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 figure reads from left to right. The blue card is instrumentation amplifier gain. The middle card applies this page's relationship. The green card is adc step. 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.

Instrumentation amplifier gain changes adc step An input card leads through the page relationship to the adc step result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. All four readouts use the same gain, division, slope, and first-order filter formulas derived on this page.

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_span/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 page ledger gives adc step as 0.732 mV.

  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 adc step. Move one control, calculate, then check your prediction.

150
Chapter baseline
ADC step

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 adc step.

Explain Only instrumentation amplifier gain 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 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 amplified span3.00 V/4,096 = 0.000732 V = 0.732 mV/count.

3

Refer the step back to temperature0.732/(150×1.00 mV/C) = 0.00488 C/count.

4. Try the practical gain

V_span=G V_sensor; q=V_span/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 0.732 mV, the referred step is 0.00488 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.

This arithmetic assumes the amplifier stays linear and the 1 mV/C slope is valid.

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 12-bit step across 3.00 V?
Answer: 3.00/4,096 = 0.732 mV/count.
Does the buffer provide anti-alias filtering?
Answer: No. It prevents loading; the low-pass stage removes fast content.
Honesty boundary.

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

20.0 mV span
Voltage or voltage-step value
gains 165
Time, interval, or service-life value
150
Chapter input or worked result
3.00 V
Voltage or voltage-step value
4,096 codes
Sensor scale, pressure, or digital result
0.732 mV/count
Voltage or voltage-step value
0.00488 C/count
Sensor scale, pressure, or digital result
16 kohm
Resistance or impedance value
1 uF
Chapter input or worked result
9.95 Hz
Frequency, sample rate, or event rate
−15.7 dB
Gain, loss, margin, or level ratio
loading examples reproduce the chapter's ideal calculations
Named teaching assumption

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

Return to the chapter with the maths unpacked →