Characterizing AD7746 Error Sources at −80°C: What to Fix Before Calibration
Characterizing AD7746 Error Sources at −80°C: What to Fix Before Calibration
Why Do This Before Calibration?
The previous post showed our AD7746 + P14 Rapid humidity system achieves 0.09% RH repeatability in a salt-solution chamber at room temperature. The next logical step is multi-point humidity calibration across the full flight temperature range.
But calibration built on unknown foundations is worthless. If the measurement circuit adds drift that changes with temperature, your calibration curve absorbs that drift and produces garbage when conditions change. Before spending weeks generating calibration data against salt-solution references, you need to know which error sources exist, which dominate, and which need to be fixed in hardware first.
This post is a systematic decomposition of the measurement chain — testing each layer in isolation to find where the errors come from.
The First Test: Something Is Wrong
We connected a C0G/NP0 ceramic 140 pF reference capacitor — chosen because it’s close to the P14 Rapid’s nominal capacitance (~148 pF) — through the AN-1585 range extender circuit, placed the board in a Stirling free-piston cooler, and swept from +14.8°C to −81.4°C over 10.2 hours (27,671 samples).
C0G ceramics have a specified tempco of ±30 ppm/°C. For a 140 pF capacitor over an 89°C temperature swing, we expected to see about 4.2 fF/°C of drift from the ceramic alone. Here’s what we actually measured:
| Parameter | Expected | Measured |
|---|---|---|
| Drift rate | 30 ppm/°C (C0G spec) | 84.6 ppm/°C |
| fF per °C | 4.2 fF/°C | 11.9 fF/°C |
| Total ΔC over 89°C | ~375 fF | 1,065 fF |
| Ratio | 1.0× | 2.8× |
The measured drift is 2.8× higher than the ceramic capacitor alone should produce. There are 54.6 ppm/°C of unexplained drift — somewhere in the measurement chain, something is adding temperature-dependent error that the calibration would blindly absorb.
Where does it come from? There are two suspects: the AD7746 ADC, and the range extender circuit.
Ruling Out the ADC: Open-Circuit Test at −78°C
To test whether the AD7746 itself degrades at cryogenic temperature, we disconnected the capacitor entirely and ran open-circuit at −78°C for 50 minutes (603 samples). If the ADC noise is anomalously high, or if it has temperature-dependent offset drift, that would explain the excess.
The AD7746 datasheet specifies 4.2 fF rms noise per single conversion at room temperature. Thermal noise in the Σ-Δ modulator scales as √T:
\[\sigma_{predicted} = 4.2 \times \sqrt{\frac{195\text{K}}{298\text{K}}} = 3.40 \text{ fF}\]| Metric | Value |
|---|---|
| Measured single-conversion σ | 3.38 fF |
| √T predicted | 3.40 fF |
| Ratio | 0.99× |
The ADC matches the √T prediction to within 1%. The ADC is not the source of the excess drift. It’s thermally limited, follows physics exactly, and actually gets quieter at cold temperatures. No anomalous noise sources, no flicker noise surprises, no cryogenic degradation.
The Range Extender: Found It
With the ADC ruled out, the excess drift must come from the range extender circuit. The AD7746’s native ±4 pF range is extended to ±48.5 pF using a resistor voltage divider on the excitation output (per Analog Devices AN-1585). The gain factor is:
\[F = \frac{R_1 + R_2}{R_2 - R_1} = \frac{100\text{k} + 118\text{k}}{118\text{k} - 100\text{k}} = 12.11\]This F factor multiplies the effective capacitance — but it also means any change in R1 or R2 with temperature directly changes the measured result. The denominator is only 18 kΩ, a small difference between two large numbers, making F catastrophically sensitive to mismatch.
The current resistors are thick-film parts rated at ±100 ppm/°C each. In the worst case (R1 and R2 drift in opposite directions), the gain factor F drifts by:
\[\frac{\Delta F}{F} \approx \frac{(TC_1 + TC_2) \times \Delta T \times (R_1 + R_2)}{R_2 - R_1}\]For ±100 ppm/°C resistors over 89°C, this produces approximately 55 ppm/°C of apparent capacitance drift — which accounts for the 54.6 ppm/°C excess we measured. The numbers match.
Random Noise: Also the Range Extender, But Manageable
The range extender also dominates the random noise budget. De-trending the 140 pF data and computing residual noise across the temperature sweep:
| Condition | Single-conv noise σ | Notes |
|---|---|---|
| Open circuit at −78°C | 3.38 fF | ADC only, matches √T |
| 140pF + range extender (warm, +9°C) | 55.3 fF | 16× ADC floor |
| 140pF + range extender (cold, −77°C) | 57.7 fF | 17× ADC floor |
The ~55 fF noise is essentially temperature-independent (cold/warm ratio 1.04), consistent with the F = 12.11 gain amplifying excitation and Johnson noise from the divider resistors. After 32× averaging this becomes ±10 fF → ±0.027 %RH — acceptable. The random noise is manageable. It’s the systematic drift that kills you.
What Needs to Change: The Resistors
The 54.6 ppm/°C excess drift comes from the thick-film range extension resistors. This drift is not calibratable because it changes with temperature during flight.
Tempco Drift Over a Flight Envelope
Board temperature swing: 65°C (from +25°C ground to −40°C in-flight):
| Scenario | R1 TC | R2 TC | Worst-case ΔRH |
|---|---|---|---|
| Current (thick/thick) | ±100 ppm | ±100 ppm | ±50 %RH |
| Both thin (±25 ppm) | ±25 ppm | ±25 ppm | ±12 %RH |
| Both precision (±5 ppm) | ±5 ppm | ±5 ppm | ±2.3 %RH |
The 118 kΩ Problem
100 kΩ is readily available in precision thin film (±0.1%, ±5 ppm/°C) at 0402. But 118 kΩ is an oddball E96 value — only thick film in 0402.
| Option | Precision? | F value | Flight drift |
|---|---|---|---|
| 118 kΩ single (current) | ❌ Thick only | 12.11 | ±50 %RH |
| 120 kΩ single | ✅ Thin film | 11.00 | ±2.3 %RH |
| 100 kΩ + 18 kΩ series | ✅ Both standard | 12.11 | ±2.3 %RH |
The Path to Calibration
This error decomposition tells us the order of operations:
-
✅ ADC is trustworthy. Open-circuit √T match (0.99×) confirms the AD7746 is not a source of calibration error at any temperature down to −80°C.
-
✅ Random noise is manageable. ±0.027 %RH after averaging — well below our target accuracy.
-
❌ Thick-film resistors must be replaced first. With ±50 %RH of uncalibratable drift, any multi-point calibration performed now would be invalidated the moment the board temperature changes. Fix the hardware, then calibrate.
-
After the fix: swap R1/R2 for precision thin film → re-verify noise → proceed with multi-point humidity cal against salt-solution references at multiple temperatures. The uncalibratable error drops from ±50 %RH to ±2.3 %RH.
The lesson: characterize the electronics before calibrating the sensor. Two $0.01 thick-film resistors would have silently destroyed every calibration point.
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