Math Bridge: Sampling energy for an MCU

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Math BridgePrototypingStruggle-friendly runway

How does a vibration signal become a battery-budget number?

Start with the fastest vibration worth keeping. Then turn the sample rate into ADC steps, energy per conversion, and energy per day.

Voltage Vera, the prototyping guideVoltage Vera guides
The one targetTurn a signal frequency into a daily ADC-energy line.
The chapter caseA 200 Hz vibration, 12-bit ADC, and 3.3 V rail.
What it buys youA controller decision backed by signal and power evidence.

A field team faces an unresolved physical question: How does a vibration signal become a battery-budget number? They must answer it before changing nyquist sample rate on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is nyquist sample rate. The middle card applies this page's relationship. The green card is adc levels. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Nyquist sample rate changes adc levels An input card leads through the page relationship to the adc levels result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The signal controls how often the converter must work; the converter's electrical design controls what each reading costs.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for nyquist sample rate is 200.

  2. 2

    Name the relationship. f s =2x200=400 Hz q=3.3/4096=0.806 mV SNR≈6.02x12+1.76=74.0 dB E sample =3.3x0.005x10 us=165 nJ E day =165 nJx400x86,400=5.70 J/day

  3. 3

    Substitute the chapter fixture. Set nyquist sample rate to 200. The page ledger gives adc levels as 4096.

  4. 4

    Read the result. Keep the stated output unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change nyquist sample rate

Try Predict the direction of adc levels. Move one control, calculate, then check your prediction.

200
Chapter baseline
ADC levels

Observe The signal controls how often the converter must work; the converter's electrical design controls what each reading costs. Reset the control to 200 and compare adc levels.

Explain Only nyquist sample rate moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only nyquist sample rate moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only nyquist sample rate. It replays the chapter fixture; it is not a field prediction.

1. Start with what the sensor must preserve

A vibration signal changes over time. Sampling means taking separate voltage readings. If the readings arrive too slowly, a fast vibration can masquerade as a slower one. The Nyquist rule sets the lowest safe starting rate: sample at least twice as fast as the highest frequency you need to keep.

Voltage Vera: Choose the signal first. The sample-rate number follows from it; it is not a board-menu preference.

2. Name every algebra move

1

Double the highest frequencyUse fs=2fmax for the Nyquist minimum.

2

Count the ADC levelsAn N-bit ADC has 2N levels.

3

Divide the voltage spanq=Vref/2N.

4

Multiply power by timeEsample=VIactivetconv.

5

Count a whole dayDaily energy is energy per sample × samples per second × 86,400 seconds.

3. Reproduce the chapter case

fs=2×200=400 Hz
q=3.3/4096=0.806 mV
SNR≈6.02×12+1.76=74.0 dB
Esample=3.3×0.005×10 µs=165 nJ
Eday=165 nJ×400×86,400=5.70 J/day

The same conversion repeated only 288 times each day costs 47.5 µJ/day. Dividing the two daily totals gives 120,000×. That large gap comes from how often the ADC must wake, before radio and processor energy are added.

4. Try the vibration limit

TryMove the highest vibration frequency while the ADC and conversion conditions stay fixed.

Nyquist sample rate
ADC levels
ADC step
Ideal ADC SNR
Energy per conversion
Continuous ADC power
Continuous daily energy
Door-sensor daily energy
Daily energy ratio

ObserveDoubling the signal limit doubles the minimum sample rate and daily conversion energy. The ADC step stays fixed because bit depth and reference voltage did not move.

ExplainThe signal controls how often the converter must work; the converter's electrical design controls what each reading costs.

Technical boundaries.

This is a minimum-rate and catalog-current ledger, not a complete acquisition design.

Anti-alias filter
A real input needs analogue filtering and margin above the theoretical 2× limit
ADC quality
Noise, non-linearity, reference error, and effective bits lower real performance
System energy
CPU wake, sensor settling, memory, clock, and radio energy are not included

Measure the target board under the real sample schedule before freezing the controller.

5. Use the number in the controller decision

An MCU can fit when it can wake, sample, process, and sleep inside the energy and timing budget. An MPU may fit when the evidence needs a full operating system or heavier local processing. The 5.70 J/day figure does not pick a controller by itself; it exposes one cost that both candidates must meet.

6. Record the evidence state

Keep the vibration bandwidth, filter, sample rate, ADC mode, reference, active current, conversion time, board voltage, measured average power, processor workload, and firmware version. If any of these changes, rerun the ledger and measurement.

7. Check yourself

Why does a 200 Hz signal start at 400 samples per second?
Answer: The Nyquist minimum is twice the highest signal frequency: 2×200=400 Hz.
Why is one 12-bit step 0.806 mV on 3.3 V?
Answer: Twelve bits give 4,096 levels, so 3.3 V/4,096=0.000806 V.
Does 5.70 J/day describe the whole endpoint?
Answer: No. It includes the stated ADC-active conversion energy only, not the sensor, processor, memory, filter, radio, or sleep path.
Honesty boundary.

The arithmetic reproduces the chapter's standard or catalog-typical vibration example.

400 Hz
A theoretical minimum, not a finished sampling plan
74.0 dB
An ideal quantisation result, not measured ADC SNR
5.70 J/day
ADC conversion energy only, not endpoint energy

Correct, not complete: this ledger does not select an MCU or MPU without measured system evidence.

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