Confused about Power Spectral Density (and relation with rms noise voltage)

[nonequilibrium]

Hello,

In the chapter on thermal noise it says that the spectral distribution per Ohm of resistance is a certain shape (constant till a certain f and then ~ 1/f²).

This means that, in the frequency range below the cutoff value, the noise power in the bandwidth $$\Delta f$$ is equal to $$4k_B T \Delta f$$ per Ohm of resistance.

What is this expression? Is it physically the power dissipitating in the resistor, at least a certain part of it corresponding to a specific selection of frequencies in the Fourier transform? So if you'd integrate over all the frequencies, you'd have the power dissipitated by the current in a resistor? Isn't this usually I or V-dependent? Or is this is a different phenomenon additional to the Joule heating?

The next sentence I find the most puzzling:

The root mean square (rms) noise voltage on a resistor R will then be equal to $$V_{rms} = \sqrt{4k_B T (\Delta f )R}$$. It can give large disturbances in broadband measurements.

So the rms voltage is a measure for the noise on the regular intended voltage, correct? But how is this dependent on a frequency-interval? What does it mean to say that if my interval of frequencies is bigger, the noise on my voltage is larger? I don't understand what interval of frequencies we're talking about. I would think it would only make sense if we integrated it over all the frequencies, then I would think the rms voltage would stand for the mean noise on the voltage, but what does it mean for a selected range of frequencies? What is a "broadband measurement"? Basically: what's the physical relevance of $$\Delta f$$? For example, the next line is "At 300K on 1 megaOhm resistor and Delta f = 10 MHz [...]" I get what it means to say "300K" and "1 megaOhm resistor", but not what "Delta f = 10 MHz" means...

 

[f95toli]

Delta f is simply the bandwidth of your system. Every electrical system has a finite bandwidth, and the rms voltage generated across the resistor depends on that.

Note that this also works with e.g. a bandpass filter. One way to generate white(=thermal) noise in certain bandwidth is to simply put a bandpas filter after the resistor.

 

[nonequilibrium]

Hm, so could I say that the quoted expression for the rms voltage in the OP (the one with Delta f) is actually integrated over all f, because the frequencies outside of the interval Delta f correspond to a zero amplitude in the signal?

And when you say "Delta f is simply the bandwidth of your system.", would it be just as okay to say "Delta f is simply the bandwidth of your signal", or is there an important difference?

Thank you

 

[DickL]

Delta f isn't always the bandwidth of the system. It may be wider or narrower than the system's bandwidth. It frequently depends on what you are doing or interested in. The delta f may be selected to concentrate on a particular noise signal or source, or to examine what is happening at a particular harmony of the primary signal function. It is not uncommon to only be concerned with part of the total system bandwidth, you maybe looking for the noise power or Vrms within that portion of the bandwidth.

 

[sardar]

Dear reader,

The power spectral density (PSD) is a measure of the power of a signal as a function of frequency. In the context of thermal noise, it represents the distribution of noise power per unit frequency in a resistor. The shape of the PSD, as mentioned, is constant until a certain frequency (known as the cutoff frequency) and then decreases with increasing frequency according to a 1/f² relationship. This is a characteristic of thermal noise and is a result of the random motion of electrons in the resistor.

The expression you mentioned, 4k_B T \Delta f, represents the noise power in a bandwidth \Delta f at a temperature T. This is the power dissipated in the resistor due to the thermal noise. This is a separate phenomenon from Joule heating, which is the power dissipated in a resistor due to a current passing through it.

The root mean square (rms) noise voltage is a measure of the average noise voltage in a resistor. It is calculated by taking the square root of the noise power in a given bandwidth and multiplying it by the resistance of the resistor. This means that the rms noise voltage is dependent on both the temperature and the bandwidth of the measurement. The larger the bandwidth, the more frequencies are included in the measurement and thus the higher the rms noise voltage will be. This is why it is important to consider the bandwidth in measurements, as it can greatly affect the results.

In the example you mentioned, "Delta f = 10 MHz" means that the measurement is being taken over a frequency range of 10 MHz. This is a fairly large bandwidth and would result in a higher rms noise voltage compared to a measurement with a smaller bandwidth.

In summary, the power spectral density and rms noise voltage are important concepts in understanding and measuring thermal noise in resistors. The PSD represents the distribution of noise power per unit frequency, while the rms noise voltage is a measure of the average noise voltage in a resistor. Both are dependent on temperature and bandwidth, and must be considered in measurements to accurately determine the level of noise present. I hope this helps clarify your confusion about power spectral density and its relation to rms noise voltage.