Johnson–Nyquist noise

Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens regardless of any applied voltage. Thermal noise is present in all electrical circuits, and in sensitive electronic equipment (such as radio receivers) can drown out weak signals, and can be the limiting factor on sensitivity of electrical measuring instruments. Thermal noise is proportional to absolute temperature, so some sensitive electronic equipment such as radio telescope receivers are cooled to cryogenic temperatures to improve their signal-to-noise ratio. The generic, statistical physical derivation of this noise is called the fluctuation-dissipation theorem, where generalized impedance or generalized susceptibility is used to characterize the medium. 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.

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. Thermal noise is distinct from shot noise, which consists of additional current fluctuations that occur when a voltage is applied and a macroscopic current starts to flow.

History of thermal 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 studied the problem in 1918, while studying thermal noise using Einstein's theories, experimentally discovered another kind of noise, the shot noise.

Frits Zernike working in electrical metrology, found unusual aleatory deflections while working with high-sensitive galvanometers. He rejected the idea that the noise was mechanical, and concluded that it was of thermal nature. In 1927, he introduced the idea of autocorrelations to electrical measurements and calculated the time detection limit. His work coincided with de Haas-Lorentz' prediction.

The same year, working independently without any knowledge of Zernike's work, John B. Johnson working in Bell Labs found the same kind of noise in communication systems, but described it in terms of frequencies. 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.

Noise of ideal resistors for moderate frequencies
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 ($1.381$ joules per kelvin). 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.

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 $$\sqrt{R}$$ in units of $nanovolts⁄√hertz$. A 10 kΩ resistor, for example, would have approximately 13 $nanovolts⁄√hertz$ at room temperature.

RMS noise voltage
The 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 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 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 noise.

The mean-square and RMS noise voltage generated in such a filter are:

\overline {V_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, 100% 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 $kT⁄2$ per degree of freedom. Using the formula for energy on a capacitor (E = $1⁄2$CV2), mean noise energy on a capacitor can be seen to also be $1⁄2$C$kT⁄C$ = $kT⁄2$. Thermal noise on a capacitor can be derived from this relationship, without consideration of resistance.

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_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 $$R_\text{S}$$. In this case, each of the two resistors dissipates noise in both itself and in the other resistor. Since only half of the source voltage drops across any one of these resistors, this maximum noise power transfer is:
 * $$P_\text{max} = k_\text{B} \,T \Delta f \, .$$

This maximum is independent of the resistance and is called the available noise power from a resistor.

Available noise power in decibel-milliwatts
Signal power is often measured in dBm (decibels relative to 1 milliwatt). Available noise power would thus be $$10\ \log_{10}(\tfrac{k_\text{B} T \Delta f}{\text{1 mW}})$$ in dBm. At room temperature (300 K), the available noise power can be easily approximated as $$10\ \log_{10}(\Delta f) - 173.8$$ in dBm for a bandwidth in hertz. Some example available noise power in dBm are tabulated below:

Nyquist's derivation of ideal resistor noise
Nyquist's 1928 paper "Thermal Agitation of Electric Charge in Conductors" used concepts about potential energy and harmonic oscillators from the equipartition law of Boltzmann and Maxwell 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:


 * $$P_\text{1} = I_1^2 R_2 = I_1^2 R_1 = (\tfrac{V_1}{2R_1})^2 R_1 = \tfrac{V_1^2}{4R_1} \, .$$

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. Such a component can be described by a frequency-dependent complex electrical impedance $$Z(f)$$. The formula for the power spectral density of the series noise voltage is

S_{v_n v_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Z(f)]. $$ The function $$\eta(f)$$ is approximately 1, except at very high frequencies or near absolute zero (see below).

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

Alternatively, a parallel noise current can be used to describe Johnson noise, its power spectral density being

S_{i_n i_n}(f) = 4 k_\text{B} T \eta(f) \operatorname{Re}[Y(f)]. $$ where $$Y(f) {=} \tfrac{1}{Z(f)}$$ is the electrical admittance; note that $$\operatorname{Re}[Y(f)] {=} \tfrac{\operatorname{Re}[Z(f)]}{|Z(f)|^2} \, .$$

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 $$\eta(f)$$ mentioned earlier is in general given by:
 * $$\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}$$), the function $$\eta(f)$$ starts to exponentially decrease to zero. At room temperature this transition occurs in the terahertz, far beyond the capabilities of conventional electronics, and so it is valid to set $$\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.