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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Temperature coefficient</span></span>

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</p><p>A <b>temperature coefficient</b> describes the relative change of a physical property that is associated with a given change in <a href="Temperature" title="Temperature">temperature</a>. For a property <i>R</i> that changes when the temperature changes by <i>dT</i>, the temperature coefficient α is defined by the following equation:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dR}{R}}=\alpha \,dT}</annotation>

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</math></span><img src="./23f32c15388c6d284ef75ed4485975c54af73fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:11.641ex; height:5.509ex;" alt="{\displaystyle {\frac {dR}{R}}=\alpha \,dT}" loading="lazy"></span></dd></dl>

<p>Here α has the <a href="Dimension" title="Dimension">dimension</a> of an inverse temperature and can be expressed e.g. in 1/K or K<sup>−1</sup>.

</p><p>If the temperature coefficient itself does not vary too much with temperature and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \Delta T\ll 1}">

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<annotation encoding="application/x-tex">{\displaystyle \alpha \Delta T\ll 1}</annotation>

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</math></span><img src="./5f789ef303672b4a861f81dd5aa9742332b477ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.836ex; height:2.176ex;" alt="{\displaystyle \alpha \Delta T\ll 1}" loading="lazy"></span>, a <a href="Linear" class="mw-redirect" title="Linear">linear</a> approximation will be useful in estimating the value <i>R</i> of a property at a temperature <i>T</i>, given its value <i>R</i><sub>0</sub> at a reference temperature <i>T</i><sub>0</sub>:

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<annotation encoding="application/x-tex">{\displaystyle R(T)=R(T_{0})(1+\alpha \Delta T),}</annotation>

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</math></span><img src="./5d8b9a8e9705a685d6630275866a2400aa631875.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.812ex; height:2.843ex;" alt="{\displaystyle R(T)=R(T_{0})(1+\alpha \Delta T),}" loading="lazy"></span></dd></dl>

<p>where Δ<i>T</i> is the difference between <i>T</i> and <i>T</i><sub>0</sub>.

</p><p>For strongly temperature-dependent α, this approximation is only useful for small temperature differences Δ<i>T</i>.

</p><p>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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<div class="mw-heading mw-heading2"><h2 id="Negative_temperature_coefficient">Negative temperature coefficient</h2></div>

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<p>Most <a href="Ceramic" title="Ceramic">ceramics</a> exhibit negative temperature dependence of resistance behaviour. This effect is governed by an <a href="Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a> over a wide range of temperatures:

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<annotation encoding="application/x-tex">{\displaystyle R=Ae^{\frac {B}{T}}}</annotation>

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</math></span><img src="./9efa0c88f6f1505276051e500f8384a309ca44bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.77ex; height:3.509ex;" alt="{\displaystyle R=Ae^{\frac {B}{T}}}" loading="lazy"></span></dd></dl>

<p>where <i>R</i> is resistance, <i>A</i> and <i>B</i> are constants, and <i>T</i> is absolute temperature (K).

</p><p>The constant <i>B</i> is related to the energies required to form and move the <a href="Charge_carrier" title="Charge carrier">charge carriers</a> responsible for electrical conduction – hence, as the value of <i>B</i> increases, the material becomes insulating. Practical and commercial NTC <a href="Resistor" title="Resistor">resistors</a> aim to combine modest resistance with a value of <i>B</i> that provides good sensitivity to temperature. Such is the importance of the <i>B</i> constant value, that it is possible to characterize NTC <a href="Thermistor" title="Thermistor">thermistors</a> using the B parameter equation:

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<annotation encoding="application/x-tex">{\displaystyle R=r^{\infty }e^{\frac {B}{T}}=R_{0}e^{-{\frac {B}{T_{0}}}}e^{\frac {B}{T}}}</annotation>

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</math></span><img src="./92e1d4f3663351de5d0d027b86f93a6f9fe083a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.043ex; height:4.343ex;" alt="{\displaystyle R=r^{\infty }e^{\frac {B}{T}}=R_{0}e^{-{\frac {B}{T_{0}}}}e^{\frac {B}{T}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{0}}</annotation>

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</math></span><img src="./9b8916196f182fcbaaca54f931176a4a4f5769cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}" loading="lazy"></span> is resistance at temperature <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation>

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</math></span><img src="./55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" loading="lazy"></span>.

</p><p>Therefore, many materials that produce acceptable values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle R_{0}}</annotation>

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</math></span><img src="./9b8916196f182fcbaaca54f931176a4a4f5769cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}" loading="lazy"></span> include materials that have been alloyed or possess variable <b>negative temperature coefficient</b> (NTC), which occurs when a physical property (such as <a href="Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a> or <a href="Electrical_resistivity" class="mw-redirect" title="Electrical resistivity">electrical resistivity</a>) of a material lowers with increasing temperature, typically in a defined temperature range. For most materials, electrical resistivity will decrease with increasing temperature.

</p><p>Materials with a negative temperature coefficient have been used in <a href="Floor_heating" class="mw-redirect" title="Floor heating">floor heating</a> since 1971. The negative temperature coefficient avoids excessive local heating beneath carpets, <a href="Bean_bag" title="Bean bag">bean bag</a> chairs, <a href="Mattress" title="Mattress">mattresses</a>, etc., which can damage <a href="Wood_flooring" title="Wood flooring">wooden floors</a>, and may infrequently cause fires.

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<div class="mw-heading mw-heading2"><h2 id="Reversible_temperature_coefficient">Reversible temperature coefficient</h2></div>

<p><a href="Remanence" title="Remanence">Residual magnetic flux density</a> or <b>B</b><sub>r</sub> changes with temperature and it is one of the important characteristics of magnet performance. Some applications, such as inertial <a href="Gyroscope" title="Gyroscope">gyroscopes</a> and <a href="Traveling-wave_tube" title="Traveling-wave tube">traveling-wave tubes</a> (TWTs), need to have constant field over a wide temperature range. The <b>reversible temperature coefficient</b> (RTC) of <b>B</b><sub>r</sub> is defined as:

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<annotation encoding="application/x-tex">{\displaystyle {\text{RTC}}={\frac {|\Delta \mathbf {B} _{r}|}{|\mathbf {B} _{r}|\Delta T}}\times 100\%}</annotation>

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</math></span><img src="./c6744f62564eeeac5015155d62ed30cdd50d6e2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.006ex; height:6.509ex;" alt="{\displaystyle {\text{RTC}}={\frac {|\Delta \mathbf {B} _{r}|}{|\mathbf {B} _{r}|\Delta T}}\times 100\%}" loading="lazy"></span></dd></dl>

<p>To address these requirements, temperature compensated magnets were developed in the late 1970s.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> For conventional <a href="Samarium%E2%80%93cobalt_magnet" title="Samarium–cobalt magnet">SmCo magnets</a>, <b>B</b><sub>r</sub> decreases as temperature increases. Conversely, for GdCo magnets, <b>B</b><sub>r</sub> increases as temperature increases within certain temperature ranges. By combining <a href="Samarium" title="Samarium">samarium</a> and <a href="Gadolinium" title="Gadolinium">gadolinium</a> in the alloy, the temperature coefficient can be reduced to nearly zero.

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<div class="mw-heading mw-heading2"><h2 id="Electrical_resistance">Electrical resistance</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Electrical_resistivity_and_conductivity#Resistivity_and_conductivity_of_various_materials" title="Electrical resistivity and conductivity">Table of materials' resistivities</a></div>

<p>The temperature dependence of <a href="Electrical_resistance" class="mw-redirect" title="Electrical resistance">electrical resistance</a> and thus of electronic devices (<a href="Wire" title="Wire">wires</a>, resistors) has to be taken into account when constructing devices and <a href="Electrical_network" title="Electrical network">circuits</a>. The temperature dependence of <a href="Electrical_conductor" title="Electrical conductor">conductors</a> is to a great degree linear and can be described by the approximation below.

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {\rho } (T)=\rho _{0}\left[1+\alpha _{0}\left(T-T_{0}\right)\right]}</annotation>

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</math></span><img src="./e6592d9ec18d3c1c33ee3448336a1565d9e83754.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.312ex; height:2.843ex;" alt="{\displaystyle \operatorname {\rho } (T)=\rho _{0}\left[1+\alpha _{0}\left(T-T_{0}\right)\right]}" loading="lazy"></span></dd></dl>

<p>where

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{0}={\frac {1}{\rho _{0}}}\left[{\frac {\delta \rho }{\delta T}}\right]_{T=T_{0}}}</annotation>

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</math></span><img src="./83381dbf76253a0bb27a4650c45381c6d49ef947.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.168ex; height:6.509ex;" alt="{\displaystyle \alpha _{0}={\frac {1}{\rho _{0}}}\left[{\frac {\delta \rho }{\delta T}}\right]_{T=T_{0}}}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \rho _{0}}</annotation>

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</math></span><img src="./d9c04a9d26b86af8c6205ba2a6287fd655b6b714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.256ex; height:2.176ex;" alt="{\displaystyle \rho _{0}}" loading="lazy"></span> just corresponds to the specific resistance temperature coefficient at a specified reference value (normally <i>T</i> = 0 °C)<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>That of a <a href="Semiconductor" title="Semiconductor">semiconductor</a> is however exponential:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {\rho } (T)=S\alpha ^{\frac {B}{T}}}</annotation>

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</math></span><img src="./2b58fd67f99cd0e8ee04031bb96b5b12f37707a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.814ex; height:4.009ex;" alt="{\displaystyle \operatorname {\rho } (T)=S\alpha ^{\frac {B}{T}}}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

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<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle B}</annotation>

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</p><p>For both, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha }">

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</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is referred to as the <i>temperature coefficient of resistance</i> (TCR).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>This property is used in devices such as thermistors.

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<div class="mw-heading mw-heading3"><h3 id="Positive_temperature_coefficient_of_resistance">Positive temperature coefficient of resistance</h3></div>

<p>A <b>positive temperature coefficient</b> (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.

</p><p>Some materials even have exponentially increasing temperature coefficient. Example of such a material is <a href="PTC_rubber" title="PTC rubber">PTC rubber</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Negative_temperature_coefficient_of_resistance">Negative temperature coefficient of resistance</h3></div>

<p>A <b>negative temperature coefficient</b> (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), <a href="Temperature_sensor" class="mw-redirect" title="Temperature sensor">temperature sensors</a> and <a href="Thermistor" title="Thermistor">thermistors</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Negative_temperature_coefficient_of_resistance_of_a_semiconductor">Negative temperature coefficient of resistance of a semiconductor</h3></div>

<p>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.

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<div class="mw-heading mw-heading2"><h2 id="Temperature_coefficient_of_elasticity">Temperature coefficient of elasticity</h2></div>

<p>The <a href="Elastic_modulus" title="Elastic modulus">elastic modulus</a> of elastic materials varies with temperature, typically decreasing with higher temperature.

</p>

<div class="mw-heading mw-heading2"><h2 id="Temperature_coefficient_of_reactivity">Temperature coefficient of reactivity</h2></div>

<p>In <a href="Nuclear_engineering" title="Nuclear engineering">nuclear engineering</a>, 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

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{T}={\frac {\partial \rho }{\partial T}}}</annotation>

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</math></span><img src="./3f0e1aa432399a0e1689c952ae9f9b2c3e0598fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.766ex; height:5.676ex;" alt="{\displaystyle \alpha _{T}={\frac {\partial \rho }{\partial T}}}" loading="lazy"></span></dd></dl>

<p>Where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

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</math></span><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> is <a href="Nuclear_chain_reaction#Effective_neutron_multiplication_factor" title="Nuclear chain reaction">reactivity</a> and <i>T</i> is temperature. The relationship shows that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{T}}</annotation>

</semantics>

</math></span><img src="./1bae4c2d7a1f5097997a285df47dd01fb03ac1dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.877ex; height:2.009ex;" alt="{\displaystyle \alpha _{T}}" loading="lazy"></span> is the value of the <a href="Partial_differential" class="mw-redirect" title="Partial differential">partial differential</a> of reactivity with respect to temperature and is referred to as the "temperature coefficient of reactivity". As a result, the temperature feedback provided by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{T}}</annotation>

</semantics>

</math></span><img src="./1bae4c2d7a1f5097997a285df47dd01fb03ac1dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.877ex; height:2.009ex;" alt="{\displaystyle \alpha _{T}}" loading="lazy"></span> has an intuitive application to <a href="Passive_nuclear_safety" title="Passive nuclear safety">passive nuclear safety</a>. A negative <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{T}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha _{T}}</annotation>

</semantics>

</math></span><img src="./1bae4c2d7a1f5097997a285df47dd01fb03ac1dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.877ex; height:2.009ex;" alt="{\displaystyle \alpha _{T}}" loading="lazy"></span> 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.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p><p>

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 <a href="Water_(properties)" class="mw-redirect" title="Water (properties)">as temperature increases</a>, causing longer neutron travel times during <a href="Neutron_moderator" title="Neutron moderator">moderation</a>, fuel material will not expand appreciably. Changes in reactivity in fuel due to temperature stem from a phenomenon known as <a href="Doppler_broadening" title="Doppler broadening">doppler broadening</a>, where resonance absorption of fast neutrons in fuel filler material prevents those neutrons from thermalizing (slowing down).<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Fuel_temperature_coefficient_of_reactivity" title="Fuel temperature coefficient of reactivity">Fuel temperature coefficient of reactivity</a></div>

<div class="mw-heading mw-heading2"><h2 id="Mathematical_derivation_of_temperature_coefficient_approximation">Mathematical derivation of temperature coefficient approximation</h2></div>

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

</p>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {dR}{dT}}=\alpha \,R}</annotation>

</semantics>

</math></span><img src="./90e797ea3a24e7ceb40c3a74e6da4a8bc2b0312b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.553ex; height:5.509ex;" alt="{\displaystyle {\frac {dR}{dT}}=\alpha \,R}" loading="lazy"></span></dd></dl>

<p>Where is defined:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R_{0}=R(T_{0})}</annotation>

</semantics>

</math></span><img src="./36ad01ad14eec4d81007d03668dda7f2fe7a4f8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.902ex; height:2.843ex;" alt="{\displaystyle R_{0}=R(T_{0})}" loading="lazy"></span></dd></dl>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation>

</semantics>

</math></span><img src="./b79333175c8b3f0840bfb4ec41b8072c83ea88d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" loading="lazy"></span> is independent of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

</semantics>

</math></span><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span>.

</p><p>Integrating the temperature coefficient differential law:

</p>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\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})}}</annotation>

</semantics>

</math></span><img src="./939249aab8c9c196e032751dc04ff3bffa4b8f72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:104.799ex; height:7.343ex;" alt="{\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})}}" loading="lazy"></span></dd></dl>

<p>Applying the <a href="Taylor_series" title="Taylor series">Taylor series</a> approximation at the first order, in the proximity of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{0}}</annotation>

</semantics>

</math></span><img src="./55b9e7d7b96196b5a6a26f4349caa3ac82fd67e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.412ex; height:2.509ex;" alt="{\displaystyle T_{0}}" loading="lazy"></span>, leads to:

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R(T)=R_{0}(1+\alpha (T-T_{0}))}</annotation>

</semantics>

</math></span><img src="./ea0cff736a5bf714ce27470277cf8f48efdeb8b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.124ex; height:2.843ex;" alt="{\displaystyle R(T)=R_{0}(1+\alpha (T-T_{0}))}" loading="lazy"></span></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Units">Units</h2></div>

<p>The thermal coefficient of <a href="Electrical_network" title="Electrical network">electrical circuit</a> parts is sometimes specified as <a href="Parts_per_notation" class="mw-redirect" title="Parts per notation">ppm</a>/°<a href="Celsius" title="Celsius">C</a>, or <a href="Parts_per_notation" class="mw-redirect" title="Parts per notation">ppm</a>/<a href="Kelvin" title="Kelvin">K</a>. This specifies the fraction (expressed in parts per million) that its electrical characteristics will deviate when taken to a temperature above or below the <a href="Operating_temperature" title="Operating temperature">operating temperature</a>.

</p>

<ul><li><a href="Microbolometer" title="Microbolometer">Microbolometer</a> (used to measure TCRs)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20091029044111/http://www.electronenergy.com/about-us/about-us.htm">"About Us"</a>. Electron Energy Corporation. Archived from <a rel="nofollow" class="external text" href="http://www.electronenergy.com/about-us/about-us.htm">the original</a> on October 29, 2009.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFKasap2006" class="citation book cs1">Kasap, S. O. (2006). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/principlesofelec0000kasa"><i>Principles of Electronic Materials and Devices</i></a></span> (Third ed.). Mc-Graw Hill. p. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofelec0000kasa/page/126/mode/2up">126</a>.</cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFAlenitsynButikov,_Eugene_I.Kondraryez,_Alexander_S.1997" class="citation book cs1">Alenitsyn, Alexander G.; Butikov, Eugene I.; Kondraryez, Alexander S. (1997). <i>Concise Handbook of Mathematics and Physics</i>. CRC Press. pp. <span class="nowrap">331–</span>332. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-8493-7745-5</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Duderstadt & Hamilton 1976, pp. 259–261</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Duderstadt & Hamilton 1976, pp. 556–559</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFDuderstadtHamilton,_Louis_J.1976" class="citation book cs1"><a href="James_Johnson_Duderstadt" class="mw-redirect" title="James Johnson Duderstadt">Duderstadt, Jame J.</a>; Hamilton, Louis J. (1976). <i>Nuclear Reactor Analysis</i>. Wiley. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-22363-8</bdi>.</cite></li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2023-12-08" href="https://en.wikipedia.org/wiki/?title=Temperature_coefficient&oldid=1188875389">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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