Johnson–Nyquist noise

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. 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. Walter H. Schottky discovered shot noise in 1918, while studying Einstein's theories of thermal noise. == Noise of ideal resistors for moderate frequencies ==

with bandwidth \Delta f {=} f_\text{upper} {-} f_\text{lower} passes only the shaded area of height 4 k_\text{B} T R 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. Johnson's experiment (Figure 1) found that the thermal noise from a resistance R at kelvin temperature T and bandlimited to a frequency band of bandwidth \Delta f (Figure 3) has a mean square voltage of: : \overline {V_n^2} = 4 k_\text{B} T R \, \Delta f where k_{\rm B} is the Boltzmann constant (). While this equation applies to ideal resistors (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures, a more accurate accounts for complex impedances and quantum effects. Conventional electronics generally operate over a more limited bandwidth, so Johnson's equation is often satisfactory. Power spectral density The mean square voltage per hertz of bandwidth is 4 k_\text{B} T R and may be called the power spectral density (Figure 2). Its square root at room temperature (around 300 K) approximates to 0.13 \sqrt{R}, which has the unit . A 10 kΩ resistor, for example, would have approximately 13 nV/ at room temperature. RMS noise voltage 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.|303x303pxThe square root of the mean square voltage yields the root mean square (RMS) voltage observed over the bandwidth \Delta f : : V_\text{rms} = \sqrt{\overline {V_n^2}} = \sqrt{ 4 k_\text{B} T R \, \Delta f } \, . A resistor with thermal noise can be represented by its Thévenin equivalent circuit (Figure 4B) consisting of a noiseless resistor in series with a gaussian noise voltage source with the above RMS voltage. Around room temperature, 3 kΩ provides almost one microvolt of RMS noise over 20 kHz (the human hearing range) and 60 Ω·Hz for R \, \Delta f corresponds to almost one nanovolt of RMS noise. RMS noise current A resistor with thermal noise can also be converted into its Norton equivalent circuit (Figure 4C) consisting of a noise-free resistor in parallel with a gaussian noise current source with the following RMS current: : I_\text{rms} = {V_\text{rms} \over R} = \sqrt {{4 k_\text{B} T \Delta f } \over R}. == Thermal noise on capacitors ==

Ideal capacitors, as lossless devices, do not have thermal noise. However, the combination of a resistor and a capacitor (an RC circuit, a common low-pass filter) has what is called kTC noise. The equivalent noise bandwidth of an RC circuit is \Delta f {=} \tfrac{1}{4RC}. When this is substituted into the thermal noise equation, the result has an unusually simple form as the value of the resistance (R) drops out of the equation. This is because higher R decreases the bandwidth as much as it increases the spectral density of the noise in the passband. The mean-square and RMS noise voltage generated in such a filter are: : \overline {V_\text{n}^2} = {4 k_\text{B} T R \over 4 R C} = {k_\text{B} T \over C} : V_\text{rms} = \sqrt{4 k_\text{B} T R \over 4 R C} = \sqrt{ k_\text{B} T \over C }. The noise charge Q_n is the capacitance times the voltage: : Q_n = C \, V_n = C \sqrt{ k_\text{B} T \over C } = \sqrt{ k_\text{B} T C } : \overline{Q_n^2} = C^2 \, \overline{V_n^2} = C^2 {k_\text{B} T \over C} = k_\text{B} T C This charge noise is the origin of the term "kTC noise". Although independent of the resistor's value, all of the kTC noise arises in the resistor. Therefore, it would incorrect to double-count both a resistor's thermal noise and its associated kTC noise, and the temperature of the resistor alone should be used, even if the resistor and the capacitor are at different temperatures. Some values are tabulated below: Reset noise An extreme case is the zero bandwidth limit called the reset noise left on a capacitor by opening an ideal switch. Though an ideal switch's open resistance is infinite, the formula still applies. However, now the RMS voltage must be interpreted not as a time average, but as an average over many such reset events, since the voltage is constant when the bandwidth is zero. 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. The noise is not caused by the capacitor itself, but by the thermodynamic fluctuations of the amount of charge on the capacitor. Once the capacitor is disconnected from a conducting circuit, the thermodynamic fluctuation is frozen at a random value with standard deviation as given above. The reset noise of capacitive sensors is often a limiting noise source, for example in image sensors. Any system in thermal equilibrium has state variables with a mean energy of per degree of freedom. Using the formula for energy on a capacitor (E = 2), mean noise energy on a capacitor can be seen to also be C = . Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance. == Thermometry ==

The Johnson–Nyquist noise has applications in precision measurements, in which it is typically called "Johnson noise thermometry". 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. The voltage is measured over a period of 100 days and integrated. This was done in 2017, when the triple point of water's temperature was 273.16 K by definition, and the Boltzmann constant was experimentally measurable. Because the acoustic gas thermometry reached 0.2 ppm in uncertainty, and Johnson noise 2.8 ppm, this fulfilled the preconditions for a redefinition. After the 2019 redefinition, the kelvin was defined so that the Boltzmann constant has an exact value (), and the triple point of water became experimentally measurable. == Thermal noise on inductors ==

Inductors are the dual of capacitors. Analogous to kTC noise, a resistor with an inductor L results in a noise current that is independent of resistance: : \overline {I_\text{n}^2} = {k_\text{B} T \over L} \, . == Maximum transfer of noise power ==

The noise generated at a resistor R_\text{S} can transfer to the remaining circuit. The maximum power transfer happens when the Thévenin equivalent resistance R_{\rm L} of the remaining circuit matches R_\text{S}. Some example available noise power levels are tabulated below: == Nyquist's derivation of ideal resistor noise ==

1928 thought experiment 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 (R_1 {=} R_2). According to the conclusion of Figure 5, the total average power transferred over bandwidth \Delta f from R_1 and absorbed by R_2 was determined to be: : \overline {P_1} = k_{\rm B} T \, \Delta f \, . Simple application of Ohm's law says the current from V_1 (the thermal voltage noise of only R_1) through the combined resistance is I_1 {=} \tfrac{V_1}{R_1 + R_2} {=} \tfrac{V_1}{2R_1}, so the power transferred from R_1 to R_2 is the square of this current multiplied by R_2, which simplifies to: : 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} \, . Setting this P_\text{1} equal to the earlier average power expression \overline {P_1} allows solving for the average of V_1^2 over that bandwidth: : \overline{V_1^2} = 4 k_\text{B} T {R_1} \, \Delta f \, . Nyquist used similar reasoning to provide a generalized expression that applies to non-equal and complex impedances too. And while Nyquist above used k_{\rm B} T according to classical theory, Nyquist concluded his paper by attempting to use a more involved expression that incorporated the Planck constant h (from the new theory of quantum mechanics). == Generalized forms ==

The 4 k_\text{B} T R 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 passive and linear. Complex impedances Nyquist's original paper also provided the generalized noise for components having partly reactive response, e.g., sources that contain capacitors or inductors. : \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} \, . At very high frequencies (f \gtrsim \tfrac{k_\text{B} T}{h}), the spectral density S_{v_n v_n}(f) 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 \eta(f)=1 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 of blackbody radiation. 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, and further connected it to properties of antennas, particularly the fact that the average antenna aperture over all different directions cannot be larger than \tfrac{\lambda^2}{4\pi}, 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 passive electrical networks, including non-reciprocal devices such as circulators and isolators. 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, : S_{v_m v_n}(f) = 2 k_\text{B} T \eta(f) (Z_{mn}(f) + Z_{nm}(f)^*) where the Z_{mn} are the elements of the impedance matrix \mathbf{Z}. 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 : S_{i_m i_n}(f) = 2 k_\text{B} T \eta(f) (Y_{mn}(f) + Y_{nm}(f)^*) where \mathbf{Y} = \mathbf{Z}^{-1} is the admittance matrix. == Notes ==