Resistor Noise Calculator

What the resistor noise calculator does

Every resistor generates a small random voltage noise simply because it is warm, called thermal or Johnson noise. This calculator finds that noise voltage from the resistance, the temperature, and the bandwidth, and can also solve backward for any of those.

Below is what thermal noise is, the equation behind it, what makes it larger or smaller, and a worked example.

How to use it

1. Choose what to calculate: the noise voltage, resistance, temperature, or bandwidth.
2. Enter the other values, such as the resistance, temperature, and bandwidth.
3. Press Calculate for the result, or Reset to clear it.

What thermal noise is

Thermal noise is a small, random fluctuating voltage that appears across any resistor purely because of its temperature. It arises from the ceaseless thermal motion of the charge carriers inside the resistor: the electrons jiggle about randomly as a result of heat, and this random motion creates a tiny, constantly changing voltage even when no current is deliberately flowing. Because it depends only on temperature and not on any applied signal, it is present in every resistor, all the time, and cannot be switched off short of cooling the resistor to absolute zero.

This noise was first measured by John Johnson and explained by Harry Nyquist in the 1920s, which is why it is also called Johnson noise or Johnson-Nyquist noise. It is fundamental, set by the laws of thermodynamics rather than by any imperfection in the resistor, so even a perfect resistor produces it. While usually extremely small, it sets a basic limit on how faint a signal a circuit can handle, because a signal weaker than the noise is lost in it. This calculator computes that noise voltage from the resistor's properties and conditions.

The equation it uses

The thermal noise voltage is the square root of a product of four things:

E = √(4 k T R Δf)

Here E is the noise voltage, k is the Boltzmann constant, T is the absolute temperature, R is the resistance, and Δf is the bandwidth over which the noise is measured. The Boltzmann constant is the fundamental link between temperature and energy, which is why it appears. The noise grows with the square root of the temperature, the resistance, and the bandwidth, so each of these raises it, but only gently because of the square root. The calculator evaluates this relationship, and can rearrange it to find any of the quantities from the others.

What raises and lowers the noise

Three things set how much thermal noise a resistor produces, and understanding them shows how to reduce it. A larger resistance gives more noise, which is one reason low-noise circuits favour smaller resistor values where possible. A higher temperature gives more noise, since the noise comes from heat, which is why the most sensitive instruments are sometimes cooled to very low temperatures to quieten their resistors. And a wider bandwidth gives more noise, because a wider range of frequencies admits more of the random fluctuation.

Because all three enter under a square root, their effect is softened: to double the noise voltage you must quadruple the resistance, the temperature, or the bandwidth. This also means the noise is often quoted per unit of bandwidth, as a noise density, since the bandwidth depends on the circuit rather than the resistor. The bandwidth is frequently the easiest factor to control, by limiting a circuit to only the range of frequencies it needs, which keeps out noise from the rest. The calculator lets you see how each factor moves the noise, which is the starting point for designing a quiet circuit.

The noise floor and why it matters

Thermal noise sets what engineers call the noise floor, the baseline level of random voltage below which real signals cannot be distinguished. Any signal stronger than the noise floor can be picked out and amplified, but a signal weaker than it is buried, mixed inseparably with the random fluctuation. This makes thermal noise a fundamental limit on the sensitivity of amplifiers, sensors, radio receivers, and any circuit that must detect faint signals.

The practical consequence is that designers of sensitive electronics treat thermal noise as an enemy to be minimised. They choose low resistor values, sometimes cool critical components, and restrict the bandwidth to only what is needed, all to push the noise floor as low as possible and let weaker signals be recovered. Knowing the thermal noise of the resistors in a circuit tells you where that floor sits and whether a given signal will rise above it. This calculator provides that figure, which is why it is useful for anyone working at the edge of what a circuit can detect.

Units and precision

The calculator takes the resistance in ohms and its multiples, the temperature in Celsius, Fahrenheit, or kelvin, and the bandwidth in hertz and its multiples, returning the noise voltage in volts and its smaller multiples. It uses the standard value of the Boltzmann constant and converts the temperature to its absolute scale internally, since the physics depends on absolute temperature. It also reports the noise level in decibel-based units. The relationship is exact, and the calculator can solve for any of the four quantities.

A worked example

Suppose a 1-kilohm resistor sits at room temperature, about 300 kelvin, in a circuit with a 10-kilohertz bandwidth.

The thermal noise voltage is E = √(4 k T R Δf) ≈ 0.41 microvolts. Spread over the full bandwidth, this is the random voltage the resistor adds to the circuit. Expressed per unit of bandwidth, the same resistor produces about 4 nanovolts for each square root of a hertz, a figure often quoted for low-noise design. Cooling the resistor or narrowing the bandwidth would lower the noise.

Questions people ask

How do you calculate thermal noise?

Use E = √(4 k T R Δf), from the Boltzmann constant, the absolute temperature, the resistance, and the bandwidth. The noise grows with the square root of each.

Why do resistors produce noise?

Because the charge carriers inside move randomly due to heat, creating a tiny fluctuating voltage. It depends only on temperature, so every warm resistor has it.

How can thermal noise be reduced?

By using smaller resistances, lowering the temperature, or narrowing the bandwidth. Each appears under a square root, so large changes are needed to make a big difference.

What is the noise floor?

The baseline random voltage below which real signals cannot be distinguished. Thermal noise sets it, limiting how faint a signal a circuit can detect.

References

A quick note on where the physics comes from. Thermal noise, the Johnson-Nyquist relationship, and the Boltzmann constant are standard physics, set out in Georgia State University's HyperPhysics and in the original work of Johnson and Nyquist. The Boltzmann constant follows NIST. The HyperPhysics link is worth a quick click to confirm it lands where you expect.

1. HyperPhysics, Johnson Noise. http://hyperphysics.phy-astr.gsu.edu/hbase/electronic/johnson.html
2. National Institute of Standards and Technology (NIST), Fundamental Physical Constants, Boltzmann constant. https://physics.nist.gov/cgi-bin/cuu/Value?k
3. Wikipedia, Johnson-Nyquist noise. https://en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise