Math Bridge: Rail to ADC Code

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Math BridgeElectronicsStruggle-friendly runway

How does one sensor voltage become a trustworthy ADC code?

Follow a rail through a resistor, divider, amplifier, and converter without losing the physical meaning.

Eddie, the electronics guideEddie guides
The one targetCarry units and limits from a 3.3 V circuit into a 12-bit code.
The chapter caseA 2.0 V LED, 68 ohm resistor, matched divider, gain 2, and 10 mV/°C sensor.
What it buys youA traceable signal chain instead of disconnected formula answers.

A field team faces an unresolved physical question: How does one sensor voltage become a trustworthy ADC code? They must answer it before changing rail 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 rail. The middle card applies this page's relationship. The green card is resistor voltage. 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.

Rail changes resistor voltage An input card leads through the page relationship to the resistor voltage result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Ratiometric stages can hold code position steady, but they do not freeze physical current, heat, or absolute voltage resolution.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for rail is 3.3.

  2. 2

    Name the relationship. VR=3.30-2.00=1.30 V ILED=1.30/68=19.1 mA; PR=24.9 mW Vdivider=3.30(10k/(10k+10k))=1.65 V VADC=2(1.65)=3.30 V q=3.30/4095=0.806 mV; q/(2x10)=0.0403 °C

  3. 3

    Substitute the chapter fixture. Set rail to 3.3. The page ledger gives resistor voltage as 1.30 V.

  4. 4

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

Predict, then change rail

Try Predict the direction of resistor voltage. Move one control, calculate, then check your prediction.

3.3
Chapter baseline
Resistor voltage

Observe Ratiometric stages can hold code position steady, but they do not freeze physical current, heat, or absolute voltage resolution. Reset the control to 3.3 and compare resistor voltage.

Explain Only rail 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 rail moves. Field effects named in the page's technical boundary stay fixed.

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

1. Start with the physical story

Every component receives only the voltage across its own terminals. A divider creates a fraction of the rail, gain scales that fraction, and the ADC replaces the result with one of finitely many codes.

Eddie: Keep asking what voltage is here, what current it causes, and how many sensor units one count represents.

2. Name every algebra move

1

Allocate voltageSubtract LED forward voltage from the rail.

2

Find currentDivide the resistor voltage by resistance.

3

Divide the railUse Vout=Vin Rlower/(Rupper+Rlower).

4

Apply gainMultiply the divider output by amplifier gain.

5

Price one countUse q=Vref/(2^N−1), then divide q by amplified sensor sensitivity.

3. Reproduce the chapter case

VR=3.30−2.00=1.30 V
ILED=1.30/68=19.1 mA; PR=24.9 mW
Vdivider=3.30(10k/(10k+10k))=1.65 V
VADC=2(1.65)=3.30 V
q=3.30/4095=0.806 mV; q/(2×10)=0.0403 °C

The full-scale code is useful only because the preceding stages preserve the intended range without clipping.

4. Try one real input

TryChange the common circuit/reference rail and predict each downstream quantity.

Rail
Resistor voltage
LED current
Resistor power
Divider output
After gain
ADC step
ADC code
Temperature per count
Ideal half-count

ObserveBecause the divider and reference share the rail, full-scale code stays fixed while current, power, volts per count, and sensor units per count move.

ExplainRatiometric stages can hold code position steady, but they do not freeze physical current, heat, or absolute voltage resolution.

Technical boundaries.

This ideal chain deliberately exposes the chapter's arithmetic.

Components
LED forward voltage, resistor values, reference, and gain all have tolerance and temperature drift.
Amplifier
Input range, output swing, offset, noise, bandwidth, and settling are omitted.
ADC
INL, DNL, reference noise, loading, calibration, and sensor accuracy remain outside one-count maths.

Correct, not complete: this ledger does not qualify an LED path, analogue front end, reference, sensor, or ADC channel.

5. Use the result in the design

Budget every stage against its actual rail, then leave headroom for component tolerance, reference error, amplifier swing, and sensor extremes.

6. Record the evidence state

Record rail and reference measurements, resistor tolerances, LED current, divider loading, gain, ADC code, temperature, calibration points, and uncertainty.

7. Check yourself

Why does the 68 ohm resistor not receive 3.3 V?
Answer: The LED already takes about 2.0 V, leaving about 1.3 V across the resistor.
Why can the code stay full scale when the rail changes?
Answer: The matched divider and gain create the same fraction of the rail that the ratiometric ADC uses as its reference.
Does 0.0403 °C per count prove 0.0403 °C accuracy?
Answer: No. It is ideal code resolution; sensor, reference, gain, noise, and calibration errors may be much larger.
Honesty boundary.

The arithmetic reproduces the chapter chain; it is a traceable ideal model, not a measurement guarantee.

Components
LED forward voltage, resistor values, reference, and gain all have tolerance and temperature drift.
Amplifier
Input range, output swing, offset, noise, bandwidth, and settling are omitted.
ADC
INL, DNL, reference noise, loading, calibration, and sensor accuracy remain outside one-count maths.

Correct, not complete: this ledger does not qualify an LED path, analogue front end, reference, sensor, or ADC channel.

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