Johnson–Nyquist noise

Thermal noise in an ideal resistor is approximately white, meaning that its power spectral density is nearly constant throughout the frequency spectrum (Figure 2).

When limited to a finite bandwidth and viewed in the time domain (as sketched in Figure 1), thermal noise has a nearly Gaussian amplitude distribution.

[1] For the general case, this definition applies to charge carriers in any type of conducting medium (e.g. ions in an electrolyte), not just resistors.

In 1905, in one of Albert Einstein's Annus mirabilis papers the theory of Brownian motion was first solved in terms of thermal fluctuations.

The following year, in a second paper about Brownian motion, Einstein suggested that the same phenomena could be applied to derive thermally-agitated currents, but did not carry out the calculation as he considered it to be untestable.

[2] Geertruida de Haas-Lorentz, daughter of Hendrik Lorentz, in her doctoral thesis of 1912, expanded on Einstein stochastic theory and first applied it to the study of electrons, deriving a formula for the mean-squared value of the thermal current.

In 1927, he introduced the idea of autocorrelations to electrical measurements and calculated the time detection limit.

[4][5][2] He described his findings to Harry Nyquist, also at Bell Labs, who used principles of thermodynamics and statistical mechanics to explain the results, published in 1928.

While this equation applies to ideal resistors (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures, a more accurate general form accounts for complex impedances and quantum effects.

Conventional electronics generally operate over a more limited bandwidth, so Johnson's equation is often satisfactory.

Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (the human hearing range) and 60 Ω·Hz for

However, the combination of a resistor and a capacitor (an RC circuit, a common low-pass filter) has what is called kTC noise.

Some values are tabulated below: An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch.

In this sense, the Johnson noise of an RC circuit can be seen to be inherent, an effect of the thermodynamic distribution of the number of electrons on the capacitor, even without the involvement of a resistor.

Once the capacitor is disconnected from a conducting circuit, the thermodynamic fluctuation is frozen at a random value with standard deviation as given above.

Any system in thermal equilibrium has state variables with a mean energy of ⁠kT/2⁠ per degree of freedom.

[9] For example, the NIST in 2017 used the Johnson noise thermometry to measure the Boltzmann constant with uncertainty less than 3 ppm.

It accomplished this by using Josephson voltage standard and a quantum Hall resistor, held at the triple-point temperature of water.

After the 2019 redefinition, the kelvin was defined so that the Boltzmann constant is 1.380649×10−23 J⋅K−1, and the triple point of water became experimentally measurable.

[14][15]: 260 Some example available noise power in dBm are tabulated below: Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors"[6] used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell[16] to explain Johnson's experimental result.

Nyquist's thought experiment summed the energy contribution of each standing wave mode of oscillation on a long lossless transmission line between two equal resistors (

over that bandwidth: Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and complex impedances too.

according to classical theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the Planck constant

voltage noise described above is a special case for a purely resistive component for low to moderate frequencies.

All of these generalizations share a common limitation, that they only apply in cases where the electrical component under consideration is purely passive and linear.

Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors.

With proper consideration of quantum effects (which are relevant for very high frequencies or very low temperatures near absolute zero), the multiplying factor

At room temperature this transition occurs in the terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set

Richard Q. Twiss extended Nyquist's formulas to multi-port passive electrical networks, including non-reciprocal devices such as circulators and isolators.

Again, an alternative description of the noise is instead in terms of parallel current sources applied at each port.

Figure 1. Johnson 's 1927 experiment showed that if thermal noise from a resistance of ${\text{R}}$ with temperature ${\text{T}}$ is bandlimited to bandwidth $\Delta f$ , then its root mean squared voltage $(V_{\text{rms}})$ is ${\sqrt {4k_{\text{B}}{\text{T}}\,{\text{R}}\,\Delta f}}$ in general, where $k_{\text{B}}$ is the Boltzmann constant .

Figure 2. Johnson–Nyquist noise has a nearly a constant $4$ $k B$ $T$ $R$ power spectral density per unit of frequency , but does decay to zero due to quantum effects at high frequencies ( terahertz for room temperature). This plot's horizontal axis uses a log scale such that every vertical line corresponds to a power of ten of frequency in hertz .

Figure 3. While thermal noise has an almost constant power spectral density of $4k_{\text{B}}TR$ , a band-pass filter with bandwidth $\Delta f{=}f_{\text{upper}}{-}f_{\text{lower}}$ passes only the shaded area of height $4k_{\text{B}}TR$ and width $\Delta f$ . Note: practical filters don't have brickwall cutoffs , so the left and right edges of this area are not perfectly vertical.

Figure 4. These circuits are equivalent:

**(A)** A resistor at nonzero temperature with internal thermal noise;

**(B)** Its Thévenin equivalent circuit: a noiseless resistor in series with a noise voltage source ;

**(C)** Its Norton equivalent circuit: a noiseless resistance in parallel with a noise current source .

Figure 5. Schematic of Nyquist's 1928 thought experiment ^{[

6

]} ^{[

2

]} 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 $l$ . Each resistor's noise signal propagates across the line at velocity ${\text{v}}$ . All impedances are identical, so both signals are absorbed by the opposite resistor instead of being reflected .

Nyquist then imagined 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 $\Delta f$ , there are $2l\,\Delta f/{\text{v}}$ modes of oscillation . ^{[

note 3

]} Each mode provides $k_{\rm {B}}T$ of energy on average, of which $k_{\rm {B}}T/2$ is electric and $k_{\rm {B}}T/2$ is magnetic, so the total energy in that bandwidth on average is $k_{\rm {B}}T\cdot 2l\,\Delta f/{\text{v}}.$ Each resistor contributed $k_{\rm {B}}T\cdot l\,\Delta f/{\text{v}}$ (half of that total energy).

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 $l/{\text{v}}$ . Dividing the average **energy** transferred from *each* resistor to the line by the transit **time** interval results in a total **power** of $k_{\rm {B}}T\,\Delta f$ transferred over bandwidth $\Delta f$ on average from each resistor.

{\displaystyle l} — Figure 5. Schematic of Nyquist's 1928 thought experiment ^{[

6

]} ^{[

2

]} 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 $l$ . Each resistor's noise signal propagates across the line at velocity ${\text{v}}$ . All impedances are identical, so both signals are absorbed by the opposite resistor instead of being reflected .

Nyquist then imagined 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 $\Delta f$ , there are $2l\,\Delta f/{\text{v}}$ modes of oscillation . ^{[

note 3

]} Each mode provides $k_{\rm {B}}T$ of energy on average, of which $k_{\rm {B}}T/2$ is electric and $k_{\rm {B}}T/2$ is magnetic, so the total energy in that bandwidth on average is $k_{\rm {B}}T\cdot 2l\,\Delta f/{\text{v}}.$ Each resistor contributed $k_{\rm {B}}T\cdot l\,\Delta f/{\text{v}}$ (half of that total energy).

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 $l/{\text{v}}$ . Dividing the average **energy** transferred from *each* resistor to the line by the transit **time** interval results in a total **power** of $k_{\rm {B}}T\,\Delta f$ transferred over bandwidth $\Delta f$ on average from each resistor.