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Properties and Applications of Thermistors—Thermally Sensitive Resistors*
J. A. Becker, C. B. Green, G. L. Pearson
I. Properties of Thermistors
Introduction
Thermistors, or thermally sensitive resistors, are devices made of solid substances whose electrical resistance depends very strongly on temperature. Although thermistors have been known for only about fifteen years, they have already acquired importance and are widely used in telephone and military equipment. Some of them are used as delay devices, protective devices, voltage regulators, regulators in carrier-frequency systems, volume limiters, instruments for measuring UHF power, and detecting elements (in the case of sources of very low power). In all these applications thermistors are preferred to other devices serving similar purposes because, being simple in construction and having small dimensions and high strength, they remain serviceable for a long time and do not require special maintenance.
These, as well as other positive properties of thermistors, suggest that they are destined to become a new element of the electrical circuit that will find wide application in communications and in thermal, electrical, and radio instruments used in physical, chemical, and biological scientific investigations and in military technology.
Characteristic examples of applications, which will be considered in the second part of this article, are:
1) simple sensitive and fast-responding thermometers, temperature compensators, and temperature regulators;
2) switches without moving contacts;
3) volume regulators or limiters;
* J. A. Becker, C. B. Green, G. L. Pearson, Bell Syst. Techn. J. 26, 1 (1947); translated from English by A. P. Pereleshina.
4) instruments for measuring pressure and flow, and simple instruments for measuring the thermal conductivity of liquids and gases;
5) delay eliminators and stabilizers;
6) generators, modulators, and amplifiers for comparatively low frequencies.
Before proceeding to a detailed discussion of these applications, it is necessary to gain an understanding of the physical laws that determine the properties of thermistors.
A natural question arises: why have devices of this kind come into use only recently? The answer is that thermistors are made from semiconductors, and their resistance can vary by \(10^3\)—\(10^6\) times depending on the presence of a negligible amount of impurities, on the methods of heat treatment, on the nature of the contacts, and on the way they are handled in operation. The practical use of semiconductors was hindered by the fact that different specimens, even if made in apparently one and the same way, proved to be very different in their properties. Even one and the same specimen, under the influence of the current passing through it or of a moderate change in temperature, can change its resistance by a factor of 2—10. Therefore, before serious use of semiconductors in industry became possible, it was necessary to devote a large number of scientific investigations to studying the nature of conductivity, the influence of impurities and heat treatment on the conductivity of semiconductors, and also methods for making reliable contacts. Although Faraday had already observed the rapid change of resistance with temperature in silver sulfide, and thousands of other semiconductors were found possessing large negative temperature coefficients of resistance, it took about a century for the achievements of physics and chemistry to give engineers this new tool, which in importance may become comparable to the electron tube and in many cases may replace it.
In order for thermistors to become widely used in industry, the following is necessary:
1) to reproduce specimens possessing identical characteristics;
2) to obtain specimens with characteristics that do not change with time; the contacts must remain unchanged, and the specimen must be chemically inert;
3) the specimens must be mechanically strong;
4) the method of manufacture must be such that the material can be given various shapes and sizes; and, finally,
5) it must be possible to cover a wide range of values of resistance, temperature coefficient, and dissipated power.
Thermistors can be manufactured by any method by which a semiconductor can be given definite dimensions and shape and contacts can be made. These methods include: 1) melting the semiconductor, cooling it, and allowing it to solidify; 2) evaporation; 3) heating pressed semiconductor powders to a temperature at which they sinter into a hard, compact mass, and firing contacts made of metallic powder.
Although all three methods are used, the third is considered the most suitable for mass production. It is similar to the method used in ceramic production or in powder metallurgy. At the sintering temperature the powder recrystallizes, and its volume decreases by a certain amount. The process of grinding into powder makes it possible to mix two semiconducting oxides (or a larger number of them)
Fig. 1. Thermistors manufactured in the form of a bead, rod, disk, washer, and flake.
in various proportions and to obtain a homogeneous and uniform solid body. As a result, with one and the same system of oxides, it is possible over wide limits to vary the values of the resistivity, the temperature coefficient of resistance, and the power consumed, since the process of grinding into powder permits the manufacture of thermistors of the most varied shapes and sizes.
Fig. 1 is a photograph of thermistors manufactured in the form of a bead, rod, disk, washer, and flake. Beads are made by stretching two platinum wires parallel to one another (at a distance 5–10 times greater than the diameter of the wire). A certain number of droplets of an oxide mixture are applied to the wires. Surface tension gives these droplets a spherical shape. Along the wires, from 10 to 20 such beads are placed at equal intervals. The beads are dried, with slight heating,
until they become strong enough that the “bead string” can be handled by hand. After this they pass through a sintering furnace. The oxides on the platinum wires shrink in volume, and a strong and unchanging electrical contact is produced. The wires are then cut in order to separate the beads from one another. With wire diameters from 0.0025 to 0.015 cm, the bead diameters range from 0.015 to 0.15 cm.
To manufacture rod-shaped thermistors, the oxides are mixed with an organic binder and a solvent; the mixture is then pressed, dried, given the required shape, heated to drive off the binding substance and, finally, sintered at high temperature. Contacts are made by coating the ends with a silver, gold, or platinum paste, similar to that used in ceramic production, and carrying out the process of its hardening at the appropriate temperature.
The diameter of the rods usually ranges from 0.08 to 0.64 cm. Their length can be varied from 0.15 to 5 cm.
In a similar way, by pressing a mixture of powder with an organic binder, disks and washers are also produced. Possible disk diameters range from 0.15 to 3 or 5 cm; their thickness is from 0.08 to 0.64 cm.
Flakes are made by mixing the oxides with a suitable binder and solvent to the consistency of cream and spreading this mixture in a thin layer on a smooth glass surface; the resulting film is dried, removed, cut into flakes of the desired sizes and shapes, and fired at the sintering temperature on a smooth ceramic surface. Contacts are produced by the method described above. The following flake dimensions are permissible: thickness—from 0.001 to 0.004 cm, length—from 0.1 to 1 cm, width—from 0.02 to 0.1 cm.
In any of these types of thermistors, leads may be attached to the contacts by soldering or by firing on pastes of heavy metals. The limiting dimensions given above are considered readily attainable.
When designing a thermistor for particular purposes, the following data should be taken into account:
1) mechanical dimensions (including the dimensions of the supports);
2) the material from which the thermistor is made and its properties (the latter include the resistivity and the nature of its variation with temperature, the specific heat capacity, density, and coefficient of thermal expansion);
3) the dissipation constant and the sensitivity to power. The dissipation constant is equal to the number of watts dissipated in the thermistor divided by the increase of its temperature, expressed in °C, relative to the ambient temperature.
Sensitivity to power is equal to the number of watts that must be dissipated in order for the resistance to decrease by 1%. These constants are determined by the nature of the surface, its area, the surrounding medium, and the thermal conductivity of the supports;
4) heat capacity, the value of which is determined by the specific heat, dimensions, and density;
5) time constant. This constant determines the rate of heating or cooling of the thermistor. It is equal to the time required for the temperature of the thermistor to be lowered by 63% of the temperature difference between the thermistor and the surrounding medium. The time constant is expressed in seconds and is equal to the value of the heat capacity in joules per degree Celsius divided by the dissipation constant in watts per degree Celsius;
6) the maximum permissible value of the power that can be dissipated under continuous operation and nonstationary processes, while the characteristics of the thermistors remain stable and its service life remains long. The maximum permissible power can be calculated if the dissipation constant and the maximum permissible temperature rise are known. It, together with the dependence of resistance on temperature, determines the maximum decrease in resistance.
Properties of Semiconductors
Most thermistors are made from semiconductors, and therefore it is very important to consider the properties of the latter. A semiconductor may be defined as a substance whose electrical conductivity at room (or near-room) temperature is much less than that of typical metals, but considerably exceeds the electrical conductivity of typical insulators. Although there is no sharp boundary between these classes of conductors, it may be assumed that at room temperature the values of specific resistance for semiconductors lie within the range from \(0.1\ \Omega\,\text{cm}\) to \(10^{9}\ \Omega\,\text{cm}\). As a rule, semiconductors have large negative values of the temperature coefficient of resistance. For example, when the temperature is raised from \(0^\circ\) to \(300^\circ\text{C}\), the resistance of a semiconductor may decrease by a factor of 1000. In a typical metal (for example, platinum), the resistance in this temperature interval increases twofold. Fig. 2 shows how the logarithm of the specific resistance of three typical semiconductors and of platinum depends on the temperature \(T\).
Curves 1 and 2 correspond to the materials No. 1 and No. 2, which are widely used at the present time. Material No. 1 consists of oxides of manganese and nickel; material No. 2, of oxides of manganese, nickel, and cobalt. The dashed portion of curve 2 refers to the region,
in which the dependence of resistance on temperature is known not as accurately as at lower temperatures. Curve 3 represents the experimental curve for a mixture of zinc and iron oxides taken in the ratio at which zinc ferrite is formed.
Fig. 2. Characteristics of the dependence of the logarithm of specific resistance on temperature for three thermistor materials and for platinum.
From Fig. 2 it is seen that both the resistance \(R\) and \(\lg R\) vary nonlinearly with the temperature \(T\).
In Fig. 3 are shown curves of the dependence of \(\lg \rho\) on \(\frac{1}{T}\) for materials No. 1 and No. 2. In form they are close to straight lines. Consequently,
\[ \left. \begin{aligned} \rho &= \rho_{\infty} e^{\frac{B}{T}} \\ \text{or}\qquad \rho &= \rho_{0} e^{\left(\frac{B}{T}-\frac{B}{T_{0}}\right)}, \end{aligned} \right\} \tag{1} \]
where \(T\) is the temperature in \({}^{\circ}\mathrm{K}\), \(\rho_{\infty}\) is the value of \(\rho\) at \(T=\infty\) (or at \(\frac{1}{T}=0\)), \(\rho_{0}\) is the value of \(\rho\) at \(T=T_{0}\), and \(B\) is a constant equal to 2.303 times the slope of the straight lines in Fig. 3.
The quantity \(B\) is expressed in \({}^{\circ}\mathrm{K}\) or in \({}^{\circ}\mathrm{C}\); in equation (1) it plays the same role as the work function in Richardson’s equation for thermionic emission. For material No. 1, \(B=3920^{\circ}\mathrm{C}\). This corresponds to an electron energy equal to \(3920/11600\) volt, or \(0.34\) volt.
At first glance the lines in Fig. 3 appear to be straight, but a more careful investigation shows that their steepness increases linearly with increasing temperature. Hence the more exact expression is:
\[ \left. \begin{aligned} \rho &= A T^{-c} e^{\frac{D}{T}} \\ \text{or}\qquad \lg \rho &= \lg A - c \lg T + \frac{D}{2.303\,T}. \end{aligned} \right\} \tag{2} \]
The constant \(c\) is equal to a small positive or negative number, or to zero. For material No. 1, \(\lg A = 5.563\), \(c=2.73\), and \(D=3100\).
For one of the forms of material No. 2, \(\lg A = 11.514,\ c = 4.83\) and \(D = 2064\).
If we define the temperature coefficient of resistance by the equation
\[ \alpha=\frac{1}{R}\frac{dR}{dT}, \tag{3} \]
then from equation (1) it follows that
\[ \alpha=-\frac{B}{T^2}. \tag{4} \]
For material No. 1 at temperature \(T = 300^\circ\) K,
\[ \alpha = -3920/90000 = -0.044. \]
For platinum, however, \(\alpha = +0.0037\), i.e. it is of the opposite sign and approximately 10 times smaller than for the semiconductor. From equation (2) it follows that
\[ \alpha=-\frac{D}{T^2}-\frac{c}{T}. \tag{5} \]
From equation (3) it follows that
\[ \alpha=\frac{1}{2.303}\frac{d}{dT}\lg R. \tag{6} \]
Fig. 3. Curves of the dependence of \(\lg \rho\) on \(\frac{1}{T}\) for materials Nos. 1 and 2 (see equation (1)).
In discussing the nature of the conductivity of semiconductors, it is more convenient to use the concept of conductivity \(\sigma\) than the concept of resistivity \(\rho\)
\[ \sigma=\frac{1}{\rho}\quad \text{and}\quad \lg\sigma=-\lg\rho. \tag{7} \]
The properties of semiconductors are expressed more clearly if, over a wide temperature interval, one plots the dependence of the conductivity (or its logarithm) on \(\frac{1}{T}\). Fig. 4 is such a graph for a number of silicon specimens with increasing impurity content. In the region of high temperatures all specimens have approximately the same value of conductivity. This value is called the intrinsic conductivity, since it appears to be, evidently, a property inherent in silicon. In the region of low tempe-
tures of the conductivity of different specimens differ sharply from one another. In this region silicon is, as it is customary to say, an impurity semiconductor. Ideally pure silicon possesses only intrinsic conductivity, and its resistance is expressed by equation (1). As the impurity concentration increases, the conductivity increases, and in the region of lower*) temperatures impurity conductivity begins to play the principal role. Some impurities affect the increase of conductivity much more strongly than others. If for every million parts of pure silicon there are one hundred parts of such impurities, then they can increase its conductivity at room temperature by \(10^7\) times. Other impurities, however, may be present in an amount of 10,000 parts per million and still have little effect on the magnitude of the conductivity.
Fig. 4. Plot of the dependence of the logarithm of conductivity on \(\frac{1}{T}\) for a number of silicon specimens. As the impurity concentration increases, the conductivity increases.
Moreover, two specimens may contain the same impurity concentration and yet differ sharply in their conductivity at low temperature; if the impurity is contained in a solid solution, i.e. is atomically dispersed, then its influence is great; but if the impurity is included in the form of particles whose dimensions appreciably exceed atomic dimensions, then its influence is insignificant. Since the dispersion of impurities in solids is affected by heat treatment, in the process of heat treatment the conductivity of many semiconductors changes greatly.
) In the text there is an obvious misprint: it says “high temperatures.” (Translator’s note.)*
On some other semiconductors, however, heat treatment has a rather insignificant effect.
An impurity is not always a foreign element; in the case of oxides or sulfides its role may be played by an excess or deficiency of oxygen or sulfur relative to the exact stoichiometric ratio. This excess or deficiency can be produced by thermal treatment. Fig. 5 shows how the conductivity depends on temperature for a number of samples of cuprous oxide, $\mathrm{Cu_2O}$, subjected to such heat treatment, so that the amount of excess oxygen varies from zero to approximately 1%[^1].
Fig. 5. Graph of the dependence of the logarithm of conductivity on $\dfrac{1}{T}$ for a number of samples of cuprous oxide. As the concentration of excess—in comparison with the exact stoichiometric ratio—oxygen increases, the conductivity increases. (The graph gives data available in the literature[^1].)
The greater the excess of oxygen, the greater the conductivity in the region of low temperatures. At high temperatures all samples possess approximately equal conductivity.
Depending on the nature of the current carriers, semiconductors may be divided into ionic, electronic, and mixed. Chlorides (for example, NaCl) and some sulfides are ionic; the remaining sulfides and some oxides (for example, uranium oxide) are mixed semiconductors; electronic semiconductors include most oxides (for example, NiO, $\mathrm{Mn_2O_3}$, $\mathrm{Fe_2O_3}$), carbides (for example, silicon carbide), and some elements (for example, boron, silicon, germanium, and tellurium). In ionic and mixed semiconductors, ions move through the solid, as a result of which the density of current carriers changes in different regions, and thus the conductivity also changes. Since this proves inconvenient, such semiconductors are rarely used for the manufacture
of thermistors, and therefore we shall focus our attention on electronic semiconductors.
There are two types of electronic semiconductors—“electronic” and “hole” semiconductors—depending on whether the current carriers are electrons or positive “holes” in the normally filled energy band*). In “electronic” semiconductors the current carriers are deflected in a magnetic field as negatively charged particles; in “hole” semiconductors, in the opposite direction. The direction of deflection is established by determining the sign of the Hall effect. The sign of the charge of the current carriers is also determined by the direction of the thermo-e.m.f. Knowing the resistivity, the Hall coefficient, and the thermo-e.m.f. of a given sample at a given temperature, one can determine the density of the current carriers, the sign of their charge, and their mobility or mean free path. The mobility is defined as the average velocity of displacement in a field whose strength is equal to \(1\ \mathrm{V/cm}\). The existence of these types of semiconductors is explained with the aid of the diagrams shown in Fig. 6.
Fig. 6. Energy-level diagrams of unexcited, electronic, and hole semiconductors.
In an unexcited semiconductor, at low temperatures the valence electrons completely fill all the allowed energy levels. According to the Pauli principle, in any system there can be only one electron at a given energy level. In semiconductors and insulators, just above the allowed band there is a region of forbidden energy values. Its width, expressed in electron-volts, is equal to \(\Delta E\). Above this forbidden band there is an allowed band, but at low temperature it contains no electrons. If such a semiconductor is placed in an electric field, not a single electron can acquire acceleration, since if this were to happen its energy would increase to
) In the paper by Becker, Green, and Pearson, instead of the terms generally accepted in our literature—“electronic” (or “hole”) semiconductor—the terms “\(N\)-type semiconductor” (or \(P\)-type semiconductor) are used. (Translator’s note.)*
of an energy level which is either occupied or forbidden. If, however, the temperature is raised, then some electrons acquire energy sufficient to pass, bypassing the forbidden region, into the upper allowed band. Now, owing to the applied field, these electrons can occupy a higher energy level and, consequently, can carry current.
Each electron that has passed into the “excited” state leaves a “hole” in the normally filled band. Other electrons, having somewhat smaller values of energy, can, having received acceleration owing to the applied field, fall into these “holes.”
These holes behave with respect to the applied field as if they were particles with a charge equal in magnitude but opposite in sign to the charge of the electron, and with a mass equal to the mass of the electron or somewhat greater than it. In a semiconductor with intrinsic (different from zero even in the absence of impurities) conductivity, the latter is due half to electrons and half to holes.
The value \(\Delta E\) is related to the quantity \(B\) from equation (1) in the following way:
\[ 2B = \Delta E \frac{e}{k}, \tag{8} \]
where \(B\) is expressed in \({}^{\circ}\mathrm{C}\), \(\Delta E\) in electron-volts, \(e\) is the charge of the electron in coulombs, and \(k\) is Boltzmann’s constant in joules per \({}^{\circ}\mathrm{C}\). The value of the ratio \(\frac{e}{k}\) is 11600, so that
\[ \Delta E = \frac{B}{5800}. \tag{8a} \]
The difference between metals, semiconductors, and insulators reduces to a difference in the values of \(\Delta E\). For metals this quantity is equal to zero or is very small. For semiconductors \(0.1\,\mathrm{eV} \lesssim \Delta E \lesssim 1.5\,\mathrm{eV}\). For insulators \(\Delta E > 1.5\,\mathrm{eV}\).
Some impurities present in a semiconductor with positive valences may occupy such energy levels that the energy \(\Delta E_1\) raises a valence electron of the impurity atom into the conduction band (see Fig. 6). Then the electron can take part in conduction; the donor impurity represents a positive ion, which is usually fixed in a definite position and cannot take part in conduction. These impurities create “electron” semiconductors (with an excess of electrons). The conductivity depends on the density of donors and on the values of \(\Delta E_1\) and \(T\).
In the same way, certain other impurities with negative valences may occupy an energy level situated at a distance \(\Delta E_2\,\mathrm{eV}\) above the filled band.
At room temperature or at a higher temperature the energy of an electron in the filled band may increase so much that it is captured by an impurity, which as a result becomes a negative ion, usually remaining immobile. However, the “hole” produced as a result of such a process may take part in the conductivity.
In all the cases shown in Fig. 6, the electron that has occupied a higher energy level (in comparison with a positive ion or a hole) has a certain probability of passing, within some short interval of time, to a lower level. However, during that same time there will be electrons which, owing to thermal motion, will rise to a higher energy level. When equal numbers of electrons rise and fall per second, equilibrium sets in. The conductivity \(\sigma\) is then equal to
\[ \sigma = Ne v_1 + Pe v_2, \tag{9} \]
where \(N\) and \(P\) are the concentrations of electrons and holes, respectively, \(e\) is the charge of the electron, and \(v_1\) and \(v_2\) are the mobilities of electrons and holes. The scheme given above explains the following experimental facts, which are difficult to interpret in any other way.
1) Oxides that are electron semiconductors (for example, ZnO), when heated in a neutral medium or in a medium with weak reducing properties, become good conductors, apparently because they contain an excess of zinc capable of donating electrons. If these oxides are then heated in a medium with gradually increasing oxidizing properties, their conductivity decreases until, in the end, they become unexcited semiconductors or insulators.
2) Oxides that are hole semiconductors (for example, NiO) become good conductors when subjected to heat treatment in a medium possessing strong oxidizing properties. It is very probable that they contain oxygen in excess of the stoichiometric ratio, and that this oxygen captures additional electrons. If these oxides are heated in a medium that is neutral or has weaker oxidizing properties, they become poorer conductors, semiconductors, or insulators.
3) If two oxides that are hole semiconductors are sintered with one another, the total conductivity of the mixture increases. The same is true for two oxides that are electron semiconductors. If, however, oxides are mixed of which one is an electron semiconductor and the other a hole semiconductor, the conductivity of the mixture decreases.
4) If a metal forms several oxides, then the one in which the metal has its highest valence belongs to the electronic semiconductors, whereas the oxide in which the metal has its lowest valence will be a hole semiconductor*.
For certain reasons, all semiconductors should be investigated with respect to their resistivity and temperature coefficient. One method by which this could be done consists in finding, for each specimen, the curves of the dependence of the resistivity on \(\frac{1}{T}\). However, such a graph would consist of such a tangle of intersecting lines, corresponding to different specimens, that it would be very difficult to single out any one of them and examine it. The same information can be obtained if, with the aid of equation (1), one finds the dependence of \(\lg \rho_0\) on \(B^6\).
In this case the most important characteristics of a specimen will be expressed by a single point, and on one graph many specimens can be investigated. Fig. 7 presents such a graph for a large number of semiconductors investigated in laboratories or described in the literature. For \(\rho_0\) and \(B\), values corresponding to the temperature \(T = 25^\circ\text{C}\) are given. The points form something resembling the Milky Way. Apparently, semiconductors with a large resistivity \(\rho_0\) also possess a large value of \(B\), and conversely. If, for a number of semiconductors, the points in Fig. 7 lie along a straight line with slope \(\frac{1}{2.3} T_0\), then the curves in Fig. 3 corresponding to these semiconductors have a point of intersection at \(\frac{1}{T}=0\).
Physical Properties of Thermistors
One of the most interesting properties of a thermistor is the character of the change in the voltage across it as the current increases. Fig. 8 shows this dependence for a bead, hardened in air, of diameter \(0.061\) cm, made of material No. 1. Each time the current is changed, it is necessary to wait for some time until the voltage reaches a new steady value. Therefore this curve is called the steady-state curve. At sufficiently small currents the dissipated power is too small to heat the thermistor more or less appreciably, and for the latter Ohm’s law remains valid. However, with a further increase in current the dissipated power increases, the temperature of the thermistor becomes higher than the ambient temperature, the resistance falls, and as a result the voltage proves to be smaller than it would have been at constant resistance. At a certain value of the current \(I_m\), the voltage
reaches a maximum or peak value \(V_m\). With a further increase in current, the voltage begins to decrease; the resistance of the thermistor, equal to \(\dfrac{dV}{dI}\), becomes negative. The numbers on the curve give, in °C, the value \((T - T_{\text{ambient}})\), where \(T\) is the temperature of the thermistor and \(T_{\text{ambient}}\) is the temperature of the surrounding medium.
Fig. 7. Plot of the dependence of \(\lg \rho\) on \(B\) for various semiconductors.
(The quantity \(B\), according to equation (4), is proportional to the temperature coefficient of resistance \(\alpha\).)
In view of the fact that the currents and voltages for different thermistors cover a large range of values, it is more convenient to plot the dependence of \(\lg V\) on \(\lg I\). Figure 9 shows such a plot for the same data as in Fig. 8. The magnitude of the increase in the temperature of the thermistor above the ambient temperature is indicated for different points of the curve in Fig. 9 exactly
Fig. 8. Static current-voltage characteristic of a typical thermistor. The numbers on the curve indicate, in °C, the value of the temperature difference between the thermistor and the surrounding medium.
Fig. 9. Plot of the dependence of $\lg V$ on $\lg I$ (corresponding to the same data as in Fig. 8). The diagonal lines correspond to different values of resistance and power.
likewise as in Fig. 8. On the logarithmic graph, straight lines drawn at an angle of \(+45^\circ\) correspond to different values of resistance, while lines at an angle of \(-45^\circ\) correspond to different values of power.
For an individual thermistor, the curve of the dependence of \(\lg V\) on \(\lg I\) is displaced when the dissipation constant \(C\) changes, as is shown in Fig. 10. This constant can be changed by changing the pressure of the gas surrounding the thermistor, by changing the medium, or by changing the degree
Fig. 10. Curves of the dependence of \(\lg V\) on \(\lg I\), plotted for three different values of the dissipation constant \(C\). These curves correspond to the constant values \(B\), \(R_0\), and \(T_0\) indicated in the upper part of the figure.
of thermal coupling of the thermistor with its surrounding medium. The value of \(C\) for an individual thermistor can readily be determined from the curve of the dependence of \(V\) on \(I\) (Figs. 8 or 9). For any point of the curve, the ratio
\[ \frac{V}{I} \]
is equal to the resistance, and the product \(V \cdot I = W\) is the dissipated power. The resistance values are recalculated, with the aid of the dependence of \(R\) on \(T\) specified by equation (2), into temperature. Then a graph of the dependence of \(W\) on \(T\) is constructed. For thermistors in which the greater part of the heat is dissipated, the quantity \(W\) will increase linearly with increasing \(T\), and consequently \(C\) remains a constant quantity. For thermistors mounted on thin wires in vacuum, however, \(W\) will grow faster than \(T\), and \(C\) will increase with increasing temperature \(T\). For thermistors of ordinary size and shape, the ratio of \(C\) to surface area varies in still air, depending on size and shape, from 1 to 40 mW/\({}^\circ\mathrm{C}\cdot\mathrm{cm}^2\).
In using a thermistor it is necessary to know how many watts will be dissipated before the resistance decreases by 1%. This quantity may be called the power sensitivity. It is equal to the ratio
\[ \frac{C}{\alpha \cdot 100} \]
and in still air varies approximately
tively from \(1\) to \(10\ \frac{\mathrm{mW}}{\mathrm{cm}^{2}}\). The values both of \(C\) and of the power sensitivity increase with increasing velocity of motion of the air surrounding the instrument. The dependence of \(C\) on the gas pressure and on the velocity of its motion is the basis for the use of thermistors as manometers and anemometers, or as instruments for measuring flow. Let us note that in Fig. 10 each curve, when displaced along the line of constant resistance, can be superposed on any other.
Fig. 11. Curves of the dependence of \(\lg V\) on \(\lg I\) at ambient temperature, constructed for three different values of the resistance \(R_0\). The curves are constructed for the constant values \(B\), \(C\), and \(T_0\) indicated in the upper part of the figure.
line of constant resistance can be superposed on any other. Figure 11 shows a family of curves of the dependence of \(\lg V\) on \(\lg I\) for various values of \(R_0\) and unchanged values of \(B\), \(C\), and \(T_0\). These conditions can be realized by changing—in order to change \(R_0\)—the length and width of the thermistor, as well as the density of the substance composing it, while at the same time preserving an unchanged surface area. If the resistance were changed by changing the temperature \(T_0\) of the surrounding medium, the resulting curves would differ very little from those shown. Let us note that each curve can be superposed on any other by shifting it along the line of constant power.
Figure 12 presents a family of curves of the dependence of \(\lg V\) on \(\lg I\) for 8 different values of \(B\) (the values of \(C\), \(R_0\), and \(T_0\) remain unchanged). In contrast to the curves of Fig. 10 and Fig. 11, each of which can be obtained from any other by shifting the latter along the corresponding axis, each curve in Fig. 12 is special, distinct from the others. For each curve in this figure there exist limiting values of the ohmic resistance for weak and for strong currents. At \(B=0\) the values of these resistances become equal to one another. With increa-
... \(B\) the logarithm of the ratio of the two limiting resistances increases in proportion to \(B\). Let us also note that the curves corresponding to values \(B > 1200^\circ\mathrm{K}\) have a maximum. For large values of \(B\), the position of this maximum corresponds to a small
Fig. 12. Curves of the dependence of \(\lg V\) on \(\lg I\), corresponding to eight different values of \(B\) and unchanged values of \(R_0\), \(C\), and \(T_0\).
power and, consequently, to a small value of the temperature difference \((T - T_0)\). The latter follows logically from the formula \(W = C(T - T_0)\). As \(B\) decreases, the position of the voltage maximum \(V_m\) corresponds to ever increasing values of power or temperature. For values \(B < 1200^\circ\mathrm{K}\) the maximum is absent.
The curves in Figs. 10–12 refer to the ideal case in which the resistance connected in series with the thermistor is equal to zero and the temperature limitations are not taken into account. In reality, however, there is always some small resistance (for example, the resistance of the leads) connected in series with the thermistor, and consequently parts of the curves corresponding to small resistances may not be observed. In the case of high power, the temperature may reach such a value that the thermistor begins to be destroyed; the limits of observation are thereby restricted. The dashed parts of the curves in Fig. 12 correspond to those regions of observation in which observation is impossible. The exact position of the dashed parts of the curves will, of course, depend on the con-
structures of the thermistor under consideration; allowance is also made for the limiting temperature values, upon reaching which the aging effects may become too large.
The curves in Figs. 9–12 are calculated by means of the following equations:
\[ R=R_0 e^{\left(\frac{B}{T}-\frac{B}{T_0}\right)}=\frac{V}{I}, \tag{10} \]
\[ W=C(T-T_0)=V\cdot I. \tag{11} \]
The constants \(R_0\), \(T_0\), \(B\), and \(C\) for these curves are precisely established. The values of the temperature \(T_m\), power \(W_m\), resistance \(R_m\), voltage \(V_m\), and current \(I_m\), corresponding to the maximum of the voltage-versus-current curves, are determined from the following equations, in which \(T_m\) is taken as the independent parameter. Differentiating Eqs. (10) and (11) with respect to \(I\) and setting the derivatives equal to zero, we obtain the equation
\[ T_m^2=B(T_m-T_0), \tag{12} \]
whose solution has the form
\[ T_m=\frac{B}{2}\left(1\pm\sqrt{1-\frac{4T_0}{B}}\right). \tag{13} \]
The maximum in Figs. 10–12 corresponds to the minus sign, the minimum to the plus sign. Note that \(T_m\) is a function only of \(B\) and \(T_0\) and does not depend on \(R\), \(R_0\), or \(C\).
From equations (4), (10), and (11) it follows that:
\[ -\alpha_m(T_m-T_0)=1, \tag{14} \]
\[ W_m=C(T_m-T_0), \tag{15} \]
\[ R_m=R_0 e^{-\frac{T_m}{T_0}} =R_0 e^{-1}\left[1-\frac{(T_m-T_0)}{T_0} +\frac{1}{2}\frac{(T_m-T_0)^2}{T_0^2}-\cdots\right], \tag{16} \]
\[ V_m=\left[CR_0(T_m-T_0)e^{-\frac{T_m}{T_0}}\right]^{\frac12} = \left\{CR_0(T_m-T_0)e^{-1} \left[1-\frac{(T_m-T_0)}{T_0} +\frac{1}{2}\frac{(T_m-T_0)^2}{T_0^2}-\cdots\right]\right\}^{\frac12}, \tag{17} \]
\[ I_m=\left[\frac{C}{R_0}(T_m-T_0)e^{\frac{T_m}{T_0}}\right]^{\frac12} = \left\{\frac{C}{R_0}(T_m-T_0)e \left[1+\frac{(T_m-T_0)}{T_0} +\frac{1}{2}\frac{(T_m-T_0)^2}{T_0^2}+\cdots\right]\right\}^{\frac12}. \tag{18} \]
Until now the discussion has been limited to the conditions of a steady state, when the power supplied to the thermistor is equal to the power dissipated by it, and the temperature remains constant. However, in many cases it is important also to consider non-steady-state conditions, when the temperature or other quantities that are functions of temperature vary with time. The following example is an elementary case illustrating the general concepts and constants contained in such problems: a massive thermistor is heated to approximately \(150—200^\circ\mathrm{C}\),
Fig. 13. Cooling characteristic of a massive thermistor: \(\lg (T-T_{\mathrm{amb}})\) as a function of time.
forcing it to operate in a region quite far from the peak of the volt-ampere characteristic. At the time \(t=0\), a constant current is supplied to the circuit, so small that the product \(I^2R\) is always a vanishingly small quantity. Then the voltage across the thermistor is measured as a function of time. From this, the resistance and temperature are calculated. In Fig. 13 there is shown a plot of the dependence of \(\lg (T-T_{\mathrm{amb}})\) on \(t\) for a rod thermistor made of material No. 1, whose length is approximately \(1.2\ \mathrm{cm}\), diameter \(0.3\ \mathrm{cm}\), and weight \(0.38\ \mathrm{g}\). During a certain time \(\Delta t\), \(C(T-T_{\mathrm{amb}})\Delta t\) joules are dissipated; as a result the temperature decreases by an amount \(\Delta T\), determined by
by the equation:
\[ -H\Delta T = C(T - T_{\mathrm{amb}})\Delta t \]
or
\[ (T - T_{\mathrm{amb}}) = -\frac{H}{C}\frac{\Delta T}{\Delta t}, \tag{19} \]
where \(H\) is the heat capacity in joules per \({}^{\circ}\mathrm{C}\). The solution of this equation is the expression
\[ (T - T_{\mathrm{amb}}) = (T_0 - T_{\mathrm{amb}})e^{-\frac{t}{\tau}}, \tag{20} \]
in which \(T_0 = T\) at \(t = 0\), and
\[ \tau = \frac{H}{C}, \tag{21} \]
where the quantity \(\tau\) is expressed in seconds. The quantity \(\tau\) is usually called the time constant.
According to equation (20), a plot of \(\lg (T - T_{\mathrm{amb}})\) versus \(t\) gives a straight line with a slope equal to \(-\dfrac{\tau}{2.303}\). If \(H\) and \(C\) vary little with temperature, then \(\tau\) will also vary little with \(T\) and \(t\). The line will not be exactly straight, but its slope for any values of \(t\) or \((T - T_{\mathrm{amb}})\) will give the corresponding values of \(\tau\) (or \(H/C\)) for this \(T\). As indicated above, the quantity \(C\) can be determined from the graph of the dissipated power versus temperature. For the given thermistor the steepness of this curve increased with increasing temperature, i.e., the value of \(C\) also increased. Table I gives the corresponding values of \(C\), \(\tau\), \(H\), as functions of \(T\) for a thermistor approximately \(1.2\ \mathrm{cm}\) long, \(0.3\ \mathrm{cm}\) in diameter and \(0.38\ \mathrm{g}\) in weight, made of material No. 1. \(T_{\mathrm{amb}} = 24^{\circ}\mathrm{C}\).
Table I
| \(T\ (^{\circ}\mathrm{C})\) | \(C\left(\dfrac{\mathrm{W}}{\mathrm{deg}\cdot{}^{\circ}\mathrm{C}}\right)\) | \(\tau\ (\mathrm{s})\) | \(H\left(\dfrac{\mathrm{J}}{\mathrm{deg}\cdot{}^{\circ}\mathrm{C}}\right)\) | \(h\left(\dfrac{\mathrm{J}}{\mathrm{g}\cdot\mathrm{deg}\cdot{}^{\circ}\mathrm{C}}\right)\) |
|---|---|---|---|---|
| 44 | 0.0037 | 76 | 0.28 | 0.75 |
| 64 | 0.0037 | 74 | 0.27 | 0.72 |
| 84 | 0.0038 | 71 | 0.27 | 0.71 |
| 104 | 0.0037 | 69 | 0.26 | 0.68 |
| 124 | 0.0038 | 68 | 0.26 | 0.67 |
| 144 | 0.0038 | 67 | 0.26 | 0.67 |
| 164 | 0.0039 | 67 | 0.26 | 0.69 |
| 184 | 0.0041 | 66 | 0.27 | 0.71 |
| 204 | 0.0042 | 66 | 0.28 | 0.73 |
temperature values of \(C\), \(\tau\), and \(H\) for a sample located in air.
When a thermistor is heated by a current passing through it, the conditions become somewhat more complicated, since in this case the power \(I^2R\) will be a function of time. At any instant of the heating cycle, the released thermal energy will be determined by the sum of the dissipated power, equal to \(C(T - T_{\text{sur}})\), and the power going into raising the temperature, equal to \(H \dfrac{dT}{dt}\). The magnitude of the released thermal energy will depend on the conditions in the circuit. In a circuit like that shown in the upper corner of Fig. 14, the change of current with time is characterized by six curves for six values of the battery voltage \(E\). If, at a certain current value, a relay operates in the circuit, then a considerable range of time-delay values can be obtained.
Fig. 14. Curves characterizing the change of current with time in the circuit shown in the figure for six different values of the battery voltage.
When the ambient temperature changes, the appearance of this family of curves changes, and where more precise time delays are required it is necessary to regulate or equalize the ambient temperature. The values \(H\), \(C\), and \(\tau\) may be changed, since the dimensions and shape of thermistors and the thermal conductivity of the surrounding medium can be varied over wide limits.
Thus, for example, the time constant \(\tau\) may vary approximately from \(10^{-3}\) sec to 10 min, i.e., by approximately a million times.
One of the most important characteristics is the aging characteristic of the thermistor, showing how constant its resistance remains over time at a given temperature. For
in order to obtain a stable thermistor, the following is necessary:
1) as starting materials, select only those semiconductors which have purely electronic conductivity;
2) select semiconductors that do not undergo chemical change when the temperature is raised under the action of a gaseous medium;
3) select semiconductors that are insensitive to impurities which may appear in the process of manufacturing the thermistor or during its use;
4) treat the semiconductors serving as the starting material in such a way that the degree of dispersion of the critical impurities
Fig. 15. Aging characteristics of thermistors made from materials Nos. 1 and 2 and aged in an oven at \(105^\circ\) C. Graph of the dependence of the increase in resistance (expressed as a percentage) on time (on a logarithmic scale)
would be at equilibrium, or else so that equilibrium at operating temperatures would be reached very slowly;
5) make contacts such that they would be firmly sintered with the instrument, have a coefficient of expansion approximately equal to the coefficient of thermal expansion of the semiconductor, and be sufficiently stable in the gaseous medium in which the thermistor will be located;
6) in some cases coat the thermistor with a thin layer of glass or another material impermeable to gases and liquids and having a suitable coefficient of expansion;
7) over the course of several days or weeks carry out so-called preliminary artificial aging, subjecting the thermistor to temperatures somewhat exceeding its operating temperatures.
If these conditions are observed, very good stability can be achieved.
Fig. 15 represents the aging characteristics of disks made from materials No. 1 and No. 2, with silver contacts and soldered-in leads (the disk diameter is 19.05 mm). Immediately after fabrication the resistance of these disks was measured; they were then “aged” in a furnace at a temperature of 105° C, and the resistance was measured periodically at 24° C. The curve of the percentage change in resistance, relative to its initial value, was plotted as a function of the logarithm of the time spent in the furnace. It should be noted that most of the aging takes place on the very first day or week. If these disks are subjected to a preliminary aging process for a week or a month, and the subsequent change in resistance is referred to the resistance value obtained at the end of this process, then they would age only by approximately 0.2% per year. In a thermistor thermometer this change in resistance would correspond to a shift of the temperature scale by 0.05° C. Thermistors installed in an evacuated tube, or coated with a thin layer of glass, age even less than those shown in the figure. For some applications such high stability of thermistors is unimportant, and there is no need to subject them to special treatment. At higher temperatures thermistors with good aging characteristics are used.
Pressed rods of material No. 1 were tested for stability by exposing them for 2 months to a temperature of 300° C. Typical specimens aged by an amount equal to 0.5–1.5% of their initial resistance. Similar thermistors, over 700 temperature cycles (each of which lasted half an hour), were exposed alternately to temperatures of +300° C and −75° C. The resistance of typical specimens changed by less than 1%. In some applications of thermistors, small temperature changes give rise to small voltage changes across the thermistor, which are then amplified by powerful amplifiers. If at the same time the resistance accidentally changes as well, even by \(1 \cdot 10^{-6}\) of its value, then the voltage across the thermistor will also change by an amount directly proportional to the current. This varying voltage is called noise, or—since it depends on the current—current noise. To obtain the best signal-to-noise ratio, it is necessary that, under operating conditions, the current noise be less than the noise due to thermal fluctuations of voltage \(^{7,8}\). To manufacture noiseless specimens, special attention must be paid to the raw material, the sizes of the smallest particles, the degree of sintering, the method of making contacts, and in general to any phases of the process as a result of which cracks or gaps might arise in the surface. All the thermistors described above were heated either by the current flowing through them or thanks to changes—
PROPERTIES AND APPLICATIONS OF THERMISTORS
changes in the ambient temperature. In thermistors with indirect heating, however, the temperature and resistance are regulated chiefly by the power supplied to the heater, which is in thermal contact with the thermistor.
An example of a specimen of this type is a sphere made of material No. 2, with a diameter of \(0.038\ \text{cm}\), placed in a small glass cylinder approximately \(0.38\ \text{cm}\) long and \(0.076\ \text{cm}\) in diameter. A small nichrome heating coil, whose resistance is \(100\ \Omega\), is wound on the cylinder and sealed with glass. Figure 16 shows a graph
Figure inscriptions: “Resistance in ohms”; “Power in ohms”; “voltage across the thermistor in volts”; “current passing through the thermistor in mA”; “heater current—0 mA”; “5 mA”; “10 mA”; “15 mA”; “20 mA”; “25 mA”; “ambient temperature \(25^\circ\text{C}\).”
Fig. 16. Curves of the dependence of \(\lg V\) on \(\lg I\) for a thermistor with indirect heating, plotted for six different values of the heater current. The values of resistance and power are given by diagonal lines.
of the dependence of \(\lg V\) on \(\lg I\) for an element made in the form of a sphere, at different values of the heater current. By this method it is possible to change the resistance of the sphere from \(3000\ \Omega\) to approximately \(10\ \Omega\). Thermistors with indirect heating are usually used where it is possible to electrically isolate the controlled and controlling circuits from one another, and to supply energy to the heater with constant resistance from the controlling circuit.
II. APPLICATIONS OF THERMISTORS
The exceptional versatility, small size, and durability of thermistors were the reason why, only five years after the first experience with the use of thermistors in the field of communications, they began to be used in large quantities in radio circuits.
Subsequently thermistors found still other important applications. Among the most important of these are temperature regulators, delay devices, automatic gain controls in feedback amplifiers, loudness limiters, and instruments for measuring microwave power.
It is remarkable that such versatility may be due to the character of the dependence of resistance on temperature. Nevertheless, it is precisely this dependence that creates a very advantageous nonlinear relation between current and voltage. This latter circumstance, together with the possibility of manufacturing sensitive elements of the most diverse shapes and sizes, is the reason why thermistors are used in the most varied fields. The most important applications of thermistors—namely, elements of electrical circuits and measuring and regulating instruments—should be considered by grouping them according to their principal characteristics: the dependence of resistance on temperature, of voltage on current, and of current on time (dynamic dependence).
Dependence of Resistance on Temperature
In the first part it was noted that the temperature coefficient of resistance of thermistors is negative, and in magnitude at room temperature is several times greater than the temperature coefficient of resistance of ordinary metals. For the commonly used thermistor material No. 1, the temperature coefficient at \(25^\circ\text{C}\) is equal to \(-4.4\%\) per \(^\circ\text{C}\), i.e., more than 10 times greater than the temperature coefficient of copper, which at the same temperature is \(+0.39\%\) per \(^\circ\text{C}\). The resistance of an electrical-circuit element made from this material at \(0^\circ\text{C}\) is 9 times greater than at \(50^\circ\text{C}\). By way of comparison, we note that the resistance of a copper wire at \(50^\circ\text{C}\) is 1.21 times greater than its value at \(0^\circ\text{C}\).
The characteristic of the dependence of the resistance of thermistors on temperature makes it possible to use them as sensitive thermometers, temperature regulators, and compensators for the effect of changes in the ambient temperature on other elements of electrical circuits.
Thermometry
The use of thermistors for measuring temperature follows logically from the basic principles of resistance thermometry. However, the large magnitude of the temperature coefficient of thermistors makes it possible to obtain a new (higher) order of sensitivity. This, together with the small size, simplicity, and ruggedness of thermistors, makes it possible to use them for the most varied temperature measurements. Intended for this
PROPERTIES AND APPLICATIONS OF THERMISTORS
...if thermistor-thermometers possess high stability, good—up to \(300^\circ\text{C}\)—and excellent—at more moderate temperatures. A thermistor subjected to good artificial aging, used for two months for precision temperature measurements at its various values (up to \(100^\circ\text{C}\)), retained its calibration unchanged to an accuracy of \(0.01^\circ\text{C}\). The stability of thermistor-thermometers may be considered (since work on this question is still continuing) as approaching the stability of precision platinum thermometers.
As a rule, thermistors are used in ordinary bridge or other circuits serving for the measurement of resistance. In order that the resistance of a thermistor-thermometer be determined solely by the temperature of the surrounding medium, it is necessary, as in any resistance thermometer, to keep the measuring current so small that it does not produce noticeable heating. Since thermistors usually have a higher resistance than metallic resistance thermometers or thermocouples, then, unlike the latter, the resistance of the leads, as a rule, is not a hindrance. This makes it possible to place the element sensitive to changes in temperature at some distance from the measuring circuit connected with it. As a result, thermistor-thermometers can be used in the most varied ways; for example, it becomes possible to transmit temperature readings to control points by wires.
Table II
Dependence of resistance on temperature for a typical thermistor-thermometer
| \(T\) (\(^\circ\text{C}\)) | Resistance (ohms) | Temperature coefficients \(B\) (\(^\circ\text{C}\)) | Temperature coefficients \(a\) (% per \(1^\circ\text{C}\)) |
|---|---|---|---|
| \(-25\) | 580 000 | 3780 | \(-6.1\) |
| 0 | 145 000 | 3850 | \(-5.2\) |
| 25 | 46 000 | 3920 | \(-4.4\) |
| 50 | 16 400 | 3980 | \(-3.8\) |
| 75 | 6 700 | 4050 | \(-3.3\) |
| 100 | 3 200 | 4120 | \(-3.0\) |
| 150 | 830 | 4260 | \(-2.4\) |
| 200 | 305 | 4410 | \(-2.0\) |
| 275 | 100 | 4600 | \(-1.5\) |
Table II gives the characteristics of a typical thermistor-thermometer.
As a sensitive thermometer, this thermistor with a simple Wheatstone bridge and a galvanometer whose sensitivity is \(2\cdot10^{-10}\ \text{a}/\text{mm}\) will make it easy to detect a temperature change of \(0.0005^\circ\text{C}\). For comparison we note that the minimum temperature change determined by a precision platinum resistance thermometer with a bridge of a special type required for it (for example, a Mueller bridge) will, with the very same galvanometer, be equal to \(0.003^\circ\text{C}\).
For this thermistor the dissipation constant*) in still air is approximately equal to \(4\ \dfrac{\text{mW}}{{}^\circ C}\).
The thermal time constant in still air is approximately equal to \(70\) sec. The dimensions of the thermistor are: diameter approximately equal to \(2.794\) mm, length approximately equal to \(13.716\) mm.
Some of the types of thermistors used in thermometry are shown in Fig. 17. These include specimens suitable
Fig. 17. Some types of thermistors that are used as resistance thermometers.
for applications as diverse as intravascular blood thermometry and measurement of the temperature of a heavily overloaded rotor. The letter \(A\) in Fig. 17 denotes a tiny bead with a response time in air of less than \(1\) sec; the letter \(B\) denotes a probe-type specimen used for measurements in gas streams and in liquids; the letter \(C\) denotes a meteorological thermometer used for the automatic radio transmission of weather data from free-flight aerostats; the letters \(D\) denote a specimen made in the form of a rod; \(E\) denotes a disk or bead used in the bulb of a metal thermometer. Such disks are soldered to metal plates in order to connect a small thermal resistance with the object whose temperature is to be measured. The letter \(F\) denotes a large, paint-coated disk for use in moist media. The characteristics of these types of thermistors are given in Table III.
*) The dissipation constant is the ratio of the supplied power, in watts, dissipated in the thermistor, to the resulting rise in temperature in \({}^\circ C\). The time constant is equal to the time required to change the temperature of the thermistor by \(63\%\) of the difference between its initial temperature and the temperature of the surrounding medium.
Table III
Thermistor thermometers
| Temperature (in °C) | A | B | C | D | E | F |
|---|---|---|---|---|---|---|
| Nominal resistance in ohms | Nominal resistance in ohms | Nominal resistance in ohms | Nominal resistance in ohms | Nominal resistance in ohms | Nominal resistance in ohms | |
| −25 | — | — | 87500 | 610000 | — | 13000 |
| 0 | 5000 | 325000 | 37000 | 153000 | 490 | 3200 |
| 25 | 2000 | 100000 | 18000 | 48500 | 175 | 950 |
| 50 | 900 | 33000 | 9700 | 17300 | 71 | 340 |
| 75 | 460 | 13000 | 5500 | 7100 | 32 | 145 |
| 100 | 250 | 6000 | 3700 | 3400 | 16 | 70 |
| 150 | 95 | 1600 | — | 870 | 4.5 | — |
| 200 | — | 500 | — | — | 1.6 | — |
| 300 | — | 80 | — | — | — | — |
| Temperature coefficient α at 25°C (in % per 1°C) | −3.4 | −4.4 | −2.8 | −4.4 | −3.8 | −4.4 |
| Maximum permissible temperature (in °C) | 150 | 300 | 100 | 150 | 200 | 100 |
| Dissipation constant C (in mW per 1°C) in still air | 0.1 | 1 | 7 | 7 | — | 20 |
| Dissipation constant C (in mW per 1°C) in still water | — | 7 | — | — | — | — |
| Thermal time constant (in sec) in still air | 1 | 30 | 25 | 60 | — | — |
| Thermal time constant (in sec) in still water | — | 4 | — | — | — | — |
| Type | bead | probe | rod | rod | disk | disk |
| Dimensions in inches: diameter | 0.015 | 0.1 | 0.05 | 0.15 | 0.2 | 0.56 |
| Dimensions in inches: thickness (or length) (without leads) | 0.02 | 0.6 | 1.2 | 0.7 | 0.1 | 0.3 |
The temperature of inaccessible objects (moving or too hot for contact thermometry) can be determined by focusing their radiation, by means of an elliptical mirror, onto a suitable thermistor. Such a thermistor may have the form of a thin flake fastened to a solid support. Its advantages in comparison with a thermocouple and a resistance bolometer are a more favorable value of the resistance, robustness, and a large temperature coefficient of resistance. A thermistor can be made so small that, owing to the reduction in heat capacity, changes in its resistance will coincide with changes in temperature. For flake thermistors the time constants may vary from 1 millisecond to 1 sec. To measure small changes in the resistance of the flake, sensitive measuring instruments (or vacuum-tube amplifiers) are needed, since the magnitude of the radiation power incident on the thermistor may be very small. Where there are no rapid changes in temperature, thermistors with large time constants and with a simpler circuit arrangement may be used.
Temperature regulation
The use of thermistors for temperature regulation is closely connected with their use as temperature-measuring instruments. In an ideal temperature-regulating element, the sensitivity to changes in temperature must be high, and the resistance at that temperature must have a value suitable for the circuit being used. The temperature rise of the regulating element caused by heating of the circuit must be small, and the stability of its calibration must be good. The shape and dimensions of the sensitive element are determined by several factors, namely: the available space, the required speed of response to changes in temperature, and the value of the power dissipated in the element of the regulating circuit that is necessary for controlling relays, motors, and vacuum tubes.
Owing to their high temperature sensitivity, thermistors are very promising as regulating elements. Their stability and the possibility of using them at comparatively high temperatures have led, for example, to the use of a rod thermistor in the control device of an aircraft engine.^9 The thermistor, mounted in the bulb of a standard thermometer 6.35 mm in diameter, operated at approximately 275° C. It was connected to a differential relay and to the controlling motor of an aircraft twenty-eight-volt direct-current system. The power dissipated in the thermistor was equal to
2 watts. The resistance of a typical thermistor at such a high temperature remained unchanged for months with an accuracy of up to $\pm 1.5\%$. The corresponding change in calibration was approximately $\pm 1^\circ$C. Several other analogous designs were developed, using the same control device with other thermistors intended to operate both at higher and at lower temperatures. Typical thermistors used at lower temperatures retained their calibration unchanged with an accuracy of up to several tenths of a degree. In general, when thermistors are used in electron-tube circuits, less power is dissipated in them than when they are used in relay circuits. This ensures a smaller temperature rise of the thermistor and, consequently, more accurate regulation. In calculations, the mean value of this temperature rise may be taken into account; various versions of installations require individual calibration in order to correct errors when they are large. The corrections may differ depending on whether they are the result of time-to-time changes in the thermal conductivity of the surrounding medium or arise in passing from one installation to another. The greater the power dissipated in the thermistor, the greater, for a given change in thermal conductivity, the magnitude of the absolute error in the value of the regulated temperature. This follows from the relation:
\[ \Delta T=\frac{W}{C}, \tag{22} \]
where $\Delta T$ is the temperature rise, $W$ is the dissipated power, and $C$ is the dissipation constant, depending on the thermal connection with the surrounding medium.
For the very same reason, the temperature determined by a resistance thermometer immersed in a moving medium will depend on the flow velocity if the temperature-sensitive element operates while being several degrees warmer than the surrounding medium.
The calculation of a thermistor for a thermostat with a ventilation tube may be carried out (since this is connected with a temperature rise) in the following way:
1) determine the dissipated power (its magnitude depends on the chosen circuit and on the required value of the limiting sensitivity);
2) estimate the permissible temperature rise of the thermistor, the anticipated change in air velocity, and the necessary accuracy of temperature regulation;
3) solve equation (22) with respect to the dissipation constant and choose a thermistor whose dimensions and design corre-
correspond to the value of this constant at the nominal air velocity.
In the case where several types of thermistors are suitable for a given circuit, the choice of one of them is determined by the required value of the time constant.
Compensators
A natural and obvious application of thermistors is their use for compensating changes in the resistance of electrical circuits caused by fluctuations in the ambient temperature. A simple example of this may be the compensation of changes in the resistance of a line made of copper wire, which increases by approximately 0.4% per degree Celsius. It would be sufficient to connect in series with this line a thermistor having a resistance equal to approximately 0.1 of the resistance of the copper, and a temperature coefficient equal to \(-4\%\) per \(1^\circ\text{C}\), in order to compensate changes in the line resistance over a small temperature interval. In practice, however, an ohmic resistance is connected in parallel (and sometimes in series) with the compensating thermistor so that the changes in resistance of such a combination are almost equal in magnitude and opposite in sign to the changes in the resistance of the circuit being compensated, as a result over a wide temperature interval (see Fig. 18).
From Fig. 18 it is seen how, with the aid of a thermistor (whose resistance at \(25^\circ\text{C}\) is \(566\ \Omega\)), connected in parallel with an ohmic resistance of \(445\ \Omega\), one can compensate changes in the resistance of a copper winding (equal to \(1000\ \Omega\) at \(25^\circ\text{C}\)). Together with the compensator, the winding in the temperature interval from \(-25^\circ\text{C}\) to \(+75^\circ\text{C}\) maintains a constant resistance value, with an accuracy up to \(\pm 1.6\%\), equal to \(1250\ \Omega\). The resistance of the copper alone changes in this same temperature interval from \(807.5\) to \(1192.5\ \Omega\), i.e., by \(\pm 19\%\) relative to its mean value. The total resistance of the circuit in the presence of the compensator, however, increases at the maximum temperature value by only 1.6%. Such a slight increase in resistance is explained by the fact that the compensating thermistor has a large temperature coefficient of resistance. The characteristics of such a thermistor are so stable that in 10 years its resistance would change by less than 1%, even if it were kept at any temperature up to \(100^\circ\text{C}\). Fig. 15 shows the aging characteristics of typical thermistors suitable for use in compensators. These curves also include the changes that occur during the aging period (several days) in the manufacture of thermistors.
so that the aging that occurs when a thermistor is used represents only part of the overall aging. In many circuits intended for operation with strict tolerances and, at the same time, with large changes in the ambient temperature, the resistance value of one (or several) circuit elements may vary with temperature within impermissible limits. The resulting change in resistance can often be reduced by connecting a simple thermistor at the appropriate point in the circuit. This proves especially effective if the circuit contains electron-tube amplifiers. In this case, frequency and amplitude variations in coupling circuits are eliminated, and temperature-caused errors in instrument readings are prevented. The change in the inductance of a coil caused by a change in the magnetic characteristics of the core material with temperature is prevented by saturating the coil with a direct current, the magnitude of which is regulated directly by the resistance of a thermistor inserted into the core.
Fig. 18. Compensation, with the aid of a thermistor, of changes in the resistance of a copper conductor caused by temperature fluctuations.
By this method the magnitude of the direct-current magnetic flux is regulated so that the inductance of the coil becomes independent of temperature.
When calculating the compensator, attention must be paid to ensuring that both the thermistor and the compensated element are subjected to the action of the same temperature. It is necessary to take into account the amount of power dissipated in the thermistor and to limit it to a value that will not produce a significant rise of the thermistor temperature above the temperature of the surrounding medium.
Voltage-current characteristics
Figure 9 showed the nonlinear static characteristic, typical in form for a thermistor, relating the values of voltage, current, resistance, and power. The part of the curve to the right of the maximum voltage value has a negative slope, which can be used in electrical circuits. At low dissipated power, this characteristic begins at a resistance value approximately equal to 50,000 ohms. Additional power dissipation raises the temperature of the thermistor and reduces its resistance. At the maximum voltage value, the resistance decreases to 17,000 ohms, i.e., to approximately \(1/3\) of its “cold” value, while the dissipated power becomes equal to 13 mW. When the dissipated power reaches 100 mW, the resistance becomes approximately 300 ohms. The characteristics of resistance as a function of power make it possible to use thermistors as sensitive instruments for measuring power and as automatically varying resistances for amplitude regulators at the output of generators and amplifiers. The nonlinearity of the voltage-current characteristic also makes it possible to use thermistors as voltage regulators, volume controls, expanders, remote-control devices, and contactless switches. In order that thermistors may be used with equal success for these purposes in both direct-current and alternating-current circuits, they are made (with the exception of two early types) exclusively from nonpolarizing semiconductors.
Instruments for measuring power
In test installations, in the range of superhigh and ultrahigh frequencies, thermistors are widely used as power-measuring elements. The special advantages of the thermistors used for this purpose are that they can be made small in size and with low electrical capacitance, they can be heavily overloaded (with the calibration remaining unchanged), and, finally, they can easily be calibrated by direct-current power or by low-frequency power. A thermistor for measuring power is used as a power-absorbing terminating resistance in transmission lines of the Lecher type and of the waveguide or coaxial types. Mounting methods have been developed that reduce the reflection of high-frequency energy from the terminating load to vanishingly small values and ensure accurate power measurement over a broad range of the frequency spectrum. According to the specifications, ther-
The thermistor is connected into one arm of a Wheatstone bridge and is subjected to low-frequency energy or direct-current energy until its resistance reaches the operating value (for example, 125 ohms or 250 ohms) in the absence of the power that is to be measured. This subsequently measured power reduces the resistance of the thermistor, the bridge becomes unbalanced, and the deviation from equilibrium is recorded by the measuring instrument of the bridge. For the test setup being described, full scale corresponds to a measured power of 1 mW, although, by using bead thermistors of different sizes (as in Fig. 19), power values from 0.1 mW to 200 mW can be measured.
Fig. 19. Thermistors used for measuring power, made in the form of beads of various sizes.
Continuous tests of these thermistors under operating conditions show that they remain stable for an indefinitely long time. A group of eight thermistor wattmeters, ordinarily operated at 10 mW and having a maximum rated power value of 20 mW, was in service for more than 3000 hours at 30 mW of input power. Their resistance at room temperature remained equal to its initial value with an accuracy of up to 1.5%, and such an important characteristic as power sensitivity changed by less than 0.5%. When power-measuring test setups are intended to operate in the presence of large fluctuations in the ambient temperature, it is necessary to compensate for changes in the thermistor temperature. According to the specifications, this is achieved by introducing two other thermistors into the bridge circuit. The latter are designed so that they are insensitive to the currents in the bridge, but readily
responded to changes in ambient temperature. One of the compensators maintains an unchanged zero point, while the other preserves the calibration of the measuring-instrument scale, independent of the effect of temperature changes on the characteristics of the measuring thermistor.
Automatic regulator of generator amplitude
Meacham\(^{10}\) and Sheffer and Wais\(^{11}\) used thermistors to stabilize the amplitude of both low- and high-frequency generators. These circuits generate because of the presence of positive feedback in the electron tube. The feedback circuit is a bridge in which at least one arm contains a thermistor heated by the output power of the generator. Owing to this, the feedback depends, in phase and amplitude, on the output power, and there is only one value of the thermistor resistance which balances the bridge and reduces the amplitude of oscillations to zero. It is evident that there is no point in trying to satisfy this condition; instead, the operating point differs from it by just enough that the bridge remains slightly unbalanced and the oscillations at the output reach a predetermined constant value. Generators whose oscillation amplitude is determined by the nonlinearity of the thermistor have many advantages in comparison with generators whose amplitude is limited by the nonlinearity of electron tubes. The distortion factor at the output proves to be smaller, and the operating characteristics depend much less on the electron tube and on fluctuations in the supply voltage. The thermal inertia of the thermistor must be sufficient to prevent its resistance from changing at the frequency of generation. This condition is easily fulfilled for all frequencies down to small fractions of a hertz. Fig. 20 shows a thermistor often used for regulating a generator, and its static characteristic. This thermistor proves suitable for generators with frequencies of the order of 100 cycles. Similar types of thermistors have been developed for lower frequencies and other values of resistance and power.
Where the high temperature sensitivity of the thermistor in the generator regulator proves disadvantageous, thermostating it together with a heater is used, or a compensating thermistor circuit similar to that shown in Fig. 21 is employed.
Automatic gain regulator in an amplifier
Since the resistance of thermistors of suitable design changes appreciably with the magnitude of the power dissipated in them or in heaters closely coupled to them, such thermistors
Fig. 20. a — A thermistor used for amplitude control. The length of the glass bulb is 38.1 mm. b — Steady-state characteristic obtained for the thermistor shown in Fig. a, used for amplitude control.
Fig. 21. Circuit in which, to compensate for the influence of ambient-temperature changes on the regulating thermistor, an auxiliary disk-shaped thermistor is used.
proved very valuable as automatic gain regulators (especially when used in amplifiers with negative feedback). A similar device is widely used in wire-communication circuits for regulating the transmission level[^12][^13][^14]. In one type of such device, a directly heated thermistor is inserted into the feedback circuit of an amplifier in such a way that the magnitude of the feedback voltage, by varying, compensates for any change of the signal at the output. To regulate changes in the overall losses caused by atmospheric conditions and other factors, the gain of each amplifier of the transmitting system is continuously adjusted by this device so that a time-constant transmission level is established from the beginning to the end of the channel. It is in precisely this way that gain is regulated in the widely used transmitting system of the K-2 type. In this system, changes in transmission losses caused by temperature fluctuations are not the same in different parts of the passband. At certain repeater stations along the transmission line, the losses are regulated by two additional thermistor gain regulators arranged so that the slope of the characteristic is proportional to frequency, while its convexity reaches a maximum at some definite frequency. These thermistors are heated by their own heaters, actuated by energy dependent on the amplitude of separate control signals that are specially sent for this purpose from the transmitting station.
In devices of this type, the thermistor responds both to a change in the ambient temperature and to the current passing through it. Where this is important, it is possible, by using a heated thermistor (similar to that shown in Fig. 21), to eliminate the response to changes in ambient temperature. The heater is included in an auxiliary circuit containing a thermistor that compensates for temperature changes. This circuit is arranged so that, for any value of the ambient temperature, the power supplied to the heater of the thermistor regulating the gain is just sufficient to regulate and maintain unchanged the temperature value in the vicinity of the gain-regulating thermistor.
In another interesting type of thermistor gain regulator, a heated thermistor is used with a heater controlled by the power at the amplifier output, and with a thermistor in the input circuit, as shown in Fig. 22. In such a device the feedback is thermal rather than electrical. A thermistor of this kind regulates the broadband transmitting system of the L-1 type. A current of the control frequency, and a current of this same frequency sent by a four-terminal network, are supplied to this system,
enters the regulator, actuates the thermistor heater, making smooth and continuous gain control possible.
Using indirectly heated thermistors with different characteristics, it is possible to protect the circuit with the load shown in Fig. 22 against overloads. The sensitivity and resistance of the thermistor are chosen for this purpose so that the thermistor forms a shunt with so high a resistance that its influence on the amplifier is negligible at any value of output power corresponding to normal operation. If, however, the output power exceeds the normal value, the thermistor heats up and its resistance decreases. It shunts the amplifier input and thus limits the output power. The choice of a thermistor with an appropriate value of the time constant makes it possible to delay the onset of the limiting effect for any time approximately from one second to several minutes.
Fig. 22. Circuit of “thermal” feedback for gain control. This circuit is also used as overload protection.
Regulators and Limiters
A group of related applications of thermistors is based on their nonlinear volt-ampere characteristic in the steady-state regime. Such applications include a voltage regulator, an audio-volume limiter, a compressor, and an expander. A compressor and an expander are devices for changing the amplitude range of signals. The purpose of a compressor is to reduce the amplitude range, whereas an expander increases it. In Fig. 23, curve 1 represents the static characteristic of a typical thermistor with negative slope to the right of the maximum voltage value. Curve 2 depicts the characteristic of an ohmic resistance \(R\) with a positive and constant value of the slope. Curve 3 corresponds to the characteristic obtained for a thermistor and an ohmic resistance connected in series. Over a considerable part of it, the voltage is almost independent of the current. This makes it possible to create a voltage regulator or limiter. If an ohmic resistance of larger value is used (curve 4), then its series connection with the thermistor gives, as a result, curve 5—the characteristic of a compressor. In these
applications the thermistor regulator is connected in parallel with the load resistance, so that in the circuit shown in Fig. 23,
\[ E = E_0 = E_1 - I R_{\mathrm{in}} . \tag{23} \]
Here \(E\) is the voltage drop across the thermistor and the ohmic resistance \(R\), \(E_0\) is the output voltage, and \(E_1\), \(I\), and \(R_{\mathrm{in}}\) are the input voltage, current, and resistance.
If the thermistor and the ohmic resistance associated with it are connected in series with the generator and the load resistance, then an expander is obtained and
\[ E_0 = E_1 - E . \tag{24} \]
When the ohmic resistance \(R\), connected in series with the thermistor, is increased, the amplitude range decreases, and conversely.
Fig. 23. Characteristics of a simple thermistor voltage regulator; circuit of a limiter or compressor.
Up to this point it has been assumed that the change of the operating point occurs slowly enough to follow the static characteristics. With a sufficiently rapid change the operating point departs from the curve of the static characteristic and tends to move along a line intersecting the static curve, corresponding to the ohmic resistance. Then, for sufficiently rapid oscillations, the regulating action may arise owing to changes in the value of the resistance, determined by the effective value of the power dissipated in the thermistor. In sound-volume limiters the thermistor is designed for such a reaction rate that it exerts a limiting effect on changes in sound volume of syllabic or lower frequency and does not reproduce more rapid
sound oscillations with the consequent, when they are present, change in the shape of the signal or nonlinear distortion. Volume limiters of this type can reconcile large changes in loudness with the absence of distortion of the signal shape $^{13,15}$.
Remote-control switches
Contactless switches and rheostats are a natural addition to the applications just considered. The thermistor is used as an element of the circuit that must be regulated, while the resistance of the thermistor depends on the energy dissipated in it directly or indirectly by the control circuit.
By taking advantage of the nonlinearity of the thermistor’s static volt-ampere characteristic, in some applications one can provide instantaneous and locking actions.
Manometer
Some interesting and useful applications—such as, for example, manometers, gas analyzers, instruments for measuring liquid level, thermal conductivity, and flow—which possess high sensitivity and a low operating temperature, are based on the dependence of the dissipation constant of a thermistor on the thermal conductivity of the medium in which it is located. As can be seen from Fig. 10, when this constant changes, the static characteristic is displaced relative to the coordinate axes. The undesirable sensitivity of the thermistor to changes in the ambient temperature in these applications can in many cases be eliminated or reduced by introducing into the measuring circuit a second thermistor with similar characteristics. The compensating thermistor is subjected to the same ambient temperature, but this action is shielded from the influence of the flow or gas pressure, which are the quantities being measured. The two thermistors are connected into adjacent arms of a Wheatstone bridge, which is balanced when the measured action is zero, and thrown out of balance when the effective thermal conductivity of the medium increases. In measurements of gas-flow velocity, the minimum value of the velocity that can be measured is limited (as in all instruments of this type) by the convection currents produced by the heated thermistor.
The manometer, which is the most typical of these applications, should be described in somewhat greater detail. The sensitive element of the thermistor manometer is a small bead 0.508 mm in diameter, suspended on two thin wire leads in a hollow sphere connected to the chamber in which it is necessary
measure the gas pressure. In Fig. 24 are shown the volt-ampere characteristics of a typical laboratory model of the manometer, taken in air at several values of the absolute pressure from \(10^{-6}\) mm Hg to atmospheric. The operating point is to the right of the peak of these curves. This thermistor and a similar one, but located in an evacuated bulb, are connected into adjacent arms of a Wheatstone bridge (the ratio of whose arms is equal to unity), as
Fig. 24. Characteristics of a typical thermistor manometer, illustrating the influence of gas pressure on the dependence of voltage on current and of resistance on power.
is shown in the diagram of Fig. 25. The calibration of such a manometer is also given there. The characteristic shifts when a gas is used whose thermal conductivity differs from the thermal conductivity of air. Such a manometer is considered the most suitable for pressure measurements from \(10^{-5}\) to \(10\) mm Hg. The lower limit of the measurable pressure is determined by such experimental conditions as, for example, the sensitivity of the measuring instrument or the ability of the instrument to maintain the zero setting for a sufficiently long time in the presence of changes in the applied voltage and in the temperature of the surroundings. The upper limit of pressure measurement is due to the fact that at high pressures the voltage that brings the bridge out of equilibrium depends ever more weakly on the pressure. This “saturation” is due to the fact that the mean free path of the free—
of the mean free path of the gas molecules becomes small in comparison with the distance between the bead thermistor and the inner surface of the manometer bulb, so that the influence of cooling becomes almost independent of pressure. The thermistor manometer has particular advantages when it is used in gases which can be thermally decomposed into their constituent parts. In this
Fig. 25. Operating circuit and calibration of a manometer using a thermistor whose characteristics are given in Fig. 24.
case the temperature of the thermistor must exceed the temperature of the surrounding medium by no more than 30°C. Under ordinary conditions, however, raising the temperature in vacuum to approximately 200°C permits measurements over a wide range of pressure values. Special models are also manufactured for use in corrosive gases. In these models only glass and a platinum alloy are exposed to the action of the gas.
Delay devices
In communications engineering, thermistors are used mainly for obtaining time delays. The physical basis for such an application is the slow increase of current after the application of voltage, considered in the first part of the article for the case
immediately heated thermistor. This type of operation is characterized by the power controlling the time delay.
By using a control power and circuit suited to the operating conditions, time delays from several milliseconds to several minutes can be obtained. Thermistors also have the advantage over other similar devices that they are small in size, light in weight, rugged, have a definitely long service life, and have no contacts, moving parts, or pneumatic control requiring careful handling.
Power-controlled delay thermistors are best suited for such applications where there is no need for close tolerances on time intervals. For some applications in the field of communications it is sufficient for the maximum and minimum time values to be in the ratio of 6/1 (as a result of simultaneous deviations from the nominal value of all the factors affecting the time delay, namely: the operating voltage by $\pm 5\%$, the operating current of the relay by $\pm 25\%$, the resistance of the relay by $\pm 5\%$, a change in ambient temperature from $20^\circ\mathrm{C}$ to $40^\circ\mathrm{C}$, and such variations in the thermistor as occur in specimens of one and the same type).
After a delay operation, the power-controlled delay thermistor must be allowed time to cool before the second operation begins. If this is not done, the second delay interval will be shorter than the first. The cooling time depends on the particular conditions in the circuit and on the construction of the thermistor, but, as a rule, is several times greater than the operating holding time. In telephone relay circuits which, soon after preliminary use, require a delay operation, the thermistor is usually connected in such a way that it is short-circuited by the relay contacts during the operating delay interval. This allows the thermistor to cool during the period when the relay is locked. If this period is sufficiently long, the thermistor becomes suitable for use as soon as the relay releases. Delay thermistors have operated in service-life tests more than half a million times, with the change in their delaying action being insignificant. In order to avoid the limitations associated with a power-controlled delay thermistor and to introduce wide limits of holding time and a long cooling period between operations, a slow-cooling method was used. In this arrangement two relays are used, and the thermistor is heated to a low resistance value by passing through it a comparatively strong current during a time interval short in comparison with the expected holding time. The current then automatically decreases to a smaller value, and
the thermistor cools until its resistance has increased enough to weaken the current still further and to switch off the operating relay. This part of the operating cycle accounts for most of the expected time interval. With this device the thermistor is ready for reuse immediately after the end of the delay interval, or after any part of it. With the proper choice of operating current and circuit parameters, possible changes in voltage and ambient temperature have little effect on the time interval. The principal variable remains the cooling time of the thermistor itself. For a given specimen it remains constant, but, as noted above, it may vary from specimen to specimen depending on the value of the dissipation constant and the heat capacity. In addition to delay devices, thermistors can be used in several similar applications. Current (voltage) surges can be prevented by means of operating relays or other apparatus sensitive to disturbances, by connecting a thermistor in series with the part of the circuit that is to be protected. In the case of a current surge, the high initial resistance of the thermistor reduces this surge to a small value, provided that it lasts long enough to overcome the thermal inertia of the thermistor. On the other hand, the normal operating voltage is applied for a time long enough to reduce the resistance of the thermistor to an insignificant value, so that the normal operating current will flow after a short interval. Thus the thermistor enables the circuit to distinguish accurately between an undesirable signal of short duration and a desired, longer signal, even when the undesirable voltage signal is several times larger than the desired one.
Generators, modulators, and amplifiers
A group of applications that have already been investigated in the laboratory, but are not yet used in engineering, includes generators, modulators, and amplifiers for the low- and audio-frequency range. If a thermistor operates in the steady state on a portion of the current-voltage characteristic having a negative slope, and if a small alternating voltage is then superposed on the direct voltage, a small alternating current will flow. If the thermistor has a small time constant $\tau$ and if the applied frequency is sufficiently low, then the alternating current-voltage characteristic will follow the steady-state curve, and the value
\[ \frac{dV}{dI} \]
will be negative. When the frequency of the applied alternating voltage increases, the magnitude of the negative resistance decreases. At some critical frequency $f_c$—
the resistance is equal to zero, and the current is shifted in phase relative to the voltage by \(90^\circ\). Near the frequency value \(f_c\), the thermistor behaves like an inductance whose magnitude is of the order of a henry. When the frequency exceeds the value \(f_c\), the resistance becomes positive and increases uniformly until it approaches the value corresponding to direct current, when the current and voltage are in phase with one another.
The critical frequency is approximately determined by the expression
\[ f_c=\frac{1}{2\tau}. \]
If \(\tau\) is a quantity of the order of \(5\cdot 10^{-5}\) sec, then the value of \(f_c\) becomes equal to 10,000 cps, and up to one half of this frequency value the thermistor has an approximately constant negative resistance. Point-contact thermistors having such or even higher values of the critical frequency are made in many laboratories. However, so far not one of them has been made sufficiently stable to find application in engineering. It has been shown theoretically and experimentally that any device with negative resistance can be used as a generator, modulator, and amplifier. In the future thermistors will apparently find application in this field as well.
REFERENCES CITED
- V. P. Zhuze and B. V. Kurchatov, Phys. Zeits. Sow. 2, 453 (1932).
- A. H. Wilson, Semiconductors and Metals, Cambridge, 1939.
- F. Seitz, The Modern Theory of Solids, Gostekhizdat, 1950.
- N. F. Mott and R. W. Gurney, Electronic Processes in Ionic Crystals, IL, 1950.
- M. Le-Blanc and H. Sachse, Phys. Zeits. 32, 887 (1931).
- W. Meyer, Zeits. f. Physik 85, 278 (1933).
- J. B. Johnson, Phys. Rev. 32, 97 (1928).
- C. J. Christensen and G. L. Pearson, Bell Syst. Techn. J. 15, 197 (1936).
- R. A. Gund, AIEE Trans. 64, 730 (1945).
- L. A. Meacham, Proc. IRE 26, 1278 (1938).
- R. L. Shepherd and R. O. Wise, Proc. IRE 31, 256 (1943).
- J. H. Bollman, Bell Labor. Record 20, No. 10, 258 (1942).
- J. E. Tweeddale, Western Electric Oscillator, December 1945, pp. 3 and 34.
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TRANSLATOR’S AFTERWORD
The work of Becker, Green, and Pearson represents the most complete of the existing surveys of the physical properties of thermistors and their diverse technical applications.
In analyzing the physical properties of thermistors, the authors proceed from the principles of band theory. A number of works by Soviet scientists (Vol’kenshtein, Davydov,
PROPERTIES AND APPLICATIONS OF THERMISTORS
Ioffe, Pekar, Frenkel, Adirovich, and others), but in the list of literature appended to the article only one of these works is indicated. In addition, attention is drawn to the fact that the authors of the translated article consider the band theory to be infallible, whereas Soviet scientists have revealed a number of shortcomings in it. First of all, the band theory applies only to crystalline bodies, and the idea of electronic bands becomes doubtful in the transition to liquid and amorphous bodies. But among semiconductors there are also amorphous and liquid compounds possessing properties common to all semiconductors, although the assumptions of the band theory do not extend to them.
It should also be noted that, according to the band theory, the role of thermal treatment is reduced to the creation of local levels characterizing various defects in the structure of the crystal; but the band theory is not able to explain the dependence of the magnitude of the conductivity of a thermistor on the thermal-treatment regime (for example, on the temperature and duration of annealing).
Finally, the band theory is as yet unable to explain the existence of semiconductors with a metallic type of conductivity.
The second part of the work is more valuable, in which various technical applications of thermistors are considered. In our periodical and book literature such applications of thermistors as manometers, gas analyzers, instruments for measuring flow velocity, etc., have not yet been described.
In the translated article these kinds of applications are also considered, and the corresponding circuits and characteristics are given.
It seems to us that these applications of thermistors deserve serious attention, and their development should be continued.
In the second part, too, not a single one of the works of our scientists devoted to the use of thermistors in technology is noted.
It should be added that at the present time thermistors are used as amplifiers not only in the form of laboratory models: a technology has been developed for manufacturing so-called transistors—crystalline amplifiers made from germanium, which is a semiconducting element25, 26.
In conclusion we give a brief list of the principal works of Soviet scientists in this field:
- F. F. Volkenshtein, Electrical Conductivity of Semiconductors, Gostekhizdat (1947).
- F. F. Volkenshtein, Electronic Processes in Real Crystals, Uspekhi Fizicheskikh Nauk 28, issue 4 (1946).
- Ya. I. Frenkel, Contemporary Theory of Metallic Bodies, Uspekhi Fizicheskikh Nauk 30, issues 1–2 (1946).
- S. I. Pekar, A New View of the Electronic Conductivity of Ionic Crystals, ZhETF 18, issue 2 (1948).
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G. I. Skanavi, “Dielectric polarization and losses in solid inorganic dielectrics,” Elektrichestvo, No. 8 (1947).
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B. T. Kolomiets, “Thermistors,” Elektrichestvo, No. 3 (1947).
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S. I. Pekar, “The crystal as a many-electron problem and the one-electron approximation,” ZhETF 18, no. 6 (1948).
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L. D. Landau and S. I. Pekar, “Effective mass of a polaron,” ZhETF 18, no. 5 (1948).
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Ya. I. Frenkel, “Theory of electrical contacts between metals,” ZhETF 16, no. 4 (1946).
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V. A. Arkhangel’skaya and A. M. Bonch-Bruevich, “Change in the conductivity of cadmium sulfide upon irradiation by electrons,” DAN 77, No. 2 (1951).
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F. F. Vol’kenshtein, “Zone theory of the solid state and the limits of its applicability,” UFN 43, no. 1 (1951).
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Ya. I. Frenkel, “Modern conceptions of the nature of dielectrics,” Elektrichestvo, No. 8 (1947).
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S. I. Pekar, “Autolocalization of an electron in a dielectric inertial-polarizing medium,” ZhETF 16, no. 4 (1946).
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S. I. Pekar, “Local quantum states of an electron in an ideal ionic crystal,” ZhETF 16, no. 4 (1946).
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B. I. Davydov and I. Shmushkevich, “Theory of electronic semiconductors,” UFN 24, no. 1 (1940).
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V. P. Zhuze and S. N. Ryvkin, “The ‘exciton’ character of light absorption and impurity photoconductivity,” DAN 77, No. 2 (1951).
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A. F. Ioffe, Semiconductors in Modern Physics and Technology, Elektrichestvo, No. 6 (1939).
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Ya. M. Ksendzov, “Electrical properties of TiO₂,” ZhTF 20, no. 1 (1950).
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I. D. Kononenko, “Modern bolometers,” ZhTF 20, no. 6 (1950).
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B. I. Boltaks, “On the dependence of the coefficient of thermoelectromotive force in semiconductors,” ZhTF 20, no. 9 (1950).
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V. I. Pruzhinina-Granovskaya, “Ceramic rheostats and resistors,” ZhTF 14, nos. 7–8 (1944).
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V. I. Pruzhinina-Granovskaya, “A new type of linear volume resistors, ‘Defar,’” Elektrichestvo, No. 11 (1946).
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M. V. Savost’yanova, “The smallest metal particles inside a crystal lattice,” UFN 22, no. 1 (1939).
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B. I. Davydov, “On the contact resistance of semiconductors,” ZhETF 9, no. 4 (1939).
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V. S. Vavilov, “Amplification of high-frequency currents by crystalline germanium triodes,” UFN 40, no. 1 (1950).
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V. S. Vavilov, “Operation of crystalline triodes in the frequency range 5–25 megacycles,” UFN 42, no. 3 (1950).