SMD Resistors

What is a resistor?
In a fundamental sense, electrical resistance is a quantity that describes the current that a current-carrying material allows at a specified voltage. The physical reason for the effect can be explained like this: Free charge carriers (electrons) are prevented from being freely accelerated in the conductor by collisions with atoms. The word resistor is derived from the Latin "resistere" for resist, which describes the physical effect quite well. The value of the resistor is given in ohms (Ω), the circuit symbol is either an "empty rectangle" according to EN60617 or has the shape of a zigzag according to ANSI.
An electrical resistor in the sense of an electronic component is used to limit, measure, divide electrical currents and much more. The basic rule for all resistors is that the material used, as well as the thickness and length of the resistive element, determine the current flow. Accordingly, a piece of wire can be considered the simplest electrical resistor, although its resistance value is infinitesimal and can usually be neglected when calculating a circuit. Simply described, the resistance value of an electrical conductor is determined by dividing the electrical voltage by the current:

R = U / I (R...resistance, U...voltage, I...current).

Ideally, the resistance value is constant. So, it does not change its value depending on current or voltage applied:

R = U / I = const.

Then it is also called "Ohmic" resistance and the relationship is known as Ohm's law. In practice, this is usually taken for granted when calculating circuits.

Noise behavior of resistors
Basically, there are two types of noise sources when operating resistors, thermal noise and current noise.

Thermal noise
Thermal noise is a fundamental physical effect that occurs even in an idealized, "theoretical" component. It is also referred to as "white noise". The effect becomes stronger with increasing temperature and it is therefore recommended to keep the temperature as low as possible to minimize this noise component. The noise also does not depend on the frequency when operating at AC voltage, but only on the measurement bandwidth ∆f. White noise occurs independently of an applied voltage and the noise voltage can be calculated or estimated as follows:

U_R=(√4∗k_b∗(TC+273.15)∗Δf

Whereby
k_b = 1.38*10-23 (Boltzmann constant).
TC = temperature in °C
R = resistance value
∆f = bandwidth in Hz (frequency range being considered).

For a value of R = 1 MΩ, a bandwidth of ∆f = 20 kHz (range of the human ear) results in a noise voltage of about 0.018 mV or 18 µV.
This value is not large but can be a problem in applications where very small signals are to be measured. Especially in audio technology, this noise is the constant companion of the developers of high-quality devices, because basically everything that is electrically conductive generates noise.

Current noise
Unlike white noise, current noise only occurs when a voltage is actually applied to a resistor. The current flowing in the element (the desired current) is overlaid by the current noise. However, the cause of the effect is precisely the desired current, so the dependence is on the value of the desired current itself. Here the behavior of the device depends very much on the resistor material used. The unit for the current noise is usually given in µV/V (typically as the maximum value). This means that this value can simply be multiplied by the voltage to get the (maximum) noise level to expect. The values depend on the resistance value itself and usually become higher at higher resistance values.

Load capacity and power loss of resistors
Resistors generate power loss by converting electrical energy into thermal energy. This heat must be dissipated to prevent the component from overheating. Maximum temperatures are indicated on the data sheets. On higher-end models, there are graphs showing the dependence of maximum power on temperature, the power derating curve. When the maximum permissible temperature is exceeded, the component is damaged and can either fail completely or its properties deteriorate or change. For resistors, which can absorb a lot of power, there are hints in the data sheet how the heat dissipation has to be done. In some cases, there are heat sinks that require not only convection and radiation but also the dissipation of heat via contact to a larger heat sink. Depending on the installation situation, the specifications may differ or the installation recommendations may be mentioned for the different situations. The power loss of resistors, i.e. the power which is converted into heat, is calculated for direct current with

P (power) = U (voltage) *I (current)

For AC current, RMS values of current and voltage are used. It is strongly recommended not to go too close to the specified power limits when selecting the component. A resistor that is operated in the limit range becomes hot and then also heats up the environment, which can have side effects on other components. These side effects can cause a circuit that is precisely designed with high quality components to lose the entire advantage of precision due to these heat effects, which will lead the project ad absurdum. In addition, one has hardly any reserves in case something "goes wrong" and the load becomes higher than initially calculated. It can sometimes be useful to connect two power resistors in parallel to avoid using a single more powerful usually more expensive resistor. If on data sheets the temperature is related to ambient air, then one should not take the room temperature around the device as reference, but the temperature that actually flows around the component (ideally cooling air). Here the value can be much higher. If there is still a need to meet higher power requirements, there are models that can help: Some resistor models allow pulsed operation at much higher loads for short periods. For this, see the guide for power resistors.

Aging and stability behavior
Basically, electrical resistors change their resistance value over their lifetime without external influences. Here, especially at the beginning of the lifetime, stronger changes can occur than later during operation. Therefore, it is obvious to artificially age the resistor in an accelerated process in case of special stability requirements. Generally, the term "stability" is used in this context. In the case of precision resistors, one usually finds graphs on data sheets showing the change in resistance at constant (specified) power as a function of the resistance value and the operating time.
Resistors change their values when the temperature changes. This dependence of the total value on the temperature depends strongly on the design and the material used. In addition, the operation of the resistor itself   generates heat. Therefore, in order to minimize this effect, sufficient ventilation must be provided. In addition, it is recommended not to operate the components close to the maximum permissible power dissipations, since in this case in particular a lot of heat generation by the component itself is to be expected.
The temperature coefficient (so-called TC value) is given as a measure of the change in resistance value. The value is specified in units of ppm / ˚C (ppm = parts per million, 1 ppm = "1 millionth" of the total value).
If a TC value of ±50 ppm / °C is specified for a resistor, then a temperature change of 1°C will change the resistor by a maximum multiplicative factor of ±0.000050, or "50 millionths" of the total value. Assume that a component has a specified resistance value of 1 MΩ. Then, if the temperature increases or decreases by 1°C, the resistance value according to the specification can change by up to 50 Ω, i.e., become smaller or larger by up to 50 Ω. In general, the following formula can be used:

R_T=R_ref *[1+TC*(T-_Tref)]

Whereby
T_ref...reference temperature (usually 20°C, sometimes also 0°C or 25°C)
R_ref...reference resistance
TC...temperature coefficient
T... operating temperature
R_T...resistance value in operation

This applies to the example in the text above:
T_ref = 20°C
R_ref = 1 MΩ = 1 000 000 Ω
TC = ±50 ppm / °C = ±0.000050 / °C

The formula gives the following values for the changes of 1°C and 10°C:
R(T=20°C) = 1 MΩ * [1 ± 0.000050 * (20 - 20)] = 1 MΩ * [1 ± 0] = 1 MΩ
R(T=21°C) = 1 MΩ * [1 ± 0.000050 * (21 - 20)] = 1 MΩ * [1 ± 0.000050] = 1 000 000 Ω ± 50 Ω
R(T=40°C) = 1 MΩ * [1 ± 0.000050 * (40 - 20)] = 1 MΩ * [1 ± 0.001000] = 1 000 000 Ω ± 1000 Ω