Editing Johnson–Nyquist noise (section)

== Nyquist's derivation of ideal resistor noise ==
[[File:Nyquist-transmission-line-derivation-of-johnson-noise-with-voltage-sources.svg|thumb|396x396px|Figure 5. Schematic of [[Harry Nyquist|Nyquist's]] 1928 [[thought experiment]]<ref name="Nyquist" /><ref name=":0" /> using two noisy resistors (each represented here by a noise-free resistor in series with a noise voltage source) connected via a long lossless [[transmission line]] of length <math>l </math>. Each resistor's noise [[Analog signal|signal]] propagates across the line at velocity <math>\text{v} </math>. All impedances are identical, so both signals are absorbed by the opposite resistor instead of being [[signal reflections|reflected]].<br><br>Nyquist then imagined [[Short circuit|shorting]] both ends of the line, thereby trapping in-flight energy on the line. Because all in-flight energy is now completely reflected (due to the now-mismatched impedance), the in-flight energy can be represented as a summation of sinusoidal [[standing waves]]. For a band of frequencies <math>\Delta f </math>, there are <math>2l \, \Delta f / \text{v}</math> [[Normal mode#Standing waves|modes of oscillation]].{{NoteTag|A standing wave occurs with frequency equal to every integer multiple of <math>\tfrac{\text{v}}{2l}</math>. The line is sufficiently long to make the number of modes within the bandwidth very large, such that the modes will be close enough in frequency to approximate a continuous frequency spectrum.}} Each mode provides <math>k_{\rm B} T</math> [[KT (energy)|of energy]] on average, of which <math>k_{\rm B} T / 2</math> is electric and <math>k_{\rm B} T / 2</math> is magnetic, so the total energy in that bandwidth on average is <math>k_{\rm B} T \cdot 2l \, \Delta f / \text{v} . </math> Each resistor contributed <math>k_{\rm B} T \cdot l \, \Delta f / \text{v} </math> (half of that total energy).<br><br>But since before the shorting there were originally no reflections, the value of that total in-flight energy also equals the combined energy that was transferred from both resistors to the line during the transit time interval of <math>l / \text{v} </math>. Dividing the average '''energy''' transferred from ''each'' resistor to the line by the transit '''time''' interval results in a total '''power''' of <math>k_{\rm B} T \, \Delta f </math> transferred over bandwidth <math>\Delta f </math> on average from each resistor.]]
Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors"<ref name="Nyquist" /> used concepts about [[Equipartition theorem#Potential energy and harmonic oscillators|potential energy and harmonic oscillators from the equipartition law]] of [[Ludwig Boltzmann|Boltzmann]] and [[James Clerk Maxwell|Maxwell]]<ref>{{Cite book |last=Tomasi |first=Wayne |url=https://books.google.com/books?id=LXPWxmakFVgC |title=Electronic Communication |date=1994 |publisher=Prentice Hall PTR |isbn=9780132200622 |language=en}}</ref> to explain Johnson's experimental result. Nyquist's [[thought experiment]] summed the energy contribution of each [[Normal mode#Standing waves|standing wave mode of oscillation]] on a long lossless [[transmission line]] between two equal resistors (<math>R_1 {=} R_2</math>). According to the conclusion of Figure 5, the total average power transferred over bandwidth <math>\Delta f </math> from <math>R_1</math> and absorbed by <math>R_2</math> was determined to be:

: <math>\overline {P_1} = k_{\rm B} T \, \Delta f \, . </math>

Simple application of [[Ohm's law]] says the current from <math>V_1</math> (the thermal voltage noise of only <math>R_1</math>) through the combined resistance is <math display="inline">I_1 {=} \tfrac{V_1}{R_1 + R_2} {=} \tfrac{V_1}{2R_1}</math>, so the power transferred from <math>R_1</math> to <math>R_2</math> is the square of this current multiplied by <math>R_2</math>, which simplifies to:<ref name="Nyquist" />

: <math>P_\text{1} = I_1^2 R_2 = I_1^2 R_1 = \left( \frac{V_1}{2R_1} \right)^2 R_1 = \frac{V_1^2}{4R_1}  \, .</math>

Setting this <math display="inline">P_\text{1}</math> equal to the earlier average power expression <math display="inline">\overline {P_1}</math> allows solving for the average of <math display="inline">V_1^2</math> over that bandwidth:

: <math>\overline{V_1^2} = 4 k_\text{B} T {R_1} \, \Delta f \, .</math>

Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and [[Johnson–Nyquist noise#Complex impedances|complex impedances]] too. And while Nyquist above used <math>k_{\rm B} T</math> according to [[Classical physics|classical]] theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the [[Planck constant]] <math>h</math> (from the new theory of [[quantum mechanics]]).<ref name="Nyquist" />