# Temperature measurements and instrumentation

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## Current revision as of 22:56, 8 July 2011

Temperature measurement using modern scientific thermometers and temperature scales goes back at least as far as the early 18th century, when Gabriel Fahrenheit adapted a thermometer (switching to mercury) and a scale both developed by Ole Christensen Røemer. Fahrenheit's scale is still in use, alongside the Celsius scale and the Kelvin scale.

## Technologies

Many methods have been developed for measuring temperature. Most of these rely on measuring some physical property of a working material that varies with temperature. One of the most common devices for measuring temperature is the glass thermometer. This consists of a glass tube filled with mercury or some other liquid, which acts as the working fluid. Temperature increases cause the fluid to expand, so the temperature can be determined by measuring the volume of the fluid. Such thermometers are usually calibrated so that one can read the temperature simply by observing the level of the fluid in the thermometer.

Important devices for measuring temperature include:

• Thermometers
• Thermocouples
• Thermistors
• Resistance Temperature Detector (RTD)
• Pyrometers
• Langmuir probes (for electron temperature of a plasma)
• Infrared

One must be careful when measuring temperature to ensure that the measuring instrument (thermometer, thermocouple, etc.) is really the same temperature as the material that is being measured. Under some conditions heat from the measuring instrument can cause a temperature gradient, so the measured temperature is different from the actual temperature of the system. In such a case the measured temperature will vary not only with the temperature of the system, but also with the heat transfer properties of the system. An extreme case of this effect gives rise to the wind chill factor, where the weather feels colder under windy conditions than calm conditions even though the temperature is the same. What is happening is that the wind increases the rate of heat transfer from the body, resulting in a larger reduction in body temperature for the same ambient temperature.

The theoretical basis for thermometers is the zeroth law of thermodynamics which postulates that if you have three bodies, A, B and C, if A and B are at the same temperature, and B and C are at the same temperature then A and C are at the same temperature. B, of course, is the thermometer.

The practical basis of thermometry is the existence of triple point cells. Triple points are conditions of pressure, volume and temperature such that three phases (matter) are simultaneously present, for example solid, vapor and liquid. For a single component there are no degrees of freedom at a triple point and any change in the three variables results in one or more of the phases vanishing from the cell. Therefore, triple point cells can be used as universal references for temperature and pressure. (See Gibb's phase rule)

Under some conditions it becomes possible to measure temperature by a direct use of the Planck's law of black body radiation. For example, the cosmic microwave background temperature has been measured from the spectrum of photons observed by satellite observations such as the WMAP. In the study of the quark-gluon plasma through heavy-ion collisions, single particle spectra sometimes serve as a thermometer.

## Thermometer

A clinical mercury thermometer
Thermometer

A thermometer (from the Greek θερμός (thermo) meaning "warm" and meter, "to measure") is a device that measures temperature or temperature gradient using a variety of different principles. A thermometer has two important elements: the temperature sensor (e.g. the bulb on a mercury thermometer) in which some physical change occurs with temperature, plus some means of converting this physical change into a value (e.g. the scale on a mercury thermometer). Thermometers increasingly use electronic means to provide a digital display or input to a computer.

### Primary and secondary thermometers

Thermometers can be divided into two separate groups according to the level of knowledge about the physical basis of the underlying thermodynamic laws and quantities. For primary thermometers the measured property of matter is known so well that temperature can be calculated without any unknown quantities. Examples of these are thermometers based on the equation of state of a gas, on the velocity of sound in a gas, on the thermal noise (see Johnson–Nyquist noise) voltage or current of an electrical resistor, and on the angular anisotropy of gamma ray emission of certain radioactive nuclei in a magnetic field. Primary thermometers are relatively complex.

Secondary thermometers are most widely used because of their convenience. Also, they are often much more sensitive than primary ones. For secondary thermometers knowledge of the measured property is not sufficient to allow direct calculation of temperature. They have to be calibrated against a primary thermometer at least at one temperature or at a number of fixed temperatures. Such fixed points, for example, triple points and superconducting transitions, occur reproducibly at the same temperature.

### Temperature

While an individual thermometer can measure degrees of hotness, the readings on two thermometers cannot be compared unless they conform to an agreed scale. There is today an absolute thermodynamic temperature scale. Internationally agreed temperature scales are designed to approximate this closely, based on fixed points and interpolating thermometers. The most recent official temperature scale is the International Temperature Scale of 1990. It extends from 0.65 K (−272.5 °C; −458.5 °F) to approximately 1,358 K (1,085 °C; 1,985 °F).

### Calibration

Mercury-in-glass thermometer

Thermometers can be calibrated either by comparing them with other calibrated thermometers or by checking them against known fixed points on the temperature scale. The best known of these fixed points are the melting and boiling points of pure water. (Note that the boiling point of water varies with pressure, so this must be controlled.)

The traditional method of putting a scale on a liquid-in-glass or liquid-in-metal thermometer was in three stages:

1. Immerse the sensing portion in a stirred mixture of pure ice and water at 1 Standard atmosphere (101.325 kPa ; 760.0 mmHg) and mark the point indicated when it had come to thermal equilibrium.
2. Immerse the sensing portion in a steam bath at 1 Standard atmosphere (101.325 kPa ; 760.0 mmHg) and again mark the point indicated.
3. Divide the distance between these marks into equal portions according to the temperature scale being used.

Other fixed points were used in the past are the body temperature (of a healthy adult male) which was originally used by Fahrenheit as his upper fixed point (96 °F (36 °C) to be a number divisible by 12) and the lowest temperature given by a mixture of salt and ice, which was originally the definition of 0 °F (−18 °C).(Benedict, 1984) (This is an example of a Frigorific mixture). As body temperature varies, the Fahrenheit scale was later changed to use an upper fixed point of boiling water at 212 °F (100 °C).(Lord, 1994)

These have now been replaced by the defining points in the International Temperature Scale of 1990, though in practice the melting point of water is more commonly used than its triple point, the latter being more difficult to manage and thus restricted to critical standard measurement. Nowadays manufacturers will often use a thermostat bath or solid block where the temperature is held constant relative to a calibrated thermometer. Other thermometers to be calibrated are put into the same bath or block and allowed to come to equilibrium, then the scale marked, or any deviation from the instrument scale recorded.(Benedict, 1984) For many modern devices calibration will be stating some value to be used in processing an electronic signal to convert it to a temperature.

### Precision, accuracy, and reproducibility

The "Boyce MotoMeter" radiator cap on a 1913 Car-Nation automobile, used to measure temperature of vapor in 1910s and 1920s cars.

The precision or resolution of a thermometer is simply to what fraction of a degree it is possible to make a reading. For high temperature work it may only be possible to measure to the nearest 10°C or more. Clinical thermometers and many electronic thermometers are usually readable to 0.1°C. Special instruments can give readings to one thousandth of a degree. However, this precision does not mean the reading is true.

Thermometers which are calibrated to known fixed points (e.g. 0 and 100°C) will be accurate (i.e. will give a true reading) at that point. Most thermometers are originally calibrated to a constant-volume gas thermometer.[citation needed] In between a process of interpolation is used, generally a linear one. (Benedict, 1984) This may give significant differences between different types of thermometer at points far away from the fixed points. For example the expansion of mercury in a glass thermometer is slightly different from the change in resistance of a platinum resistance of the thermometer, so these will disagree slightly at around 50°C.(Duncan, 1973) There may be other causes due to imperfections in the instrument, e.g. in a liquid-in-glass thermometer if the capillary varies in diameter.(Duncan, 1973)

For many purposes reproducibility is important. That is, does the same thermometer give the same reading for the same temperature (or do replacement or multiple thermometers give the same reading)? Reproducible temperature measurement means that comparisons are valid in scientific experiments and industrial processes are consistent. Thus if the same type of thermometer is calibrated in the same way its readings will be valid even if it is slightly inaccurate compared to the absolute scale.

An example of a reference thermometer used to check others to industrial standards would be a platinum resistance thermometer with a digital display to 0.1°C (its precision) which has been calibrated at 5 points against national standards (-18, 0, 40, 70, 100°C) and which is certified to an accuracy of ±0.2°C. (Reference Thermometer).

According to a British Standard, correctly calibrated, used and maintained liquid-in-glass thermometers can achieve a measurement uncertainty of ±0.01°C in the range 0 to 100°C, and a larger uncertainty outside this range: ±0.05°C up to 200 or down to -40°C, ±0.2°C up to 450 or down to -80°C. (Temperature Measurement)

## Thermocouple

Thermocouple plugged to a multimeter displaying room temperature in °C.

A thermocouple is a junction between two different metals that produces a voltage related to a temperature difference. Thermocouples are a widely used type of temperature sensor for measurement and control and can also be used to convert heat into electric power. They are inexpensive and interchangeable, are supplied fitted with standard connectors, and can measure a wide range of temperatures. The main limitation is accuracy: system errors of less than one degree Celsius (C) can be difficult to achieve.(Technical Notes)

Any junction of dissimilar metals will produce an electric potential related to temperature. Thermocouples for practical measurement of temperature are junctions of specific alloys which have a predictable and repeatable relationship between temperature and voltage. Different alloys are used for different temperature ranges. Properties such as resistance to corrosion may also be important when choosing a type of thermocouple. Where the measurement point is far from the measuring instrument, the intermediate connection can be made by extension wires which are less costly than the materials used to make the sensor. Thermocouples are usually standardized against a reference temperature of 0 degrees Celsius; practical instruments use electronic methods of cold-junction compensation to adjust for varying temperature at the instrument terminals. Electronic instruments can also compensate for the varying characteristics of the thermocouple, and so improve the precision and accuracy of measurements.

Thermocouples are widely used in science and industry; applications include temperature measurement for kilns, gas turbine exhaust, diesel engines, and other industrial processes.

### Principle of operation

In 1821, the German–Estonian physicist Thomas Johann Seebeck discovered that when any conductor is subjected to a thermal gradient, it will generate a voltage. This is now known as the thermoelectric effect or Seebeck effect. Any attempt to measure this voltage necessarily involves connecting another conductor to the "hot" end. This additional conductor will then also experience the temperature gradient, and develop a voltage of its own which will oppose the original. Fortunately, the magnitude of the effect depends on the metal in use. Using a dissimilar metal to complete the circuit creates a circuit in which the two legs generate different voltages, leaving a small difference in voltage available for measurement. That difference increases with temperature, and is between 1 and 70 microvolts per degree Celsius (µV/°C) for standard metal combinations.

### Practical use

A thermocouple measuring circuit with a heat source, cold junction and a measuring instrument.

#### Voltage–temperature relationship

 n Type K 1 25.08355 2 7.860106x10−2 3 -2.503131x10−1 4 8.315270x10−2 5 -1.228034x10−2 6 9.804036x10−4 7 -4.413030x10−5 8 1.057734x10−6 9 -1.052755x10−8

For typical metals used in thermocouples, the output voltage increases almost linearly with the temperature difference (ΔT) over a bounded range of temperatures. For precise measurements or measurements outside of the linear temperature range, non-linearity must be corrected. The nonlinear relationship between the temperature difference (ΔT) and the output voltage (mV) of a thermocouple can be approximated by a polynomial:

$\Delta T = \sum_{n = 0}^N a_n v^n$

The coefficients an are given for n from 0 to between 5 and 13 depending upon the metals. In some cases better accuracy is obtained with additional non-polynomial terms(Thermocouple Database). A database of voltage as a function of temperature, and coefficients for computation of temperature from voltage and vice-versa for many types of thermocouple is available online(Thermocouple Database).

In modern equipment the equation is usually implemented in a digital controller or stored in a look-up table; older devices use analog circuits.

Piece-wise linear approximations are an alternative to polynomial corrections(Thermocouple Calibration).

#### Cold junction compensation

Thermocouples measure the temperature difference between two points, not absolute temperature. To measure a single temperature one of the junctions—normally the cold junction—is maintained at a known reference temperature, and the other junction is at the temperature to be sensed.

Having a junction of known temperature, while useful for laboratory calibration, is not convenient for most measurement and control applications. Instead, they incorporate an artificial cold junction using a thermally sensitive device such as a thermistor or diode to measure the temperature of the input connections at the instrument, with special care being taken to minimize any temperature gradient between terminals. Hence, the voltage from a known cold junction can be simulated, and the appropriate correction applied. This is known as cold junction compensation.

It is worth noting that the EMF (or voltage) is NOT generated at the junction of the two metals of the thermocouple but rather along that portion of the length of the two dissimilar metals that is subjected to a temperature gradient.

Alternatively cold junction compensation can be performed by computation using look-up tables (Baker) and polynomial interpolation.

### Power Production

A thermocouple can produce current, which means it can be used to drive some processes directly, without the need for extra circuitry and power sources. For example, the power from a thermocouple can activate a valve when a temperature difference arises. The electrical energy generated by a thermocouple is converted from the heat energy which must be supplied to the hot side to maintain the electric potential. A continuous flow of heat is necessary because the current flowing through the thermocouple tends to cause the hot side to cool down and the cold side to heat up (the Peltier effect).

Thermocouples can be connected in series to form a thermopile, where all the hot junctions are exposed to a higher and all the cold junctions to a lower temperature. The output is the sum of the voltages across the individual junctions, giving larger voltage and power output. Using the radioactive decay of transuranic elements as a heat source, this arrangement has been used to power spacecraft on missions too far from the Sun to utilize solar power.

Thermocouple wire is available in several different metallurgical formulations per type, typically, in decreasing levels of accuracy and cost: special limits of error, standard, and extension grades.

#### Extension wire

Extension grade wires made of the same metals as a higher-grade thermocouple are used to connect it to a measuring instrument some distance away without introducing additional junctions between dissimilar materials which would generate unwanted voltages; the connections to the extension wires, being of like metals, do not generate a voltage. In the case of platinum thermocouples, extension wire is a copper alloy, since it would be prohibitively expensive to use platinum for extension wires. The extension wire is specified to have a very similar thermal coefficient of EMF to the thermocouple, but only over a narrow range of temperatures; this reduces the cost significantly.

The temperature-measuring instrument must have high input impedance to prevent any significant current draw from the thermocouple, to prevent a resistive voltage drop across the wire. Changes in metallurgy along the length of the thermocouple (such as termination strips or changes in thermocouple type wire) will introduce another thermocouple junction which affects measurement accuracy.

### Types

Certain combinations of alloys have become popular as industry standards. Selection of the combination is driven by cost, availability, convenience, melting point, chemical properties, stability, and output. Different types are best suited for different applications. They are usually selected based on the temperature range and sensitivity needed. Thermocouples with low sensitivities (B, R, and S types) have correspondingly lower resolutions. Other selection criteria include the inertness of the thermocouple material, and whether it is magnetic or not. Standard thermocouple types are listed below with the positive electrode first, followed by the negative electrode.

#### K

Type K (chromel–alumel) is the most common general purpose thermocouple with a sensitivity of approximately 41 µV/°C, chromel positive relative to alumel.(ASTM, 1974) It is inexpensive, and a wide variety of probes are available in its −200 °C to +1350 °C / -328 °F to +2462 °F range. Type K was specified at a time when metallurgy was less advanced than it is today, and consequently characteristics vary considerably between samples. One of the constituent metals, nickel, is magnetic; a characteristic of thermocouples made with magnetic material is that they undergo a step change in output when the magnetic material reaches its Curie point (around 354 °C for type K thermocouples).

#### E

Type E (chromel–constantan) has a high output (68 µV/°C) which makes it well suited to cryogenic use. Additionally, it is non-magnetic.

#### J

Type J (iron–constantan) has a more restricted range than type K (−40 to +750 °C), but higher sensitivity of about 55 µV/°C. The Curie point of the iron (770 °C)(Buschow) causes an abrupt change in the characteristic, which determines the upper temperature limit.

#### N

Type N (Nicrosil–Nisil) (Nickel-Chromium-Silicon/Nickel-Silicon) thermocouples are suitable for use at high temperatures, exceeding 1200 °C, due to their stability and ability to resist high temperature oxidation. Sensitivity is about 39 µV/°C at 900 °C, slightly lower than type K. Designed to be an improved type K, it is becoming more popular.

#### Platinum types B, R, and S

Types B, R, and S thermocouples use platinum or a platinum–rhodium alloy for each conductor. These are among the most stable thermocouples, but have lower sensitivity than other types, approximately 10 µV/°C. Type B, R, and S thermocouples are usually used only for high temperature measurements due to their high cost and low sensitivity.

B

Type B thermocouples use a platinum–rhodium alloy for each conductor. One conductor contains 30% rhodium while the other conductor contains 6% rhodium. These thermocouples are suited for use at up to 1800 °C. Type B thermocouples produce the same output at 0 °C and 42 °C, limiting their use below about 50 °C.

R

Type R thermocouples use a platinum–rhodium alloy containing 13% rhodium for one conductor and pure platinum for the other conductor. Type R thermocouples are used up to 1600 °C.

S

Type S thermocouples are constructed using one wire of 90% Platinum and 10% Rhodium (the positive or "+" wire) and a second wire of 100% platinum (the negative or "-" wire). Like type R, type S thermocouples are used up to 1600 °C. In particular, type S is used as the standard of calibration for the melting point of gold (1064.43 °C).

#### T

Type T (copper–constantan) thermocouples are suited for measurements in the −200 to 350 °C range. Often used as a differential measurement since only copper wire touches the probes. Since both conductors are non-magnetic, there is no Curie point and thus no abrupt change in characteristics. Type T thermocouples have a sensitivity of about 43 µV/°C.

#### C

Type C (tungsten 5% rhenium – tungsten 26% rhenium) thermocouples are suited for measurements in the 0 °C to 2320 °C range. This thermocouple is well-suited for vacuum furnaces at extremely high temperatures. It must never be used in the presence of oxygen at temperatures above 260 °C.

#### M

Type M thermocouples use a nickel alloy for each wire. The positive wire contains 18% molybdenum while the negative wire contains 0.8% cobalt. These thermocouples are used in vacuum furnaces for the same reasons as with type C. Upper temperature is limited to 1400 °C. It is less commonly used than other types.

#### Chromel-gold/iron

In chromel-gold/iron thermocouples, the positive wire is chromel and the negative wire is gold with a small fraction (0.03–0.15 atom percent) of iron. It can be used for cryogenic applications (1.2–300 K and even up to 600 K). Both the sensitivity and the temperature range depends on the iron concentration. The sensitivity is typically around 15 µV/K at low temperatures and the lowest usable temperature varies between 1.2 and 4.2 K.

### Thermocouple comparison

The table below describes properties of several different thermocouple types. Within the tolerance columns, T represents the temperature of the hot junction, in degrees Celsius. For example, a thermocouple with a tolerance of ±0.0025×T would have a tolerance of ±2.5 °C at 1000 °C.

Type Temperature range °C (continuous) Temperature range °C (short term) Tolerance class one (°C) Tolerance class two (°C) IEC Color code BS Color code ANSI Color code
K 0 to +1100 −180 to +1300 ±1.5 between −40 °C and 375 °C
±0.004×T between 375 °C and 1000 °C
±2.5 between −40 °C and 333 °C
±0.0075×T between 333 °C and 1200 °C
J 0 to +700 −180 to +800 ±1.5 between −40 °C and 375 °C
±0.004×T between 375 °C and 750 °C
±2.5 between −40 °C and 333 °C
±0.0075×T between 333 °C and 750 °C
N 0 to +1100 −270 to +1300 ±1.5 between −40 °C and 375 °C
±0.004×T between 375 °C and 1000 °C
±2.5 between −40 °C and 333 °C
±0.0075×T between 333 °C and 1200 °C
R 0 to +1600 −50 to +1700 ±1.0 between 0 °C and 1100 °C
±[1 + 0.003×(T − 1100)] between 1100 °C and 1600 °C
±1.5 between 0 °C and 600 °C
±0.0025×T between 600 °C and 1600 °C
Not defined.
S 0 to 1600 −50 to +1750 ±1.0 between 0 °C and 1100 °C
±[1 + 0.003×(T − 1100)] between 1100 °C and 1600 °C
±1.5 between 0 °C and 600 °C
±0.0025×T between 600 °C and 1600 °C
Not defined.
B +200 to +1700 0 to +1820 Not Available ±0.0025×T between 600 °C and 1700 °C}} No standard use copper wire No standard use copper wire Not defined.
T −185 to +300 −250 to +400 ±0.5 between −40 °C and 125 °C
±0.004×T between 125 °C and 350 °C
±1.0 between −40 °C and 133 °C
±0.0075×T between 133 °C and 350 °C
E 0 to +800 −40 to +900 ±1.5 between −40 °C and 375 °C
±0.004×T between 375 °C and 800 °C
±2.5 between −40 °C and 333 °C
±0.0075×T between 333 °C and 900 °C
Chromel/AuFe −272 to +300 n/a Reproducibility 0.2% of the voltage; each sensor needs individual calibration.

## Thermistor

NTC thermistor, bead type, insulated wires

A thermistor is a type of resistor whose resistance varies with temperature. The word is a portmanteau of thermal and resistor. Thermistors are widely used as inrush current limiters, temperature sensors, self-resetting overcurrent protectors, and self-regulating heating elements.

Thermistors differ from resistance temperature detectors (RTD) in that the material used in a thermistor is generally a ceramic or polymer, while RTDs use pure metals. The temperature response is also different; RTDs are useful over larger temperature ranges, while thermistors typically achieve a higher precision within a limited temperature range [usually −90 °C to 130 °C].

Thermistor symbol

Assuming, as a first-order approximation, that the relationship between resistance and temperature is linear, then:

$\Delta R=k\Delta T \,$

where

ΔR = change in resistance
ΔT = change in temperature
k = first-order temperature coefficient of resistance

Thermistors can be classified into two types, depending on the sign of k. If k is positive, the resistance increases with increasing temperature, and the device is called a positive temperature coefficient (PTC) thermistor, or posistor. If k is negative, the resistance decreases with increasing temperature, and the device is called a negative temperature coefficient (NTC) thermistor. Resistors that are not thermistors are designed to have a k as close to zero as possible(smallest possible k), so that their resistance remains nearly constant over a wide temperature range.

Instead of the temperature coefficient k, sometimes the temperature coefficient of resistance α (alpha) or αT is used. It is defined as(US, Sensor)

$\alpha_T = \frac{1}{R(T)} \frac{dR}{dT}.$

For example, for the common PT100 sensor, α = 0.00385 or 0.385 %/°C. This αT coefficient should not be confused with the α parameter below.

### Steinhart-Hart Equation

In practice, the linear approximation (above) works only over a small temperature range. For accurate temperature measurements, the resistance/temperature curve of the device must be described in more detail. The Steinhart-Hart equation is a widely used third-order approximation:

$\frac{1}{T}=a+b\,\ln(R)+c\,\ln^3(R)$

where a, b and c are called the Steinhart-Hart parameters, and must be specified for each device. T is the temperature in kelvins and R is the resistance in ohms. To give resistance as a function of temperature, the above can be rearranged into:

$R=e^{{\left( x-{y \over 2} \right)}^{1\over 3}-{\left( x+{y \over 2} \right)}^{1\over 3}}$

where

$y={{a-{1\over T}}\over c}$ and $x=\sqrt{{{{\left({b\over{3c}}\right)}^3}+{{y^2}\over 4}}}$

The error in the Steinhart-Hart equation is generally less than 0.02 °C in the measurement of temperature. As an example, typical values for a thermistor with a resistance of 3000 Ω at room temperature (25 °C = 298.15 K) are:

$a = 1.40 \times 10^{-3}$
$b = 2.37 \times 10^{-4}$
$c = 9.90 \times 10^{-8}$

### B parameter equation

NTC thermistors can also be characterised with the B parameter equation, which is essentially the Steinhart Hart equation with c = 0 and B = 1/b.

$\frac{1}{T}=\frac{1}{T_0} + \frac{1}{B}\ln \left(\frac{R}{R_0}\right)$

where the temperatures are in kelvins and R0 is the resistance at temperature T0 (usually 25 °C = 298.15 K). Solving for R yields:

$R=R_0e^{B(1/T-1/T_0)}$

or, alternatively,

$R=r_\infty e^{B/T}$

where $r_\infty=R_0 e^{-{B/T_0}}$. This can be solved for the temperature:

$T={B\over { {\ln{(R / r_\infty)}}}}$

The B-parameter equation can also be written as $\ln R=B/T + \ln r_\infty$. This can be used to convert the function of resistance vs. temperature of a thermistor into a linear function of lnR vs. 1 / T. The average slope of this function will then yield an estimate of the value of the B parameter.

### Conduction model

Many NTC thermistors are made from a pressed disc or cast chip of a semiconductor such as a sintered metal oxide. They work because raising the temperature of a semiconductor increases the number of electrons able to move about and carry charge - it promotes them into the conduction band. The more charge carriers that are available, the more current a material can conduct. This is described in the formula:

$I = n \cdot A \cdot v \cdot e$

I = electric current (amperes)
n = density of charge carriers (count/m³)
A = cross-sectional area of the material (m²)
v = velocity of charge carriers (m/s)
e = charge of an electron ($e=1.602 \times 10^{-19}$ coulomb)

The current is measured using an ammeter. Over large changes in temperature, calibration is necessary. Over small changes in temperature, if the right semiconductor is used, the resistance of the material is linearly proportional to the temperature. There are many different semiconducting thermistors with a range from about 0.01 kelvin to 2,000 kelvins (−273.14 °C to 1,700 °C).

Most PTC thermistors are of the "switching" type, which means that their resistance rises suddenly at a certain critical temperature. The devices are made of a doped polycrystalline ceramic containing barium titanate (BaTiO3) and other compounds. The dielectric constant of this ferroelectric material varies with temperature. Below the Curie point temperature, the high dielectric constant prevents the formation of potential barriers between the crystal grains, leading to a low resistance. In this region the device has a small negative temperature coefficient. At the Curie point temperature, the dielectric constant drops sufficiently to allow the formation of potential barriers at the grain boundaries, and the resistance increases sharply. At even higher temperatures, the material reverts to NTC behaviour. The equations used for modeling this behaviour were derived by W. Heywang and G. H. Jonker in the 1960s.

Another type of PTC thermistor is the polymer PTC, which is sold under brand names such as "Polyswitch" "Semifuse", and "Multifuse". This consists of a slice of plastic with carbon grains embedded in it. When the plastic is cool, the carbon grains are all in contact with each other, forming a conductive path through the device. When the plastic heats up, it expands, forcing the carbon grains apart, and causing the resistance of the device to rise rapidly. Like the BaTiO3 thermistor, this device has a highly nonlinear resistance/temperature response and is used for switching, not for proportional temperature measurement.

Yet another type of thermistor is a silistor, a thermally sensitive silicon resistor. Silistors are similarly constructed and operate on the same principles as other thermistors, but employ silicon as the semiconductive component material.

### Self-heating effects

When a current flows through a thermistor, it will generate heat which will raise the temperature of the thermistor above that of its environment. If the thermistor is being used to measure the temperature of the environment, this electrical heating may introduce a significant error if a correction is not made. Alternatively, this effect itself can be exploited. It can, for example, make a sensitive air-flow device employed in a sailplane rate-of-climb instrument, the electronic variometer, or serve as a timer for a relay as was formerly done in telephone exchanges.

The electrical power input to the thermistor is just:

$P_E=IV\,$

where I is current and V is the voltage drop across the thermistor. This power is converted to heat, and this heat energy is transferred to the surrounding environment. The rate of transfer is well described by Newton's law of cooling:

$P_T=K(T(R)-T_0)\,$

where T(R) is the temperature of the thermistor as a function of its resistance R, T0 is the temperature of the surroundings, and K is the dissipation constant, usually expressed in units of milliwatts per degree Celsius. At equilibrium, the two rates must be equal.

$P_E=P_T\,$

The current and voltage across the thermistor will depend on the particular circuit configuration. As a simple example, if the voltage across the thermistor is held fixed, then by Ohm's Law we have I = V / R and the equilibrium equation can be solved for the ambient temperature as a function of the measured resistance of the thermistor:

$T_0=T(R) -\frac{V^2}{KR}\,$

The dissipation constant is a measure of the thermal connection of the thermistor to its surroundings. It is generally given for the thermistor in still air, and in well-stirred oil. Typical values for a small glass bead thermistor are 1.5 mW/°C in still air and 6.0 mW/°C in stirred oil. If the temperature of the environment is known beforehand, then a thermistor may be used to measure the value of the dissipation constant. For example, the thermistor may be used as a flow rate sensor, since the dissipation constant increases with the rate of flow of a fluid past the thermistor.

### Applications

• PTC thermistors can be used as current-limiting devices for circuit protection, as replacements for fuses. Current through the device causes a small amount of resistive heating. If the current is large enough to generate more heat than the device can lose to its surroundings, the device heats up, causing its resistance to increase, and therefore causing even more heating. This creates a self-reinforcing effect that drives the resistance upwards, reducing the current and voltage available to the device.
• PTC thermistors are used as timers in the degaussing coil circuit of CRT displays and televisions. When the unit is initially switched on, current flows through the thermistor and degauss coil. The coil and thermistor are intentionally sized so that the current flow will heat the thermistor to the point that the degauss coil shuts off in under a second.
• NTC thermistors are used as resistance thermometers in low-temperature measurements of the order of 10 K.
• NTC thermistors can be used as inrush-current limiting devices in power supply circuits. They present a higher resistance initially which prevents large currents from flowing at turn-on, and then heat up and become much lower resistance to allow higher current flow during normal operation. These thermistors are usually much larger than measuring type thermistors, and are purposely designed for this application.
• NTC thermistors are regularly used in automotive applications. For example, they monitor things like coolant temperature and/or oil temperature inside the engine and provide data to the ECU and, indirectly, to the dashboard. They can be also used to monitor temperature of an incubator.
• Thermistors are also commonly used in modern digital thermostats and to monitor the temperature of battery packs while charging.

### History

The first NTC thermistor was discovered in 1833 by Michael Faraday, who reported on the semiconducting behavior of silver sulfide. Faraday noticed that the resistance of silver sulphide decreased dramatically as temperature increased. Because early thermistors were difficult to produce and applications for the technology were limited, commercial production of thermistors did not begin until the 1930s.(McGee, 1988)

## Pyrometer

An optical pyrometer
A sailor checking the temperature of a ventilation system

A pyrometer is a non-contacting device that intercepts and measures thermal radiation, a process known as pyrometry. This device can be used to determine the temperature of an object's surface.

The word pyrometer comes from the Greek word for fire, "πυρ" (pyro), and meter, meaning to measure. Pyrometer was originally coined to denote a device capable of measuring temperatures of objects above incandescence (i.e. objects bright to the human eye).

### Principle of operation

A pyrometer has an optical system and detector. The optical system focuses the thermal radiation onto the detector. The output signal of the detector (Temperature T) is related to the thermal radiation or irradiance j* of the target object through the Stefan–Boltzmann law, the constant of proportionality σ, called the Stefan-Boltzmann constant and the emissivity ε of the object.

$j^{\star} = \varepsilon\sigma T^{4}$

This output is used to infer the object's temperature. Thus, there is no need for direct contact between the pyrometer and the object, as there is with thermocouple and Resistance temperature detector (RTDs).

## References

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