Johnson–Nyquist_noise

# Johnson–Nyquist noise **Johnson–Nyquist noise**, also called thermal noise or simply Johnson noise, is the random voltage or current that appears across any electrical conductor purely from the thermal agitation of its charge carriers, present at equilibrium whether or not any current is deliberately flowing. It is proportional to absolute temperature and, for an ideal resistor, spread almost perfectly flat across frequency, which is why it sets an unavoidable noise floor beneath every other signal a circuit carries. The primary microsim on this page lets the reader drag a resistor's temperature, resistance and measurement bandwidth and watch the resulting noise voltage's scope trace, histogram and available power move exactly as the underlying law predicts. As one of the mechanisms behind [[Noise_(electronics)|electronic noise]] in every real circuit, thermal noise is usually the ultimate limit on a system's [[Signal-to-noise_ratio|signal-to-noise ratio]]: the floor beneath which no amount of amplification alone can help, because amplifying the signal amplifies the noise riding on it by exactly the same factor. ## History of thermal noise Physicist John B. Johnson, working at Bell Labs, first measured the noise experimentally and published his results in 1928, finding a voltage that grew with temperature and did not depend on which metal the resistor was made from.[^johnson1928] His colleague Harry Nyquist supplied the theoretical explanation in a companion paper the same year, deriving the noise voltage from thermodynamic reasoning rather than from any specific model of the metal itself.[^nyquist1928] The two papers, published back to back in the same journal, are usually cited together as the origin of the noise's two conventional names. ## Noise of ideal resistors for moderate frequencies For an ideal [[Resistor|resistor]] of resistance R at absolute temperature T, observed over a bandwidth Δf, the noise's power spectral density is flat: `S_v = 4*k_B*T*R` (volts squared per hertz), so the mean-square open-circuit voltage is `<v_n^2> = 4*k_B*T*R*deltaf` and the RMS voltage is `v_rms = sqrt(4*k_B*T*R*deltaf)`.[^ell096] A 1 kΩ resistor at 293 K, close to room temperature, works out to 4.02 nV per root-hertz, or 4.02 µV of RMS noise measured over a 1 MHz bandwidth, a figure large enough to matter in any amplifier with real gain ahead of it.[^ell096] The equivalent RMS noise current, from the same resistor treated as a Norton source, is `i_rms = sqrt(4*k_B*T*deltaf/R)`, the same physical fluctuation expressed as a current rather than a voltage. Because the spectral density does not depend on frequency at all, up into the terahertz range where quantum corrections finally roll it off, thermal noise is called white. Evaluating the available-power expression at T = 290 K, the reference temperature radio engineers use by convention, gives a power spectral density of about −174 dBm per hertz, the figure every radio receiver's own noise floor is ultimately measured against. None of this reasoning is special to metals: the same law governs thermal noise in any conducting medium, including an electrolyte carrying current as moving [[Ion|ions]] rather than as electrons. ## Thermal noise on capacitors A [[Capacitor|capacitor]] is a reactive element with no resistance of its own and so generates no thermal noise directly; the noise appears only once a capacitor is charged or discharged through some real resistance, whether a deliberate resistor or the on-resistance of a switch. Integrating the resistor's flat noise spectrum through the single-pole low-pass response that the resistor and capacitor form together gives a total mean-square voltage of `k_B*T/C`, in which the resistance value cancels out completely: a larger resistance admits noise over a narrower bandwidth in exact proportion to how much more of it there is, and the two effects offset each other perfectly, leaving only the capacitance to set the final noise level. The practical name for this effect is reset noise, or kTC noise: whenever a capacitor is disconnected, or "reset," from a charging path through a resistive switch, the voltage left on the capacitor carries an RMS uncertainty of `sqrt(k_B*T/C)`, unavoidable however small the switch's own resistance is made. The effect is a limiting noise source in switched-capacitor circuits and in the pixels of many image sensors, where each pixel's own small sense capacitor is reset once per frame. ## Thermometry Because the available noise power from a resistor depends on temperature and Boltzmann's constant alone, and not on the resistance or the material a resistor is made from, measuring that power gives a primary thermometer: one calibrated directly from a fundamental constant rather than against a reference material whose own properties must first be calibrated against something else. Johnson noise thermometry exploits exactly this, and is particularly valuable at cryogenic temperatures, where many conventional thermometers based on material properties become unreliable or need extensive individual calibration. Since the 2019 redefinition of the SI base units fixed the numerical value of the Boltzmann constant exactly, a Johnson-noise measurement is now, in principle, a direct realization of the kelvin itself rather than a measurement referred to some other calibrated standard. ## Thermal noise on inductors An ideal [[Inductor|inductor]], like an ideal capacitor, has no resistance of its own and generates no thermal noise directly; any noise associated with a real inductor comes from the resistance of the wire used to wind it. By the same reasoning that gives a capacitor's reset noise as kT/C, integrating a series resistor's noise through the single-pole response that the resistance and inductance form together gives a total mean-square current noise of `k_B*T/L`, again independent of the resistance value itself, with only the inductance setting the final level. The result is the magnetic analogue of kTC noise, and it appears in exactly the same circumstance: whenever a lossy resistive element and a reactive energy-storage element are considered together as a single noisy filter. ## Maximum transfer of noise power Maximum power transfer theory applies to a noisy resistor just as it does to any other source: the largest possible noise power a resistor can deliver to a load is achieved when the load's resistance matches the source's exactly, at which point exactly half of the open-circuit noise voltage appears across the load. Remarkably, that maximum available power, `P = k_B*T*deltaf`, does not depend on the resistance value at all, only on the temperature and the bandwidth, because a larger source resistance produces a proportionally larger open-circuit voltage but also a proportionally worse match to any fixed load resistance, and the two effects cancel exactly.[^ell096] This is the same expression, evaluated per hertz, that gives the −174 dBm/Hz figure quoted above; framed as an available power rather than a voltage, it is the form used whenever noise from several different-impedance sources must be added on a common footing, as in the equivalent-noise-temperature calculations used to specify a receiver.[^ell098] ## Nyquist's derivation of ideal resistor noise Nyquist's own derivation reached the same result from thermodynamics rather than from any detailed model of electrons colliding inside a resistor. He imagined a lossless [[Transmission_line|transmission line]] of characteristic impedance R, terminated at each end by a resistor of that same resistance R, with the whole system held at a common temperature T; because the line is matched at both ends, no power reflects, and in thermal equilibrium the two resistors must be exchanging power at exactly equal rates, or the line would carry a net heat flow between two bodies at the same temperature, forbidden by the second law of thermodynamics.[^nyquist1928] Modeling the confined line as a one-dimensional cavity supporting standing-wave modes and invoking the equipartition theorem, the statistical-mechanics result associated with [[Ludwig_Boltzmann|Ludwig Boltzmann]] and [[James_Clerk_Maxwell|James Clerk Maxwell]] and familiar from the classical derivation of black-body radiation, each mode in equilibrium carries an average energy of `k_B*T`. Counting how many such modes fall within a bandwidth Δf and converting that count into a power flowing in each direction along the line reproduces `P = k_B*T*deltaf` exactly, with no reference anywhere in the argument to what the resistor at either end is actually made of. That the answer depends only on temperature, not on any material property, is the derivation's real content: the same argument that fixes the black-body spectrum fixes the noise of every resistor. ## Generalized forms The result generalizes beyond a pure resistor to any passive circuit element or network with a real, resistive part to its impedance. For a general complex impedance Z(f), the noise voltage's power spectral density is proportional not to the impedance itself but to its real part alone, `S_v(f) = 4*k_B*T*Re(Z(f))`; a purely reactive element, having no real part at any frequency, therefore contributes no thermal noise of its own, consistent with the separate results for an ideal capacitor and an ideal inductor above. The classical law `P = k_B*T*deltaf` is itself only an approximation, valid when the photon energy at the frequency of interest is small compared with the thermal energy available, `h*f << k_B*T`. At sufficiently high frequency or sufficiently low temperature this classical approximation breaks down, and the available power per unit bandwidth must be replaced with a quantum expression that rolls off at high frequency instead of staying flat forever, `h*f / (exp(h*f/(k_B*T)) - 1)`, structurally the same factor that appears in Planck's law for the spectrum of black-body radiation, per mode rather than per unit area. At room temperature this correction only becomes noticeable in the terahertz range, far above the frequencies of ordinary electronics, which is why the flat, classical law is an excellent approximation everywhere circuits normally operate. For a network with more than two terminals, the single noise source of a two-terminal resistor generalizes to a full matrix of correlated noise sources, one for every port, related to the network's own impedance or admittance matrix by the same reasoning that gives the two-terminal result; ports that are only reactively coupled to each other can still show correlated noise even though neither port alone is noisy in isolation. ## Microsims The primary microsim, *Thermal noise*, puts three sliders directly on the law it demonstrates: temperature T from 1 K to 1000 K (default 300 K), resistance R on a logarithmic scale from 10 Ω to 1 MΩ (default near 100 kΩ), and bandwidth Δf on a logarithmic scale from 1 kHz to 10 MHz (default 1 MHz). Each frame the sketch computes v_rms from the Johnson–Nyquist law, fills a scope trace with Gaussian samples of that standard deviation, accumulates them into a histogram overlaid with the matching bell curve, and prints the derived RMS voltage, the flat spectral density and the available power alongside a small inset of a resistor with its electrons agitating. A freeze button holds one noise realization steady, and a second button toggles the spectral density between the flat classical prediction and the quantum roll-off at very high frequency. A three.js companion sim, built on the [[Spectral_density]] model, displays this same thermal-noise spectrum as the flat case among a family of coloured noises. *Try:* Drag the temperature slider from its default of 300 K down toward 1 K and watch the scope trace and the histogram both shrink, then raise the bandwidth slider back up and watch the same trace widen again. ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Johnson–Nyquist_noise) : [Wikitube](https://en.wikitube.io/wiki/Johnson–Nyquist_noise) Skeleton mirrored at revision 1368660154. Prose, emphasis and the microsims are Wikitube's own. ## Notes The primary microsim's Gaussian noise samples, generated by a Box–Muller transform, are an ILLUSTRATIVE stand-in for the astronomical number of individual electron impulses that a real resistor sums; the resistor-with-jittering-electrons inset is a cartoon of the mechanism, not a literal picture of it. Page numbers in the references below are PDF pages of the linked open edition. ## See also - [[Signal-to-noise_ratio]] - [[Noise_(electronics)]] - [[Resistor]] - [[Capacitor]] - [[Inductor]] - [[Spectral_density]] - [[Thermal_radiation]] - [[Quantum_mechanics]] - [[Transmission_line]] - [[Voltage]] ## References [^johnson1928]: Johnson, J. B. "Thermal Agitation of Electricity in Conductors." *Physical Review*, vol. 32, no. 1, 1928, pp. 97–109. [^nyquist1928]: Nyquist, H. "Thermal Agitation of Electric Charge in Conductors." *Physical Review*, vol. 32, no. 1, 1928, pp. 110–113. [^ell096]: Ellingson, S. *Radio Systems Engineering — Revised First Edition*. 2023, p. 96 (Eqs. 4.3–4.5, thermal noise power and RMS voltage). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. [^ell098]: Ellingson, S. *Radio Systems Engineering — Revised First Edition*. 2023, pp. 98–99 (Eq. 4.7, equivalent noise temperature). Open Textbook Library: https://open.umn.edu/opentextbooks/textbooks/radio-systems-engineering . CC BY-NC. ## External links - Live sketch: https://editor.p5js.org/sciencenibber/full/u1DcFfTzy - Editor (fork): https://editor.p5js.org/sciencenibber/sketches/u1DcFfTzy