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Noise corner calculation.

Contents


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.

Typical amplifier noise graph.
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:

fc=(fenf*sqrt(enf^4-en^4)/(en^2).
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.

Table 1: Increase in wideband noise as a function of fc for a linear amplifier and a RIAA equalized amplifier. The RIAA equalizer has 0 dB gain at 1 kHz.
 Linear amplifierRIAA equalized amplifier
fcNoise, 20 Hz..20 kHzIncrease over fc=1 HzNoise, 20 Hz..20 kHzIncrease over fc=1 Hz
1 Hz1.41 µV-114.8 dBu0 dB934 nV-118.4 dBu0 dB
20 Hz1.42 µV-114.7 dBu0 dB1.03 µV-117.5 dBu0.9 dB
50 Hz1.43 µV-114.7 dBu0.1 dB1.17 µV-116.4 dBu1.9 dB
100 Hz1.44 µV-114.6 dBu0.1 dB1.37 µV-115.1 dBu3.3 dB
200 Hz1.46 µV-114.5 dBu0.3 dB1.70 µV-113.2 dBu5.2 dB
500 Hz1.53 µV-114.1 dBu0.7 dB2.43 µV-110.1 dBu8.3 dB
1 kHz1.64 µV-113.5 dBu1.3 dB3.30 µV-107.4 dBu11.0 dB
2 kHz1.84 µV-112.5 dBu2.3 dB4.58 µV-104.6 dBu13.8 dB
5 kHz2.33 µV-110.4 dBu4.4 dB7.15 µV-100.7 dBu17.7 dB
10 kHz2.98 µV-108.3 dBu6.5 dB10.1 µV-97.7 dBu20.6 dB
20 kHz3.98 µV-105.8 dBu9.0 dB14.2 µV-94.7 dBu23.6 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:

Table 2: Same as table.1, but A-weighted.
 Linear amplifierRIAA equalized amplifier
fcNoise, 20 Hz..20 kHzIncrease over fc=1 HzNoise, 20 Hz..20 kHzIncrease over fc=1 Hz
1 Hz1.12 µV-116.8 dBu0 dB567 nV-122.7 dBu0 dB
20 Hz1.12 µV-116.8 dBu0 dB572 nV-122.6 dBu0.1 dB
50 Hz1.12 µV-116.8 dBu0.1 dB579 nV-122.5 dBu0.2 dB
100 Hz1.13 µV-116.7 dBu0.1 dB591 nV-122.4 dBu0.4 dB
200 Hz1.15 µV-116.6 dBu0.2 dB614 nV-122.0 dBu0.7 dB
500 Hz1.19 µV-116.2 dBu0.6 dB678 nV-121.2 dBu1.6 dB
1 kHz1.27 µV-115.7 dBu1.1 dB774 nV-120.0 dBu2.7 dB
2 kHz1.40 µV-114.9 dBu2.0 dB936 nV-118.4 dBu4.3 dB
5 kHz1.74 µV-113.0 dBu3.9 dB1.31 µV-115.5 dBu7.2 dB
10 kHz2.20 µV-110.9 dBu5.9 dB1.76 µV-112.9 dBu9.8 dB
20 kHz2.90 µV-108.5 dBu8.3 dB2.42 µV-110.1 dBu12.6 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:

en=sqrt(4*k*T*R).

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=en*10^(NF/20).

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=sqrt(ent^2-en^2).

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


Downloads.

Spread-sheet with noise corner calculations.


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