# Thermal Noise Calculator Formulae & Equations

### Thermal Noise Includes

As noise is a very important factor in many electronics / RF applications and circuits, it is important to be able to calculate the values for noise under the conditions.

There are many equations that enable the prediction of thermal noise levels.

The thermal noise calculations can be undertaken relatively easily in most instances. While some integration may be required in some instances, there are straightforward equations for most calculations required.

## Thermal noise calculator

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

## Enter Values:

Temperature:   °Celsius
Resistance:   Ohms, Ω (not required for dBm calculation).
Bandwidth:   Hz

## Results:

Noise voltage:   μV RMS
Noise power:   dBm

## 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:

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.

## 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.

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).

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

By Ian Poole

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