Thermal noise increases with resistance

 

General

Every practitioner knows that resistors rustle; especially the unspeakable carbon mass resistances are notorious for this. But anyone who now believes that high-quality resistors are noise-free is wrong. Because even the highest quality resistor has a noise caused by thermal effects, which is called thermal noise according to its cause. This occurs with every resistor without exception, regardless of whether it is a component or a parasitic resistor. With ideal resistances, thermal noise is the only source of noise, but with real resistances there are additional causes.

The following explains in simple terms where the thermal noise comes from, what other sources of noise there are, and how to select a resistor that is suitable for the respective application.


Thermal noise of a resistor

As already said, all resistors rustle, even in theory; you can even calculate the noise voltage. This applies to every ohmic resistance without exception, even if it is an undesirable transition resistance or the like. The reason is that the free electrons in an electrical conductor are always in motion when the temperature is higher than absolute zero. Due to the completely random movement of the extremely numerous free electrons, it happens that from one snapshot to the next the electrons move on average either in the direction of one or the other connection. When electrons move, this is identical to a current flow, and thus, depending on the mean direction of movement, a positive or negative current occurs between the snapshots, the magnitude of which depends on the number of electrons that causes this current flow.

This can be illustrated by a very simple picture with one or two electrons: If a single electron flies through a conductor at a certain speed (shown in copper color, Figure 11.Example), this is equivalent to a certain current flow, with the polarity depends on the direction of flight. In example 2 it is reversed compared to example 1 due to the opposite flight direction. If two electrons fly in the same direction at the same speed instead of one, the current is logically twice as high (example 3). But if these two electrons fly in opposite directions, the resulting current is zero (example 4).


Figure 1: Free electrons in the electrical conductor

Since the movement is completely random, not only is the polarity of the current purely random, but also its height and frequency. In the larger average over time, of course, no current flows, but from snapshot to snapshot it does. A signal with random polarity, random amplitude and random frequency is perceived as noise when it is amplified and reproduced through a loudspeaker, which is why the current generated in this way is called noise current. Although this is quite small, it can be noticeable in the event of a small useful signal. The noise energy and thus noise voltage of thermal noise is the same for all frequencies, which is why one speaks of white noise in technical jargon.

The noise current is temperature-dependent and increases with temperature. In the absolute zero point (0 K = -273 ° C) the noise current is zero because the free electrons have no energy to move in the conductor without an externally forced current flow. Starting from the zero point, the noise energy increases linearly with the temperature. A higher temperature results in a higher energy of the free electrons in the material, so that the amplitude of the noise current increases. Because of the thermal dependency, this effect is called thermal noise. This noise can be greatly reduced by cooling with e.g. liquid nitrogen or liquid helium, which is used in highly sensitive special measuring devices. However, this effort is far too high for normal devices.

One can visualize it in such a way that the atoms of the resistance vibrate more and more violently with increasing temperature (see also Current flow in the conductor) and thus push the free electrons more and more. The thermal energy is independent of the resistance of the material, so that the noise current is also independent of its resistance. The noise voltage occurring at the connections is, however, dependent on the ohmic resistance because it is caused by the flowing noise current. If the noise current is known, it can easily be calculated according to Ohm's law. This relationship also explains why the noise voltage is greater with a large resistance than with a small resistance.

But how can you calculate the noise current? There is a constant of proportionality which describes how much energy per Kelvin temperature increase is supplied to the electrons as mean thermal energy and is then converted into noise by their chaotic movement. This proportionality constant is called the Boltzmann constant in honor of the Austrian scientist Ludwig Boltzmann, who is considered one of the founders of statistical mechanics. With the help of this Boltzmann constant one can calculate the noise current (and thus of course the noise voltage). Incidentally, thermal noise encompasses a frequency range which, according to the classical concept, is "infinitely high" (this is not entirely correct due to quantum mechanical effects but is limited to approx. 100 THz), with the mean noise energy being the same for every frequency. One speaks of white noise. Usually, however, the noise is only of interest in a certain frequency range, for example with audio signals between 16 Hz and 20 kHz, because the human ear does not hear frequencies above 20 kHz anyway (usually well below). This is why the limited bandwidth is taken into account when calculating the noise voltage. The formula for this is (without mathematical derivation):

The constant k is the Boltzmann constant, T0 the temperature in Kelvin, B the bandwidth and R the resistance value.

The result is, and this is important, the Effectivelyvalue of thermal noise. In principle, you cannot specify peak values ​​for noise, as these can theoretically be infinitely high. The probability of occurrence of high peaks is very low: with white noise, 68% of the peaks are not higher than the rms value, only 0.37% of the peaks are greater than three times the rms value, and only 0.0000002% have an amplitude of more than six times the rms value.


Further sources of noise with resistors

Thermal noise is not the only source of noise in real resistors. The technical structure of the resistors can result in additional sources of noise that by far exceed the thermal noise. An inglorious because very noisy example are the so-called carbon mass resistors, which consist of small, compressed grains of coal. The numerous transitions from grain to grain are decisive for the resistance, since the cross-section is greatly reduced at the tips and edges and thus the resistance increases. If the grains change their position in relation to one another only minimally, the moment a contact breaks off or a new contact is formed, the resistance value suddenly changes a little, which is noticeable as noise. Unfortunately, this happens permanently due to the flow of electricity: If electricity flows over a grain of coal, it becomes warm and expands. This slightly changes the position of the granules in the neighborhood. As soon as new contacts are formed with noise, the current flow through the hot granule is reduced, whereby it cools down a little and its volume again decreases a little and thus affects its neighbors, which in turn results in newly formed or torn contacts. This process never stops, which is why it ultimately rustles permanently. In addition, there are micro-voltage flashovers due to the tiny distances between the grains. The noise generated in this way is rather low-frequency, i.e. the amplitude decreases with increasing frequency, because the contacts cannot tear off at will and cannot be re-formed. This noise cannot be calculated on the basis of theoretical considerations, as it is production-dependent and is influenced, among other things, by the grain size, grain shape and the pressing pressure. This parameter would be a hot candidate to be listed in a data sheet. However, coal mass resistances are all but extinct.

This type of noise also occurs in a strongly attenuated form with normal carbon film resistors. The reason is that the carbon layer does not consist "of one piece", but rather has a certain amount of imperfections, which means that mechanical micro-movements can also occur here. Even with metal film resistors, the resistance layer does not consist of homogeneous metal. For this reason, this type of noise also occurs here, e.g. at the grain boundaries of the alloys used. The thicker the layer, the more stable it is, which explains why metal film resistors and especially wirewound resistors have a further reduced noise compared to metal film resistors.

Furthermore, the flowing current alone causes noise due to the movement of the charge carriers. This is due to the fact that the electrons flow more or less chaotically through a conductor and even then do not come out in a row at the back even if they are sent in strictly ordered at the front. If you count the number of electrons per unit of time, it therefore always fluctuates a little around the mean value. This fluctuation around the mean value is nothing more than white noise.


Resistance noise in practice / data sheets

It is very helpful to have a rough idea of ​​the causes of the noise, but fortunately in practice you do not need to deal with it in detail when designing circuits. You can (and must) easily calculate the thermal noise yourself, while a quantitative summary of the other noise sources can be found in the data sheets of competent manufacturers.

Most of the time, information on non-thermal noise is found in the form "x μV / V", i.e. a so-called current noise of x μV occurs per volt of applied voltage. The total noise of a resistor can be calculated by calculating this current noise on the basis of the data sheet information and also calculating the thermal noise using the formula given above. Since these are so-called stochastically independent signals, they add up orthogonally, as the saying goes. This means that all noise contributions are first squared, then added and finally the square root is taken from the sum. This has the following background: Statistically independent noise sources rustle lonely and alone. If, in the simplest case, two independent noise signals with the same amplitude are superimposed, it rarely happens that the maxima and minima of the two signals coincide. Fortunately, you do not get twice the noise voltage, but only one around higher rms value.