance, temperature, age, storage interval, segment currents, cutoff, reserve, route, payload, wind, owner, and abort trigger.

7. Check yourself

Why is 10 Ah not yet an energy budget?
Answer: Amp-hours measure charge. Multiplying by 22.2 V gives the 222 Wh nameplate energy.
Why does climb sag exceed cruise sag?
Answer: With the same 0.040 Ω resistance, 45 A creates 1.80 V drop while 25 A creates 1.00 V.
Does 177.6 Wh prove the mission is safe?
Answer: No. It is a reserved energy bound; loaded voltage, route uncertainty, temperature, wind, and measured flight performance still matter.
Honesty boundary.

The pack values are the chapter's explicitly labelled catalog-typical teaching case.

22.2 V and 10.0 Ah
Catalog-typical 6S mapping-drone pack
25/45 A and 40 mΩ
Catalog-typical cruise, climb, and pack resistance assumptions
2% and 20%
Illustrative monthly self-discharge and design reserve

Correct, not complete: qualify the actual pack and aircraft before flight.

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