Temperature coefficient

Temperature dependency is a forwarding to this article. For the temperature dependence of chemical reaction rates, see Arrhenius equation .

A temperature coefficient ( temperature coefficient ) describes the relative change in a particular physical quantity when the temperature changes compared to a specified reference temperature . The variable of interest is usually, but not always, a material property .

Temperature coefficients are considered for different quantities such as the length , the volume (see expansion coefficient ), the pressure , the electrical resistance or the voltage on a semiconductor diode . A more or less linear relationship between the respective variable and the temperature, i.e. an approximately constant temperature coefficient, is generally only present in a limited temperature range.

Basics

If the variable of interest is dependent on the temperature without hysteresis and without jumps , i.e. unambiguously , its temperature dependency can be described on the basis of the reference temperature . In the simplest case, an approximation function with a single temperature coefficient is sufficient: ${\displaystyle \xi }$ ${\displaystyle T}$ ${\displaystyle T_{0}}$

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

20 ° C is often chosen as the reference temperature.

${\displaystyle \xi (T)=\xi (20\,^{\circ }\mathrm {C} )\cdot \left[1+\alpha _{20}\,\left(T-20\,^{\circ }\mathrm {C} \right)\right]}$

In general, every temperature characteristic can be described by a Taylor series :

${\displaystyle {\xi (T)=\xi (T_{0}+\Delta T)=\xi (T_{0})\cdot (1+\alpha _{T_{0}}\cdot {\Delta T}+\beta _{T_{0}}\cdot {\Delta T}^{2}+\gamma _{T_{0}}\cdot {\Delta T}^{3}+\dots +k_{n,T_{0}}\cdot {\Delta T}^{n}+\dots )}}$

The approximation results from a Taylor polynomial of the nth degree:

${\displaystyle {\xi (T)=\xi (T_{0}+\Delta T)=\xi (T_{0})\cdot (1+\alpha _{T_{0}}\cdot {\Delta T}+\beta _{T_{0}}\cdot {\Delta T}^{2}+\gamma _{T_{0}}\cdot {\Delta T}^{3}+\dots +k_{n,T_{0}}\cdot {\Delta T}^{n})}}$

The most commonly used linear approximation results for: ${\displaystyle n=1}$

${\displaystyle \xi (T)=\xi (T_{0}+\Delta T)=\xi (T_{0})\cdot (1+\alpha _{T_{0}}\cdot \Delta T)}$

It is

• ${\displaystyle \Delta T}$the temperature difference to the reference temperature ( ) ${\displaystyle T-T_{0}}$
• ${\displaystyle \alpha _{T_{0}}}$ the 1st order temperature coefficient at the reference temperature
• ${\displaystyle \beta _{T_{0}}}$ the 2nd order temperature coefficient at the reference temperature
• ${\displaystyle \gamma _{T_{0}}}$ the temperature coefficient of the 3rd order at the reference temperature
• ${\displaystyle k_{n,{T_{0}}}}$ the temperature coefficient of the nth order at the reference temperature

The temperature coefficients can be calculated as follows by deriving the known function : ${\displaystyle \xi (\tau )}$

${\displaystyle \alpha _{T_{0}}={\frac {1}{1\,\xi (T_{0})}}\cdot \left.{\frac {\mathrm {d} \xi (\tau )}{\mathrm {d} \tau }}\right|_{\tau =T_{0}}}$

${\displaystyle \beta _{T_{0}}={\frac {1}{2!\,\xi (T_{0})}}\cdot \left.{\frac {\mathrm {d} ^{2}\xi (\tau )}{\mathrm {d} \tau ^{2}}}\right|_{\tau =T_{0}}}$

${\displaystyle \gamma _{T_{0}}={\frac {1}{3!\,\xi (T_{0})}}\cdot \left.{\frac {\mathrm {d} ^{3}\xi (\tau )}{\mathrm {d} \tau ^{3}}}\right|_{\tau =T_{0}}}$

${\displaystyle k_{n,T_{0}}={\frac {1}{n!\,\xi (T_{0})}}\cdot \left.{\frac {\mathrm {d} ^{n}\xi (\tau )}{\mathrm {d} \tau ^{n}}}\right|_{\tau =T_{0}}}$

It should be noted that the temperature coefficients depend on the reference temperature.

Examples

Temperature coefficient for the ideal gas

For the ideal gas , the temperature coefficients for pressure change and volume change are the same . ${\displaystyle {\frac {1}{273{,}15}}\,\mathrm {K} ^{-1}}$

With the idealizing assumptions, pressure change and volume change are linear.

Temperature coefficient of electrical resistance

The temperature dependence of the electrical resistance of components ( lines , resistors ) must always be taken into account when designing assemblies and designing circuits . On the other hand, this property is also used, e.g. B. with resistance thermometers .

Strictly speaking, since the temperature coefficient of the electrical resistance is not constant, there are polynomials for calculating the resistance from the present temperature, for example standardized for the Pt100 . Linear functions are often required for control engineering applications. The linear temperature coefficient indicates the relative change in the resistance value per change in temperature to a reference temperature; this is often chosen to be 0 ° C or 25 ° C instead of 20 ° C. For the conductor materials copper and aluminum, which are important in electrical engineering, a value of 0.4% per Kelvin can be used for estimates in the temperature range 0 ° C to 50 ° C. Commercially available low-power resistors, which should have as constant a resistance value as possible over the entire operating temperature range, have customary temperature coefficients in the range from 100 ppm per Kelvin to 200 ppm per Kelvin, precision resistors  in the range around 50 ppm per Kelvin. In this case, the linear temperature coefficient is specified with the prefix TK , for example TK100 for a resistance with 100 ppm per Kelvin. ${\displaystyle \alpha }$

Linear resistance-temperature coefficients of some substances at 20 ° C

Pure metals	${\displaystyle \alpha }$in K −1	Alloys	${\displaystyle \alpha }$in K −1	Non-metals	${\displaystyle \alpha }$in K −1
Aluminum (99.5%)	4.0 · 10 −3	Aldrey (AlMgSi)	3.6 · 10 −3	carbon	−0.5 · 10 −3
lead	4.2 · 10 −3	Beryllium bronze (SnBe4Pb)	0.5 · 10 −3	graphite	−0.2 · 10 −3
Iron (pure)	6.57 · 10 −3	Manganin (Cu84Ni4Mn12)	± 0.04 · 10 −3	Arc Coal	0.5 · 10 −3
gold	3.7 · 10 −3	Constantan (CuNi44)	± 0.01 · 10 −3	Germanium	−48 · 10 −3
Copper (99.9%)	3.93 · 10 −3	Isaohm	± 0.003 · 10 −3	silicon	−75 · 10 −3
nickel	6.0 · 10 −3	Brass (CuZn37)	1.6 · 10 −3
platinum	3.92 · 10 −3	Soft iron (4% Si)	0.9 · 10 −3
mercury	0.9 · 10 −3	Steel C15	5.7 · 10 −3
silver	3.8 · 10 −3
Tantalum	3.3 · 10 −3
tungsten	4.4 · 10 −3

Further examples

The temperature coefficient of a quartz oscillator describes the temperature dependence of the natural frequency.

The temperature coefficient of a nuclear reactor describes the temperature dependence of the reactivity (see also reactivity coefficient ).

Individual evidence

1. ↑ ^{a b c d e f g h i j k l} Friedrich table book electrical engineering / electronics . 582nd edition. Bildungsverlag EINS, Cologne 2007
2. ↑ ^{a b c} Specific resistances and temperature coefficients. Archived from the original on January 21, 2005 ; Retrieved December 27, 2011 . 
3. ↑ Electrical engineering book of tables . Europe teaching materials, Wuppertal 1966.
4. ↑ ^{a b} H. H. Gobbin: constants of nature . Wittwer, Stuttgart 1962.
5. ↑ isabellenhuette.de: Isaohm (PDF; 371 kB).
6. ^ Frank Bernhard: Technical temperature measurement . Springer, 2004, ISBN 3-642-18895-8 , pp. 609 ( limited preview in Google Book search).