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Averaging reduces uncorrelated random noise, but it does not reliably remove resistor temperature drift, op-amp offset drift, thermal gradients, warm-up movement, hysteresis, aging, or slowly varying 1/f noise. Those effects are correlated with temperature, time, power, or device behavior, so averaging eventually reaches a floor—or produces a very precise estimate of a value that is still drifting.
For a precision DC or low-frequency measurement, treat drift and noise as separate error sources. Calculate resistor-ratio and op-amp offset errors, inspect low-frequency noise specifications rather than only 1 kHz noise, control thermal gradients, and use a zero-drift amplifier only when its switching artifacts and impedance requirements suit the circuit.
Why a stable measurement can still move
Suppose a sensor, supply, and load appear constant, but an ADC reading wanders slowly. The cause may be ordinary broadband noise, but it may also be resistor self-heating, op-amp offset drift, PCB leakage, reference drift, warm-up, flicker noise, or a thermal gradient across the board.
A time-domain plot alone does not identify the cause. White noise appears as random scatter. Temperature drift often follows a temperature change. Flicker noise can look like random slow wandering. Warm-up may produce a large transient before the circuit reaches equilibrium, while hysteresis can leave a residual offset after the temperature returns to its starting value.
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That distinction determines the remedy: filtering can reduce bandwidth-limited random noise, while thermal design, component selection, calibration, or a different amplifier architecture is needed for drift.
Temperature drift: the terms that matter
- Temperature coefficient of resistance (TCR): the fractional resistance change per degree, commonly expressed in ppm/°C.
- Offset-voltage drift: change in an op amp’s input offset voltage with temperature, usually specified in µV/°C or nV/°C.
- Bias-current drift: change in input bias current with temperature.
- Gain drift: change in closed-loop gain caused by resistor-ratio changes and amplifier behavior.
- Warm-up drift: output movement while the die, package, PCB, and nearby components approach thermal equilibrium.
- Thermal hysteresis: a residual value change after a temperature excursion, even when the measured temperature returns to its original value.
- Aging: change with time rather than temperature.
It is misleading to call every slow movement “noise.” Drift is usually correlated with temperature, time, power, or mechanical conditions. Noise is a random process described statistically or spectrally, although very-low-frequency noise can be difficult to distinguish from drift in a short measurement.
Resistor temperature coefficient
Near a reference temperature, a resistor can be approximated as:
R(T) ≈ R0[1 + αR(T − T0)]
Here, R0 is the resistance at T0, αR is the temperature coefficient in fractional change per degree, and T − T0 is the temperature change.
A ±50 ppm/°C resistor exposed to a 40°C change moves by approximately:
50 ppm/°C × 40°C = 2,000 ppm = 0.2%
That is before considering tolerance, aging, self-heating, nonlinear temperature behavior, voltage coefficient, humidity, or mechanical stress. The calculation is first-order, but it is often enough to show that a resistor can dominate a supposedly precise measurement. See the temperature-drift overview for an illustrative treatment of resistor and amplifier drift.
Absolute TCR versus ratio tracking
In a gain-setting network, divider, difference amplifier, bridge, or instrumentation amplifier, the difference between resistor temperature coefficients is often more important than either absolute coefficient.
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G = 1 + RF/RG
The approximate gain change caused by resistor tempcos is:
ΔG ≈ (RF/RG)(αF − αG)ΔT
If both resistors change by nearly the same fraction, their ratio remains comparatively stable. If one resistor is near a regulator or power device and the other is several centimetres away, a stable ambient temperature does not guarantee a stable ratio.
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Matched resistor networks help because the elements share a package and usually a more similar thermal environment. In its example discussion, Analog Devices notes that discrete difference-amplifier resistor ratios must match in value and relative drift; even initial mismatch can severely limit common-mode rejection. Its application note is a useful reference for resistor matching and instrumentation-amplifier noise.
Resistor selection and layout
- Use a matched network when ratio tracking or common-mode rejection matters.
- Use low-TCR parts when absolute resistance matters.
- Place matched elements close together and give them similar copper areas and thermal paths.
- Keep them away from regulators, power transistors, hot ICs, connectors, and strong airflow.
- Consider self-heating. The dissipated power is
P = I²R = V²/R; a changing signal or supply can therefore change the resistor temperature even when ambient temperature is constant. - Avoid unnecessarily high values. Higher resistance increases Johnson noise and makes input bias current, leakage, contamination, and PCB-surface effects more significant.
- Check voltage coefficient, aging, humidity, soldering stress, mechanical stress, and package thermal behavior in high-precision designs.
Op-amp offset and noise gain
An op amp’s input offset voltage is multiplied at the output by the circuit’s noise gain, not necessarily by the signal gain. For an ordinary voltage-feedback inverting or non-inverting stage:
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The output offset is approximately:
VOUT,OS = GNVOS
Its temperature-dependent component is approximately:
ΔVOUT,OS ≈ GN × TCVOS × ΔT
For example, with a noise gain of 101, offset drift of 0.5 µV/°C, and a 20°C change:
ΔVOUT,OS ≈ 101 × 0.5 µV/°C × 20°C = 1.01 mV
That is large compared with many thermocouple, bridge, shunt-current, and low-level sensor signals. An amplifier with a low initial offset but poor drift can be worse over temperature than one with a slightly larger initial offset and much lower drift. Analog Devices provides a useful comparison of op-amp DC error characteristics.
Bias current and resistance
Input bias current flowing through a source or feedback resistance creates an error:
V = IBR
The resulting voltage is then multiplied by the relevant noise gain. This is why an excellent voltage-offset specification does not make a circuit accurate if it uses very high resistances.
A compensation resistor at the opposite input can reduce bias-current error when input currents are sufficiently matched. However, it adds Johnson noise and can introduce its own temperature coefficient, capacitance, and leakage. Include both the benefit and the new error sources in the budget.
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A complete low-frequency error budget may need to include VOS, TCVOS, IB, bias-current drift, source resistance, feedback resistance, resistor-ratio error and drift, CMRR and PSRR versus temperature, protection leakage, PCB leakage, reference drift, ADC gain drift, and sensor-excitation stability.
Flicker noise and thermal noise
Flicker noise, also called 1/f noise, generally becomes stronger as frequency decreases. A useful engineering model is:
en(f) = √(ewhite² + K/f)
This is an approximation over a relevant frequency range, not a promise that every amplifier follows an exact 1/f law at every frequency. The 1/f corner is the frequency where flicker-noise density equals the approximately flat broadband-noise density.
Flicker noise and thermal drift are different physical effects, but both can appear as slow output movement. A short record may look like an offset shift. A longer record may reveal random wandering rather than a monotonic relationship with temperature. Correlating the output with temperature, changing the bandwidth, repeating the experiment, and examining the spectrum can help separate them.
Resistors also generate Johnson noise:
eR = √(4kTR)
At room temperature, a 1 kΩ resistor produces approximately 4 nV/√Hz. Johnson noise is broadband, while flicker noise can dominate at sufficiently low frequencies in semiconductor amplifiers. Independent noise sources combine by root-sum-square:
eTOTAL = √(e1² + e2² + …)
Input-referred noise is then multiplied by the circuit’s noise gain to obtain output-referred noise. Analog Devices discusses Johnson noise, flicker noise, popcorn noise, noise gain, and RSS combination.
What averaging actually improves
For N independent samples with random noise standard deviation σ:
σAVG = σ/√N
Reducing white-noise RMS by 10 times therefore requires approximately 100 times as many independent samples. This improvement applies to uncorrelated random noise, not to a changing offset or a slowly varying thermal condition.
| Error source | Does averaging normally reduce it? | Why |
|---|---|---|
| Independent white noise | Yes | Variance falls approximately as 1/N. |
| Temperature drift | No | Samples are correlated with temperature and time. |
| Warm-up movement | No | The mean itself is changing. |
| Flicker noise | Not indefinitely | Low-frequency samples are correlated and can create an averaging floor. |
| Reference or supply drift | No | The measurement chain is moving systematically. |
| Fixed resistor-ratio error | No | Averaging cannot correct an inaccurate gain. |
In practice, RMS error may initially fall close to 1/√T as averaging time increases. It then flattens when flicker noise, drift, or environmental variation dominates. At longer times the apparent mean can wander with temperature, aging, mechanical motion, or self-heating.
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Averaging a drifting signal can produce a precise estimate of the wrong, time-dependent value. It improves repeatability only when the limiting error is random and uncorrelated; it does not automatically improve absolute accuracy.
Bandwidth, oversampling, and aliasing
Repeated-sample averaging, a moving-average filter, analog integration, oversampling, and decimation are related but not identical:
- Repeated-sample averaging combines separate readings.
- A moving average is a digital low-pass filter with a finite window.
- Analog integration provides continuous-time low-pass behavior.
- Oversampling collects samples faster than the minimum signal bandwidth.
- Decimation lowers the sample rate after filtering.
A narrower bandwidth reduces integrated white noise, but it does not automatically remove 1/f noise or drift. Oversampling without suitable analog and digital anti-alias filtering can fold out-of-band noise into the measurement band. A finite observation period also sets an effective lower frequency limit; extending the measurement can expose more low-frequency noise instead of producing indefinite 1/√N improvement.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Conventional precision versus zero-drift amplifiers
Zero-drift amplifiers use auto-zeroing, chopping, or related correction techniques to reduce offset and offset drift and suppress low-frequency flicker noise in the relevant baseband. They are often strong candidates for DC, sub-hertz, bridge, thermocouple, weigh-scale, and precision current-sense circuits.
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Auto-zeroing
An auto-zero amplifier periodically samples and corrects its DC error. This can provide very low offset and drift, but the sampling process can fold noise into the baseband. The result may be excellent DC accuracy with artifacts that matter in a wider-band design.
Chopping
A chopper modulates the signal and demodulates it, moving low-frequency error away from baseband. Ripple, clock feedthrough, intermodulation, charge injection, or components at the chopping frequency and its harmonics may remain. Filtering and layout must be designed around those artifacts.
Zero-drift does not mean zero noise. The architecture can suppress low-frequency offset and flicker behavior while introducing switching-related errors, limited bandwidth, settling behavior, or input-current effects.
When conventional precision is preferable
A conventional precision amplifier may be the better choice when the signal bandwidth is wide, settling and spectral purity are critical, the signal is large enough that offset drift is not dominant, or chopper ripple would interfere with the measurement. If its 1/f corner is well below the signal band, its absence of switching artifacts may outweigh its poorer sub-hertz behavior.
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High-impedance nodes require particular care with zero-drift devices. For example, the TI OPAx383 datasheet warns that input series resistances above 100 kΩ can increase output-referred clock noise because of internal clocking and charge injection. When high values are unavoidable, matching input impedances is recommended.
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How to read amplifier specifications
Do not select an amplifier using only a “low-noise” number at 1 kHz. Compare:
- Typical and maximum VOS.
- Typical and maximum TCVOS.
- 0.1–10 Hz peak-to-peak noise.
- Voltage-noise density at the frequencies that matter.
- Current-noise density and input bias current.
- 1/f corner.
- CMRR and PSRR over temperature.
- Common-mode range, output swing, gain-bandwidth product, settling time, input capacitance, supply range, and supply current.
- Chopping or auto-zero artifacts and stability with capacitive loads.
- Specified temperature range and package behavior.
0.1–10 Hz noise is commonly specified peak-to-peak, while wider-band noise is commonly specified as RMS over a stated bandwidth. These figures are not interchangeable. A low 1 kHz noise density does not prove good 0.1–10 Hz performance.
As examples of the type of specification to examine, the AD8628 product page lists 1 µV offset, 0.002 µV/°C input offset drift, 0.5 µV peak-to-peak noise over 0.1–10 Hz, and operation specified from −40°C to +125°C. Specifications and availability can change, so verify the current datasheet for a production design: AD8628 official product page.
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Worked low-frequency error example
Consider a sensor amplifier with noise gain 101, a 20°C temperature change, op-amp offset drift of 0.5 µV/°C, and 100 nV RMS of uncorrelated input-referred white noise per measurement.
Offset-drift term
The output drift is:
101 × 0.5 µV/°C × 20°C = 1.01 mV
This term does not fall when 10,000 readings are averaged. If the temperature continues changing during the averaging window, the result is a time-weighted value of the moving output.
White-noise term
The input-referred white-noise RMS after averaging is approximately:
- 1 sample: 100 nV RMS
- 100 samples: 10 nV RMS
- 10,000 samples: 1 nV RMS
At the output, multiply those figures by the noise gain of 101: approximately 10.1 µV RMS, 1.01 µV RMS, and 0.101 µV RMS respectively.
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Practical test for separating drift from noise
- Short the amplifier input or connect a known stable source.
- Allow the complete circuit—not only the IC—to reach thermal equilibrium.
- Log the output and local temperature simultaneously.
- Choose a sampling rate appropriate to the signal bandwidth and apply suitable anti-alias filtering.
- Repeat the measurement at several controlled temperatures.
- Plot output against temperature and time.
- Calculate temperature slope, short-term standard deviation, 0.1–10 Hz peak-to-peak noise, warm-up shift, and hysteresis after a temperature cycle.
- For long-duration stability, calculate Allan deviation as well as ordinary standard deviation.
- Repeat with several averaging windows and compare the measured improvement with the ideal 1/√N prediction.
A temperature-correlated component is likely thermal drift or thermal coupling. A stationary random component is more consistent with noise. A slope that changes after temperature cycling may indicate hysteresis, aging, mechanical stress, or self-heating. Abrupt millisecond-scale offset steps can be popcorn noise rather than ordinary white or flicker noise.
Design checklist
- Define the signal bandwidth, observation time, temperature range, and required absolute accuracy.
- Calculate noise gain separately from signal gain.
- Budget VOS, offset drift, bias current, bias-current drift, resistor noise, resistor-ratio drift, CMRR, PSRR, reference, excitation, and ADC errors.
- Use matched resistor networks where ratio tracking matters.
- Place matched components close together and away from heat sources.
- Minimize resistor self-heating and avoid unnecessary resistance.
- Measure component and die temperature where possible; ambient temperature is not always the relevant temperature.
- Compare 0.1–10 Hz noise with broadband noise density.
- Choose zero-drift architecture when sub-hertz offset and flicker performance dominate and artifacts can be controlled.
- Choose conventional precision architecture when wide bandwidth, clean spectrum, fast settling, or low ripple matters more.
- Check high-value source and feedback resistors against the chosen amplifier’s input-current and clock-feedthrough behavior.
- Filter switching artifacts without compromising stability or settling.
- Allow for warm-up before calibration.
- Do not assume calibration removes unpredictable noise or hysteresis.
- Verify typical versus guaranteed specifications over the actual operating temperature.
Bottom line
Signal averaging is powerful against independent white noise: N samples provide roughly a √N improvement. It is not a cure for temperature drift, inaccurate resistor ratios, op-amp offset drift, thermal gradients, warm-up, hysteresis, aging, or correlated 1/f noise.
For a stable low-frequency measurement, first identify the limiting error. Control the thermal environment and resistor ratios, calculate op-amp errors using noise gain, inspect low-frequency specifications, and test the complete signal chain over time and temperature. A zero-drift amplifier can be the right solution for DC and sub-hertz work, but its ripple, charge injection, impedance, bandwidth, and settling trade-offs must be part of the design.
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