Math Bridge: Resolution and Response Fit

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Math BridgeSensorsStruggle-friendly runway

When “range passes” but resolution fails

One thread through Candidate C's 165 C span, eight-bit step size, and the fridge cycle's timing check.

Phoebe, the physics guidePhoebe guides
The one targetTurn value and time discretisation into pass/fail gates.
The chapter case−40 to +125 C, 8 bits, 0.5 C need, 15 min ripple.
What it buys youReject a candidate for the right measurable reason.

A field team faces an unresolved physical question: When “range passes” but resolution fails They must answer it before changing candidate c adc bit depth 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 candidate c adc bit depth. The middle card applies this page's relationship. The green card is rms code noise. 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.

Candidate C ADC bit depth changes rms code noise An input card leads through the page relationship to the rms code noise result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The value outputs use range/2^N. The timing output compares the chapter's one-minute sample plan with twice the illustrative 15-minute ripple frequency.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for candidate c adc bit depth is 8.

  2. 2

    Name the relationship. q=range/2^N; σ_q=q/√12; SNR=6.02N+1.76; timing margin=f_s/(2f_signal)

  3. 3

    Substitute the chapter fixture. Set candidate c adc bit depth to 8. The page ledger gives rms code noise as 0.19 degrees C.

  4. 4

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

Predict, then change candidate c adc bit depth

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

8
Chapter baseline
RMS code noise

Observe The value outputs use range/2^N. The timing output compares the chapter's one-minute sample plan with twice the illustrative 15-minute ripple frequency. Reset the control to 8 and compare rms code noise.

Explain Only candidate c adc bit depth 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 candidate c adc bit depth moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only candidate c adc bit depth. It replays the chapter fixture; it is not a field prediction.

1. A range can hide coarse steps

Candidate C covers the needed temperatures, but the ADC must divide the whole −40 to +125 C span into codes. Covering both endpoints does not prove the gaps between codes are small enough.

Phoebe: Range asks “can it reach the value?” Resolution asks “can it tell two nearby values apart?” They are different gates.

2. Count the temperature span and codes

1

Subtract endpointsrange=125−(−40)=165 C.

2

Count eight-bit codes2^8=256.

3

Divideq=165/256=0.6445 C, rounded to 0.645 C.

3. Compare with the requirement

The fridge monitor needs 0.500 C resolution. Candidate C's 0.645 C step is larger, so it fails. The excess is (0.645−0.500)/0.500=29.0%. Its ideal quantisation noise is 0.645/√12=0.186 C, and its ideal eight-bit SNR is 49.9 dB.

4. Try the converter depth

q=range/2^N; σ_q=q/√12; SNR=6.02N+1.76; timing margin=f_s/(2f_signal)

TryRaise the ADC from 6 to 12 bits while the 165 C range, 0.500 C requirement, 15 min cycle, and 1 min sampling stay fixed.

Temperature step
RMS code noise
Ideal SNR
Meets 0.500 C?
Nyquist timing margin

ObserveAt 8 bits, the 0.64 C display fails resolution while the independent 7.50× timing margin passes.

ExplainThe value outputs use range/2^N. The timing output compares the chapter's one-minute sample plan with twice the illustrative 15-minute ripple frequency.

Technical boundaries.

The ADC step is only one part of accuracy.

Sensor tolerance, reference error, noise, nonlinearity, response lag, and installation effects remain
Needs separate evidence
The 15-minute compressor ripple is an illustrative catalog-typical case, not a guaranteed chapter sensor waveform
Needs separate evidence

Use field evidence or a deeper model before release.

5. Turn the 15-minute cycle into a sample limit

1

Signal frequencyf=1/(15×60)=0.00111 Hz.

2

Nyquist rate2f=0.00222 Hz.

3

Longest interval1/(2f)=450 s=7.5 min.

6. Read the matrix honestly

A one-minute sample interval is 7.5 times faster than the bare Nyquist rate for this slow ripple, so the timing gate passes. Candidate C still fails because its eight-bit value steps are too coarse. A pass in one column cannot cancel a fail in another must-have column.

7. Check yourself

Why is the span 165 C?
Answer: The upper endpoint minus the lower endpoint is 125−(−40)=165 C.
Does 0.645 C meet a 0.500 C resolution need?
Answer: No. The code step is larger than the smallest required change.
Why does a one-minute interval pass the timing example?
Answer: Nyquist permits at most 7.5 minutes for the 15-minute ripple, and one minute is shorter.
Honesty boundary.

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

−40 to +125 C span
Sensor scale, pressure, or digital result
eight bits
Digital resolution or converter setting
0.500 C need
Chapter input or worked result
0.0625 C Candidate A resolution
Current or responsivity value
“moves slowly” chapter decision are preserved
Chapter input or worked result
15.0-minute ripple is explicitly illustrative
Named teaching assumption

The 165 C, 0.645 C, 29.0%, 0.186 C, 49.9 dB, 7.5 min, and 7.5× results are direct calculations, not claims of total sensor accuracy.

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