Thermal White Noise - Johnson–Nyquist noise

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SUMMARY

The discussion centers on measuring thermal white noise, specifically Johnson–Nyquist noise, generated by chemical batteries. Participants debate the correct formulas to use, emphasizing that classical thermodynamics leads to a power spectral density independent of frequency, while quantum effects introduce frequency dependency. The conversation highlights the importance of using the correct equations, particularly the one derived from Nyquist's work, which is applicable at low frequencies. The complexities of calculating frequency in relation to thermal noise and the implications of using the wrong formula are also addressed.

PREREQUISITES
  • Understanding of Johnson–Nyquist noise and its derivation from classical thermodynamics.
  • Familiarity with the concept of power spectral density and its measurement.
  • Knowledge of quantum effects in thermal noise calculations.
  • Basic grasp of complex frequency and its implications in signal processing.
NEXT STEPS
  • Study the derivation of Johnson–Nyquist noise from classical thermodynamics.
  • Learn about the application of the Lambert W equation in solving complex frequency problems.
  • Research the differences between classical and quantum approaches to thermal noise.
  • Examine the role of measurement bandwidth in noise analysis, particularly in low-frequency applications.
USEFUL FOR

Electrical engineers, physicists, and researchers involved in noise measurement and analysis in electronic systems, particularly those working with thermal noise in batteries and low-noise RF applications.

  • #31
the_emi_guy said:
This is where you are getting off track, noise does not add in the manner you are describing.


We need to know the *bandwidth* of the 20KHz signal, or more specifically, the bandwidth of the receiver that will be trying to recover this signal. I'll explain why shortly.



Here you are saying that the thermal noise is both white and limited to the very small 20-1000Hz band. These are mutually exclusive statements. White means its power spectral density at 1MHz is the same as at 20Hz.

So let's say that the power spectral density is 1nW per Hz of bandwidth. This means that there will be 1nW of noise power in every 1Hz of bandwidth. If I build a perfect (doesn't create any noise of its own) receiver that tunes to 20KHz with a bandwidth of 1Hz (receives everything between 20,000Hz and 20,001Hz, rejects everything else) then I will have 1nW of noise power coming out of this receiver.

On the other hand, let's say your receiver has 1KHz of bandwidth (20KHz - 21KHz received) then we will have 1000*1nW = 1uW of noise power received.

If your 20KHz signal is 1W, then our SNR is 1W/1uW.

Hope this clarifies things, we need to know the bandwidth of the 20KHz receiver, and the noise power in whatever units (dBm/Hz, nW/Hz).

I'm sorry if things seem to be off track, but I think we're close to solving our problem.
I wanted to add that I calculated the bandwidth using the second equation in the pdf with value; 300 K, (r+R) 1000,2 Ohm, -204dBV/Hz and the Boltzmann's constant of 1,38*10^-3 which gave me a bandwidth of 2,51 *10^21 (Zetta Hz), which in the electromagnetic specter would be gamma radiation (if it was a frequency), which doesn't seem very reasonable to me. And as previously mentioned, the f=KbK/h of 6,2 THz didn't make much sense either. So I'm really, really confused.
 
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  • #32
So you're saying that heat generated from a battery does not output an alternating current in a circuit? The transmitter generates a square wave, so it has infinite bandwidth. However, the oscillator indicated that the transmitter wasn't generating a perfect 20 kHz signal, you could see some oscillations on top of the square wave, and the oscillator indicated it was swinging between 20 000 - 20400 Hz. I understand now the relationship between noise power and bandwidth.
Have you even considered that the variations in the osc freq is due to phase noise being generated by the oscillator and has nothing to do with the battery at all ?

cheers
Dave
 
  • #33
Closed pending moderation.
 
  • #34
Thanks Evo! Thread re-opened after merging two threads on the same subject.
 
  • #35
@Mechatron -- it sounds like the ringing or noise you are seeing is caused by something else. Batteries are not noisy. You can always follow the battery with a low-dropout linear voltage regulator to be extra sure that your power supply is quiet.
 
  • #36
berkeman said:
@Mechatron -- it sounds like the ringing or noise you are seeing is caused by something else. Batteries are not noisy. You can always follow the battery with a low-dropout linear voltage regulator to be extra sure that your power supply is quiet.

Glad you could join berkeman. I can't wait to hear from the_emi_guy again. We're getting close to solve this mysterious problem. It is very clearly stated in the 1995 IEEE International Frequency Control Symposium that batteries generate thermal noise. It's very small, but it matters. Phototransistors are also noisy when they get hot. That is why we add heatsinks to them, to cool them off. But disturbances from voltage supplies can be filtered through a filter capacitor, which acts as a high pass filter. It seems logic to me, to use the signal-to-noise approach. I'll get back to this in the morning and hopefully the_emi_guy will reply soon.
 
  • #37
Mechatron said:
Glad you could join berkeman. I can't wait to hear from the_emi_guy again. We're getting close to solve this mysterious problem. It is very clearly stated in the 1995 IEEE International Frequency Control Symposium that batteries generate thermal noise. It's very small, but it matters. Phototransistors are also noisy when they get hot. That is why we add heatsinks to them, to cool them off. But disturbances from voltage supplies can be filtered through a filter capacitor, which acts as a high pass filter. It seems logic to me, to use the signal-to-noise approach. I'll get back to this in the morning and hopefully the_emi_guy will reply soon.

Adding a bypass capacitor is a good idea, and in combination with a series resistor that forms a low-pass filter, not a high-pass filter.

What do you think about adding a low-dropout linear voltage regulator? that may be your best bet. What level of noise voltage are you shooting for? Are you making uV level measurements? It's hard to believe that the battery noise is creating an issue in a radio transmitted waveform... :smile:
 
  • #38
Mechatron said:
It is very clearly stated in the 1995 IEEE International Frequency Control Symposium that batteries generate thermal noise.

Of course batteries generate white noise, everything with a resistance does. But since the noise is white it does not have a specific "frequency"; the noise contains ALL frequencies within the BW of you measurement; all you see on an oscilloscope is a wide "band", not e.g. a sinewave or any form of pattern If you look at a spectrum analyzer all you see it a flat line. If your noise has "structure", it is either interference or noise with a non-white spectrum (flicker noise, random walk etc).

Hence, whereas the white noise might make the signal from your transmiter slight wider (althought this effect is nearly always extremely tiny compared to other sources of noise) it won't really distort the shape of you signal (unless there are other non-linear effects involved).

The point is, it is extremely unlikely that the noise from the battery is the source of your problem.

I occasionlly do measurement where we both generate and measure frequencies with rather high accuracy (albeit in the GHz range); white noise is almost never a problem as such (unless you count the noise tempeature of the amplifiers), you are mostly fighting with non-white noise (1/f noise in the electronics) and -more importantly- systematic noise from badly filtered power supplies etc.
 
  • #39
berkeman said:
Adding a bypass capacitor is a good idea, and in combination with a series resistor that forms a low-pass filter, not a high-pass filter.

What do you think about adding a low-dropout linear voltage regulator? that may be your best bet. What level of noise voltage are you shooting for? Are you making uV level measurements? It's hard to believe that the battery noise is creating an issue in a radio transmitted waveform... :smile:

A battery can reach temperatures of over 120 Fahrenheit (322 Kelvin). The more current that passes through your circuit, the hotter your battery gets. When you need to power a machine consistently, things can get pretty hot, and this would generate significant thermal noise, which would be very bad for a very sensitive transmitter circuit. Why do I care so much about such low frequencies? Because I want a receiver to focus on a specific signal, so the bandwidth of the bandpass filter will be very small, so the transmitter must generate a very clean signal. So I have to compensate for white noise generated from light, such as generated from 60 Hz lamps and thermal noise from electronic components. I would not go for the voltage regulator, as the voltage regulator would consume more current. And it speaks for itself, it's a voltage regulator. But it's a frequency regulator I need. I'd rather use a rectifier diode. But a filter capacitor does just that, rectifies the DC voltage. A DC battery is actually 99% DC and a fraction AC mixed.
 
  • #40
f95toli said:
Of course batteries generate white noise, everything with a resistance does. But since the noise is white it does not have a specific "frequency"; the noise contains ALL frequencies within the BW of you measurement; all you see on an oscilloscope is a wide "band", not e.g. a sinewave or any form of pattern If you look at a spectrum analyzer all you see it a flat line. If your noise has "structure", it is either interference or noise with a non-white spectrum (flicker noise, random walk etc).

Hence, whereas the white noise might make the signal from your transmiter slight wider (althought this effect is nearly always extremely tiny compared to other sources of noise) it won't really distort the shape of you signal (unless there are other non-linear effects involved).

The point is, it is extremely unlikely that the noise from the battery is the source of your problem.

I occasionlly do measurement where we both generate and measure frequencies with rather high accuracy (albeit in the GHz range); white noise is almost never a problem as such (unless you count the noise tempeature of the amplifiers), you are mostly fighting with non-white noise (1/f noise in the electronics) and -more importantly- systematic noise from badly filtered power supplies etc.

A noise signal is typically considered as a linear addition to a useful information signal. Typical signal quality measures involving noise are signal-to-noise ratio.

Signal-to-noise ratio is defined as the power ratio between a signal (meaningful information) and the background noise (unwanted signal).

I want to know what the linear addition to the 20 kHz signal would be.
 
  • #41
Mechatron said:
I want to know what the linear addition to the 20 kHz signal would be.

In order to answer that you need the BW of your 20 kHz signal plus information of how any noise couples into it. If you assume that the noise voltage is simply added to your signal (which is pretty unlikely given that there must be other components in the circuit go generate the 20 kHz in the first place) you can just calculate the noise per root Hz, multiply by the the BW and then calculate the RMS voltage noise that will be added to our signal. If that voltage is much smaller than the signal amplitude you can eliminate Johnson noise from the battery as being a significant source of noise.

This is all pretty much standard stuff, and several of the people who have already replied in this thread has experience with this (I am one of them).
When we say that the battery is unlikely to be the source of your noise, we are speaking from experience.
 
  • #42
f95toli said:
In order to answer that you need the BW of your 20 kHz signal plus information of how any noise couples into it. If you assume that the noise voltage is simply added to your signal (which is pretty unlikely given that there must be other components in the circuit go generate the 20 kHz in the first place) you can just calculate the noise per root Hz, multiply by the the BW and then calculate the RMS voltage noise that will be added to our signal. If that voltage is much smaller than the signal amplitude you can eliminate Johnson noise from the battery as being a significant source of noise.

This is all pretty much standard stuff, and several of the people who have already replied in this thread has experience with this (I am one of them).
When we say that the battery is unlikely to be the source of your noise, we are speaking from experience.

I can calculate the noise per root Hz, you mean V/sqrt(Hz)? And if I multiply that with Hz, I get VHz/sqrt(Hz). Can you please illustrate this calculation with a mathematical expression?

But the RMS noise voltage is in volts. But the noise that will be added to my signal should be in Hz.
 
  • #43
f95toli said:
In order to answer that you need the BW of your 20 kHz signal plus information of how any noise couples into it. If you assume that the noise voltage is simply added to your signal (which is pretty unlikely given that there must be other components in the circuit go generate the 20 kHz in the first place) you can just calculate the noise per root Hz, multiply by the the BW and then calculate the RMS voltage noise that will be added to our signal. If that voltage is much smaller than the signal amplitude you can eliminate Johnson noise from the battery as being a significant source of noise.

This is all pretty much standard stuff, and several of the people who have already replied in this thread has experience with this (I am one of them).
When we say that the battery is unlikely to be the source of your noise, we are speaking from experience.

Look at the diagram in the link below:
http://s13.postimg.org/m6bl81wtz/snr.png

I need that noise in Hz, not dbV/Hz. I can't see how calculating the RMS voltage noise that will be added to my signal is in Hz. I like the steps you provided, I want to use this method, but the final value must be in Hz.
 
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  • #44
Mechatron said:
Look at the diagram in the link below:
http://s13.postimg.org/m6bl81wtz/snr.png

I need that noise in Hz, not dbV/Hz. I can't see how calculating the RMS voltage noise that will be added to my signal is in Hz. I like the steps you provided, I want to use this method, but the final value must be in Hz.

OK, now I see what you are trying to do.

In order to do this calculation you need more information. The book you are referring to (which I believe I have somewhere) is talking about Johnson noise affecting the feedback signal in the electronics in the oscillator loops.
In a typical "simple" circuit you would have are reference resonator and then feedback electronics creating a loop which oscillates at some frequency (set by the resonator). Now, since the feedback signal is just a voltage any type of noise will interfer it, meaning the output frequency becomes noisy.

Hence, in this situation the voltage noise is "translated" to frequency noise(or phase noise).

However, how this happens depends on the details of the circuit, there is no general answer and iit can become quite complicated, especially if there are non-linear processes involved (which is often the case at RF frequencies). In any real life situation you would probably use SPICE etc to do the calculation.

Moreover, a properly designed circuit will be more of less insensitive to things like noise in the batteries: the white noise you see when measure an high frequency oscillator is typically dominated by the noise temperature of the feedback amplifier. That said, for frequencies as low as 20 kHz most of the "problematic" noise will probably be 1/f noise from both the amplfier and the rest of the electronics (white noise is rarely a real problem since it averages out over long times).


If you want to read more about this I would recommend Enrico Rubiola's book on phase- and frequency noise in oscillators. Enrico is THE guy when it comes to stuff like this.

You can also find plenty of free information on his website
http://rubiola.org/index.html
 
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  • #45
Mechatron,
Can you confirm f95toli's idea that you are concerned about how thermal noise translates to phase noise on your 20KHz clock? If so, be aware that phase noise is measured in time units (ps, UI etc), not Hz.

There is a procedure for doing this but this kind of analysis is used on clocks that are in the hundreds of MHz and GHz. I really doubt that it would make any sense to apply it to a 20KHz signal.

Sounds like you have a 20KHz clock and it does not look clean on your scope?

Maybe you should just send us a screenshot of your actual signal, you may just have a signal integrity issue.
 
  • #46
I did mention osc phase noise many posts ago but it was ignored


Dave
 
  • #47
davenn said:
I did mention osc phase noise many posts ago but it was ignored


Dave

You're right, you may have nailed it on post #32. He may be dealing with a frequency stability or drift issue.
 
  • #48
How about a circuit diagram or block diagram of exactly what is being done, what frequencies are involved, and roughly how the battery noise affects (gets transformed into something that affects) the final performance?
 
  • #49
f95toli said:
OK, now I see what you are trying to do.

In order to do this calculation you need more information. The book you are referring to (which I believe I have somewhere) is talking about Johnson noise affecting the feedback signal in the electronics in the oscillator loops.
In a typical "simple" circuit you would have are reference resonator and then feedback electronics creating a loop which oscillates at some frequency (set by the resonator). Now, since the feedback signal is just a voltage any type of noise will interfer it, meaning the output frequency becomes noisy.

Hence, in this situation the voltage noise is "translated" to frequency noise(or phase noise).

However, how this happens depends on the details of the circuit, there is no general answer and iit can become quite complicated, especially if there are non-linear processes involved (which is often the case at RF frequencies). In any real life situation you would probably use SPICE etc to do the calculation.

Moreover, a properly designed circuit will be more of less insensitive to things like noise in the batteries: the white noise you see when measure an high frequency oscillator is typically dominated by the noise temperature of the feedback amplifier. That said, for frequencies as low as 20 kHz most of the "problematic" noise will probably be 1/f noise from both the amplfier and the rest of the electronics (white noise is rarely a real problem since it averages out over long times).


If you want to read more about this I would recommend Enrico Rubiola's book on phase- and frequency noise in oscillators. Enrico is THE guy when it comes to stuff like this.

You can also find plenty of free information on his website
http://rubiola.org/index.html

That was really useful. I agree that it is rather pink noise than white noise. I agree the noise contain all frequencies within any given bandwidth. But I've been given instructions on how to calculate signal-to-noise ratio by you guys. If only you instructed me how to use this to calculate the distortion of the signal in Hz, that would be great. These little spikes must have some frequency of them selves. Anyway, I do want to calculate this phase noise.

Would you please tell me how I can find the phase noise of a linear process? Taking only the battery and its internal resistance into consideration?
 
  • #50
meBigGuy said:
How about a circuit diagram or block diagram of exactly what is being done, what frequencies are involved, and roughly how the battery noise affects (gets transformed into something that affects) the final performance?

Then I'd have to sue my self for copyright infringement. Since phase noise is measured in dBc/Hz, that still messes up my problem. No matter from what perspective I look at it. I think the phase noise and/or pink noise distort the signal, making it longer, 20Khz+noise (Hz). But if that's not the case... I'm lost.
 
  • #51
f95toli said:
OK, now I see what you are trying to do.

In order to do this calculation you need more information. The book you are referring to (which I believe I have somewhere) is talking about Johnson noise affecting the feedback signal in the electronics in the oscillator loops.
In a typical "simple" circuit you would have are reference resonator and then feedback electronics creating a loop which oscillates at some frequency (set by the resonator). Now, since the feedback signal is just a voltage any type of noise will interfer it, meaning the output frequency becomes noisy.

Hence, in this situation the voltage noise is "translated" to frequency noise(or phase noise).

However, how this happens depends on the details of the circuit, there is no general answer and iit can become quite complicated, especially if there are non-linear processes involved (which is often the case at RF frequencies). In any real life situation you would probably use SPICE etc to do the calculation.

Moreover, a properly designed circuit will be more of less insensitive to things like noise in the batteries: the white noise you see when measure an high frequency oscillator is typically dominated by the noise temperature of the feedback amplifier. That said, for frequencies as low as 20 kHz most of the "problematic" noise will probably be 1/f noise from both the amplfier and the rest of the electronics (white noise is rarely a real problem since it averages out over long times).


If you want to read more about this I would recommend Enrico Rubiola's book on phase- and frequency noise in oscillators. Enrico is THE guy when it comes to stuff like this.

You can also find plenty of free information on his website
http://rubiola.org/index.html

Look what I found!

"Wien's displacement law determines the most likely frequency of the emitted thermal radiation"
http://en.wikipedia.org/wiki/Thermal_radiation
 
  • #52
Mechatron said:
Look what I found!

"Wien's displacement law determines the most likely frequency of the emitted thermal radiation"
http://en.wikipedia.org/wiki/Thermal_radiation

Did you happen to notice the wavelengths involved (horizontal axis of first illustration)?
Convert that to frequency and tell me how that has anything to do with your problem.
 
  • #53
Mechatron said:
That was really useful. I agree that it is rather pink noise than white noise. I agree the noise contain all frequencies within any given bandwidth. But I've been given instructions on how to calculate signal-to-noise ratio by you guys. If only you instructed me how to use this to calculate the distortion of the signal in Hz, that would be great. These little spikes must have some frequency of them selves. Anyway, I do want to calculate this phase noise.

Would you please tell me how I can find the phase noise of a linear process? Taking only the battery and its internal resistance into consideration?


Distortion and phase noise are not typically characterized in Hz. Let's say you have a frequency drift caused by some white/pink battery noise. The amplitude of this noise will be random. The corresponding frequency drift to be random, the frequency is bouncing all over the place. We typically characterize this in units of time-rms (ps rms, ns-rms, UI-rms).

Is this an academic exercise, or are you trying to fix something that isn't working?

Have you tried powering your device from a clean bench power supply?
 
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  • #54
the_emi_guy said:
Distortion and phase noise are not typically characterized in Hz. Let's say you have a frequency drift caused by some white/pink battery noise. The amplitude of this noise will be random. The corresponding frequency drift to be random, the frequency is bouncing all over the place. We typically characterize this in units of time-rms (ps rms, ns-rms, UI-rms).

Is this an academic exercise, or are you trying to fix something that isn't working?

Have you tried powering your device from a clean bench power supply?

This is not academic exercise. I'm researching and developing a transmitter.
So far I know that I'm looking for the flicker noise generated from thermal radiation.
I have found an equation for spectral phase noise:

http://s27.postimg.org/y8m7mo68j/Frequency.png

The flicker noise has a cut-off at the flicker corner frequency of 1/f.
So the corner frequency is measured in 1/f; 1/Hz; Hz ^ - 1 ? Yet is says the unit for corner frequency is in Hz.
But then I read on the following link:
http://en.wikipedia.org/wiki/Corner_frequency
The cutoff frequency or corner frequency is given by angular frequency.
Angular frequency is measured by 2∏f in Hz and not 1/Hz.


So I'm thinking the flicker noise has a frequency similar to the 20kHz signal, with a cut off frequency. So the frequency I'm interested is the difference between the flicker noise and the desired signal. So if the oscillator measures 20400 hz, the cut off frequency is 400 hz. I am thinking correctly?

Reference:
http://www.ieee.li/pdf/essay/phase_noise_basics.pdf
 
  • #55
A simplified block diagram of an analagous system would allow us to talk about it coherently. As it is, it is all arm waving and mis-understanding. You words can be interpreted in many ways, and we (well, they, actually) are slowly honing in on the narrow set of assumtions you are unconsciously making. The field is very broad, and you are viewing it with a set of hidden assumptions. An architectural diagram would help immensely.
 
  • #56
meBigGuy said:
A simplified block diagram of an analagous system would allow us to talk about it coherently. As it is, it is all arm waving and mis-understanding. You words can be interpreted in many ways, and we (well, they, actually) are slowly honing in on the narrow set of assumtions you are unconsciously making. The field is very broad, and you are viewing it with a set of hidden assumptions. An architectural diagram would help immensely.

Come on man, I'm almost there. Look at what I posted just recently.
 
  • #57
meBigGuy said:
A simplified block diagram of an analagous system would allow us to talk about it coherently. As it is, it is all arm waving and mis-understanding. You words can be interpreted in many ways, and we (well, they, actually) are slowly honing in on the narrow set of assumtions you are unconsciously making. The field is very broad, and you are viewing it with a set of hidden assumptions. An architectural diagram would help immensely.

This problem really hertz
 
  • #58
Mechatron said:
This is not academic exercise. I'm researching and developing a transmitter.
So far I know that I'm looking for the flicker noise generated from thermal radiation.
I have found an equation for spectral phase noise:

http://s27.postimg.org/y8m7mo68j/Frequency.png

The flicker noise has a cut-off at the flicker corner frequency of 1/f.
So the corner frequency is measured in 1/f; 1/Hz; Hz ^ - 1 ? Yet is says the unit for corner frequency is in Hz.
But then I read on the following link:
http://en.wikipedia.org/wiki/Corner_frequency
The cutoff frequency or corner frequency is given by angular frequency.
Angular frequency is measured by 2∏f in Hz and not 1/Hz.


So I'm thinking the flicker noise has a frequency similar to the 20kHz signal, with a cut off frequency. So the frequency I'm interested is the difference between the flicker noise and the desired signal. So if the oscillator measures 20400 hz, the cut off frequency is 400 hz. I am thinking correctly?

Reference:
http://www.ieee.li/pdf/essay/phase_noise_basics.pdf

You need to understand that noise does not "have" a frequency. Look at the diagram in your first link. It is showing you the frequency distribution of flicker noise; extending from DC up to the point where other noise sources dominate (and beyond). The typical process is to integrate the phase noise over a frequency band of interest to obtain total noise power (in that band). That can then be converted to, say, picoseconds *RMS* of phase deviation of your clock (emphasis on RMS, the instantaneous phase deviation will be bouncing all over the place since it is caused by noise).

You have a 20KHz clock, and it is measuring 20400Hz with a frequency counter?
Have you tried replacing the suspect "hot" battery with a bench power supply?
 
  • #59
the_emi_guy said:
Distortion and phase noise are not typically characterized in Hz. Let's say you have a frequency drift caused by some white/pink battery noise. The amplitude of this noise will be random. The corresponding frequency drift to be random, the frequency is bouncing all over the place. We typically characterize this in units of time-rms (ps rms, ns-rms, UI-rms).

Is this an academic exercise, or are you trying to fix something that isn't working?

Have you tried powering your device from a clean bench power supply?

Tell me something. This noise voltage. Don't you agree that it's AC voltage? AC noise voltage? And AC voltages are generated with frequencies? Yes?
 
  • #60
Mechatron said:
Tell me something. This noise voltage. Don't you agree that it's AC voltage? AC noise voltage? And AC voltages are generated with frequencies? Yes?

I haven't been following this thread for the last page or so. Have you posted o'scope pictures of the ringing/noise and a schematic of your setup?
 

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