Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the voltage or current 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 discovered shot noise in 1918, while studying Einstein's theories of thermal noise. Frits Zernike working in electrical metrology, found unusual random 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's 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 {\displaystyle R} at kelvin temperature T {\displaystyle T} and bandlimited to a frequency band of bandwidth Δ f {\displaystyle \Delta f} (Figure 3) has a mean square voltage of:
V n 2 ¯ = 4 k B T R Δ f {\displaystyle {\overline {V_{n}^{2}}}=4k_{\text{B}}TR\,\Delta f}
where k B {\displaystyle k_{\rm {B}}} is the Boltzmann constant (1.380649×10−23 J⋅K−1). While this equation applies to ideal resistors (i.e. pure resistances without any frequency-dependence) at non-extreme frequency and temperatures, a more accurate Johnson–Nyquist noise § Generalized forms 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 B T R {\displaystyle 4k_{\text{B}}TR} and may be called the power spectral density (Figure 2). Its square root at room temperature (around 300 K) approximates to 0.13 R {\displaystyle {\sqrt {R}}} , which has the unit nanovolts/√hertz. A 10 kΩ resistor, for example, would have approximately 13 nV/√Hz 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 Δ f {\displaystyle \Delta f} :
V rms = V n 2 ¯ = 4 k B T R Δ f . {\displaystyle V_{\text{rms}}={\sqrt {\overline {V_{n}^{2}}}}={\sqrt {4k_{\text{B}}TR\,\Delta f}}\,.}
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