Thermal Noise Calculator

°C K °F

Hz kHz MHz GHz

Calculation Results

RMS Noise Voltage (Vn) –

Noise Power (P) –

Noise Power Density –

Noise Power (dBm) –

function calculateThermalNoise() { var R = parseFloat(document.getElementById('resistance').value); var T_val = parseFloat(document.getElementById('temperature').value); var T_unit = document.getElementById('tempUnit').value; var B_val = parseFloat(document.getElementById('bandwidth').value); var B_unit = parseFloat(document.getElementById('bwUnit').value); if (isNaN(R) || isNaN(T_val) || isNaN(B_val) || R < 0 || B_val < 0) { alert("Please enter valid positive numbers for Resistance and Bandwidth."); return; } // Constants var k = 1.380649e-23; // Boltzmann constant // Convert Temperature to Kelvin var T; if (T_unit === "C") { T = T_val + 273.15; } else if (T_unit === "F") { T = (T_val – 32) * 5 / 9 + 273.15; } else { T = T_val; } if (T 0) { P_dbm = 10 * Math.log10(P_total / 0.001); } // Formatting Voltage for readability var Vn_disp; if (Vn < 1e-6) { Vn_disp = (Vn * 1e9).toFixed(3) + " nV"; } else if (Vn < 1e-3) { Vn_disp = (Vn * 1e6).toFixed(3) + " µV"; } else if (Vn < 1) { Vn_disp = (Vn * 1e3).toFixed(3) + " mV"; } else { Vn_disp = Vn.toFixed(6) + " V"; } // Display results document.getElementById('resVn').innerText = Vn_disp; document.getElementById('resPower').innerText = P_total.toExponential(4) + " W"; document.getElementById('resDensity').innerText = P_density.toExponential(4) + " W/Hz"; document.getElementById('resDbm').innerText = P_dbm === -Infinity ? "-∞" : P_dbm.toFixed(2) + " dBm"; document.getElementById('resultsArea').style.display = 'block'; }

Understanding Thermal Noise (Johnson-Nyquist Noise)

Thermal noise, also known as Johnson-Nyquist noise, is the electronic noise generated by the thermal agitation of charge carriers (usually electrons) inside an electrical conductor at equilibrium. This phenomenon occurs regardless of any applied voltage and is intrinsic to all resistive components.

The Physics Behind the Noise

At any temperature above absolute zero, atoms in a conductor vibrate. These vibrations cause the free electrons to move randomly. While the average current is zero over time, these instantaneous random fluctuations create a measurable noise voltage across the terminals of the resistor.

The Formula

The Root Mean Square (RMS) noise voltage is calculated using the following equation:

Vn = √(4 · k · T · B · R)

• k: Boltzmann Constant (≈ 1.38 × 10⁻²³ J/K)
• T: Absolute Temperature (Kelvin)
• B: Bandwidth (Hertz)
• R: Resistance (Ohms)

Practical Example Calculation

Suppose you have a 50 Ω resistor (standard in RF systems) operating at room temperature (25°C or 298.15 K) over a bandwidth of 1 MHz.

1. Identify variables: R = 50, T = 298.15, B = 1,000,000.
2. Calculate Vn: √(4 × 1.38e-23 × 298.15 × 1,000,000 × 50).
3. Result: Vn ≈ 0.907 µV (microvolts).
4. Noise Power: The available noise power is kTB ≈ 4.11 × 10⁻¹⁵ Watts, or -113.86 dBm.

Key Takeaways for Engineers

• Temperature Impact: Cooling a circuit (e.g., using liquid nitrogen or cryogenic coolers) is a common way to reduce noise in sensitive radio telescopes or scientific instruments.
• Bandwidth Limitation: The wider the bandwidth of your measurement, the more noise you capture. This is why narrow-band filters are used to improve Signal-to-Noise Ratio (SNR).
• Impedance Matters: Higher resistance values generate higher noise voltages. In low-noise amplifier (LNA) design, managing input resistance is critical.