165. Noise: addition, resistors, the amplifier, 1/f and the ADC budget

Noise is the only error that cannot be calibrated out: it is random. All you can do is keep it out of the path — choose the resistances, the amplifier and the bandwidth. This section is about how to calculate how much noise reaches the ADC input, how to decide whether it is worth fighting, and when noise actually helps.

Terms used in the documentation

Term	Meaning
thermal (Johnson) noise	noise of any resistor due to temperature
voltage noise density (en)	amplifier noise voltage per root hertz
current noise density (in)	amplifier noise current per root hertz
source impedance	internal resistance of the signal source
1/f (flicker) noise	noise that rises towards low frequencies
corner frequency	where 1/f noise equals white noise
noise bandwidth	equivalent bandwidth for noise calculation
root sum of squares (RSS)	square root of the sum of squares
input-referred noise	noise recalculated to the input
peak-to-peak noise	full span of the noise
quantization noise	error from rounding to a code
oversampling / averaging	taking extra samples / averaging them
dither	noise added on purpose
signal-to-noise ratio (SNR)	ratio of signal to noise

165.1 Noise adds as a root

Independent noise sources do not add directly — their squares add: U = √(U₁² + U₂² + …). Two equal sources of 2 µV each give 2.83 µV, not 4. A 10 µV source and a 1 µV source together give 10.05 µV: the smaller source is almost invisible.

Hence the first rule of fighting noise: find the strongest source and work only on it. Reducing a weak one tenfold gains less than a percent.

165.2 Thermal noise of a resistor

Every resistor makes noise: U = √(4 · k · T · R · Δf). In the 20 Hz…20 kHz band at 25 °C the ngspice model gives:

A resistance a hundred times larger gives ten times more noise: it grows as the square root of the resistance and of the bandwidth.

165.3 Amplifier and stage noise

Amplifier noise is specified as a spectral density referred to the input — for example, 20 nV/√Hz. In a ×11 stage it reaches the output together with the noise of the feedback resistors. Model, 20 Hz…20 kHz band:

With 100 kΩ / 10 kΩ resistors their noise is already a noticeable share; by reducing the resistances tenfold at the same gain, we get almost the noise of the amplifier alone. Below a certain level the noise is set only by the part, and from there only a part with lower density helps.

165.4 Current noise and source resistance

An amplifier has a second noise — current noise in. By itself it is invisible: it becomes a voltage only across a resistance it flows through. A follower with 20 nV/√Hz voltage noise on a sensor with an internal resistance of 1 MΩ, 20 Hz…20 kHz band:

For the bipolar input the current noise on a megohm gives 0.5 pA/√Hz × 1 MΩ = 500 nV/√Hz — 25 times its own voltage noise, and four times the noise of the FET input. For a FET input the current noise is negligible, and only the noise of the sensor itself remains — it cannot get any lower.

The boundary between "voltage noise matters" and "current noise matters" is the ratio en / in: here 20 nV / 0.5 pA = 40 kΩ. For a source well below this resistance, choose the amplifier by voltage noise; above it, by current noise — and for high-impedance sensors (photodiode, piezo sensor, electrochemical sensor) a FET input is used.

165.5 1/f noise and the corner frequency

At low frequencies amplifier noise grows as 1/√f — this is 1/f noise. The frequency at which it equals the white noise is the corner frequency fc. The total noise in the band fmin…fmax:

U = en · √(fc · ln(fmax / fmin) + fmax − fmin)

For a 20 nV/√Hz amplifier with a 100 Hz corner, in the 0.1…10 Hz band — the one where slow sensors live: 434 nV. White noise in this band is only 63 nV — 1/f noise is seven times larger. That is why precise slow measurements use amplifiers with a low corner frequency or with auto-zero, and a filter cannot remove it: it is in the same band as the signal.

165.6 Bandwidth decides everything

Noise grows as the square root of the bandwidth, and without a filter the bandwidth is limited only by the amplifier itself. A ×11 stage on 10 kΩ / 1 kΩ:

In the model with an ideal amplifier the filter removes noise twentyfold. A real amplifier with 1 MHz bandwidth (sections 162–164) itself cuts everything above 91 kHz, and the filter gains about sevenfold. In both cases a filter in front of the ADC that cuts everything above the signal band is cheaper than any low-noise part.

165.7 Noise budget for the ADC

One code of a 12-bit ADC with a 3.3 V scale is 806 µV. Peak-to-peak noise is roughly 6.6 times its RMS value: 11.2 µV after the filter is about 74 µV peak-to-peak, ten times less than a code. Such a path shows stable codes, and there is no need to deal with noise.

The calculation goes like this: 1. refer all sources to one point (usually the ADC input); 2. add them as a root sum of squares and multiply by 6.6 — this gives the peak-to-peak value; 3. compare with the ADC code: if the peak-to-peak noise is less than a code, the noise is not visible; if it is more — first narrow the bandwidth, then reduce the resistances, and only then change the amplifier.

165.8 Quantization and averaging

The ADC itself also adds an error: rounding to a code is uniformly distributed within a code, and its RMS value is code / √12, 232.6 µV. The signal-to-noise ratio of an ideal 12-bit ADC is 74.0 dB. The quiet path from 165.7 is invisible against this background: √(11.2² + 232.6²) = 232.8 µV.

Averaging N samples reduces the random error by √N: 16 samples — fourfold, that is two more bits. But only if there is noise. An experiment: an ideal 12-bit ADC, a constant input from 0 to 3 codes, the average of 16 samples:

The quiet path gives 16 identical samples, and the average repeats the staircase. With half a code of noise the samples spread over neighbouring codes, and their average follows the input smoothly; the random error is √((0.5² + 1/12) / 16) = 0.144 code: the squares of noise and quantization add up and are divided by 16, so the RMS value drops fourfold.

A real ADC also has its own noise, and it mixes itself in: the samples even of a quiet path usually spread over neighbouring codes. Averaging gains √N only against white noise; it barely touches 1/f noise from 165.5 and drift.

Hence the dithering trick: noise or a sawtooth with a span of one or two codes is added on purpose to a path that is too quiet, so that averaging yields extra bits. The reverse is also true: if averaging is needed, the filter in front of the ADC must not cut the noise down to fractions of a code.

165.9 Pitfalls

A low-noise amplifier was fitted, and the noise did not drop. The noise was set by the resistors or the bandwidth, not by the part.

Megohm feedback "is noisy". Thermal noise grows as the square root of resistance: 10 MΩ is ten times noisier than 100 kΩ.

A high-impedance sensor is noisier than calculated. Amplifier current noise across the source resistance; a bipolar input on megohms — use a FET input.

A filter does not help a slow sensor. In a band of fractions of a hertz 1/f noise dominates; it cannot be filtered out — you need a part with a low corner.

ADC codes "jump" with a steady signal. The peak-to-peak noise is larger than a code; most often the filter in front of the ADC is missing.

Averaging did not add bits. The noise at the ADC input is much less than a code — all samples are identical; add noise or a sawtooth.

Noise sources were added directly and overestimated. Independent sources add as the root of the sum of squares.

165.10 Summary

• Independent noise adds as a root: 2 and 2 µV give 2.83 µV, 10 and 1 µV give 10.05 µV; only the strongest source is worth fighting.
• A 100 kΩ resistor in the 20 Hz…20 kHz band makes 5.74 µV of noise, 10 MΩ 57.4 µV.
• A ×11 stage with a 20 nV/√Hz amplifier: 36.5 µV on 100 kΩ / 10 kΩ and 31.7 µV on 10 kΩ / 1 kΩ.
• On a 1 MΩ source a bipolar input gives 73.0 µV, a FET input 18.4 µV; the selection boundary is en / in = 40 kΩ.
• 1/f noise with a 100 Hz corner in the 0.1…10 Hz band is 434 nV versus 63 nV of white noise.
• Without a filter, in the band up to 1 MHz — 219 µV with an ideal amplifier and 82 µV with a 1 MHz amplifier; a 1.59 kHz RC filter leaves 11.2 µV, with an ADC code of 806 µV.
• Quantization is 232.6 µV; averaging 16 samples works only with noise: without it a staircase up to 0.49 code, with half a code of noise 0.14 code.

Next: section 166 "Resistors in detail: TCR, pulse load, shunt, battery divider".