Thermal Noise Formulas & Calculator

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Particularly in radio frequency, RF design and development it is necessary to make thermal noise calculations. Radio receiver applications, RF thermal noise is a key attribute, limiting the sensitivity of the radios.

Calculating the thermal noise and knowing the value can help improve the performance of the whole system, enabling the right steps to be taken to optimise performance and adopt the best approaches.

To calculate the thermal noise levels, there are formulas or equations that are relatively straightforward. In addition to this there is an online calculator to provide additional assistance.

Basic thermal noise calculation and equations.

Thermal noise is effectively white noise and extends over a very wide spectrum. The noise power is proportional to the bandwidth. It is therefore possible to define a generalised equation for the noise voltage within a given bandwidth as below:

${\displaystyle V^2=4 k T{\int }_{\mathrm{f1}}^{\mathrm{f2}}R df}$

Where:
V = integrated RMS voltage between frequencies f1 and f2
R = resistive component of the impedance (or resistance) Ω
T = temperature in degrees Kelvin
(Kelvin is absolute zero scale thus Kelvin = Celsius + 273.16)
f1 & f2 = lower and upper limits of required bandwidth

For most cases the resistive component of the impedance will remain constant over the required bandwidth. It therefore possible to simplify the thermal noise equation to:

Where:
B = bandwidth in Hz

Thermal noise calculations for room temperature

It is possible to calculate the thermal noise levels for room temperature, 20°C or 290°K. This is most commonly calculated for a 1 Hz bandwidth as it is easy to scale from here as noise power is proportional to the bandwidth. The most common impedance is 50 Ω.

${\displaystyle V=\sqrt{4 (1.3803 {10}^{-23}) 290 50 1}}$

${\displaystyle V=0.9 nV}$

Thermal noise power calculations

While the thermal noise calculations above are expressed in terms of voltage, it is often more useful to express the thermal noise in terms of a power level.

To model this it is necessary to consider the noisy resistor as an ideal resistor, R connected in series with a noise voltage source and connected to a matched load.

${\displaystyle P=\frac{V^2}{4R}}$

${\displaystyle P=\frac{{\left(\sqrt{4 k T B R}\right)}^2}{4 R}}$

${\displaystyle P=k T B}$

Note: it can be seen that the noise power is independent of the resistance, only on the bandwidth.

This figure is then normally expressed in terms of dBm.

Thermal noise in a 50 Ω system at room temperature is -174 dBm / Hz.

It is then easy to relate this to other bandwidths: because the power level is proportional to the bandwidth, twice the bandwidth level gives twice the power level (+3dB), and ten times the bandwidth gives ten times the power level (+10dB).

Thermal noise calculator

The thermal noise calculation below provides an easy method of determining the various thermal noise values that may be required.

Thermal Noise Calculated for Common Bandwidths

The table below provides the thermal noise floor calculations for various common bandwidths and common applications.

Bandwidth and Thermal Noise Power
Bandwidth (Δf) Hz	Thermal Noise Power dBm
1	-174
10	-164
100	-154
1k	-144
10k	-134
100k	-124
200k (2G GSM channel)	-121
1M (Bluetooth channel)	-114
5M (3G UMTS channel)	-107
10M	-104
20M (Wi-Fi channel)	-101

These values for thermal noise power are easy to calculate from the online calculator or the formulas, but the table provides a handy reference.

Watch the video: Examples on Thermal Noise in Communication Engineering by Engineering Funda (July 2022).

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