## Noise corner calculation.

### Introduction.

Amplifier input voltage and current noise is normally specified in the data-sheets as typical values at 2 or 3 frequencies or as a graph.
For simulation purposes, you will often want to specify the noise as a mid-band value and as a frequency where the 1/f noise dominates.

### Noise corner calculation.

Fig.1: Graph of amplifier voltage noise vs. frequency.

Fig.1 shows a simulation of an amplifier with 10 nV/√Hz noise and a 1 kHz 1/f noise corner.
The graph is plotted over a very large frequency range for clarity.
The blue line is the thermal noise, the cyan line is the 1/f noise with a slope of 10 dB/decade and the lime plot is the sum.

The 1/f noise corner is:

where:
fc: The noise corner.
en: The thermal noise level ( 10 nV/√Hz ).
enf: The 1/f noise level ( 300 nV/√Hz ).
fenf: The frequency where enf is measured ( 1 Hz ).

If you put these numbers ( 10 nV/√Hz, 300 nV/√Hz and 1 Hz ) into the formula, the result is 900 Hz.
If you put the exact numbers ( 10 nV/√Hz, 316 nV/√Hz and 1 Hz ) into the formula, the result is 999 Hz.
This is an error of 10%, but it can be very difficult to read a graph more accurately.
If the component data-sheet specify the noise at 2 different frequencies, use those numbers.
If you design components for use in Spice, use this calculation as an initial guess and trim the value of fc until the model matches your data-sheet.
For linear audio applications, an error of 10% is often not a big issue as long as fc <1 kHz.
Table 1 below shows the increase in wide-band ( 20 Hz to 20 kHz ) noise for different values of fc for a linear amplifier and a RIAA equalized amplifier.
For a linear amplifier, an increase of fc from 50 Hz to 500 Hz will increase the wide-band noise by 0.6 dB which is acceptable for many applications.
For a RIAA equalized amplifier, an increase of fc from 50 Hz to 500 Hz will increase the wide-band noise by 6.4 dB which may or may not be acceptable for a given application.

 Linear amplifier RIAA equalized amplifier fc Noise, 20 Hz..20 kHz Increase over fc=1 Hz Noise, 20 Hz..20 kHz Increase over fc=1 Hz 1 Hz 1.41 µV -114.8 dBu 0 dB 934 nV -118.4 dBu 0 dB 20 Hz 1.42 µV -114.7 dBu 0 dB 1.03 µV -117.5 dBu 0.9 dB 50 Hz 1.43 µV -114.7 dBu 0.1 dB 1.17 µV -116.4 dBu 1.9 dB 100 Hz 1.44 µV -114.6 dBu 0.1 dB 1.37 µV -115.1 dBu 3.3 dB 200 Hz 1.46 µV -114.5 dBu 0.3 dB 1.70 µV -113.2 dBu 5.2 dB 500 Hz 1.53 µV -114.1 dBu 0.7 dB 2.43 µV -110.1 dBu 8.3 dB 1 kHz 1.64 µV -113.5 dBu 1.3 dB 3.30 µV -107.4 dBu 11.0 dB 2 kHz 1.84 µV -112.5 dBu 2.3 dB 4.58 µV -104.6 dBu 13.8 dB 5 kHz 2.33 µV -110.4 dBu 4.4 dB 7.15 µV -100.7 dBu 17.7 dB 10 kHz 2.98 µV -108.3 dBu 6.5 dB 10.1 µV -97.7 dBu 20.6 dB 20 kHz 3.98 µV -105.8 dBu 9.0 dB 14.2 µV -94.7 dBu 23.6 dB

Some people always ask me for A-weighted figures. Unless you have something to hide, I do not know what you want with them, but here they are:

 Linear amplifier RIAA equalized amplifier fc Noise, 20 Hz..20 kHz Increase over fc=1 Hz Noise, 20 Hz..20 kHz Increase over fc=1 Hz 1 Hz 1.12 µV -116.8 dBu 0 dB 567 nV -122.7 dBu 0 dB 20 Hz 1.12 µV -116.8 dBu 0 dB 572 nV -122.6 dBu 0.1 dB 50 Hz 1.12 µV -116.8 dBu 0.1 dB 579 nV -122.5 dBu 0.2 dB 100 Hz 1.13 µV -116.7 dBu 0.1 dB 591 nV -122.4 dBu 0.4 dB 200 Hz 1.15 µV -116.6 dBu 0.2 dB 614 nV -122.0 dBu 0.7 dB 500 Hz 1.19 µV -116.2 dBu 0.6 dB 678 nV -121.2 dBu 1.6 dB 1 kHz 1.27 µV -115.7 dBu 1.1 dB 774 nV -120.0 dBu 2.7 dB 2 kHz 1.40 µV -114.9 dBu 2.0 dB 936 nV -118.4 dBu 4.3 dB 5 kHz 1.74 µV -113.0 dBu 3.9 dB 1.31 µV -115.5 dBu 7.2 dB 10 kHz 2.20 µV -110.9 dBu 5.9 dB 1.76 µV -112.9 dBu 9.8 dB 20 kHz 2.90 µV -108.5 dBu 8.3 dB 2.42 µV -110.1 dBu 12.6 dB

Although only voltage noise is mentioned above, the calculation is valid for current noise too if the current 1/f noise has a 10 dB/decade slope.

### Calculating noise voltage density from noise figure.

For simulation purposes, you will normally want the noise specification in absolute units like V or V/√Hz.
Many discrete low-noise components have the noise specified as a noise-figure ( NF ), so the noise voltage must be calculated from this.
This calculation is greatly simplified by assuming that en >> in*Rg ( as it is with FETs with low generator resistances ), where en is the voltage noise, in is the current noise and Rg is the generator resistance.

If in any way possible, try to find NF from tabular data; many graphs are on a very coarse scale showing a NF of 0 dB over a wide range. This is obviously useless as it indicates a noise voltage of 0.
For this example I use a J-FET with following specs:
NF=5 dB @ VDS=10 V, Rg=100 Ω, ID=5 mA, f=100 Hz, Ta=25 °C.
NF=1 dB @ VDS=10 V, Rg=100 Ω, ID=5 mA, f=1 kHz., Ta=25 °C.

The reference level for this specification is the thermal noise voltage density of a 100 Ω resistor at 25 °C:

where:
k is Boltzmann's constant ( 1.38e-23 J/K )
T is the temperature in K ( 25 °C = 298.15 K )
R is the resistor value ( 100 Ω )
en is the noise voltage density ( 1.28 nV/√Hz ).

The total noise level is:

ent=2.28 nV/√Hz for NF=5 dB and ent=1.44 nV/√Hz for NF=1 dB.

The amplifier noise level is:

ena=1.89 nV/√Hz for NF=5 dB and ent=652 pV/√Hz for NF=1 dB.

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