Math Bridge: MEMS sensitivity chain

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Math BridgeDesign MethodologyStruggle-friendly runway

What physical chain sits behind an accelerometer sensitivity row?

Follow acceleration through proof-mass displacement, gap fraction, capacitance, and front-end voltage.

Blueprint Bina, the design guideBlueprint Bina guides
The one targetCarry acceleration into a sensitivity estimate.
The chapter case2 g, 30 ng mass, 8 N/m spring, 2 µm gap.
What it buys youA tolerance-aware datasheet question.

A field team has a real problem to settle: What physical chain sits behind an accelerometer sensitivity row? They must decide what happens before they change acceleration on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is acceleration. The middle card uses this page's rule. The green card is acceleration. 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.

Acceleration changes acceleration An input card leads through the page rule to the acceleration result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. The fixed sensitivity comes from fixed mechanical and electrical constants; manufacturing and environment make those constants uncertain.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for acceleration is 2.

  2. 2

    Name the rule. a=2x9.81=19.6 m/s² x=(3.0x10⁻⁸x19.6)/8=73.6 nm x/d₀=3.68%; ΔC=0.0736 pF Vout=1.0x0.0736/1.0=73.6 mV Sensitivity=73.6/2=36.8 mV/g

  3. 3

    Put in the chapter value. Set acceleration to 2. The page rule gives acceleration as 19.62 m/s^2.

  4. 4

    Read the result. Keep m/s^2 next to the value. Use it only within the limits on this page.

Predict, then change acceleration

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

2
Chapter baseline
Acceleration

Observe The fixed sensitivity comes from fixed mechanical and electrical constants; manufacturing and environment make those constants uncertain. Reset to 2 and compare acceleration.

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

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

1. Begin with a mass on a spring

Acceleration creates force on a proof mass. The spring moves until its restoring force balances that inertial force. The movement is tiny but measurable as a capacitance change.

Blueprint Bina: Follow one unit at a time: g to m/s², metres, farads, then volts.

2. Name every algebra move

1

Convert ga=ag×9.81 m/s².

2

Balance forcex=ma/k.

3

Compare with the gapΔC/C₀≈x/d₀.

4

Find capacitance changeΔC=C₀x/d₀.

5

Read charge as voltageVout=VrefΔC/Cf.

3. Reproduce the 2 g case

a=2×9.81=19.6 m/s²
x=(3.0×10⁻⁸×19.6)/8=73.6 nm
x/d₀=3.68%; ΔC=0.0736 pF
Vout=1.0×0.0736/1.0=73.6 mV
Sensitivity=73.6/2=36.8 mV/g

The result is an internal analog teaching estimate, not the LIS3DH or ADXL345 digital LSB/g specification.

4. Try the acceleration

TryMove the input while the small-deflection approximation and die constants stay fixed.

Acceleration
Acceleration
Deflection
Gap fraction
Capacitance change
Front-end output
Sensitivity

ObserveIn this linear model, deflection, capacitance change, and output all scale with acceleration while mV/g stays fixed.

ExplainThe fixed sensitivity comes from fixed mechanical and electrical constants; manufacturing and environment make those constants uncertain.

Technical boundaries.

This is a small-deflection static chain, not a MEMS device model.

Mechanics
Damping, resonance, cross-axis motion, and stops are omitted
Capacitance
Comb geometry and differential readout are reduced to one gap approximation
Electronics
Noise, nonlinearity, ADC scaling, temperature, and digital calibration are omitted

Use the manufacturer's tested sensitivity, tolerance, temperature, supply, bandwidth, and noise conditions for selection.

5. Ask which tolerance dominates

Trace mass, spring, gap, rest capacitance, feedback capacitance, and reference voltage into the sensitivity error budget before accepting one summary number.

6. Preserve the evidence row

Record device, range, sensitivity condition, temperature, supply, bandwidth, axis, tolerance type, calibration state, and the bench check that can reject the choice.

7. Check yourself

Why must 2 g become 19.6 m/s²?
Answer: The force equation uses SI acceleration, and 1 g is about 9.81 m/s².
Why is 73.6 nm allowed in the linear approximation?
Answer: It is only about 3.68% of the 2 µm gap in this teaching case.
Is 36.8 mV/g the ADXL345's published digital sensitivity?
Answer: No. It is the internal analog order-of-magnitude case defined here.
Honesty boundary.

The page reproduces the chapter's catalog-typical MEMS-comb constants.

73.6 nm
Static deflection under stated ideal constants
0.0736 pF
Approximate capacitance change
36.8 mV/g
Internal analog estimate, not a digital datasheet guarantee

Correct, not complete: this chain does not qualify an accelerometer selection.

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