Temperature coefficient

Most ceramics exhibit negative temperature dependence of resistance behaviour. This effect is governed by an Arrhenius equation over a wide range of temperatures:

${\displaystyle R=Ae^{\frac {B}{T}}}$

where R is resistance, A and B are constants, and T is absolute temperature (K).

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

${\displaystyle R=r^{\infty }e^{\frac {B}{T}}=R_{0}e^{-{\frac {B}{T_{0}}}}e^{\frac {B}{T}}}$

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 variable negative temperature coefficient (NTC), which occurs when a physical property (such as thermal conductivity or electrical 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 in floor heating since 1971. The negative temperature coefficient avoids excessive local heating beneath carpets, bean bag chairs, mattresses, etc., which can damage wooden floors, and may infrequently cause fires.

Residual magnetic flux density or B_r changes with temperature and it is one of the important characteristics of magnet performance. Some applications, such as inertial gyroscopes and traveling-wave tubes (TWTs), need to have constant field over a wide temperature range. The reversible temperature coefficient (RTC) of B_r is defined as:

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

To address these requirements, temperature compensated magnets were developed in the late 1970s.[1] For conventional SmCo magnets, B_r decreases as temperature increases. Conversely, for GdCo magnets, B_r increases as temperature increases within certain temperature ranges. By combining samarium and gadolinium in the alloy, the temperature coefficient can be reduced to nearly zero.

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

${\displaystyle \operatorname {\rho } (T)=\rho _{0}\left[1+\alpha _{0}\left(T-T_{0}\right)\right]}$

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 (normally T = 0 °C)[2]

That of a semiconductor is however exponential:

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

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 the temperature coefficient of resistance (TCR).[3]

This property is used in devices such as thermistors.

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

In nuclear 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

${\displaystyle \alpha _{T}={\frac {\partial \rho }{\partial T}}}$

Where ${\displaystyle \rho }$ is reactivity and T is temperature. The relationship shows that ${\displaystyle \alpha _{T}}$ is the value of the partial 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 to passive 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 expands as temperature increases, causing longer neutron travel times during moderation, fuel material will not expand appreciably. Changes in reactivity in fuel due to temperature stem from a phenomenon known as doppler 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:

${\displaystyle \int _{R_{0}}^{R(T)}{\frac {dR}{R}}=\int _{T_{0}}^{T}\alpha \,dT~\Rightarrow ~\ln(R){\Bigg \vert }_{R_{0}}^{R(T)}=\alpha (T-T_{0})~\Rightarrow ~\ln \left({\frac {R(T)}{R_{0}}}\right)=\alpha (T-T_{0})~\Rightarrow ~R(T)=R_{0}e^{\alpha (T-T_{0})}}$

Applying the Taylor 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 of electrical circuit parts is sometimes specified as ppm/°C, or ppm/K. This specifies the fraction (expressed in parts per million) that its electrical characteristics will deviate when taken to a temperature above or below the operating temperature.

• Microbolometer (used to measure TCRs)

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