Resistor Thermal Noise (Johnson–Nyquist)
Vn = √(4 × k × T × R × BW)
Quick answer
This calculator solves Vn = √(4 × k × T × R × BW) for Noise Voltage. With Temperature 50 °C, Resistance 0.47 kΩ, Bandwidth 1 kHz, the result is 91.58 nV. It solves in 4 directions, so you can enter the values you have and get the one you need.
Calculator
Formula
Description
Johnson–Nyquist noise is the thermal noise voltage generated by any resistor due to random thermal motion of charge carriers. It is a fundamental physical limit that cannot be eliminated by better manufacturing. The noise is white (flat across frequency) and sets the noise floor in precision analog circuits. At room temperature (25 °C), a 1 kΩ resistor generates about 4 nV/√Hz. This noise is critical in low-noise amplifier design, sensor front-ends, and audio preamplifiers.
Variables
- Vn — RMS noise voltage (V)
- T — Temperature (°C, converted to Kelvin internally)
- R — Resistance (Ω)
- BW — Measurement bandwidth (Hz)
Practical Notes
k = 1.380649 × 10⁻²³ J/K (Boltzmann constant). Temperature is entered in °C and internally converted to Kelvin (T_K = T_°C + 273.15). To minimize noise, use the lowest resistance that meets circuit requirements, reduce bandwidth with filtering, and cool the circuit if necessary. Metal-film resistors have lower excess noise (current noise) than carbon resistors, but thermal noise is identical for all resistor types at the same resistance.
Symbols and units
| Symbol | Meaning | Default unit | Accepted units |
|---|---|---|---|
| Vn | Noise Voltage | nV | pV, nV, µV, mV, V, kV |
| T | Temperature | °C | °C |
| R | Resistance | Ω | mΩ, Ω, kΩ, MΩ |
| BW | Bandwidth | Hz | Hz, kHz, MHz, GHz |
Step by step
- 1. What you know: Temperature 50 °C, Resistance 0.47 kΩ, Bandwidth 1 kHz
- 2. The relation: Vn = √(4 × k × T × R × BW)
- 3. Substitute: T = 50 °C, R = 0.47 kΩ, BW = 1 kHz
- 4. Result: Noise Voltage = 91.58 nV
Worked examples
Minimum range
Given: Temperature 2.5 °C, Resistance 1 Ω, Bandwidth 5 Hz
Result: Noise Voltage = 0.2759 nV
Low range
Given: Temperature 25 °C, Resistance 10 Ω, Bandwidth 50 Hz
Result: Noise Voltage = 2.869 nV
Typical
Given: Temperature 50 °C, Resistance 0.47 kΩ, Bandwidth 1 kHz
Result: Noise Voltage = 91.58 nV
High range
Given: Temperature 85 °C, Resistance 4.7 kΩ, Bandwidth 20 kHz
Result: Noise Voltage = 1.364 µV
Maximum range
Given: Temperature 850 °C, Resistance 47 kΩ, Bandwidth 0.2 MHz
Result: Noise Voltage = 24.15 µV
Solving for each variable
Solve for Noise Voltage
Temperature 2.5 °C, Resistance 1 Ω, Bandwidth 5 Hz → 0.2759 nV
Solve for Temperature
Noise Voltage 1 nV, Resistance 1 Ω, Bandwidth 5 Hz → 3348 °C
Solve for Resistance
Noise Voltage 1 nV, Temperature 2.5 °C, Bandwidth 5 Hz → 13.14 Ω
Solve for Bandwidth
Noise Voltage 1 nV, Temperature 2.5 °C, Resistance 1 Ω → 65.69 Hz
How Noise Voltage changes with Temperature
| Temperature (input) | Noise Voltage (result) |
|---|---|
| 5 °C | 84.97 nV |
| 12.5 °C | 86.11 nV |
| 25 °C | 87.97 nV |
| 37.5 °C | 89.8 nV |
| 50 °C | 91.58 nV |
| 75 °C | 95.06 nV |
| 100 °C | 98.42 nV |
| 200 °C | 110.8 nV |
| 350 °C | 127.2 nV |
| 500 °C | 141.7 nV |
The same result in other units
| Unit | Value |
|---|---|
| pV | 91580 |
| nV | 91.58 |
| µV | 0.09158 |
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<p><a href="https://vbatpower.com/formulas/resistor-noise/">Resistor Thermal Noise (Johnson–Nyquist)</a> Calculator by vbatpower.com</p>Frequently asked questions
How do you calculate Resistor Thermal Noise (Johnson–Nyquist)?
Use Vn = √(4 × k × T × R × BW). With Temperature 50 °C, Resistance 0.47 kΩ, Bandwidth 1 kHz, Noise Voltage works out to 91.58 nV.
What is the formula for Resistor Thermal Noise (Johnson–Nyquist)?
The formula is Vn = √(4 × k × T × R × BW). Johnson–Nyquist noise is the thermal noise voltage generated by any resistor due to random thermal motion of charge carriers.
How do you find Temperature when you know the other values?
Rearrange for Temperature: with Noise Voltage 1 nV, Resistance 1 Ω, Bandwidth 5 Hz, Temperature is 3348 °C.
How do you find Resistance when you know the other values?
Rearrange for Resistance: with Noise Voltage 1 nV, Temperature 2.5 °C, Bandwidth 5 Hz, Resistance is 13.14 Ω.
What units does Resistor Thermal Noise (Johnson–Nyquist) use?
Noise Voltage is shown in pV, nV, µV. Values are held in SI base units internally, so you can mix prefixes without converting by hand.
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