Talk:Johnson–Nyquist noise
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Physics B‑class Low‑importance  

Electronics B‑class Low‑importance  

Radiation[edit]
From this paper:
"3) The radiation resistance of free space is √(μ_{0} /ε_{0}). What is the rms noise voltage at the terminals of an antenna which will compete, say, with an FM radio signal?"
Is that supposed to imply that free space generates thermal noise???  Omegatron 18:36, Apr 15, 2005 (UTC)
 You are confusing radiation resistance with the impedance of free space. Radiation resistance is a property of the antenna geometry, and is a measure of the power lost from the transmitter because it is radiated as electromagnetic waves. The impedance of free space describes the ratio of electrostatic to magnetic field in the propagating wave.
 I thought noise power was 4kTBR where B is bandwith and R is the source resistance. Am I wrong again??Light current 01:46, 1 September 2005 (UTC)
 No its Ok. I noticed they are quoting power so resistance drops out of the equation. Its noise voltage/current that depends on resistance Light current 12:29, 1 September 2005 (UTC
 Yes, but the 4 also drops out. Noise power to a matched load resistance is kTB, not 4kTB! 10Jan2006
 The page is rong. You can only state that the transfered johnson noise power to a matchin resistence is kTB, so this is the maximum transfered power. The noise can be modeled by a voltage source in series with the resistence with rms value of sqrt(4 K T B R) or by the norton equivalent of a current source in pararel with the resistence of sqrt(4 K T B / R). If no one corrects me I am editing the page.
I see a number of problems with the discussion of radio communications on this page. First of all, it is NOT true that the noise received by a receiver listening to a radio channel is kTB. It can be much higher OR lower than this. If we exclude external sources of noise (like lightning and the sun), then the dominant noise source is the receiver itself. Internal receiver noise is usually rated by Noise_figure, which, for a receiver, is the ratio of Pout/G*Pin, where Pin is the thermal noise from a resistor at 290K attached to the input of the receiver and Pout is the noise output power of the receiver (usually can consider just the first few stages) and G is the receiver gain. So, a perfect receiver would have a noise figure of 1 (0 dB). A receiver which introduced internal noise equal to thermal noise at 290K (which is what is implied in this article as being always true) would be 2 (3 dB).
UHF and microwave receivers regularly achieve noise figures of less than 1 dB (1.25).
In real use, the receiver is not attached to a resistor as its source but to an antenna with the proper feedpoint impedance. Because the resistive part of the feedpoint impedance is not a power loss but a power transformation (from an EM wave propagating in free space to a signal in the wires going to the receiver), it does not produce Johnson noise.
So, in summary, if you attach an antenna to a receiver, point it away from all radio sources (free space), then you will see simply the receiver internal noise (plus the cosmic background radiation, but that is almost surely much less than the receiver internal noise), and this internal noise may be much less than kTB. If noone responds to this, I will change the article to reflect this, along with links to other relevant articles such as Noise_figure Sbreheny (talk) 06:10, 24 December 2007 (UTC)
It seems to me the P_{receivernoise} equation should represent a closed system at equilibrium. If true, then the antennas temperature is equal to its surrounding temperature. Therefore, the amount of thermal noise received by the antenna due to blackbody radiation is relative to the antennas temperature. Correct me if I'm wrong, but power noise (KTB) is applicable to antennas; e.g., a 10MHzBW 50ohm (radiation resistance) antenna at 300K should receive K*300*10MHz = 4E14 watts. Paul Lowrance, April 13 2008. —Preceding unsigned comment added by 72.25.73.160 (talk) 16:27, 13 April 2008 (UTC)
Hi! Thanks for your interest in this topic. An antenna may not "see" surroundings as you expect, though. For example, a directive antenna pointed up into the sky is almost unaffected by the ground. The blackbody radiation it sees is just the cosmic background at 3K. Therefore, you cannot assume that the noise temperature of the antenna is the same as its physical temperature. There will be a contribution from its physical temperature, but that is determined by the amount of ohmic resistance in the antenna, NOT its entire feedpoint impedance. If the antenna is resonant (impedance is all real), for example, and has a 50 ohm radiation resistance and a 1 ohm ohmic resistance, then the noise power at the output will be attenuated because you have a 1 ohm noise source driving an unmatched load, namely 100 ohms (antenna plus receiver)) Sbreheny (talk) 06:04, 20 May 2008 (UTC)
 I agree. Although I believe my example applies since I made reference to "a closed system at equilibrium." In your example the cosmic background is 2.7K, but if the antenna is 293K then we have an example that is not at equilibrium. BTW, one would be hard pressed to find a 3K night sky on Earth due to atmosphere. I'd hate to step outside with a 3K night sky. Kind of like being in outer space without an insulated spacesuit a bit chilly. :) When considering the cosmic background we must use the universe as the closed system. If both the antenna and surrounding ambient temperature is 3K then the Johnson noise from the antennas radiation resistance is Vn=sqrt(4 k 3Kelvin R BW). Although, if the antennas temperature is different than the surrounding ambient temperature then the problem is more complex. This is an interesting topic and reminds me of the "Dew effect," which is essentially all objects pointing up toward a clear night sky will drop in temperature at a faster rate than other objects, and therefore reaching the dew point temperature before other objects thereby forming dew. For example, the top part of a car will form dew before the sides of a car. A flat surface at 300K radiates 459 watts/m^2, and at 200K radiates 91W/m^2. So if such a surface is facing a 200K night sky the object is losing 368 watts per square meter.PaulLowrance (talk) 14:42, 20 May 2008 (UTC)
I think that you a mixing things up a bit. Radiation is, as I'm sure you know, not the only heat transfer method. Therefore, you can have an antenna which is neither getting hotter or colder but yet "sees" a much colder temperature via radiation. The rest of the thermal input is coming from conduction and convection. The reason why you don't freeze instantly when you step outside is because of the warm blanket of air (as you say) but it is NOT because of radiation from the air, but from convection and conduction. I agree that an antenna pointing at the sky is rarely going to see anything close to 3K, but that is not because of the atmosphere  it is because of cosmic RF sources. The atmosphere's attenuation to some frequencies (for example, around 400MHz) is very low. Since the attenuation is low, this necessarily also means that the noise radiation at that frequency is also low. This is precisely why the "Dew effect" you mention happens. Objects which "see" the ground stay a bit warmer than those which "see" the sky. So, it really is not all that complicated to determine the noise output of an antenna which is pointed at a quiet portion of the sky, in a frequency range where atmospheric attenuation is negligible. It would be equal to the component received from space plus the contribution from the ohmic losses in the antenna. This is often much lower than a noise temperature of 300K (otherwise it wouldn't make sense to make preamps with very low noise figure).Sbreheny (talk) 04:35, 26 May 2008 (UTC)
 I think you're misunderstanding what I am saying. I never said you can't have an antenna that is or is not getting hotter or colder. If the antenna is at a different temperature than its surroundings such as the cosmic sky then the system is not in equilibrium. You're using examples that are not at equilibrium. As stated, my reference to the Thermal noise equation was for a closed system at equilibrium, which means if the antenna is at the same temperature as its surrounding then the amount of Thermal noise received from the antenna is equal to the well known equation Vn = sqrt(4 k T R BW). As to why people don't freeze outside in the night sky, we probably don't agree. The cosmic background is 2.7K, but the night sky temperature seen from Earth is significantly higher. I just measured it a few minutes ago where I live and it's 247K. The reason it's not 2.7K on Earth is due to the atmosphere. The thick atmosphere absorbs an appreciable amount of blackbody radiation, and therefore the atmosphere radiates back to Earth an appreciable amount of blackbody radiation. I would agree convection and conduction play a significant role as to what temperature a body would drop to and reach thermal equilibrium. So going back to a real measurement of 247K (26.2C, 15.1F) night sky, that's not all that cold. Trust me, a person would quickly freeze to death if the night sky from Earth was 2.7K (270.5C, 454.8F).PaulLowrance (talk) 06:49, 26 May 2008 (UTC)
Hi again Paul. The sky's temperature is frequency dependent. I'm sure that in the IR spectrum, it is not 3K. However, at RF frequencies, it is. Check out [1] top of page 5. Since antennas do not receive thermal noise in the IR spectrum, I was focusing on the RF sky noise temperature. As for the first part, I honestly do not know the technical definition of equilibrium (I had thought it meant "not changing temperature" but perhaps I am wrong). My main concern is that antennas are very rarely surrounded by materials which are true blackbodies. Therefore, antennas will usually NOT output noise equal to the equation you give. You may be strictly speaking correct for thermal equilibrium, but since that situation is very rare with real antennas, I don't think it should be given as a typical example in this article. Sbreheny (talk) 05:33, 28 May 2008 (UTC)
 Hi there. Aren't you mixing two of our main discussions together? My reference to the atmosphere was in reference to our discussion of a person freezing if the night sky was 2.7K from Earth perspective, which it is not note to my previous mention of the clear night sky measurement out here at 247K (26.2C, 15.1F). That discussion is relevant to blackbody radiation, which is THz radiation near 30THz the atmosphere radiates an significant amount of such radiation. Our other discussion was regarding antennas receiving thermal noise due to blackbody radiation. I was pointing out that your example was not at equilibrium, and therefore the problem is far more complex *if* one wants an accurate answer. Although your example of an antenna on Earth does indeed receive an appreciable amount of thermal noise from the ground, and *can* receive up to the full amount, sqrt(4 k T R BW). One needs to consider ground type and surroundings. Highly reflective grounds drastically decrease the received thermal noise. Now if one wishes to see an antenna receive a lot of thermal noise then place a 3+ element yagi antenna inside a deep canyon. If an antenna is placed deep enough inside a cave then the antenna will receive nearly sqrt(4 K T R BW) of thermal noise. Another example of an antenna receiving near full thermal noise is say a 10 GHz antenna inside a thick walled building. Going back to antennas facing the night sky, one method of drastically decreasing the received thermal noise is a good ground reflective system. Amateurs radio hobbyist often place grids of wires in the ground under the antenna. Such a ground reflective system can also increases the antennas effective gain. Anyhow, yes, it would be nice to add to the article that antennas receive an appreciable amount of thermal noise and can receive the full Vn = sqrt(4 k T R BW) thermal noise. Would you mind if I added a few simple sentences to the article regarding antennas receiving thermal noise? I probably qualify since in college I wrote an antenna simulation program that specialized in Quad antennas.PaulLowrance (talk) 15:57, 28 May 2008 (UTC)
 Hi Paul. Why don't you give the specific example that you want to include here and then I (and others) can comment? I think you are right that we substantially agree and may not be talking about the same thing. Making it more concrete by looking at exactly what you want to add would probably be best.Sbreheny (talk) 15:27, 31 May 2008 (UTC)
Noise voltage and power[edit]
There are two statements in this section
"The root mean square (rms) of the voltage, , is given by..."
"The root mean square (rms) of the voltage, , is given by..."
I understand the difference in the formulas (the former is per sqrt(Hz) ), but the terminology is confusing. agr 11:31, 17 October 2006 (UTC)
 The first one is incorrect. Hanspi's diff of today added the formula for volts per root hertz, and his diff comment calls it power spectral density, which is the square root of actually. But he didn't fix the text. Dicklyon 15:22, 17 October 2006 (UTC)
"Although the rms value for thermal noise is well defined, the instantaneous value can only be defined in terms of probability. The instantaneous amplitude of thermal noise has a Gaussian, or normal, distribution." (p. 203)
"The crest factor of a waveform is defined as the ratio of the peak to the rms value." ". . . a crest value of approximately 4 is used for thermal noise." (p. 204) [2]—Preceding unsigned comment added by 70.18.4.75 (talk • contribs)
 That would imply a peak that is four standard deviations from the mean, which is not how a Gaussian process works. In fact, the peak factor is essentially infinite for a Gaussian process, or is a random variable with an unboundedly increasing expectation the longer you wait. But stepping back, can you say what question your quotes here were intended to address? Dicklyon 06:43, 21 October 2006 (UTC)
Is the formula for the JohnsonNyquist noise on capacitors valid for both RC circuits in series and in parallel? 66.31.1.215 18:21, 16 February 2007 (UTC)
 Yes, if you use the effective R in both cases, they are equivalent (Norton vs Thevenin models). But if you just use the resistor value and you're hooked up to something with its own impedance that you don't take into account, you'll get the wrong answer. Dicklyon 19:00, 16 February 2007 (UTC)
Thanks for the explanation. I'm sorry though, but I do not understand your answer completely. If I understand you correctly, you say that the effective resistance is the same for both RC circuits. So that would mean you should use the square root of (4kTRf), with that effective resistor as R, right? But is this then, the same as the square root of (kT/C)? Also, I thought Norton and Thevenin only work for resistor circuits, not for RC circuits.
 No, I don't mean they have the same effective R. I mean that IF they have the same effective R, then they have the same RC, same noise bandwidth, and same noise. The R comes from a Norton or Thevenin equivalent of the circuit connected to the C. So a series RC connected to a voltage source, or a parallel RC connected to a current source have the same noise. A series RC and a parallel RC connected both to the same source, of whatever impedance, can never have the same effective R if the two circuits have the same R, because the impedance of the source modifies them in different ways. Dicklyon 00:55, 18 February 2007 (UTC)
Ok, I understand it better now. So only if the Norton and/or Thevenin equivalence hold, are they the same and does the R of the RC not contribute to the noise, right?
Switched Capacitor Filters[edit]
This article seems to imply that noise from Caps is equivalent to noise from Resistors. I'm not sure that this is true, given the earlier teachings on the benefits (noise) of switchedcapacitor filters. Maybe an explanation of the paradox is necessary. Petersk 16:41, 12 April 2007 (UTC)
 Why do you think it's not true? Check any book on switchcap filter noise analysis. Dicklyon 18:36, 12 April 2007 (UTC)
KTC noise (reset noise)[edit]
KTC noise is not fundamental noise. The source of Thermal noise across a capacitor comes from resistance. The source of KTC noise is easy to understand from .noise Spice analysis. Regardless of the parallel resistance across the cap (it could theoretically be infinite, or 1T ohm, 1p Ohm, etc.), the noise comes from parallel resistance across the capacitor. Kent H. Lundberg properly describes KTC noise, Please see page 10: http://web.mit.edu/klund/www/papers/UNP_noise.pdf —Preceding unsigned comment added by 72.25.73.160 (talk) 01:58, 18 April 2008 (UTC)
Also the KTC noise equation, Vn = sqrt(K T / C), is accurate for pure white noise with infinite bandwidth. For example, consider an amazing 220uF cap that has 100T ohm parallel internal resistance. According to the KTC equation the cap noise should be 4.3nV rms, but in reality such KTC noise would be 4.1nV rms due to the limitation of thermal noise bandwidth, which is ~ 7 THz. With a parallel resistance of 500T ohms the KTC noise is 3.4nV rms. —Preceding unsigned comment added by 72.25.73.160 (talk) 14:37, 18 April 2008 (UTC)
 It's not clear what you mean by "not a fundamental noise"; your source says it's just thermal noise, which is true. But it does not say that it comes from a resistance across the capacitor (or if it says that, I can't find it). And it's not clear where the Lundberg paper is published, so not clear that you can even rely on it as a source. Please clarify and point out what you think he says in relation to your assertions. Dicklyon (talk) 15:57, 30 April 2008 (UTC)
 Kent H. Lundberg pdf goes over the basic math how kTC noise is derived. Kent starts with the spectral density of the thermal noise of the resistor, which is V^{2}=4kTR. Kent then shows the noise bandwidth of the resistors thermal noise, which is 1/2pi * pi/2RC = 1/4RC. Kent continues, "Thus, the total output noise voltage that is measurable across the capacitor, in volts, is simply " As you can see, the kTC equation is derive from the thermal noise equation and the resistors thermal noise bandwidth The capacitor is acting as a lowpass filter for the caps parallel resistance thermal noise.PaulLowrance (talk)
 Yes, that's the conventional "hard way" to get to kTC, the long way around by assuming a white noise in a resistor. The only trouble is that when it's not clear what the resistance is, or there's an ideal switch instead of a resistor, it's not clear how it applies. One might think that using a 0resistance switch, opening to infinite resistance, would avoid the noise, if the noise comes from resistance. But, the noise is NOT avoided in this case, since it's not really a consequence of resistance. It's a consequence of electrons in a state variable in a quantum system in thermodynamic equilibrium with an environment at temperature T; the kTC falls out easily from such an analysis, in which it becomes clear that Johnson noise need not be thought of as being caused by resistors. Dicklyon (talk) 05:57, 1 May 2008 (UTC)
 You said, "It's a consequence of electrons in a state variable in a quantum system in thermodynamic equilibrium with an environment at temperature T" Yes, but such a system still has resistance. You have referred to an ideal switch across the cap, and such an ideal switch has infinite *resistance." Even a single ideal capacitor is properly modeled with infinite parallel resistance. I am unaware of how to derive the kTC noise voltage equation without using resistance.PaulLowrance (talk) 17:35, 1 May 2008 (UTC)
 It is impossible to make a switch with infinite resistance or a perfect capacitor with infinite parallel resistance. BTW, what do you mean by "that's the conventional 'hard way' to get to kTC"? Can you please show the easy way of deriving ?PaulLowrance (talk)
 You mentioned "One might think that using a 0resistance switch, opening to infinite resistance, would avoid the noise, if the noise comes from resistance." As you know this is theoretical since an infinite switch is not real, but mathematically there is no problem with this because the resistance cancels out from the noise bandwidth equation and the thermal noise equation. In nonmathematical terms this means the theoretical resistor may be infinite resistance, but the thermal noise is also infinite. What is nice about this conventional method is the use of fundamental equations that apply such as thermal noise and resistance and capacitor filtering work out mathematically on paper and in Spice simulations. In Spice we can take a real capacitor, apply realistic parallel resistance, and arrive at correct results matching the kTC equation. Not to sound repetitive, but it's important to note Spice clearly shows the noise comes from the capacitors parallel resistor.PaulLowrance (talk)
 And kTC noise from opening a switch is DC, not infinite bandwidth. Dicklyon (talk) 15:57, 30 April 2008 (UTC)
 Can you please explain this DC kTC noise? Thanks. As far as I know, kTC noise is not DC and comes from thermal noise.PaulLowrance (talk)
 It can be DC when considering the situation after a reset event. The amount of charge remaining on a capacitor when the switch is opened is a random variable that does not change with time after the event. Each time you do it, you get an independent sample; each one is a different DC value; the variance of the distribution is noise. It's a thermal noise, in that it depends on the thermal coming and going of electrons through the switch when the switch is closed; the variance is proportional to absolute temperature, based on the Boltzmann distribution of electron energies. Dicklyon (talk) 20:55, 30 April 2008 (UTC)
 I fail to see why you believe such kTC noise is DC. The thermal noise caused by the caps parallel resistance causes the voltage across the cap to vary over time, which is not DC since it is random. If you analyze kTC Spice sims you will see the fluctuating charge across the capacitor is due to the capacitors parallel resistance. I see below you believe kTC is immediate and somehow DC. If that is what you believe then perhaps we should first focus on the follow thread before returning to this thread.PaulLowrance (talk)
 If there's a parallel resistor you'll get an AC noise. That's not the situation I was referring to. I'm talking about what's called kTC noise or "reset noise", which is a DC random variable that comes from opening a switch to a voltage source. Sorry if I got us off track, as I notice that "reset noise" is only one part of what the article section discusses. The point of discussing it, however, is to show that the thermal noise in an RC circuit can be attributed to the capacitor as well as to the resistor; it's just a different point of view; instead of resistance and bandwidth, the charge variance can be directly computed from just the capacitance. The noise is not caused by the resistor; it's caused by the electrons. Dicklyon (talk) 03:58, 1 May 2008 (UTC)
 It appears we agree that reset noise is not DC noise since it is caused by nonDC noise, which randomly charges/discharges the capacitor over time.PaulLowrance (talk) 16:35, 1 May 2008 (UTC)
 You have said kTC noise is not due to resistance. Although it appears we agree conventional method derives the kTC equations from resistance and cap bandwidth. Can you show an alternative mathematical method of deriving the kTC noise voltage equation?PaulLowrance (talk) 16:35, 1 May 2008 (UTC)
 Regarding the 2nd equation in your article, charge variance noise. Can we derive this equation without resistance? In your article you derive Qn from the kT/C noise voltage equation, which as I have pointed out is derived from resistance. If true, then both equations are derived from resistance.PaulLowrance (talk) 16:35, 1 May 2008 (UTC)
 I didn't read the entire book, but read all the his section on kTC noise. BTW, I was asking for an alternative method of deriving kTC noise voltage equation, not an explanation. Perhaps we can end this thread since it's now similar to another thread below.PaulLowrance (talk) 23:42, 1 May 2008 (UTC)
 By the way, the beauty of the kTC approach is that it works the same for the RC circuit as for the reset noise. The variance of charge is kTC, which means the variance of voltage is kT/C. This is a whole lot easier than integrating the response band of a white noise with spectral density determined by a resistor. It's the same physics, just a different viewpoint on it. Dicklyon (talk) 04:10, 1 May 2008 (UTC)
 What do you mean by the kTC approach? As I understood, you said Kent H. Lundberg approach was the conventional method, which includes resistance. So far I am unaware of your approach as to how you derive the kTC noise *voltage* equation. Furthermore, as shown in your article the charge variance noise equation is derived from the kTC noise voltage equation, which as I pointed out is derived from resistance and cap bandwidth.PaulLowrance (talk) 16:44, 1 May 2008 (UTC)
 I'm not sure what you're referring to by "your article". Do you mean the Sarpeshkar et al? It recalls that each degree of freedom has mean fluctuation energy kT/2, and from that directly gets the noise voltage on the capacitor, mean square voltage kT/C. The charge variance kTC can be equally easily derived that way; I probably misremembered exactly how they had done it, but in any case resistance is not relevant to the physical argument. Dicklyon (talk) 21:45, 1 May 2008 (UTC)
 No, I'm referring to your Wikipedia article. I should have been more specific. You still have not shown mathematically where the noise voltage comes from and I don't see this in the book by Sarpeshkar. You mentioned kT/2, but that is in joules, not volts. The noise voltage I am referring to is shown in your wikipedia article on how to derive Qn. Your article says, "Using the formula for energy on a capacitor (E=1/2*C*V^{2}), noise energy on a capacitor can be seen to also be 1/2*C*(k*T/C), or also kT/2." As you can see, you replace V with the KTC noise voltage equation, which is sqrt(kT/C). So 1/2CV^{2} = 1/2C(sqrt(kT/C))^{2} = 1/2C(kT/C) = kT/2. So you're using the kTC noise voltage equation, but you don't show how it's derived. On the other hand, Kent H. Lundberg shows how the KTC noise voltage is derived, from resistance and bandwidth.PaulLowrance (talk) 23:42, 1 May 2008 (UTC)
 When a resistor is involved, the kTC noise is in whatever the bandwidth is. But it's always a charge variance of kTC. Dicklyon (talk) 15:57, 30 April 2008 (UTC)
 It's unclear what you're saying. As far as I know, there's always parallel resistance across a capacitor. Note that the transition from a real to perfect capacitor is an increase in parallel resistance. Therefore a theoretically perfect capacitor has infinite parallel resistance.PaulLowrance (talk)
 What I'm saying is that the variance of the charge of a capacitor, when it has been reset to a voltage by closing and opening a switch, is kTC. This is true immediately; it does not depend on waiting for a time constant having to do with the usually very huge parallel leakage resistance; it is true even if that resistance is infinite or missing. That leakage is too small to have any effect, and is not any part of the explanation of kTC noise. It's unclear why you think it's relevant. The more usual explanation is the switch resistance. But it's not necessary, since there are physical explanations that give the right result even if the switch goes ideally from 0 to infinite resistance. Dicklyon (talk) 20:51, 30 April 2008 (UTC)
 I'm uncertain why you thought I was talking about cap leakage. Cap leakage is indeed caused by parallel resistance, but I am talking about thermal noise caused by the caps parallel resistance. Anyhow, I don't believe your above description is kTC noise. You said the caps charged voltage changes immediately upon closing or opening a switch. Can you provide a real example so we can analyze this to find the source. Thanks. BTW, I noticed you did not comment on the above math. Do you now see how the kTC noise equation is derived from the caps parallel resistance and thermal noise??"PaulLowrance (talk)
 What parallel cap do you mean? An R across a C would normally be caused a leakage resistance, would it not? For specific situations in which kTC reset noise is important, see these books. Dicklyon (talk) 03:58, 1 May 2008 (UTC)
 Again, I was not talking about capacitor leakage. A capacitor has parallel resistance, which does indeed cause leakage, but such parallel resistance is also the cause of thermal noise. As seen in my statements, I was talking about the thermal noise caused by the caps parallel resistance. Can you please show where I even used the term "leakage?"PaulLowrance (talk) 17:09, 1 May 2008 (UTC)
 Here is a book that talks about charge variance kTC being the same as Johnson noise, and from "the Brownian motion of charge carriers in a resistor"; he would have been even more correct to omit "in a resistor", since the same effect is found with an ideal switch. Also, read the cited article by Sarpeshkar et al. (the link works to get a copy), and note on p.27 where they explain how all thermal noises can be attributed to the capacitors and inductors, whose respective voltages and currents are state variable in thermodynamic equilibrium; they unify the concepts of Johnson noise and shot noise by looking carefully at the physics of quantummechanical systems. Dicklyon (talk) 05:12, 1 May 2008 (UTC)
 Indeed, the book you mention above is correct and clarifies my point that kTC noise comes from resistance, not pure capacitance and/or inductance. As you can see he clearly shows the resistor in kTC noise model. So far I am unaware of how you are going to derive the kTC noise voltage equation without including resistance. BTW, it appears one point you are missing is that an ideal switch is modeled as infinite resistance. Perhaps an easier method of understanding what is occurring in theoretical infinite resistance is to begin with finite resistance and begin increasing the resistance toward infinity. In doing so you will note the thermal noise increases as resistance increases.PaulLowrance (talk) 17:09, 1 May 2008 (UTC)
 Yes, it's correct, but the interpretation that kTC noise "comes from resistance" is still just one possible interpretation. Sarpeshkar's is another, with a simpler tie to the underlying quantum physics, which works even when there is not relevant resistance instead of only in the limit. Dicklyon (talk) 21:45, 1 May 2008 (UTC)
 I understand Sarpeshkar offers an "interpretation," but I don't see where he derived V^{2}, the kTC noise voltage. At some level you'll need to include resistance.PaulLowrance (talk) 23:42, 1 May 2008 (UTC)
Mike Engelhardt, the creator of LTspice, gave permission to post his email, as follows.
Paul,
Yes, I fully agree with you that kTC noise is due to the resistance(even though the switch resistor specifics don't matter). This is basically lot and parcel that reactances generate no noise to the extent that they have no losses. That is why SPICE is correct not to include *any* noise whatsoever from a capacitor. There simply is none if the capacitor has no loss(resistance mechanism). BTW, LTspice will include noise from a capacitor because it supports series and parallel losses in the cap and those do generate noise. So while LTspice has noise coming from capacitors, its due to the losses in the capacitor, not the capacitance itself.
Now the problem with using SPICE or LTspice with kTC noise is that LTspice is doing a noise computation of the linear circuit. But you can't see any kTC noise in a linear circuit. It is the result of the resistance of the switch as it opens. It occurs during the nonlinear operation. It isn't that LTspice is incorrect, it just that it doesn't do nonlinear noise analysis.
Feel free to post the above as a quote of private communication with me.
Regards,
Mike
 I understand that's the conventional view. And nobody is claiming that the noise is "generated" the reactance. The noise is generated by discrete electrons behaving thermodynamically. Dicklyon (talk) 16:09, 4 May 2008 (UTC)
 You still haven't shown any method of deriving the kTC noise equations without resistance. I agree with Mike that kTC noise does indeed come from resistance. Your description of such noise being generated by discrete electrons behaving thermodynamically is a nonmathematical description of how the noise is generated, and resistance is included in such a process.
 The method is as follows: invoke thermodynamics fundamental conclusion that in a system in thermodynamic equilibrium, every state variable has a mean fluctuation energy of kT/2. Equate that to the noise energy of a capacitor, CV^{2}/2, for V being the rms voltage fluctuation. Solve for V. No resistance needs to be invoked. Dicklyon (talk) 03:09, 6 May 2008 (UTC)
 Regarding your above text, "for V being the rms voltage fluctuation", that is what I was talking about in our previous discussion. Such rms Voltage is derived from resistance. As you describe, the cap energy is 0.5CV^2. The V is kTC noise voltage, sqrt(kT/C). Now again I'll show how such noise voltage is derived >
 A capacitor has parallel resistance. Even a perfect cap has parallel resistance, infinite. We know that a resistor generates thermal noise voltage. We know that a capacitor is a filter and the resulting bandwidth is 1/4RC. That is fundamental science. In continuing, we know that Thermal noise is Vn = sqrt(4kTRB), and thus the Thermal noise voltage caused by resistance in parallel with the capacitor is Vn = sqrt(4kTR(1/4RC)) = sqrt(kT(1/C)) = sqrt(kT/C). As you can see, we know from fundamental science that a resistor of *any* resistance across a capacitor will generate sqrt(kT/C) noise.
 Note that Thermal noise begins to drop near 1 THz. The kTC noise voltage equation is not perfect in a real world. For example, consider an amazing 220uF capacitor that has 100T ohms of parallel resistance. According to the kTC noise voltage equation the noise should be 4.3nV rms, but in reality it is appreciably less due to the limitation of thermal noise bandwidth.PaulLowrance (talk) 16:20, 6 May 2008 (UTC)
 The important thing, however, is that resistance is not needed to capture some of that noise on a capacitor. At any time, the charge on a capacitor being held at voltage V has a variance kTC, and if you cut it off by opening an ideal switch, you're left that with that charge variance, and a corresponding voltage variance. No resistance is needed. Dicklyon (talk) 16:09, 4 May 2008 (UTC)
 Again, a theoretical ideal switch has infinite resistance with infinite Thermal noise. Real switches have finite resistance. This issue is simple to model in Spice and one can see where the noise comes from, resistance.
The main problem with arguments containing a resistance of zero is that the "zero" is a limit and can mean different things depending on the nature of the limit process. After reading this discussion, I talked to one of the Authors of the Sarpeshkar et. al. paper, and he told me that the derivation shown in that paper was intended as a short cut only, but had triggered intensive and almost religious discussions in the research community that have not been decided yet. The "filtered noise" view is more productive in the design of switchedcapacitor filters, while the thermodynamicequilibrium view is more productive in the design of optosensors, especially when one goes on to finding ways how to prevent the sensor from reaching thermal equilibrium by different methods (c.f. the work of Boyd Fowler).
Now Wikipedia is an encyclopedia, it is not the place to decide between correct and incorrect, but it is most certainly the place to write about a disagreement that is going on in the research community. I therefore suggest to rewrite the kTC section as follows: (1) kT/C noise viewn as filtered Johnson Nyquist noise, (1b) how this is applied to explain noise in SC circuits, (2) kT/C noise viewn as a thermal equilibrium state of a system with one degree of freedom, (2b) how this is applied in photo sensorics (c.f. Boyd Fowler), (3) physical cause under debate since Sarpeskar's paper.
Target: not longer than it is now. I also suggest to remove the table with the number of electrons; those who know what this is about really don't need that table; those who would need that table cannot make any use of it.
Dick: I apologize for saying "factual error" in my edit; I promise to not do such a rushed edit again on a Wikipedia article.
Dick, Paul, I'd welcome comments!Hanspi (talk) 08:42, 9 September 2009 (UTC)
 He said that? Interesting. It would be OK to clarify and to represent other points of view, if we can find sources; I'm not aware of any real disagreement or ongoing debate with this approach, however. I'm not clear on how you can represent the infinite R (not zero R) static noise case as filtered noise, but if you have a source that does it that way, that would be fine to include. The table is useful, I think, as it gives a good way to visually the amount of noise on small capacitors. Not sure what you mean by "those who would need that table cannot make any use of it". Dicklyon (talk) 14:56, 9 September 2009 (UTC)
 The Pain '99 paper is another that analyzes noise on a capacitor explicitly in terms of the fluctuations on numbers of electrons, without reference to resistance. Dicklyon (talk) 15:32, 9 September 2009 (UTC)
 I won't admit which of the authors said it :) Oh yes, I have several sources describing sampled noise in SC filters where the exact noise power spectral density is derived from the aliased lowpassfiltered white noise and where it is shown that the sampled noise is only white if the RC time constant while the switch is lower than a certain fraction of the sample time, for example Goette/Gobet 1989, "Exact Noise Analysis of SC Circuits and an Approximate Computer Implementation" in TCASI. As I said, there seems to be a different viewpoint taken by SC people and opto people, so maybe the debate is not really going on, but we have just two "clubs" holding two opinions?
 I'm not sure what you mean by static noise. Are you talking about the ensemble average and standard deviation generated by a random model of a single experiment? Can you describe that formally?
 Well, whichever one it was, tell him and her hi for me, and I'll tell the others hi for you. "Static" just means the DC value after the switch opening event; you can think of it an ensemble issue, or you can make it a time problem. I agree the SC problem is a bit different, as things happen fast and the events are not independent. It can still be analyzed as electrons with a Boltzmann distribution coming and going; should be equivalent to the more resistororiented conventional analysis, right? I'll be happy to see what you've got, and how you'd rewrite it. Dicklyon (talk) 18:46, 9 September 2009 (UTC)
 I'll say hi to whomever it was! One technical nit to pick: The problem I have is with your talking about resistances going to zero or infinity, for in both limits the system will never reach thermal equilibrium: perpetual oscillation in the first case, never settling in the second case. So using a thermalequilibrium derivation is inconsistent with assuming or . Now I'm not sure whether this is so, or whether I misunderstand thermal equilibrium in a circuit with a capacitor. What's your comment on this?Hanspi (talk) 21:08, 9 September 2009 (UTC)
 The equilibrium applies when the C is connected to a finite R, or to a voltage source, or an infinite capacitor. The opening a switch to freeze the state of the capacitor at some number of electrons will take a sample of the electron count distribution. Dicklyon (talk) 03:53, 10 September 2009 (UTC)
 So a finite R is indeed needed to talk about equilibrium! Does it then make sense at all to discuss what would happen without a finite R? It seems that the equilibriumstate derivation is, after all, just a short cut, making the value of R disappear as in the long derivation, with the advantage that it disappears immediately and the disadvantage that the reason for its disappearance from the formula remains unseen.
 By the way, the paper describing how you can use capacitive sensors without letting them run into a thermal equilibrium, giving noise far below kT/C, is Boyd Fowler's "Reset Noise Reduction in Capacitive Sensors", IEEE Trans. CASI, vol. 53, no. 8, August 2006. He claims (convincingly) that kT/(20C) is feasible in practice. If you want a copy of any paper I mention, write to hanspeter dot schmid at fhnw dot ch. I consider Wikipedia an education project, i.e., can give coworkers in education copies according to my country's copyright law. Hanspi (talk) 12:20, 10 September 2009 (UTC)
 Yes, I'm familiar with that paper; it was the basis for the devices he designed at Pixel Devices Intl; I talked with him today – he gave a talk this morning at the HDR symposium at Stanford, covering a range of sometimesrelated techniques for HDR sensors. Dicklyon (talk) 19:06, 10 September 2009 (UTC)
Noise at very high frequencies[edit]
So resistors emit white light? What's the connection to blackbody radiation? — Omegatron (talk) 00:34, 4 May 2008 (UTC)
 Resistors emit blackbody radiation; but that radiation has a completely different spectrum from the electrical noise that is due to discrete charges moving in thermal equilibrium. Dicklyon (talk) 16:03, 4 May 2008 (UTC)
 Dicklyon, we are in sync. I clicked "Save page" to see someone else had already edited the page.
 Anyhow, it depends what you mean by white light. All objects emit blackbody radiation, and in a sense the amount of such radiation in the visible spectrum would to some degree resemble "white light," but such blackbody radiation in the visible spectrum is so weak that you cannot sense it with your naked eyes. Here's a blackbody radiation calculator,
 As far as Johnson noise, the normalized power density begins to drop off at around 1THz. See the graph at the bottom of the following pdf,
 PaulLowrance (talk) 16:09, 4 May 2008 (UTC)
 A year or so ago I worked out the comparison between Thermal noise and Blackbody radiation. Both are caused by Thermal energy, but we must remember that Blackbody radiation energy varies relative to frequency E = hf. For example, the Thermal noise current in material is appreciably equal at 2KHz and 1HHz, but such noise current @ 2KHz emits twice the blackbody radiation compared to 1KHz noise current. That's why the blackbody radiation spectrum is not flat like Thermal noise. Thermal noise is appreciably flat till near 1THz.PaulLowrance (talk) 16:20, 6 May 2008 (UTC)
I quote "The power spectral density, or voltage variance (mean square) per hertz of bandwidth, is given by
\bar v_{n}^2 = 4 k_B T R
where kB is Boltzmann's constant in joules per kelvin, T is the resistor's absolute temperature in kelvins, and R is the resistor value in ohms." Where does 'Hz' come from? [k_B in Joules/Kelvin]x[T in Kelvin]x[R in ohms] = [J][ohms] I can't find anything that leads to 'per Hertz or 1/time. Looking other books in transmission medium noise, they describe another apparently inconsistent result (No = 4*Kb*T) and also verbally express the units as Joules/Hertz!? What happened to [Ohms' in those cases? Loucosta (talk) 17:16, 15 June 2008 (UTC)loucosta
 Your equation is missing frequency. It should be V^2 = 4 k_B T R B, where B is the bandwidth in Hz. If you want the noise voltage then it's Vn = sqrt(4 k_B T R B). PaulLowrance (talk) 15:42, 16 June 2008 (UTC)
"The cause of Johnson noise is blackbody radiation within the conductor. This explains
why the noise power is independent of the resistance – it only depends on the
temperature. However the resistance does affect the observed voltage. The actual charges
in the conductor move so as to try to nullify the electrical fluctuations caused by the
blackbody radiation. But the resistance to their movement limits the nullification, so a net
random time varying potential can be observed across the conductor."
http://www.claysturner.com/dsp/JohnsonNyquist%20Noise.pdf —Preceding unsigned comment added by 71.167.60.17 (talk) 06:13, 3 November 2009 (UTC)
The previous comment is correct  Johnson noise definitely is blackbody emission. To be specific, it is a 1D blackbody in the low frequency limit (hf << kT, also known as the RayleighJeans limit). To check this, take Planck's law, which is the classical result for a 3D blackbody, multiply by an area of lambda squared (recalling that lambda = the speed of light divided by the frequency), and expand the exponential to first order in the limit hf << kT. This gives the familiar Johnson noise result. Tls60 (talk) 12:39, 8 September 2011 (UTC)
Very large resistances[edit]
Do very large resistances have very large noise voltages?
The resistivity of air is about 10^{13} Ω. So for a short distance, a typical resistance is about 10 PΩ. Does this then produce 2 V of thermal noise? What about long distances? —Preceding unsigned comment added by 71.167.60.17 (talk) 05:57, 3 November 2009 (UTC)
 Yes you are right. You may measure this voltage.
6MHz BW Receiver Noise Power Calculations[edit]
What am I missing here? As previously stated the JohnsonNyquist noise at room temperature in a 6MHz bandwidth is 106dBm. With a receiver noise figure of 3dB the receiver noise power would then be 103dBm. With a receiver noise figure of 1dB the noise power would be 105dBm. Where do you get 112dBm?
"For example a 6 MHz wide channel such as a television channel received signal would compete with the tiny amount of power generated by room temperature in the input stages of the receiver, which, for a TV receiver with a noise figure of 3 dB would be −106 dBm, or one fortieth of a picowatt. For a TV with a noise figure of 1 dB, the noise power would be −112 dBm." Golden Eternity (talk) 21:13, 28 May 2010 (UTC)
Confusing Text[edit]
From the "Noise voltage and power" section, the following statement is confusing: "For the general case, the above definition applies to". There is not a definition above the sentence (it is the first section). If it's referring to shot noise, it should just say the "For the general case, the above definition of shot noise applies to" to remain clear. —Preceding unsigned comment added by 70.133.83.60 (talk) 21:06, 16 June 2010 (UTC)
 Agreed. Also, the page starts defining Johnson noise by saying it isn't shot noise. This is confusing  most/many readers are unlikely to know what shot noise is, if they're looking up Johnson noise. Then there is that confusing section about radiation resistance further down  why are we trying to define radiation resistance here, when it has a perfectly good page of it's own? this page needs work  if nobody else gets there first, I'll try to tidy it up a bit at the weekend. GyroMagician (talk) 11:18, 17 June 2010 (UTC)
 If you look back to 2005, you can see that that statement followed closely after the definition "the noise generated by the equilibrium fluctuations of the electric current inside an electrical conductor." It made less sense as it drifted further away from what it was referring to. Dicklyon (talk) 03:09, 18 June 2010 (UTC)
Date of discovery[edit]
I think that the date (1928) given for Johnson's first thermal noise measurements might be incorrect here  though his detailed Physical Review paper was published (backtoback with Nyquist's theory paper) in 1928, he published a Nature paper (Nature 119, p. 50, Jan. 8, 1927) in early January 1927 summarizing that work, and an abstract for a paper presented by Johnson at an American Physical Society Annual Meeting (held December 2830, 1926) is published in a conference proceedings in a February 1927 Physical Review issue (Phys. Rev. 29, p. 367, Feb. 1927).
I suspect that since I had to dig this all up, this could be running afoul of WP:NOR, but it's clear to me that Johnson must have discovered thermal noise no later than December 1926, earlier than the 1928 date given. Does anyone disagree that the February 1927 abstract publication  for a December 1926 conference  is sufficient evidence? Should I change it to 1926 (with the reference given)? 00:54, 8 September 2011 (UTC) — Preceding unsigned comment added by Lambda(T) (talk • contribs)
 Good find! No, I think the OR all belongs to Johnson  Nature is respected enough for us to use it as a source ;) That gives January 1927. If you have a reference for the APS meeting, that is also perfectly valid. Please go ahead and make the changes. It always seemed amazing to me that Nyquist had time put together the theory paper, when Johnson had only just discovered the effect. It all makes more sense now. GyroMagician (talk) 07:26, 8 September 2011 (UTC)
 Done! I left off the Nature paper (Nature 119, p. 50, Jan. 8, 1927) for now, as it's only a brief letter, the conference took place before its publication, and the 1928 Phys. Rev. papers are far more detailed. However, all of these citations  including the one I added  are in Nyquist's 1928 Physical Review paper already cited here (Phys. Rev. 32, 97), which is where I discovered the 1926 date.
 You can still be somewhat amazed at Nyquist's quick theoretical response to Johnson's experiments; in his aforementioned paper (Phys. Rev. 32, 97) he also cites conference proceedings (Phys. Rev. 29, 614 (April 1927)) containing an abstract for a paper entitled "Thermal agitation in conductors" he presented at the February 2526, 1927 APS meeting in New York; the abstract itself already presents the expression for the noise spectral density and outlines the theoretical derivation. So Nyquist still did follow rather quickly on Johnson's heels (though I suspect that, working at Bell Labs, he may have had a bit more advance notice).  17:29, 8 September 2011 (UTC) — Preceding unsigned comment added by Lambda(T) (talk • contribs)
"From the equation above, noise power in a resistor at room temperature, in dBm, is then ..." =[edit]
This does not follow from "the equation above" (which equation?). It is for the case of a matched load. Very misleading. — Preceding unsigned comment added by Peter1c (talk • contribs) 12:19, 8 May 2013 (UTC)
Power Dissipation[edit]
The discution I start here reffers to the following statemant found on the page:
"A resistor in a short circuit dissipates a noise power of" (this line follows a formula of power).
Now I want to put the question > into what type of power/energy is transformed that power quantity (4*k*T*deltaF).
The word 'dissipation' reffer generaly at transforming work (in various forms, including electrical work) into heat (a type of energy that rises the temperature of things (or changes thier phase > melding, vaporising)).
The noise power exist all the time and it cannot "dissipate" into heat because otherwise a resistor will heat spontaneosely.
So I suggest that the words: "a circuit disipates a noise power" is not suitable because the noise power does not transform into anything (it does not dissipates, it stays into the device in the same form all the time).
A more intuitive phrase would be: "the circuit present at the terminals a noise power of ... ". — Preceding unsigned comment added by 84.247.79.142 (talk) 15:51, 4 September 2014 (UTC)
 In a short circuit with just a resistor, Johnson noise turns heat into electric current, while Joule heating simultaneously turns electric current into heat. These two processes are equal and opposite. ...But anyway I agree with you that the wording needs improvement... :D Steve (talk) 01:44, 6 September 2014 (UTC)
proposal to split off reset noise (kTC) to a different article[edit]
I think it would be nice to have a separate page called "reset noise" for capacitor reset noise, any objections?
Reasons: One way to derive the thermal reset noise on a capacitor (mean square voltage of kTC) is using Johnson noise, but that's not the only way to derive it. Even basic statistical mechanical arguments work so there is no need for justifying kTC noise with fluctuationdissipation theorems. Another motivation for the split is that Johnson noise is a dynamically changing noise whereas reset noise is a static fluctuation.
One possible problem is that presently "thermal noise" redirects to this article.Nanite (talk) 11:13, 15 June 2015 (UTC)
credit where it is due[edit]
I have changed the rather common and derogatory, offhanded remark that Johnson was the "first to measure" to "first to discover and measure."
Johnson did discover it in 1926 (it had clearly not been "predicted" by Nyquist in 1928, as is frequently claimed) it was formulated by Nyquist as a direct result of its discovery/observation by Johnson, who had isolated it sufficiently to understand that it was an intrinsic thermal noise (by comparison, one cannot make the same eponymous claim about Brownian motion, which was completely misunderstood by it's discoverer). Wikibearwithme (talk) 03:19, 16 September 2016 (UTC)
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Faulty formula 'Noise power in decibels'[edit]
this is wrong
 , mit in Hz
corrected
 , mit in Hz
values in the table are 6.2 dB to low 26.85°C is a funny room temperature.
If no one objects I will correct the formula.
Someone did not know to handle dBm ?¿ But at dBm are also problems, there is a value stated 0.4 dB different to this here (and also wrong).
See https://www.sengpielaudio.com and many other sources.
AK45500 (talk) 21:44, 8 December 2020 (UTC)
 I just verified the numbers. 174 seems correct at 300K. By the way, my antivirus warns me against that web site.Constant314 (talk) 22:42, 8 December 2020 (UTC)
 You are wrong. Please verify explicitely your dBm calculations :
 174 dBm are 174 dBV @ 1000 Ohm == about 2 nV
 root ( 4 x 1.38E23 J/k x 300 K x 1 Hz x 1000 Ohm )  this is about 4.1 nV
 sengpielaudio.com is a VERY reliable site, it is referenced every where, your antivirus will be wrong.
 use http instead of https , it is 6 years old Eberhard_Sengpiel — Preceding unsigned comment added by AK45500 (talk • contribs) 09:04, 9 December 2020 (UTC)
 Just use one of the thousand other refereces !
 Please correct the formula and the table ! Thanks !
 Please correct also dBm , Thanks .
 AK45500 (talk) 08:52, 9 December 2020 (UTC)
 I think that I see the conceptual problem now. The thermal noise power in a resister would dissipate if it were shorted out is:
 But the thermal noise power that resister can deliver in the best case (a matched load) would be:
 The factor of 4 is accounting for the extra 6 dB in your result. The text of the article should perhaps be tweaked, but there is no math error that needs to be corrected. Constant314 (talk) 10:20, 9 December 2020 (UTC)
 sorry YOU are wrong. There is a voltage , NO power match ! which results in a power loss of 6 dB in your formula. please correct.
 Voltage , not delivered power. It will never deliver max. power with resistance matching and it will dissipate nose power itself. YOU have to state the max. total noise power . Please correct. You like to be the only one , thinking this was , do you ??!! Please give robust reliable proof. Sengpiel and all AES engineers contruct their equipment the other way. I measure noise, and YOU are wrong.
in the best case (a matched load)
 This is wrong. A matched load will have a loss of 3/4 max. power
 ( half the current at half the voltage > only 1/4 at load )
 matched load is NOT usefull as max. power is concerned , but stabilizing dissipated power (at a lower level).
 ¡ use a matched load at a nuclear power station ! Wow ! Texas will be nuked ¡
 AK45500 (talk) 12:33, 9 December 2020 (UTC)
 You are correct that a matched load will lose 3/4 of the power. That is why your calculation is incorrect by 6 dB. Power plants do not operate at a matched load. For noise calculations you are interested in the noise power that the resister can inject into the rest of the circuit. That is maximized when the rest of the circuit is matched to the resister. The maximum power that the resister can inject into the rest of the circuit is 173.8 dBm. Constant314 (talk) 23:00, 9 December 2020 (UTC)
Googled "thermal noise power formula"
[3]I found the sengpielaudio calculator. They calculate dBu and dBv, neither of which is dBm.
I also found
[4] Thermal noise in a 50 Ω system at room temperature is 174 dBm / Hz.
[5] calculator produces 173.8277942 dBm
[6] Eq. 2.121 verifies formula for power P = kTB
[7] Verifies Thermal noise power equation
[8] 173.9 dBm⁄Hz
Here is an old but good one from the IEEE: "the available thermal noise power, from a resistor, a lossy network, a lossy dielectric, or an antenna is always kTB" [9]
Constant314 (talk) 21:02, 10 December 2020 (UTC)
and some printed sources
"Noise power at 300 K = 173.83 dBm/Hz"^{[1]}^{: 260 }
"Noise power = kTB" ^{[2]}^{: 621 }
"P = kTB" ^{[3]}^{: 203 } Constant314 (talk) 22:03, 10 December 2020 (UTC)
 ^ Vizmuller, Peter (1995), RF Design Guide, Artech House, ISBN 0890067546
 ^ Ludwig, Reinhold; Bretchko, Pavel (2000), RF Circuit Design, PrenticeHall, ISBN 0130953237
 ^ Hayward, Wes (1994), Introduction to Radio Frequency Design, ARRL, ISBN 0872594920