Resistor Noise Calculator

Resistance (\( R \)):

Temperature (\( T \)):

Bandwidth (\( \Delta f \)):

RMS Noise Voltage (\( V_{\text{RMS}} \)):

Noise Level (\( L_u \)) (dB):

Noise Level (\( L_v \)) (dB):

1. What is Resistor Noise Calculator?

Definition: This calculator computes the RMS noise voltage (\( V_{\text{RMS}} \)), noise level (\( L_u \)), and noise level (\( L_v \)) generated by a resistor due to thermal noise, also known as Johnson-Nyquist noise.

Purpose: It is used in electronics to evaluate the noise contribution of resistors in circuits, which is critical for designing low-noise amplifiers, sensors, and other sensitive electronic systems.

2. How Does the Calculator Work?

The calculator uses the following formulas for resistor noise:

\( V_{\text{RMS}} = \sqrt{4 k T R \Delta f} \)
\( L_u = 20 \log_{10} \left( \frac{V_{\text{RMS}}}{V_{0,u}} \right) \), where \( V_{0,u} = 0.77459667 \, \text{V} \)
\( L_v = 20 \log_{10} \left( \frac{V_{\text{RMS}}}{V_{0,v}} \right) \), where \( V_{0,v} = 1 \, \text{V} \)

Where:

\( V_{\text{RMS}} \): RMS noise voltage (V);
\( L_u \): Noise level relative to 0.77459667 V (dB);
\( L_v \): Noise level relative to 1 V (dB);
\( k = 1.380649 \times 10^{-23} \, \text{J/K} \): Boltzmann constant;
\( T \): Temperature (K);
\( R \): Resistance (Ω);
\( \Delta f \): Bandwidth (Hz).

Steps:

Enter the resistance (\( R \)), temperature (\( T \)), and bandwidth (\( \Delta f \)) with their units.
Convert resistance to ohms, temperature to Kelvin, and bandwidth to Hertz.
Calculate the RMS noise voltage using the formula for \( V_{\text{RMS}} \).
Calculate \( L_u \) using the noise level formula with \( V_{0,u} = 0.77459667 \, \text{V} \).
Calculate \( L_v \) using the noise level formula with \( V_{0,v} = 1 \, \text{V} \).
Convert the RMS noise voltage to the selected output unit (V, mV, µV).
Display \( V_{\text{RMS}} \), \( L_u \), and \( L_v \) in scientific notation if their absolute value is less than 0.001, otherwise with 4 decimal places.

3. Importance of Resistor Noise Calculation

Calculating resistor noise and its levels is crucial for:

Low-Noise Design: Minimizing noise in high-gain amplifiers and sensitive measurement systems where thermal noise can dominate.
Signal Integrity: Ensuring that noise does not degrade the signal-to-noise ratio in communication systems or sensors.
Component Selection: Choosing resistors with appropriate resistance values and materials to reduce noise in critical applications.

4. Using the Calculator

Example 1: Calculate the RMS noise voltage, \( L_u \), and \( L_v \) for a resistor with \( R = 1 \, \text{kΩ} \), \( T = 25 \, \text{°C} \), and \( \Delta f = 1 \, \text{MHz} \):

Input Values:
\( R = 1 \, \text{kΩ} = 1000 \, \text{Ω} \);
\( T = 25 \, \text{°C} = 25 + 273.15 = 298.15 \, \text{K} \);
\( \Delta f = 1 \, \text{MHz} = 1 \times 10^6 \, \text{Hz} \);
\( k = 1.380649 \times 10^{-23} \, \text{J/K} \);
\( V_{0,u} = 0.77459667 \, \text{V} \);
\( V_{0,v} = 1 \, \text{V} \).
Inside the square root: \( 4 k T R \Delta f = 4 \times 1.380649 \times 10^{-23} \times 298.15 \times 1000 \times 1 \times 10^6 \);
\( = 1.647 \times 10^{-11} \);
RMS noise voltage: \( V_{\text{RMS}} = \sqrt{1.647 \times 10^{-11}} \approx 4.058 \times 10^{-6} \, \text{V} = 4.058 \, \text{µV} \);
\( L_u = 20 \log_{10} \left( \frac{4.058 \times 10^{-6}}{0.77459667} \right) = 20 \log_{10} (5.239 \times 10^{-6}) \approx 20 \times (-5.281) \approx -105.618 \, \text{dB} \);
\( L_v = 20 \log_{10} \left( \frac{4.058 \times 10^{-6}}{1} \right) = 20 \log_{10} (4.058 \times 10^{-6}) \approx 20 \times (-5.392) \approx -107.834 \, \text{dB} \);
Result: \( V_{\text{RMS}} = 4.0580 \, \text{µV} \), \( L_u = -105.6180 \, \text{dB} \), \( L_v = -107.8340 \, \text{dB} \).

Example 2: Calculate the RMS noise voltage, \( L_u \), and \( L_v \) for a resistor with \( R = 500 \, \text{Ω} \), \( T = 300 \, \text{K} \), and \( \Delta f = 10 \, \text{kHz} \):

Input Values:
\( R = 500 \, \text{Ω} \);
\( T = 300 \, \text{K} \);
\( \Delta f = 10 \, \text{kHz} = 10 \times 10^3 = 10000 \, \text{Hz} \);
\( k = 1.380649 \times 10^{-23} \, \text{J/K} \);
\( V_{0,u} = 0.77459667 \, \text{V} \);
\( V_{0,v} = 1 \, \text{V} \).
Inside the square root: \( 4 k T R \Delta f = 4 \times 1.380649 \times 10^{-23} \times 300 \times 500 \times 10000 \);
\( = 8.284 \times 10^{-13} \);
RMS noise voltage: \( V_{\text{RMS}} = \sqrt{8.284 \times 10^{-13}} \approx 2.879 \times 10^{-7} \, \text{V} = 0.2879 \, \text{µV} \);
\( L_u = 20 \log_{10} \left( \frac{2.879 \times 10^{-7}}{0.77459667} \right) = 20 \log_{10} (3.717 \times 10^{-7}) \approx 20 \times (-6.430) \approx -128.596 \, \text{dB} \);
\( L_v = 20 \log_{10} \left( \frac{2.879 \times 10^{-7}}{1} \right) = 20 \log_{10} (2.879 \times 10^{-7}) \approx 20 \times (-6.541) \approx -130.814 \, \text{dB} \);
Result: \( V_{\text{RMS}} = 2.8790e-1 \, \text{µV} \), \( L_u = -128.5960 \, \text{dB} \), \( L_v = -130.8140 \, \text{dB} \).

5. Frequently Asked Questions (FAQ)

Q: Why does resistor noise increase with temperature?
A: Thermal noise arises from the random motion of electrons, which increases with temperature. As temperature rises, electron agitation increases, leading to higher noise voltage.

Q: How can I reduce resistor noise in a circuit?
A: You can reduce noise by lowering the resistance, reducing the bandwidth, or operating at a lower temperature. However, practical constraints like circuit requirements and cooling feasibility must be considered.

Q: What do \( L_u \) and \( L_v \) represent in this context?
A: \( L_u \) is the noise level in dB relative to a reference voltage of 0.77459667 V, and \( L_v \) is the noise level in dB relative to a reference voltage of 1 V. Both indicate the noise voltage’s magnitude on a logarithmic scale.