Johnson–Nyquist RC Noise

Johnson–Nyquist noise is the electronic noise generated by the thermal agitation of charge carriers inside an electrical conductor at equilibrium. In a simple RC circuit consisting of a resistor and a capacitor, the thermal noise of the resistor is filtered by the RC low-pass characteristic. A surprising and elegant classical result is that the total integrated voltage variance across the capacitor depends only on temperature and capacitance, not on resistance.

Worked Example

Let us calculate the thermal noise properties for a standard RC circuit at room temperature (300 K) with a 10 kΩ resistor and a 1 nF capacitor. The classical one-sided resistor voltage PSD is given by 4 * kB * T * R, and the corner frequency is fc = 1 / (2 * π * R * C).

Temperature T = 300 K, Resistance R = 10 kΩ = 1e4 Ω, Capacitance C = 1 nF = 1e-9 F.
Boltzmann constant kB = 1.380649e-23 J/K.
DC Power Spectral Density: S(0) = 4 * 1.380649e-23 * 300 * 1e4 = 1.6567788e-16 V²/Hz.
Corner frequency: fc = 1 / (2 * π * 1e4 * 1e-9) ≈ 15915.494 Hz.
Probe at fc (ratio = 1): PSD = S(0) / (1 + 1²) = 8.283894e-17 V²/Hz.
All-frequency voltage variance: var = kB * T / C = 1.380649e-23 * 300 / 1e-9 = 4.141947e-12 V².
Bandwidth up to fc (ratio = 1): Band variance = var * (2/π) * atan(1) = 4.141947e-12 * (2/π) * (π/4) = 2.0709735e-12 V².
Band RMS voltage: sqrt(2.0709735e-12) ≈ 1.4390877e-06 V.

Experiment

You can interact with the Johnson–Nyquist RC Noise module to observe how different parameters affect the PSD and integrated noise variance.

Set Temperature K to "300", Resistance kiloohms to "10", Capacitance nanofarads to "1", Bandwidth over corner to "1", and Probe frequency over corner to "1". Observe that the Band voltage variance is exactly half (0.5 spectral fraction) of the All-frequency voltage variance (4.141947e-12 V²).
Set Bandwidth over corner to "All". Notice the Band voltage variance equals the All-frequency voltage variance, and the spectral fraction is 1.
Set Resistance kiloohms to "100" while keeping Bandwidth over corner to "1" and Probe frequency over corner to "1". Observe that the corner frequency drops by a factor of 10, the PSD at the probe increases, but the All-frequency voltage variance and the Band voltage variance remain exactly the same as in step 1. This demonstrates that the total noise variance is independent of resistance.
Set Temperature K to "0". Observe that the thermal noise vanishes, resulting in 0.0 for PSD and variance.

Limitations

This classical model assumes the PSD is perfectly flat at all frequencies (white noise). In reality, quantum effects (zero-point noise) cause the PSD to drop off at extremely high frequencies (when hf > kB*T). This module does not make claims about quantum zero-point noise or high-frequency validity beyond the classical RC filtering. Furthermore, the "All-frequency" variance mathematically integrates to infinity, but physical systems always have some finite parasitic inductance and capacitance that provide an ultimate high-frequency cutoff.

References

For a detailed theoretical derivation of Johnson-Nyquist noise and kT/C noise in circuits, refer to:

EE 435 Lecture 40 Spring 2021, Iowa State University - Covers noise in electronic circuits, including thermal noise and the kT/C variance result.

Johnson–Nyquist RC Noise