Editing Johnson–Nyquist noise (section)

== Generalized forms ==
The <math>4 k_\text{B} T R</math> voltage noise described above is a special case for a purely resistive component for low to moderate frequencies. In general, the thermal electrical noise continues to be related to resistive response in many more generalized electrical cases, as a consequence of the [[fluctuation-dissipation theorem]]. Below a variety of generalizations are noted. All of these generalizations share a common limitation, that they only apply in cases where the electrical component under consideration is purely [[Passivity (engineering)|passive]] and linear.

=== Complex impedances ===
Nyquist's original paper also provided the generalized noise for components having partly [[electrical reactance|reactive]] response, e.g., sources that contain capacitors or inductors.<ref name=Nyquist/> Such a component can be described by a frequency-dependent complex [[electrical impedance]] <math>Z(f)</math>. The formula for the [[power spectral density]] of the series noise voltage is
:<math>
S_{v_n v_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Z(f)].
</math>
The function <math>\eta(f)</math> is approximately 1, except at very high frequencies or near absolute zero (see below).

The real part of impedance, <math>\operatorname{Re}[Z(f)]</math>, is in general frequency dependent and so the Johnson–Nyquist noise is not white noise. The RMS noise voltage over a span of frequencies <math>f_1</math> to <math>f_2</math> can be found by taking the square root of integration of the power spectral density:
:<math> V_\text{rms} = \sqrt{\int_{f_1}^{f_2} S_{v_n v_n}(f) df}</math>.

Alternatively, a parallel noise current can be used to describe Johnson noise, its [[power spectral density]] being
:<math>
S_{i_n i_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Y(f)].
</math>
where <math>Y(f) {=} \tfrac{1}{Z(f)}</math> is the [[electrical admittance]]; note that <math>\operatorname{Re}[Y(f)] {=} \tfrac{\operatorname{Re}[Z(f)]}{|Z(f)|^2} \, .</math>

=== Quantum effects at high frequencies or low temperatures ===
With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near [[absolute zero]]), the multiplying factor <math>\eta(f)</math> mentioned earlier is in general given by:<ref>{{Cite journal |last1=Callen |first1=Herbert B. |last2=Welton |first2=Theodore A. |date=1951-07-01 |title=Irreversibility and Generalized Noise |url=https://link.aps.org/doi/10.1103/PhysRev.83.34 |journal=Physical Review |volume=83 |issue=1 |pages=34–40 |doi=10.1103/PhysRev.83.34}}</ref>
:<math>\eta(f) = \frac{hf/k_\text{B} T}{e^{hf/k_\text{B} T} - 1}+\frac{1}{2}
\frac{h f}{k_\text{B} T} \, .</math>
At very high frequencies (<math>f \gtrsim \tfrac{k_\text{B} T}{h}</math>), the spectral density <math>S_{v_n v_n}(f)</math> now starts to exponentially decrease to zero. At room temperature this transition occurs in the terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set <math>\eta(f)=1</math> for conventional electronics work.

==== Relation to Planck's law ====
Nyquist's formula is essentially the same as that derived by Planck in 1901 for electromagnetic radiation of a blackbody in one dimension—i.e., it is the one-dimensional version of [[Planck's law|Planck's law of blackbody radiation]].<ref>{{cite book |title=Fundamentals of Microwave Photonics |page=63 |url=https://books.google.com/books?id=mg91BgAAQBAJ&pg=PA63 |first1=V. J.|last1=Urick|first2=Keith J.|last2=Williams|first3=Jason D.|last3=McKinney|isbn=9781119029786 |date=2015-01-30 |publisher=John Wiley & Sons }}</ref> In other words, a hot resistor will create electromagnetic waves on a [[transmission line]] just as a hot object will create electromagnetic waves in free space.

In 1946, [[Robert H. Dicke]] elaborated on the relationship,<ref>{{Cite journal| doi = 10.1063/1.1770483| volume = 17| issue = 7| pages = 268–275| last = Dicke| first = R. H.| title = The Measurement of Thermal Radiation at Microwave Frequencies| journal = Review of Scientific Instruments| date = 1946-07-01| pmid=20991753| bibcode = 1946RScI...17..268D| s2cid = 26658623| doi-access = free}}</ref> and further connected it to properties of antennas, particularly the fact that the average [[antenna aperture]] over all different directions cannot be larger than <math>\tfrac{\lambda^2}{4\pi}</math>, where λ is wavelength. This comes from the different frequency dependence of 3D versus 1D Planck's law.

=== Multiport electrical networks ===
[[Richard Q. Twiss]] extended Nyquist's formulas to multi-[[Port (circuit theory)|port]] passive electrical networks, including non-reciprocal devices such as [[circulator]]s and [[Isolator (microwave)|isolator]]s.<ref>{{Cite journal | doi = 10.1063/1.1722048| title = Nyquist's and Thevenin's Theorems Generalized for Nonreciprocal Linear Networks| journal = Journal of Applied Physics| volume = 26| issue = 5| pages = 599–602| year = 1955| last1 = Twiss | first1 = R. Q.| bibcode = 1955JAP....26..599T}}</ref> 
Thermal noise appears at every port, and can be described as random series voltage sources in series with each port. The random voltages at different ports may be correlated, and their amplitudes and correlations are fully described by a set of [[cross-spectral density]] functions relating the different noise voltages,
: <math>S_{v_m v_n}(f) = 2 k_\text{B} T \eta(f) (Z_{mn}(f) + Z_{nm}(f)^*)</math>
where the <math>Z_{mn}</math> are the elements of the [[impedance matrix]] <math>\mathbf{Z}</math>.
Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port. Their cross-spectral density is given by
: <math>S_{i_m i_n}(f) = 2 k_\text{B} T \eta(f) (Y_{mn}(f) + Y_{nm}(f)^*)</math>
where <math>\mathbf{Y} = \mathbf{Z}^{-1}</math> is the [[Admittance parameters|admittance matrix]].