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Calculated Input Referred Voltage Thermal Noise for The Following Circuits

Reviewed by Calculator Editorial Team

Thermal noise is an inherent property of electronic components that limits the performance of analog circuits. Calculating the input-referred voltage thermal noise helps engineers optimize circuit designs for minimum noise. This guide explains how to calculate thermal noise for common circuits including MOSFETs, BJTs, and resistors, and provides practical insights for noise optimization.

Introduction

Thermal noise, also known as Johnson-Nyquist noise, is generated by the random thermal motion of charge carriers in conductive materials. It's a fundamental limit to the performance of electronic circuits, particularly at low frequencies. The input-referred voltage thermal noise represents how much noise is present at the input of a circuit after accounting for any amplification or attenuation that occurs within the circuit.

Understanding and calculating thermal noise is crucial for:

  • Designing low-noise amplifiers
  • Optimizing sensor interfaces
  • Selecting appropriate resistors and transistors
  • Understanding the fundamental limits of circuit performance

The calculator on this page provides quick calculations for common circuit configurations, while this guide explains the underlying principles and practical considerations.

Thermal Noise Basics

The Johnson-Nyquist Formula

The power spectral density of thermal noise in a resistor is given by:

Sv(f) = 4kBTR Δf

Where:

  • Sv(f) = noise power spectral density (V²/Hz)
  • kB = Boltzmann's constant (1.38 × 10-23 J/K)
  • T = absolute temperature (K)
  • R = resistance (Ω)
  • Δf = bandwidth (Hz)

For a resistor at room temperature (300K), this simplifies to approximately:

Sv(f) ≈ 4 × 10-21 R Δf

Input-Referred Noise

The input-referred voltage noise takes into account any amplification or attenuation that occurs between the noise source and the input. For a circuit with a voltage gain Av, the input-referred noise is:

Vn,in = Vn,out / Av

Where Vn,out is the output noise voltage.

Circuit-Specific Calculations

Resistor Thermal Noise

For a simple resistor, the input-referred voltage noise is simply the thermal noise of the resistor itself. The total noise voltage is:

Vn = √(4kBTRΔf)

MOSFET Thermal Noise

For a MOSFET in saturation, the input-referred voltage noise is dominated by the channel thermal noise:

Vn,in = √(4kBTγgmΔf) / gm

Where γ is the excess noise factor (typically 2/3 for long-channel devices)

BJT Thermal Noise

For a bipolar junction transistor, the input-referred voltage noise includes contributions from the base resistance and the collector current:

Vn,in = √(4kBTRbΔf + 2qIcΔf) / β

Where β is the current gain

Practical Considerations

When designing low-noise circuits, consider these practical aspects:

  1. Temperature effects: Thermal noise increases with temperature, so cooling components can help reduce noise.
  2. Frequency dependence: Thermal noise is white (frequency-independent) at low frequencies but may become significant at high frequencies.
  3. Component selection: Use high-quality resistors and transistors with low noise characteristics.
  4. Layout considerations: Minimize parasitic resistances and capacitances that can contribute to additional noise.

Note: Thermal noise is fundamental and cannot be completely eliminated, but it can be minimized through careful circuit design and component selection.

Comparison Table

This table compares the input-referred voltage thermal noise for different circuit configurations at room temperature (300K) and a 1Hz bandwidth:

Circuit Type Resistance (Ω) Noise Voltage (V)
1kΩ Resistor 1000 1.3 × 10-9
10kΩ Resistor 10000 3.9 × 10-9
MOSFET (gm = 1mS) N/A 1.2 × 10-9
BJT (β = 100, Ic = 1mA) N/A 1.5 × 10-9

FAQ

What is the difference between thermal noise and shot noise?

Thermal noise results from the random motion of charge carriers in conductive materials, while shot noise arises from the discrete nature of charge carriers. Thermal noise is present in all conductors, while shot noise is primarily associated with semiconductor devices.

How does temperature affect thermal noise?

Thermal noise increases with temperature. The noise power is directly proportional to absolute temperature, so cooling components can help reduce thermal noise.

What is the input-referred noise?

Input-referred noise is the equivalent noise voltage that would appear at the input of a circuit to produce the same output noise as the actual circuit. It accounts for any amplification or attenuation within the circuit.