Sensor Noise Sources Explained: Thermal, Shot, 1/f and More

The new op amp has half the voltage noise of the old one. The board gets rebuilt, the measurement is repeated, and the noise floor hasn’t moved. The amplifier was never the problem. The sensor noise sources in a measurement add up like the sides of a right triangle, not like a column of numbers, so the largest contributor sets the floor and halving a small one changes almost nothing. This article goes through the physical sources of random noise (thermal, shot, 1/f, mechanical-thermal and quantization), shows how they combine, and gives a quick way to find the one that’s limiting your measurement.

Key Takeaways

• Uncorrelated noise sources add as a root-sum-square. A source three to five times smaller than the largest barely changes the total.
• Thermal (Johnson) noise is set by resistance, temperature and bandwidth: a 1 kΩ resistor produces about 4 nV/√Hz at 25 °C, and no component choice removes it.
• 1/f noise dominates slow measurements below its corner frequency. Chopping removes an amplifier’s own 1/f noise but not the sensor’s.
• In small MEMS sensors, the moving mass itself is jostled by molecular agitation, which can set the floor before any electronics do.
• Interference (mains hum, switching noise) is not random noise. It doesn’t average away like noise and needs shielding, layout or filtering instead.

“Which noise am I actually seeing?”

Forum threads about noise usually start with a symptom: “my readings fluctuate by 20 counts”, “the noise got worse when I increased the gain resistor”, “the reading drifts slowly over minutes”, “noise is fine on the bench but bad in the enclosure”. Each symptom points to a different source, and the fix for one does nothing for the others.

The first split is between random noise and interference. Random noise has a broad spectrum and a statistical amplitude; it falls with the square root of the number of averaged samples. Interference is deterministic: a tone at 60 Hz, a burst at the switching frequency of a regulator, a spike when a motor starts. Averaging helps interference only by luck. If your FFT shows narrow peaks, you’re looking at interference, and the rest of this article is the second step.

The second split is by frequency. Noise that rises toward low frequencies is 1/f noise; flat noise is thermal or shot noise; noise that changes with the sensor’s mechanical design (and not with the electronics) may be mechanical-thermal.

How the main sensor noise sources actually work

Thermal noise: the floor set by resistance and temperature

In 1928, J. B. Johnson reported in Physical Review that every conductor produces a fluctuating voltage from the thermal agitation of its charge carriers, and Harry Nyquist derived the formula in a companion paper in the same issue. The voltage noise density is √(4kTR), where k is Boltzmann’s constant, T is absolute temperature and R is resistance. Analog Devices’ tutorial MT-047 puts the practical number on it: a 1 kΩ resistor produces about 4 nV/√Hz at 25 °C. The noise scales with the square root of resistance, so 100 kΩ gives about 40 nV/√Hz.

Two things follow. Every resistive sensor (strain gauge bridge, RTD, thermistor) and every resistor in the signal path contributes noise you can calculate in advance. And the only levers are lower resistance, lower temperature and narrower bandwidth.

Shot noise: the graininess of current

Walter Schottky described in 1918, in Annalen der Physik, the fluctuation that arises because current crossing a junction is made of discrete charges. MT-047 gives the density as √(2qI), with q the electron charge and I the current. It matters for photodiodes, for bias currents of bipolar and JFET amplifier inputs, and for any sensor read as a small current. MT-047 adds the practical point: an amplifier’s current noise only becomes a voltage when it flows through an impedance, so its effect grows with the source resistance.

1/f noise: why slow measurements are noisier than expected

Below a certain frequency, noise density rises as frequency falls. MT-048 calls the crossover the 1/f corner and gives typical values: about 1 to 10 Hz for the best low-noise amplifiers, up to around 100 Hz for JFET and general-purpose op amps, and higher for very fast ones. Robert Kiely’s Analog Dialogue article (2017) adds that the physical sources of 1/f noise are still debated, and gives the practical guidance: once the system bandwidth extends more than about a decade above the corner, 1/f noise contributes little; for slow measurements (weighing, temperature, DC strain), it can dominate.

Kiely also draws the line that matters most for sensor designers. Chopper-stabilized (zero-drift) amplifiers move the amplifier’s own 1/f noise up to the chopping frequency, but they don’t remove 1/f noise generated in the sensor. For that, the sensor’s excitation itself has to be modulated, for example with AC excitation of a bridge, so the signal moves away from DC while the sensor’s drift doesn’t.

Mechanical-thermal noise: when the sensor’s own mass jitters

Thermal agitation acts on moving parts too. T. B. Gabrielson’s 1993 paper in IEEE Transactions on Electron Devices showed that the small moving structures in micromachined acoustic and vibration sensors are especially susceptible to mechanical noise from molecular agitation, and that this mechanical-thermal noise is often one of the limiting noise sources in small-signal applications. No amplifier can remove it, because it’s in the physical quantity the sensor measures. If a MEMS sensor’s noise floor doesn’t improve with better electronics, this is the likely reason.

Quantization noise: the converter’s contribution

The ADC adds its own noise: an RMS value of q/√12 for a step size q, as Analog Devices’ MT-001 derives. With a well-designed front end it should be below the analog noise; if it isn’t, more gain before the converter or a converter with smaller steps helps.

How they add up

MT-047 states the combining rule: uncorrelated noise voltages add as a root-sum-square, √(V₁² + V₂² + V₃²), and a term three to five times larger than the others dominates. Kiely’s rule of thumb is similar: a source below about one-fifth of the largest adds only a percent or two to the total.

Worked example (illustrative values): a 1 kΩ bridge (≈ 4 nV/√Hz thermal), an amplifier with 3 nV/√Hz voltage noise and negligible current noise. Total density = √(4² + 3²) = 5 nV/√Hz. Over a 1 kHz single-pole bandwidth (noise bandwidth about 1.57 kHz, per MT-048), RMS noise ≈ 5 nV × √1570 ≈ 0.20 µV at the input. Replacing the amplifier with a 1.5 nV/√Hz part gives √(4² + 1.5²) ≈ 4.3 nV/√Hz, only 15% better. The bridge sets the floor.

Sensor noise sources compared

Source	Physical cause	Spectrum	Scales with	What reduces it	What doesn’t
Thermal (Johnson)	Thermal agitation of charge carriers	Flat (white)	√(R·T·bandwidth)	Lower resistance, narrower bandwidth, cooling	Better amplifier
Shot	Discrete charges crossing a junction	Flat	√(current·bandwidth)	Lower current, narrower bandwidth	Lower resistance
1/f (flicker)	Debated; device and material processes	Rises toward DC	1/f below the corner	Chopping (amplifier only), AC excitation (sensor too)	Longer averaging at DC
Mechanical-thermal	Molecular agitation of moving parts	Shaped by the mechanical response	Sensor mass and damping	Sensor design, larger proof mass	Any electronics
Quantization	Finite ADC step size	Roughly flat	Step size	More gain, finer converter, oversampling	Better sensor
Interference (not random)	Coupling from mains, switching, radio	Narrow peaks	Layout and environment	Shielding, grounding, filtering, synchronous sampling	Averaging

The “what doesn’t” column explains the opening. Changing the amplifier only helps when the amplifier is the dominant term, and in a resistive-sensor front end it often isn’t.

What to actually do

Measure the floor with the sensor replaced by its source resistance

Replace the sensor with a resistor of the same value (or short the input for a low-impedance sensor) and record the noise. Then reconnect the sensor in a quiet, stable condition. If the two are similar, your electronics set the floor. If the sensor version is much noisier, the sensor (or its environment) dominates, and electronics upgrades won’t help.

Calculate before you buy

For each source, write its density at your operating point: √(4kTR) for each significant resistance, amplifier voltage noise, current noise times source resistance, and quantization noise referred to the input. Combine them as a root-sum-square. The biggest term is where your effort goes.

Look at the spectrum, not just the RMS value

An FFT separates the cases quickly: narrow peaks are interference, a flat floor is thermal or shot noise, a floor rising toward DC is 1/f. The fix for each is different, and the RMS number alone hides which one you have.

Match the fix to the source

Narrow the bandwidth for white noise (noise falls with its square root). Modulate the excitation for sensor 1/f noise. Use a zero-drift amplifier when the amplifier’s own 1/f noise is the dominant term. Change the sensor when mechanical-thermal noise sets the floor.

What not to do

Don’t increase a bridge’s resistance to save excitation current without recalculating noise: thermal noise rises with its square root. And don’t average a slow measurement indefinitely expecting the noise to keep falling. Once 1/f noise dominates, longer averaging stops helping and drift takes over.

How to put it to work

If you have an hour

List every resistance and active component in your sensor front end, write each one’s noise density, and calculate the root-sum-square. Circle the largest term.

If you have a sprint

Take three captures: input shorted (or replaced by a matching resistor), sensor connected at rest, and sensor in its real environment. Compute the spectrum of each. The differences tell you whether the electronics, the sensor or the environment sets your floor.

If you want it to last the product’s lifetime

Keep the noise budget with the schematic and rerun the three captures when a component, enclosure or cable changes. Interference especially changes with the installation, so a field capture is worth more than a lab one.

The question to ask yourself: “Which single noise source would I have to halve to make a visible difference, and have I measured that it’s the largest?”

Frequently asked questions

What are the main sources of noise in a sensor?

The main random sources are thermal (Johnson) noise from resistance, shot noise from current crossing junctions, 1/f (flicker) noise at low frequencies, mechanical-thermal noise in small moving structures such as MEMS, and quantization noise from the ADC. Interference from mains, switching supplies and radio is a separate, deterministic category that needs different fixes.

How do you calculate thermal noise of a resistor?

Voltage noise density is √(4kTR), with k = 1.38 × 10⁻²³ J/K, T in kelvin and R in ohms. A 1 kΩ resistor gives about 4 nV/√Hz at 25 °C. Multiply by the square root of the noise bandwidth to get RMS noise, for example about 126 nV RMS over 1 kHz.

What is 1/f noise and how do I reduce it?

1/f noise is low-frequency noise whose power density rises as frequency falls, below a corner frequency that ranges from about 1 Hz to hundreds of hertz depending on the device. Zero-drift (chopper) amplifiers remove the amplifier’s own 1/f noise; modulating the sensor’s excitation (AC excitation) is needed to remove the sensor’s.

How do noise sources add together?

Uncorrelated noise sources add as a root-sum-square: total = √(n₁² + n₂² + …). A source three to five times smaller than the largest has little effect on the total, so improving it is wasted effort. Correlated sources, such as the same interference coupled into two paths, don’t follow this rule.

Why doesn’t averaging reduce my noise anymore?

Averaging reduces white noise by the square root of the number of samples, but it can’t reduce 1/f noise or drift, which grow with the length of the measurement, and it doesn’t reliably remove interference at a fixed frequency. When averaging stops helping, the dominant source has changed.

Related reading: ADC resolution vs accuracy, how to read a sensor datasheet and grounding and shielding sensor signals.

Sources

1. Johnson, J. B. (1928). “Thermal Agitation of Electricity in Conductors.” Physical Review, 32(1), 97–109. https://doi.org/10.1103/PhysRev.32.97
2. Nyquist, H. (1928). “Thermal Agitation of Electric Charge in Conductors.” Physical Review, 32(1), 110–113. https://doi.org/10.1103/PhysRev.32.110
3. Schottky, W. (1918). “Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern.” Annalen der Physik, 362(23), 541–567. https://doi.org/10.1002/andp.19183622304
4. Gabrielson, T. B. (1993). “Mechanical-Thermal Noise in Micromachined Acoustic and Vibration Sensors.” IEEE Transactions on Electron Devices, 40(5), 903–909. https://doi.org/10.1109/16.210197
5. Analog Devices (2008). “Op Amp Noise.” Tutorial MT-047, Rev. 0. https://www.analog.com/media/en/training-seminars/tutorials/MT-047.pdf
6. Analog Devices (2008). “Op Amp Noise Relationships: 1/f Noise, RMS Noise, and Equivalent Noise Bandwidth.” Tutorial MT-048, Rev. 0. https://www.analog.com/media/en/training-seminars/tutorials/MT-048.pdf
7. Kiely, R. (2017). “Understanding and Eliminating 1/f Noise.” Analog Dialogue, 51(5). Analog Devices. https://www.analog.com/media/en/analog-dialogue/volume-51/number-2/articles/understanding-and-eliminating-1-f-noise.pdf