# Dataforth Tech Note – Resistor Thermal Noise

Noise in measurement and control systems is generally considered to be EMI, electromagnetic interference. Unwanted voltages and currents induced by external magnetic fields, electric fields, ground currents, etc fall into the category of “noise” and often cause serious errors in low level instrumentation. Proper wiring, shielding, and utilizing isolated signal conditioning modules with internal multi-pole filtering will minimize if not completely eliminate this type noise.

There are, however, other subtle sources of noise that often go unrecognized. Electrons within a conducting media or a semiconductor device that are available to move are responsible for current flow (charge per second) when excited by external voltages. With no externally applied voltages, electrons are still in motion randomly interacting with other electrons and with the material’s lattice sites and/or impurities; however, their average velocity in any direction remains zero (i.e. no current flows). This statistically random electron motion creates noise voltages whether there is an applied external voltage or not. Consequently, conducting media generates internal noise without current flow.

Additional types of noise occur when current flows. The random statistical nature of trillions of electrons traveling with an average velocity in the same direction traversing random paths and interacting with material lattice sites will create several types of noise. In many instances, these noise voltages will seriously affect instrumentation. The laws of material physics and quantum mechanics which govern electron motion are random and, therefore, behavior models must be treated with statistical methods. This means that noise voltages are typically expressed as a “mean square” value.

One common noise category is resistor thermal noise, which is the noise developed in a resistor in the absence of current flow. Thermal noise was modeled by Nyquest in 1928 and experimentally measured by Johnson. This noise, often referred to as “Johnson” noise, is generated in a resistor independent of any current flow and has a mean-square voltage value of 4*k*T*R*(BW). In this expression “k” is Boltzman’s constant, “T” is temperature in degrees Kelvin, “R” is resistance in ohms, and “BW” is bandwidth, in Hz.. For example, at 100 degrees C, the noise voltage measured with an ideal true RMS 1 Meg Hz bandwidth voltmeter within a 500k ohm resistor is approximately 100 micro-volts. Clearly, this can cause serious errors when measuring low level voltages with high gain signal conditioning modules.

A Dataforth signal conditioning module with a multiple pole Low-pass 4 Hz bandwidth filter would reduce this resistor thermal noise to less than 3 nano-volts. The above expression illustrates that mean square thermal noise voltage is directly dependent on the temperature, the resistor value and bandwidth.

In practice, there is always some parasitic capacitance (C) across the leads of a resistor due to the printed circuit board or lead wire connections. For this situation, when the thermal noise in a resistor is shunted by a non-zero capacitance (C), the mean-square voltage value is given by k*T/C. For illustration purposes, consider a resistor at 100 degrees C. with a 1 pico-farad shunt capacitance. This model predicts a limit of approximately 71 micro-volts.

Temperature and resistor values can not always be minimized; however, using signal conditioning modules with small bandwidth multi-pole low-pass filters will ensure that external thermal resistor is essentially eliminated

All Dataforth signal conditioning modules are carefully tested for total internal module noise. The Dataforth signal conditioning module noise specification is a RMS representation and includes the contributions of all internal noise sources. Each Dataforth module is shipped with a test report that states total output noise (https://www.dataforth.com/catalog/pdf/Sample_81974-1.pdf).