Thermal Noise Calculator

Origin of Johnson–Nyquist Thermal Noise

Johnson–Nyquist thermal noise arises because charge carriers in every resistive element undergo random thermal motion. John B. Johnson observed the voltage fluctuations, and Harry Nyquist supplied their theoretical treatment. The effect is present even when a circuit has no applied signal or external interference, so it establishes a physical floor for low-level electronic measurements. Knowing the resistor's thermal-noise voltage helps engineers assess low-noise amplifiers, radio receivers, and precision sensor interfaces.

The Johnson–Nyquist Resistor-Voltage Equation

For this calculator, the RMS thermal-noise voltage across the resistor is $\sqrt{4kTRB}$, where $k$ is Boltzmann’s constant (1.380649×10⁻²³ J/K), $T$ is the resistor’s absolute temperature in kelvins, $R$ is resistance in ohms, and $B$ is the noise bandwidth in hertz. The RMS voltage rises with the square root of temperature, resistance, and bandwidth, rather than in direct proportion to them. This unavoidable random voltage is one component of an electronic system’s overall noise budget.

Thermal-Noise Bandwidth Considerations

In a Johnson–Nyquist calculation, bandwidth is the frequency range over which resistor noise is measured or amplified. Widening that range admits more noise power and therefore increases the integrated RMS voltage. A radio receiver limited to a narrow channel collects noise from only that slice of spectrum, whereas a wideband instrumentation amplifier integrates noise over a broader range. Choosing the appropriate effective bandwidth is consequently an important part of estimating resistor noise realistically.

Johnson–Nyquist Voltage Example Calculation

For a 50 Ω resistor at about 300 K measured over a 10 kHz bandwidth, the calculator uses:

Formula: v = sqrt(4 k 300 ⁢ 50 ⁢ 10000)

$$v=\sqrt{4k3005010000}$$

The result is about 0.091 μV RMS. Though small, this resistor-generated voltage can matter when the desired signal is similarly small, particularly before gain and filtering are applied.

Matched-Load Thermal Noise Power

For Johnson–Nyquist noise, the available power delivered from a resistor to a matched load is $P=kTB$ watts. Unlike the open-circuit voltage calculation on this page, that matched available-power expression does not depend on resistance. Communication-system designers commonly use it to establish an ideal receiver noise floor.

Thermal Noise in Low-Noise Circuit Design

Johnson–Nyquist resistor noise is especially important in radio astronomy, medical imaging, and sensitive scientific instruments. Designers may select low-excess-noise resistors, reduce detector temperature, or restrict bandwidth, but the thermal component itself remains set by temperature, resistance, and bandwidth. This calculator quantifies the resistor’s contribution so it can be compared with amplifier and sensor noise sources.

Recording Thermal-Noise Voltage Estimates

When evaluating a prototype, engineers often place the computed Johnson–Nyquist voltage beside measured noise data. The copy button transfers the displayed voltage estimate for use in a lab notebook, simulation comment, or noise-budget worksheet.

Reducing Resistor Thermal Noise

Lowering the resistor’s operating temperature reduces thermal agitation and therefore lowers its Johnson–Nyquist voltage. Reducing resistance or narrowing the measurement bandwidth also lowers the calculated RMS noise voltage, although either choice can involve signal, loading, power, or system-flexibility trade-offs. These trade-offs are central to low-noise electronic design.

Summary of Resistor Thermal Noise

Johnson–Nyquist thermal noise cannot be removed from an ordinary resistor, but its RMS voltage can be estimated from resistance, temperature, and bandwidth. By entering those three quantities here, you can assess the resistor-noise level relevant to a circuit input or measurement. That estimate is useful for students, radio builders, and engineers who need to understand how quietly a resistor-based circuit can operate.

Extended Resistor Thermal-Noise Example

Consider a sensor amplifier using a 1 kΩ thermistor at 350 K with a 2 kHz measurement bandwidth. The resistor’s Johnson–Nyquist voltage is:

Formula: v = sqrt(4 k ⁢ 350 ⁢ 1000 ⁢ 2000)

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

The result is about 0.197 μV RMS. If the target signal is only a few microvolts, this resistor term should be included with amplifier and sensor noise before deciding whether the available signal-to-noise ratio is adequate.

Resistor Thermal-Noise Comparison Table

This Johnson–Nyquist table summarizes RMS voltage for resistors at 300 K over a 1 kHz bandwidth.

At fixed temperature and bandwidth, higher resistance produces a higher RMS thermal-noise voltage. Precision-circuit designers often choose the lowest practical resistance while balancing loading, power consumption, and other circuit constraints.

Johnson–Nyquist Calculator Limitations and Assumptions

This Johnson–Nyquist calculator models the thermal voltage generated by an ideal resistor and does not add excess resistor noise, semiconductor shot noise, or 1/f noise. It reports the resistor’s RMS noise voltage from $\sqrt{4kTRB}$, not noise power delivered through a particular source-and-load network. At sufficiently high frequencies, quantum effects require a Planck correction rather than the classical expression used here.

The calculation also treats temperature as uniform across the resistor. In high-power applications, self-heating can make the resistor’s effective operating temperature higher than ambient, which increases its thermal-noise voltage. Use the actual or estimated resistor temperature whenever self-heating is significant.

Practical Johnson–Nyquist Noise-Budget Workflow

In practical circuit design, calculate resistor thermal noise as one term in a full input-referred noise budget rather than treating it as the complete noise floor. A typical workflow starts with source-resistance noise, then includes amplifier input-voltage noise and current-noise contributions through source impedance. Independent noise sources are normally combined by root-sum-square, not simple arithmetic addition.

For a precision sensor front end, changing source resistance affects more than this calculator’s resistor-voltage result. Amplifier current noise can create an additional voltage across the source impedance, so a larger resistor may worsen total noise faster than the thermal-noise estimate alone suggests. Comparing several resistance and bandwidth choices helps reveal a useful compromise between signal level, power consumption, and resolution.

Filter shape also matters when applying this thermal-noise estimate to a real instrument. The calculator uses the entered bandwidth directly, while practical low-pass and band-pass filters have an equivalent noise bandwidth that can differ from a stated -3 dB cutoff. Use the filter’s equivalent noise bandwidth when matching the calculation to measurement data.

Related Signal and Noise 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.