Johnson Noise: The Thermal Noise You Can't Escape

What Is Johnson Noise?

Johnson noise—also called Johnson-Nyquist noise or thermal noise—is the random electrical noise generated by the thermal motion of charge carriers (electrons) in a resistor or conductor. It's not a design flaw or a manufacturing defect. It's fundamental physics. Any resistor at any temperature above absolute zero radiates noise. Period.

The noise exists because electrons are constantly jiggling around due to thermal energy. That random motion creates random current fluctuations, which show up as voltage noise across the resistor. The hotter the resistor, the more violent the jiggling, and the more noise you get.

Why should you care? Because Johnson noise sets a hard floor on the signal-to-noise ratio (SNR) of your amplifier, receiver, sensor interface, or measurement circuit. You can't design it away. You can only understand it, account for it, and sometimes work around it by cooling, filtering, or averaging.

The Formula

The power spectral density of Johnson noise is given by the Nyquist formula:

where:

• $S_v$ is the voltage noise power spectral density in V²/Hz
• $k_B$ is Boltzmann's constant: $1.38 \times 10^{-23}$ J/K
• $T$ is the absolute temperature in Kelvin
• $R$ is the resistance in ohms

This is the key equation. Memorize it or bookmark it. Everything else follows from this.

Notice what's missing: frequency. Johnson noise is white—it has the same power density at 1 Hz and 1 GHz (within the limits of the Rayleigh-Jeans approximation, which holds for RF and audio). That's both good and bad. Good because it's predictable. Bad because you can't filter it away without also filtering your signal.

From Spectral Density to RMS Voltage

The spectral density tells you the noise per unit bandwidth. To get the actual RMS voltage you'd measure, you need to integrate over your measurement bandwidth $B$:

This is the form you'll use most often in practice. It shows that noise voltage scales with the square root of bandwidth, resistance, and temperature.

A Worked Example: Audio Preamp Input

Let's design a microphone preamp and figure out what Johnson noise we're dealing with.

Your microphone has an output impedance of 2 kΩ. Your preamp has an input impedance of 10 kΩ (the source sees a parallel combination of the mic impedance and the preamp input impedance, but let's focus on the preamp input stage for now). The preamp is operating at room temperature (T = 293 K). Your audio bandwidth is 20 Hz to 20 kHz, so $B = 19,980$ Hz ≈ 20 kHz.

First, calculate the Johnson noise voltage from the 10 kΩ input resistor:

Now add the microphone source impedance. The mic output impedance of 2 kΩ also generates Johnson noise:

These add in quadrature (they're uncorrelated), so the total input-referred noise is:

Now, if your preamp has a gain of 40 dB (100 V/V), the output noise is:

For a typical line-level signal of 1 V, your SNR is:

That's not bad for audio, but it's not great either. If your preamp's op-amp adds its own noise (typically 10–20 nV/√Hz), you'd need to account for that too. Use the Johnson-Nyquist Thermal Noise Calculator to plug in your own values and see what you're working with.

Why Temperature Matters

Notice that Johnson noise scales linearly with absolute temperature. At room temperature (293 K), you get a certain noise floor. Drop the temperature to 77 K (liquid nitrogen), and you reduce noise by a factor of $\sqrt{293/77} \approx 1.95$—nearly 6 dB quieter.

That's why cryogenic amplifiers are used in radio astronomy and some satellite receivers. The noise reduction is real and significant. But it's expensive and impractical for most applications.

For everyday circuits, the practical takeaway is: Johnson noise at room temperature is what it is. You're stuck with it unless you're willing to cool your hardware.

Bandwidth Matters Too

Notice the square-root dependence on bandwidth. Double your measurement bandwidth, and noise increases by $\sqrt{2} \approx 1.41$ (about 1.5 dB). Halve it, and noise drops by the same factor.

This is why narrowband filters are so useful in measurement and receiver design. A 1 kHz bandwidth filter on a sensor signal will have much less Johnson noise than a 1 MHz filter measuring the same resistance. The trade-off is that you lose signal content outside that bandwidth.

Common Mistakes and Gotchas

Mistake 1: Ignoring Johnson noise in low-impedance circuits. Some engineers think Johnson noise only matters in high-impedance circuits. Wrong. A 50 Ω resistor at room temperature generates about 0.9 nV/√Hz. That's tiny in absolute terms, but if you're amplifying it with a 60 dB gain stage, it becomes 0.9 nV × 1000 = 0.9 μV/√Hz at the output. In a wideband system (say, 1 GHz), that's 28.5 μV RMS. Not negligible. Mistake 2: Forgetting to add noise sources in quadrature. If you have multiple resistors or noise sources in your circuit, they don't add linearly—they add in quadrature because they're uncorrelated. The total noise is $\sqrt{V_1^2 + V_2^2 + V_3^2 + \ldots}$, not $V_1 + V_2 + V_3 + \ldots$. This is a huge source of error in hand calculations. Mistake 3: Using the wrong bandwidth. When you calculate Johnson noise, use the actual bandwidth your circuit responds to. If you have a 1 MHz filter but you're only looking at a 10 kHz signal, use 10 kHz, not 1 MHz. The bandwidth is set by your filter, not by the input signal frequency. Mistake 4: Confusing noise spectral density with total noise. The formula $S_v = 4 k_B T R$ gives you noise power per Hz. To get total noise, you must multiply by bandwidth. A common mistake is to quote the spectral density as if it's the total noise voltage, which underestimates noise by orders of magnitude in wideband systems. Mistake 5: Assuming Johnson noise is the only noise source. In real circuits, you also have flicker noise (1/f noise), shot noise, and quantization noise (in digital systems). Johnson noise is often the dominant noise source in resistive elements, but it's not always the biggest problem. In an ADC, quantization noise might dominate. In a photodiode, shot noise might be larger. Always check the spec sheets and do a noise budget.

Practical Design Implications

In RF and analog design, Johnson noise sets the noise figure floor of your system. Your amplifier's noise figure can never be lower than the noise generated by the source resistance itself. If your source is a 50 Ω antenna at room temperature, that generates about 0.9 nV/√Hz. Your preamp noise figure is limited by how close it can get to that number.

In sensor circuits, Johnson noise from the sensor's output impedance and the input impedance of your signal conditioning circuit directly limits your measurement resolution. If you're trying to measure a 10 mV signal with a 100 kΩ impedance sensor at room temperature, Johnson noise is about 40.5 nV/√Hz. Over a 10 kHz bandwidth, that's 1.28 μV RMS—about 0.01% of your signal. That's probably okay. But if your bandwidth is 1 MHz, noise climbs to 12.8 μV, and you've lost a decimal place of precision.

In ADC applications, Johnson noise from the input network can degrade SNR. The ADC SNR and ENOB Calculator lets you factor in Johnson noise alongside quantization noise to see the real-world SNR of your data acquisition system.

When to Worry, When to Ignore

If your signal is in the millivolt range or larger, and your impedances are under 1 kΩ, Johnson noise is probably not your biggest problem. Your circuit's op-amp noise, digital noise, or EMI is more likely to dominate.

If your signal is in the microvolt range, or your impedances are very high (megohms), or your bandwidth is very wide, Johnson noise becomes a real constraint. That's when you need to optimize: lower impedances, narrower bandwidth, lower noise op-amps, or sometimes cryogenic cooling.

Try It

Take your circuit topology, grab your component values, and plug them into the Johnson-Nyquist Thermal Noise Calculator. Enter your resistance, temperature, and bandwidth. See what noise voltage you're actually dealing with. Then compare it to your signal level and your op-amp's noise spec. That's your noise budget.

Johnson noise is not optional—it's physics. But understanding it means you can design around it, account for it in your specifications, and avoid nasty surprises in the lab.