Phoebe guidesA field team has a real problem to settle: How does a phone turn gravity into an API number? They must decide what happens before they change acceleration in metres per second squared on the device. Predict the direction first.
See the relationship first
The figure reads from left to right. The blue card is acceleration in metres per second squared. The middle card uses this page's rule. The green card is proof-mass shift. 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.
Derive the baseline in four moves
- 1
Name the input. The chapter baseline for acceleration in metres per second squared is 9.8.
- 2
Name the rule. F=ma; x=F/k; C0=ε0A/d0; ΔC≈2C0x/d0; a=kx/m
- 3
Put in the chapter value. Set acceleration in metres per second squared to 9.8. The page rule gives proof-mass shift as 0.980 nm.
- 4
Read the result. Keep nm next to the value. Use it only within the limits on this page.
Predict, then change acceleration in metres per second squared
Try Predict what happens to proof-mass shift. Move one control, calculate, then check your idea.
Observe Every stage is linear in this small-displacement model, so doubling acceleration doubles force, displacement, and differential capacitance before calibration recovers a. Reset to 9.8 and compare proof-mass shift.
Explain Only acceleration in metres per second squared moves here. The other chapter values stay fixed.
Check yourself
What should you do before you trust the result?
What does this small model leave out?
1. The API does not touch acceleration
A microscopic proof mass lags behind the phone frame. The chip first senses its displacement, then works backward through a calibrated mechanical and electrical model.
2. Balance inertia and the spring
Write inertial forceF = ma.
Write spring forceF = kx after the simple static model settles.
Set them equal and divide by kma = kx ⇒ x = ma/k.
Now inspect the proof-mass model before treating x as a number: the spring, damper, mass, and phone frame must have separate roles for the force balance to mean anything physical.
Read the diagram from the reference frame into spring k and damper b, then locate proof mass m and displacement z. Frame acceleration creates relative motion; the spring supplies the restoring force kx, and damping limits oscillation. That makes displacement x the mechanical intermediate that the electrical sensing stage can observe.
3. Turn motion into capacitance
One gap shrinks while the opposite gap grows. Their difference reinforces the displacement and rejects changes common to both sides.
Follow the capacitor diagram to connect that moving proof mass to a measurable electrical change: the gap in C = ε0A/d is the physical spacing altered by x.
Start at the voltage source and fixed plate, cross electric field E and gap g, and finish at the moving plate displaced by z. One differential gap closes as the other opens, so their capacitance changes reinforce motion while common effects tend to cancel. Calibration then inverts that electrical change to recover x and, through a = kx/m, acceleration.
4. Try the acceleration
TryMove acceleration while the chapter's illustrative proof mass, spring, plate area, and gap stay fixed.
ObserveAt 9.8 m/s², the force is 0.980 nN, displacement is 0.980 nm, C0 is 88.5 fF, and the differential change is about 0.0868 fF.
ExplainEvery stage is linear in this small-displacement model, so doubling acceleration doubles force, displacement, and differential capacitance before calibration recovers a.
This is a low-frequency, small-displacement, ideal parallel-plate approximation.
- comb fingers
- Needs separate evidence
- damping
- Needs separate evidence
- resonance
- Needs separate evidence
- feedback
- Needs separate evidence
- parasitics
- Needs separate evidence
- temperature compensation
- Needs separate evidence
- cross-axis correction
- Needs separate evidence
- analogue electronics
- Needs separate evidence
- ADCs
- Needs separate evidence
- vendor calibration
- Needs separate evidence
Use field evidence or a deeper model before release.
5. Reproduce the resting-phone chain
6. Interpret gravity honestly
A resting phone reports about 9.8 m/s² because the supported frame and proof mass have the same relative displacement as an equivalent acceleration. One raw reading cannot distinguish gravity from other sustained acceleration without orientation and motion context.
7. Check yourself
Why is x = ma/k?
Why is the electrical change differential?
Does a 9.8 m/s² raw value prove hard motion?
These are the chapter inputs, worked results, and named teaching assumptions.
- 0.1 µg mass
- Current or responsivity value
- 1 N/m spring
- Distance, wavelength, or size
- 200 × 100 µm plate
- Distance, wavelength, or size
- 2 µm gap
- Distance, wavelength, or size
- resulting 0.980 nm / 88.5 fF / 0.0868 fF
- Distance, wavelength, or size
They are not vendor specifications; Under the Hood keeps the real sensor and API limits.