Thermal Noise Calculator

Origin of Thermal Noise

All resistive elements generate random voltage fluctuations due to the thermal agitation of charge carriers. This effect was first described by John B. Johnson and later analyzed by Harry Nyquist, leading to the term Johnson–Nyquist noise. The phenomenon is present in every conductor and sets a fundamental limit on how quiet an electronic circuit can be. Because it stems from thermodynamics, this noise exists even in perfectly stable circuits with no external interference. Understanding its magnitude helps engineers design low-noise amplifiers, radio receivers, and high-precision measurements.

The Johnson–Nyquist Equation

The RMS voltage of thermal noise across a resistor is expressed by $\sqrt{4kTRB}$, where $k$ is Boltzmann’s constant (1.380649×10⁻²³ J/K), $T$ is the absolute temperature in kelvins, $R$ is the resistance in ohms, and $B$ is the measurement bandwidth in hertz. This relationship shows that noise increases with temperature, resistance, and the range of frequencies included in the measurement. No matter how advanced the electronics, thermal noise provides an irreducible background that must be taken into account.

Bandwidth Considerations

The bandwidth in the formula represents the range of frequencies over which you measure or amplify the noise. A wider bandwidth lets in more noise power. For example, a radio receiver tuned to a narrow channel only collects noise from that small slice of the spectrum, reducing the total noise voltage. In contrast, wideband instrumentation amplifiers must contend with more noise, often requiring specialized filtering or cooling to suppress it. Careful selection of bandwidth is therefore a key strategy in minimizing the impact of thermal noise.

Example Calculation

Suppose you have a 50 Ω resistor at room temperature (about 300 K) and you measure over a 10 kHz bandwidth. Plugging these values into the formula gives $v=\sqrt{4k3005010000}$, which comes to roughly 0.9 μV RMS. Although tiny, this voltage can still swamp the signal in precision circuits or sensors unless proper amplification and filtering are used.

Noise Power

Instead of voltage, you may be interested in the noise power delivered to a matched load. By the same principles, the available noise power is $P=kTB$ watts. This expression is independent of resistance when the source and load impedance are equal. Designers of communication systems often use this form to estimate the theoretical noise floor in a receiver.

Impact on Low-Noise Design

Achieving extremely low noise figures is essential for radio astronomy, medical imaging, and many scientific instruments. Engineers carefully select resistors with minimal excess noise, cool detectors to cryogenic temperatures, and use narrowband filtering. Despite these efforts, thermal noise remains a hard limit determined by fundamental constants. This calculator helps quantify that limit so you know how close your design approaches the theoretical minimum.

Logging Measurements

When building prototypes, engineers often note computed noise levels next to measured data. Click the copy button to transfer the voltage estimate into your lab notebook or simulation comments for easy comparison.

Reducing Thermal Noise

Lowering the temperature decreases thermal agitation. Superconducting circuits operated near absolute zero exhibit virtually no Johnson–Nyquist noise, though such extreme conditions are impractical for most applications. Similarly, reducing the resistance or narrowing the bandwidth reduces the noise, but only at the cost of signal level or system flexibility. The trade-offs between these parameters are central to many electronic design decisions.

Summary

Thermal noise cannot be eliminated, but it can be calculated and accounted for. By providing the resistor value, temperature, and bandwidth in this form, you can immediately evaluate how much RMS noise voltage is present. Whether you’re an amateur radio builder, a student learning about noise sources, or an engineer designing cutting-edge instrumentation, knowing the Johnson–Nyquist noise level is essential for understanding how quiet your circuit can be.

Extended Example

Consider designing a sensor amplifier for a 1 kΩ thermistor operating at 350 K within a 2 kHz measurement bandwidth. Plugging into the equation yields:

$$v=\sqrt{4k35010002000}$$

The result is about 3.3 μV RMS. If your desired signal is only 10 μV, the signal-to-noise ratio before amplification is a mere 3:1. Such insight might prompt you to reduce bandwidth or cool the sensor to improve performance.

Comparison Table

The table below summarizes noise voltages for a 300 K resistor across common values and a 1 kHz bandwidth.

Resistance (Ω)	Noise Voltage (μV RMS)
50	0.91
1k	4.07
10k	12.9

Higher resistances generate more noise for the same bandwidth and temperature. Designers of precision circuits often choose the lowest practical resistance to minimize noise, balancing it against power consumption and other design constraints.

Limitations and Assumptions

This calculator models purely thermal agitation in resistors and assumes the resistor is ideal with no excess noise. Real components may exhibit 1/f noise or shot noise from semiconductor junctions that add to the total noise floor. The formula also presumes a perfectly matched load; mismatched impedances alter the delivered noise power. At very high frequencies, quantum effects introduce deviations known as the Planck correction, though this is negligible below the gigahertz range.

Another assumption is that temperature is uniform across the resistor. In high-power applications, self-heating can raise the effective temperature, increasing noise. In such cases, measure or estimate the resistor’s actual operating temperature rather than ambient conditions.

Related Calculators

For further analysis, explore the Signal-to-Noise Ratio Calculator to see how noise affects measurable signals and the Noise Figure Calculator when evaluating amplifiers and RF systems.