Math Bridge: Gain, ADC Bins, and Filtering

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Math BridgeSensorsStruggle-friendly runway

How 20 mV becomes useful ADC resolution

One measurement chain from a tiny sensor span through gain, converter bins, and the chapter's 10 Hz filter.

Phoebe, the physics guidePhoebe guides
The one targetConnect gain to ADC step size and temperature resolution.
The chapter case20.0 mV, 150×, 12 bits, 9.95 Hz.
What it buys youKnow which stage protects which information.

A field team faces an unresolved physical question: How 20 mV becomes useful ADC resolution They must answer it before changing instrumentation amplifier gain on the real device. Predict the direction first.

See the relationship before changing it

The chapter figure below shows: Signal conditioning prepares a small sensor signal for ADC conversion through gain, filtering, level shifting, and sampling.

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.

Sensor conditioning runs through amplification, noise filtering, galvanic isolation and A/D conversion to digital output. A component table links each stage to functions and applications.
Signal conditioning prepares a small sensor signal for ADC conversion through gain, filtering, level shifting, and sampling.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for instrumentation amplifier gain is 150.

  2. 2

    Name the relationship. V_span=G V_sensor; q=V_ADC,FS/4096; ΔT=q/(G S); |H|=1/√(1+(f/f_c)²)

  3. 3

    Substitute the chapter fixture. Set instrumentation amplifier gain to 150. The ideal 3.3 V ADC code width is 0.806 mV regardless of gain.

  4. 4

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

Predict, then change instrumentation amplifier gain

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

150
Chapter baseline
Temperature per code

Observe All four readouts use the same gain, division, slope, and first-order filter formulas derived on this page. Reset the control to 150 and compare temperature per code.

Explain Only instrumentation amplifier gain moves here; ADC code width stays fixed. 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 instrumentation amplifier gain moves. Field effects named in the page's technical boundary stay fixed.

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

1. Start with a small span

The sensor moves from 10 to 30 mV, a span of 20.0 mV. Gain multiplies the difference from the chosen offset; it does not create information that the sensor never produced.

2. Size the gain

To fill 3.3 V exactly, G = 3.3/0.0200 = 165. The chapter chooses a practical 150×, so the sensor span becomes 0.0200×150 = 3.00 V and leaves headroom.

3. Turn the span into converter steps

1

Count binsA 12-bit ADC has 4,096 codes.

2

Divide the ADC full-scale span3.3 V/4,096 = 0.000806 V = 0.806 mV/count.

3

Refer the step back to temperature0.806/(150×1.00 mV/C) = 0.00537 C/count.

4. Try the practical gain

V_span=G V_sensor; q=V_ADC,FS/4096; ΔT=q/(G S); |H|=1/√(1+(f/f_c)²)

TryMove the gain and stop at the chapter's practical 150× setting.

ADC span
ADC step
Temperature/count
60 Hz attenuation

ObserveAt 150× the span is 3.00 V, the bin is 0.806 mV, the referred step is 0.00537 C, and the filter gives −15.7 dB at 60 Hz.

ExplainAll four readouts use the same gain, division, slope, and first-order filter formulas derived on this page.

Technical boundaries.

The conditioning calculation assumes a linear amplifier and a valid 1 mV/C sensor slope, with converter and filter settings held fixed. The computed span, code width, and filter attenuation still require measured evidence before they describe the installed measurement chain.

measured evidence
Needs separate evidence

Use field evidence or a deeper model before release.

5. Derive the filter answer

For 16 kohm and 1 uF, RC = 0.016 s and f_c = 1/(2πRC) = 9.95 Hz. At 60 Hz, |H| = 1/√[1+(60/9.95)²], so 20log₁₀|H| = −15.7 dB.

6. Keep loading separate

A 10 kohm source feeding 10 Mohm loses only 0.0999% through the divider. Feeding 100 kohm loses 9.09%. A buffer protects the voltage; it does not replace the gain or anti-alias filter.

7. Check yourself

Why choose 150× instead of 165×?
Answer: It maps 20.0 mV to 3.00 V and leaves headroom below 3.3 V.
What is the ideal 12-bit step across the full 3.3 V ADC range?
Answer: 3.3/4,096 = 0.806 mV/count.
Does the buffer provide anti-alias filtering?
Answer: No. It prevents loading; the low-pass stage removes fast content.
Honesty boundary.

The conditioning calculation assumes a linear amplifier and a valid 1 mV/C sensor slope, with converter and filter settings held fixed. The computed span, code width, and filter attenuation still require measured evidence before they describe the installed measurement chain.

20.0 mV span
Unamplified sensor voltage span
gains 165
Gain that exactly fills converter range
150
Practical amplifier gain leaving voltage headroom
3.00 V
Amplified sensor voltage span
4,096 codes
Twelve-bit analog-to-digital converter code count
0.806 mV/count
Converter code width across full voltage range
0.00537 C/count
Input-referred temperature change per converter count
16 kohm
Low-pass filter resistance
1 uF
Low-pass filter capacitance
9.95 Hz
Low-pass filter cutoff frequency
−15.7 dB
Filter attenuation at sixty hertz
loading examples reproduce the chapter's ideal calculations
Ideal source-loading voltage-loss comparisons

Treat these figures as teaching evidence, not as a complete release claim.

Return to the chapter with the maths unpacked →