Atemperature coefficient describes the relative change of a physical property that is associated with a given change intemperature. For a propertyR that changes when the temperature changes bydT, the temperature coefficient α is defined by the following equation:

${\displaystyle \frac{dR}{R}=\alpha dT}$

Here α has thedimension of an inverse temperature and can be expressed e.g. in 1/K or K⁻¹.

If the temperature coefficient itself does not vary too much with temperature and${\displaystyle \alpha \mathrm{\Delta }T\ll 1}$, alinear approximation will be useful in estimating the valueR of a property at a temperatureT, given its valueR₀ at a reference temperatureT₀:

${\displaystyle R(T)=R(T_0)(1+\alpha \mathrm{\Delta }T),}$

where ΔT is the difference betweenT andT₀.

For strongly temperature-dependent α, this approximation is only useful for small temperature differences ΔT.

Temperature coefficients are specified for various applications, including electric and magnetic properties of materials as well as reactivity. The temperature coefficient of most of the reactions lies between 2 and 3.

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Mostceramics exhibit negative temperature dependence of resistance behaviour. This effect is governed by anArrhenius equation over a wide range of temperatures:

${\displaystyle R=A{e}^{\frac{B}{T}}}$

whereR is resistance,A andB are constants, andT is absolute temperature (K).

The constantB is related to the energies required to form and move thecharge carriers responsible for electrical conduction – hence, as the value ofB increases, the material becomes insulating. Practical and commercial NTCresistors aim to combine modest resistance with a value ofB that provides good sensitivity to temperature. Such is the importance of theB constant value, that it is possible to characterize NTCthermistors using the B parameter equation:

${\displaystyle R=r^{\mathrm{\infty }}{e}^{\frac{B}{T}}=R_0{e}^{-\frac{B}{T_0}}{e}^{\frac{B}{T}}}$

where${\displaystyle R_0}$ is resistance at temperature${\displaystyle T_0}$.

Therefore, many materials that produce acceptable values of${\displaystyle R_0}$ include materials that have been alloyed or possess variablenegative temperature coefficient (NTC), which occurs when a physical property (such asthermal conductivity orelectrical resistivity) of a material lowers with increasing temperature, typically in a defined temperature range. For most materials, electrical resistivity will decrease with increasing temperature.

Materials with a negative temperature coefficient have been used infloor heating since 1971. The negative temperature coefficient avoids excessive local heating beneath carpets,bean bag chairs,mattresses, etc., which can damagewooden floors, and may infrequently cause fires.

Residual magnetic flux density orB_r changes with temperature and it is one of the important characteristics of magnet performance. Some applications, such as inertialgyroscopes andtraveling-wave tubes (TWTs), need to have constant field over a wide temperature range. Thereversible temperature coefficient (RTC) ofB_r is defined as:

${\displaystyle \text{RTC}=\frac{|\mathrm{\Delta }{\mathbf{B}}_r|}{|{\mathbf{B}}_r|\mathrm{\Delta }T}\times 100\mathrm{\%}}$

To address these requirements, temperature compensated magnets were developed in the late 1970s.^[1] For conventionalSmCo magnets,B_r decreases as temperature increases. Conversely, for GdCo magnets,B_r increases as temperature increases within certain temperature ranges. By combiningsamarium andgadolinium in the alloy, the temperature coefficient can be reduced to nearly zero.

The temperature dependence ofelectrical resistance and thus of electronic devices (wires, resistors) has to be taken into account when constructing devices andcircuits. The temperature dependence ofconductors is to a great degree linear and can be described by the approximation below.

${\displaystyle \rho (T)={\rho }_0\left[1+{\alpha }_0\left(T-T_0\right)\right]}$

where

${\displaystyle {\alpha }_0=\frac{1}{{\rho }_0}{\left[\frac{\delta \rho }{\delta T}\right]}_{T=T_0}}$

${\displaystyle {\rho }_0}$ just corresponds to the specific resistance temperature coefficient at a specified reference value (normallyT = 0 °C)^[2]

That of asemiconductor is however exponential:

${\displaystyle \rho (T)=S{\alpha }^{\frac{B}{T}}}$

where${\displaystyle S}$ is defined as the cross sectional area and${\displaystyle \alpha }$ and${\displaystyle B}$ are coefficients determining the shape of the function and the value of resistivity at a given temperature.

For both,${\displaystyle \alpha }$ is referred to as thetemperature coefficient of resistance (TCR).^[3]

This property is used in devices such as thermistors.

Apositive temperature coefficient (PTC) refers to materials that experience an increase in electrical resistance when their temperature is raised. Materials which have useful engineering applications usually show a relatively rapid increase with temperature, i.e. a higher coefficient. The higher the coefficient, the greater an increase in electrical resistance for a given temperature increase. A PTC material can be designed to reach a maximum temperature for a given input voltage, since at some point any further increase in temperature would be met with greater electrical resistance. Unlike linear resistance heating or NTC materials, PTC materials are inherently self-limiting. On the other hand, NTC material may also be inherently self-limiting if constant current power source is used.

Some materials even have exponentially increasing temperature coefficient. Example of such a material isPTC rubber.

Anegative temperature coefficient (NTC) refers to materials that experience a decrease in electrical resistance when their temperature is raised. Materials which have useful engineering applications usually show a relatively rapid decrease with temperature, i.e. a lower coefficient. The lower the coefficient, the greater a decrease in electrical resistance for a given temperature increase. NTC materials are used to create inrush current limiters (because they present higher initial resistance until the current limiter reaches quiescent temperature),temperature sensors andthermistors.

An increase in the temperature of a semiconducting material results in an increase in charge-carrier concentration. This results in a higher number of charge carriers available for recombination, increasing the conductivity of the semiconductor. The increasing conductivity causes the resistivity of the semiconductor material to decrease with the rise in temperature, resulting in a negative temperature coefficient of resistance.

Theelastic modulus of elastic materials varies with temperature, typically decreasing with higher temperature.

Innuclear engineering, the temperature coefficient of reactivity is a measure of the change in reactivity (resulting in a change in power), brought about by a change in temperature of the reactor components or the reactor coolant. This may be defined as

${\displaystyle {\alpha }_T=\frac{\mathrm{\partial }\rho }{\mathrm{\partial }T}}$

Where${\displaystyle \rho }$ isreactivity andT is temperature. The relationship shows that${\displaystyle {\alpha }_T}$ is the value of thepartial differential of reactivity with respect to temperature and is referred to as the "temperature coefficient of reactivity". As a result, the temperature feedback provided by${\displaystyle {\alpha }_T}$ has an intuitive application topassive nuclear safety. A negative${\displaystyle {\alpha }_T}$ is broadly cited as important for reactor safety, but wide temperature variations across real reactors (as opposed to a theoretical homogeneous reactor) limit the usability of a single metric as a marker of reactor safety.^[4]

In water moderated nuclear reactors, the bulk of reactivity changes with respect to temperature are brought about by changes in the temperature of the water. However each element of the core has a specific temperature coefficient of reactivity (e.g. the fuel or cladding). The mechanisms which drive fuel temperature coefficients of reactivity are different from water temperature coefficients. While water expandsas temperature increases, causing longer neutron travel times duringmoderation, fuel material will not expand appreciably. Changes in reactivity in fuel due to temperature stem from a phenomenon known asdoppler broadening, where resonance absorption of fast neutrons in fuel filler material prevents those neutrons from thermalizing (slowing down).^[5]

In its more general form, the temperature coefficient differential law is:

${\displaystyle \frac{dR}{dT}=\alpha R}$

Where is defined:

${\displaystyle R_0=R(T_0)}$

And${\displaystyle \alpha }$ is independent of${\displaystyle T}$.

Integrating the temperature coefficient differential law:

${\displaystyle {\int }_{R_0}^{R(T)}\frac{dR}{R}={\int }_{T_0}^T\alpha dT\Rightarrow \ln (R){|}_{R_0}^{R(T)}=\alpha (T-T_0)\Rightarrow \ln \left(\frac{R(T)}{R_0}\right)=\alpha (T-T_0)\Rightarrow R(T)=R_0{e}^{\alpha (T-T_0)}}$

Applying theTaylor series approximation at the first order, in the proximity of${\displaystyle T_0}$, leads to:

${\displaystyle R(T)=R_0(1+\alpha (T-T_0))}$

The thermal coefficient ofelectrical circuit parts is sometimes specified asppm/°C, orppm/K. This specifies the fraction (expressed in parts per million) that its electrical characteristics will deviate when taken to a temperature above or below theoperating temperature.

• Microbolometer (used to measure TCRs)

• Duderstadt, Jame J.; Hamilton, Louis J. (1976).Nuclear Reactor Analysis. Wiley.ISBN 0-471-22363-8.

• Electric and magnetic fields in matter
• Electrical engineering
• Nuclear physics

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