Math Bridge: Divider and ADC Resolution

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Math BridgeSensorsStruggle-friendly runway

How many ohms fit inside one ADC code?

One thread from a divider curve to voltage bins and local resistance resolution.

Phoebe, the physics guidePhoebe guides
The one targetTranslate voltage codes back to resistance.
The chapter case3.3 V, 12 bits, and 10 kΩ.
What it buys youKnow whether a threshold clears the ADC floor.

A field team faces an unresolved physical question: How many ohms fit inside one ADC code? They must answer it before changing thermistor resistance in kilo-ohms 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 thermistor resistance in kilo-ohms. The middle card applies this page's relationship. The green card is ideal adc code. 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.

Thermistor resistance in kilo-ohms changes ideal adc code An input card leads through the page relationship to the ideal adc code result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. At 4.0 kohm the slope rises to about 168 uV/ohm, so one code shrinks to about 4.79 ohm. The ADC stayed the same; the divider curve changed the resistance meaning.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for thermistor resistance in kilo-ohms is 10.

  2. 2

    Name the relationship. V=VsRt/(Rf+Rt); q=Vref/2^N; ΔR=q/(dV/dRt)

  3. 3

    Substitute the chapter fixture. Set thermistor resistance in kilo-ohms to 10. The page ledger gives ideal adc code as 2048 counts.

  4. 4

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

Predict, then change thermistor resistance in kilo-ohms

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

10
Chapter baseline
Ideal ADC code

Observe At 4.0 kohm the slope rises to about 168 uV/ohm, so one code shrinks to about 4.79 ohm. The ADC stayed the same; the divider curve changed the resistance meaning. Reset the control to 10 and compare ideal adc code.

Explain Only thermistor resistance in kilo-ohms 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 thermistor resistance in kilo-ohms moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only thermistor resistance in kilo-ohms. It replays the chapter fixture; it is not a field prediction.

1. See a curved ruler

A divider does not map equal resistance changes to equal voltage changes everywhere. The ADC then rounds that curved voltage into equal code bins.

Phoebe: One code means a different number of ohms at each operating point.

2. Find the voltage bin

q=Vref/2^N; σq=q/√12

At 3.3 V and 12 bits, 4096 codes give q = 0.806 mV and an ideal RMS floor of 0.233 mV.

3. Find the local divider slope

Vout=VsRt/(Rf+Rt)
dVout/dRt=VsRf/(Rf+Rt)²; ΔRmin=q/(dVout/dRt)

The derivative is the local conversion ruler. Dividing one voltage bin by that slope turns the ADC floor into ohms.

4. Try the thermistor resistance

V=VsRt/(Rf+Rt); q=Vref/2^N; ΔR=q/(dV/dRt)

TryMove the thermistor while the chapter's 10 kΩ fixed resistor, 3.3 V rail, and 12-bit ADC stay fixed.

Divider voltage
Ideal ADC code
Code step
RMS floor
Local slope
Ohms per code
Ideal ADC ceiling

ObserveAt 10 kΩ, the divider is 1.650 V and code 2048. Its slope is 82.5 µV/Ω, so one 0.806 mV code spans about 9.77 Ω.

ExplainAt 4.0 kΩ the slope rises to about 168 µV/Ω, so one code shrinks to about 4.79 Ω. The ADC stayed the same; the divider curve changed the resistance meaning.

Technical boundaries.

This is an ideal unloaded divider and ADC.

Reference tolerance, integral nonlinearity, effective bits, source impedance, sample-capacitor settling, resistor tolerance, thermistor self-heating, and analogue noise can raise the real floor
Needs separate evidence

Use field evidence or a deeper model before release.

5. Compare with the chapter move

The chapter's warm point moves from code 2048 to about 1170, a separation of 878 codes. That is far larger than the one-code floor calculated here.

6. Turn the result into a gate

Choose an operating range, calculate the worst local ohms per code, add measured noise and tolerances, then demand a threshold separation that still clears that combined floor.

7. Check yourself

Why is one code 0.806 mV?
Answer: Divide 3.3 V by 2^12 = 4096 and convert volts to millivolts.
Why do ohms per code change along the divider curve?
Answer: The local derivative dV/dR changes with resistance even though the ADC voltage step does not.
Does 9.77 Ω guarantee real 10 Ω resolution?
Answer: No. It is the ideal quantisation-only value; circuit and converter errors must be added.
Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

the chapter's 3.3 V
Time, interval, or service-life value
12-bit
Digital resolution or converter setting
10 kΩ/10 kΩ balance point
Resistance or impedance value
4.0 kΩ comparison
Resistance or impedance value

It exposes the ideal quantisation floor, not the complete accuracy of a physical thermistor channel.

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