Question

Calculate the rms thermal noise associated with$1.0-\text{M}\Omega$ load resistor operated at room temperature if an oscilloscope with a 1-MHz bandwidth is used. If the bandwidth is reduced to 100 Hz by what factor will the noise be reduced?

Short answer

The rms of themal noise at 1 Mhz bandwidth is${\overline{\upsilon }}_{rms}=1.28\times {10}^{-4}\text{V}$

If the bandwidth reduced by 100 Mhz, the noise reduced by factor 100 times.

Step-by-step solution

Given information

The room temperature,$T=298\text{\hspace{0.17em}K}$

The frequency bandwidth,$\Delta f=1\text{MHz}$

The resistance of the resistor,$R=1.0\text{M}\Omega$

Step 2. Explanation

Thermal noise for a resistive circuit element can be given as

${\overline{\upsilon }}_{rms}=\sqrt{4kTR\Delta f}$

$\begin{array}{l}\text{Here,\hspace{0.17em}\hspace{0.17em}}k\text{=Boltzmann'sconstant=1.38}\times {\text{10}}^{-23}\text{J/K}\\ T\text{=TemperatureinK,\hspace{0.17em}}R\text{=Resistancein}\Omega ,\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}\Delta f\text{=Bandwidth}\end{array}$

Substitutes the value in the above expression

$\begin{array}{l}{\overline{\upsilon }}_{rms}=\sqrt{4kTR\Delta f}\\ =\sqrt{4\times \text{1.38}\times {\text{10}}^{-23}\times 298\times 1\times {10}^6\times 1.0\times {10}^6}\\ {\overline{\upsilon }}_{rms}=1.28\times {10}^{-4}\text{V}\end{array}$

For$\Delta f=100\text{\hspace{0.17em}Hz}$

$\begin{array}{l}{\overline{\upsilon }}_{rms,2}=\sqrt{4kTR\Delta f}\\ {\overline{\upsilon }}_{rms,2}=\sqrt{4\times \text{1.38}\times {\text{10}}^{-23}\times 298\times 1\times {10}^6\times 100}\\ {\overline{\upsilon }}_{rms,2}=1.28\times {10}^{-6}\text{V}\end{array}$

Noise reduction factor is

$\frac{{\overline{\upsilon }}_{rms}}{{\overline{\upsilon }}_{rms,2}}=\frac{1.28\times {10}^{-4}\text{V}}{1.28\times {10}^{-6}\text{V}}=100$

Therefore, the rms of themal noise at 1 Mhz bandwidth is${\overline{\upsilon }}_{rms}=1.28\times {10}^{-4}\text{V}$

If the bandwidth reduced by 100 Mhz, the noise reduced by factor 100 times.