What is the temperature coefficient of resistivity. Temperature coefficient of electrical resistance for copper. Temperature coefficient of electrical resistance. Alloy resistivity

TKS is a value characterizing the relative change in the resistance of the resistor when the temperature changes by one degree. TKS characterizes reversible changes in the resistance of the resistor due to changes in ambient temperature or changes in the electrical load on the resistor. A change in the resistance of the resistor under the influence of external influences (temperature, load, etc.) leads to changes in the parameters of electrical circuits, and in critical cases, to their breakdown. Therefore, changes in the resistor resistance value must be taken into account when constructing electrical circuits.

In practice, they use the average TCR value, which is determined in the operating temperature range at a given electrical load of the resistor using a TCR meter, or by measuring three values of resistor resistance at normal temperature (+20°C) and at extreme temperatures (maximum positive temperature and minimum negative temperature). Based on the measured resistance values of the resistor, TCR is determined using the following formula

Where TKS temperature coefficient of resistor resistance when the temperature changes by 1 / °C;

algebraic difference between the resistor resistance measured at given positive and negative temperatures and the resistor resistance measured at normal temperature (+ 20 ° C);

R resistor resistance measured at normal (+20°C) temperature;

algebraic difference between a given positive and a given negative temperature and normal (+20°C) temperature.

Description of laboratory work and measuring stand

The test object used in this work is inductive-resistive voltage dividers, the diagram of which is shown in Fig. 8.

The functional diagram of the measuring stand is shown in Fig. 9.

The following equipment is used to carry out measurements:

Gi pulse generator (type G5-54);

Gn low frequency generator (type GZ-112, GZ-118);

OS oscilloscope (type S1-65);

V1, V2 voltmeter (type VZ-38);

PC switch (type PG-5P2N);

thermostat (SNOL type);

Bl. 1 block of resistors and inductances, consisting of the following elements:

MLT 1.1 kOhm ±1%;

BC 5.1 kOhm + 1%;

MLT 10 kOhm ±1%;

MLT 51 kOhm ±5%;

MLT 100 kOhm ±5%;

MLT 75 kOhm ± 5%;

MLT 1.1 kOhm±5%;

Bl. 2 block of resistors, consisting of the following elements:

MLT 100 Ohm ± 5%;

MLT 10 kOhm ±5%;

MLT 1.1 kOhm ±5%.

Rice. 8. Diagram of inductive-resistive voltage dividers

Rice. 9. Functional diagram of the measuring stand.

Preparing for measurements.

Measurements are carried out in a laboratory under normal climatic conditions in accordance with GOST 11478-75.

ATTENTION! Before starting measurements, you must familiarize yourself with the safety rules when working with devices. It is also necessary to familiarize yourself with the descriptions of measuring instruments and these guidelines. It is necessary to check that all instruments included in the measuring installation are turned on, and it is also necessary to check that the measuring instruments and the laboratory stand are grounded. In addition, it is necessary to assemble a stand diagram in accordance with Fig. 9. It is necessary to place the control knobs of the measuring instruments in a position in which there is no signal at the input of the inductive-resistor dividers and no supply voltage. After which it is necessary to turn on all measuring instruments and allow them to warm up for at least 15 minutes. Then it is necessary to adjust the measuring instruments in accordance with the operating instructions.

Free electron concentration n in a metal conductor with increasing temperature remains practically unchanged, but their average speed of thermal movement increases. The vibrations of the crystal lattice nodes also increase. The quantum of elastic vibrations of the medium is usually called phonon. Small thermal vibrations of the crystal lattice can be considered as a collection of phonons. With increasing temperature, the amplitudes of thermal vibrations of atoms increase, i.e. the cross section of the spherical volume occupied by the vibrating atom increases.

Thus, as the temperature increases, more and more obstacles appear in the path of electron drift under the influence of an electric field. This leads to the fact that the average free path of an electron λ decreases, the mobility of electrons decreases and, as a consequence, the conductivity of metals decreases and the resistivity increases (Fig. 3.3). The change in the resistivity of a conductor when its temperature changes by 3K, related to the resistivity value of this conductor at a given temperature, is called the temperature coefficient of resistivity TK ρ or . The temperature coefficient of resistivity is measured in K -3. The temperature coefficient of resistivity of metals is positive. As follows from the definition given above, the differential expression for TK ρ has the form:

(3.9)

According to the conclusions of the electronic theory of metals, the values of pure metals in the solid state should be close to the temperature coefficient (TK) of expansion of ideal gases, i.e. 3: 273 = 0.0037. In fact, most metals have ≈ 0.004. Some metals have higher values, including ferromagnetic metals - iron, nickel and cobalt.

Note that for each temperature there is a temperature coefficient TK ρ. In practice, for a certain temperature range, the average value is used TK ρ or :

, (3.10)

Where ρ3 And ρ2- resistivity of conductor material at temperatures T3 And T2 respectively (in this case T2 > T3); there is a so-called average temperature coefficient of resistivity of this material in the temperature range from T3 to T2.

In this case, when the temperature changes in a narrow range from T3 to T2 accept a piecewise linear approximation of the dependence ρ(T):

(3.11)

Reference books on electrical materials usually give values at 20 0 C.

Fig.3.1 Dependence of resistivity ρ metal conductors depending on temperature T. Jump ρ (branch 5) corresponds to the melting point T PL.

Fig.3.2. Dependence of copper resistivity on temperature. The jump corresponds to the melting temperature of copper 1083 0 C.

As follows from formula (3.33), the resistivity of conductors depends linearly on temperature (branch 4 in Fig. 3.3), with the exception of low temperatures and temperatures above the melting point T>T PL.

As the temperature approaches 0 0 K, an ideal metal conductor has a resistivity ρ tends to 0 (branch 3). For technically pure conductors (with a very small amount of impurities) over a small area of several kelvins, the value ρ ceases to depend on temperature and becomes constant (branch 2). It is called “residual” resistivity ρ OST. Magnitude ρ OST determined only by impurities. The purer the metal, the less ρ OST .

Near absolute zero, another dependence is possible ρ on temperature, namely, at a certain temperature T S resistivity ρ drops abruptly to almost zero (branch 3). This state is called superconductivity, and conductors with this property are called superconductors. The phenomenon of superconductivity will be discussed below in 3.3.

Example 3. 6. The temperature coefficient of resistivity of copper at room temperature is 4.3 30-3 -3 K. Determine how many times the electron free path will change when the copper conductor is heated from 300 to 3000 K.

Solution. The electron mean free path is inversely proportional to the resistivity. Therefore, by how many times the resistivity of copper increases when heated, by how many times the electron free path will decrease. The resistivity of copper will increase several times. Consequently, the electron free path will decrease by 3 times.

Change in the resistivity of metals during melting.

When metals transition from solid to liquid, most of them experience an increase in resistivity ρ , as shown in Fig. 3.3 (branch 5). Table 3.2 shows the values showing the relative change in the resistivity of various metals during melting. The resistivity increases during melting for those metals (Hg, Au, Zn, Sn, Na) that increase in volume during melting, i.e. reduce density. However, some metals, such as gallium (Ga) and bismuth (Bi), reduce ρ 0.58 and 0.43 times, respectively. For most metals in the molten state, the resistivity increases with increasing temperature (branch 6 in Fig. 3.3), which is associated with an increase in their volume and a decrease in density.

Table 3.2. Relative change in resistivity of various metals during melting.

Change in resistivity of metals during deformation.

Change ρ during elastic deformations of metal conductors is explained by a change in the amplitude of vibrations of the nodes of the metal crystal lattice. When stretched, these amplitudes increase, and when compressed, they decrease. An increase in the amplitude of oscillations of nodes leads to a decrease in the mobility of charge carriers and, as a consequence, to an increase in ρ.

A decrease in the oscillation amplitude, on the contrary, leads to a decrease in ρ. However, even significant plastic deformation, as a rule, increases the resistivity of metals due to distortion of the crystal lattice by no more than 4-6%. The exception is tungsten (W), ρ which increases by tens of percent with significant compression. In connection with the above, it is possible to use plastic deformation and the resulting hardening to increase the strength of conductor materials without compromising their electrical properties. During recrystallization, the resistivity can be reduced again to its original value.

Specific resistance of alloys.

As already indicated, impurities disrupt the correct structure of metals, which leads to an increase in their resistivity. Figure 3.3 shows the dependence of resistivity ρ and conductivity γ copper concentration N various impurities in fractions of a percent. We emphasize that any alloying leads to an increase in the electrical resistivity of the alloyed metal compared to the alloyed one. This also applies to cases when a metal with a lower ρ. For example, when alloying copper with silver ρ there will be more copper-silver alloy than ρ copper, despite the fact that ρ less silver than ρ copper, as can be seen from Fig. 3.3.

Fig.3.3. Resistivity Dependence ρ and conductivity γ copper from the content of impurities.

Significant increase ρ observed when two metals are fused if they form with each other solid solution, in which atoms of one metal enter the crystal lattice of another. Curve ρ has a maximum corresponding to a certain specific ratio between the content of components in the alloy. Such a change ρ from the content of alloy components can be explained by the fact that due to its more complex structure compared to pure metals, the alloy can no longer be likened to a classical metal.

The change in the specific conductivity of the γ alloy in this case is caused not only by a change in the mobility of carriers, but in some cases also by a partial increase in the concentration of carriers with increasing temperature. An alloy in which the decrease in mobility with increasing temperature is compensated by an increase in carrier concentration will have a zero temperature coefficient of resistivity. As an example, Fig. 3.4 shows the dependence of the resistivity of a copper-nickel alloy on the composition of the alloy.

Heat capacity, thermal conductivity and heat of fusion of conductors.

Heat capacity characterizes the ability of a substance to absorb heat Q when heated. Heat capacity WITH of any physical body is a value equal to the amount of thermal energy absorbed by this body when it is heated by 3K without changing its phase state. Heat capacity is measured in J/K. The heat capacity of metallic materials increases with increasing temperature. Therefore, the heat capacity WITH determined with an infinitesimal change in its state:

Fig.3.4. Dependence of resistivity of copper-nickel alloys on composition (in percent by weight).

Heat capacity ratio WITH to body weight m called specific heat capacity With:

Specific heat capacity is measured in J/(kg? K). The values of the specific heat capacity of metals are given in table. 3.3. As can be seen from Table 3.3, refractory materials are characterized by low specific heat capacity values. So, for example, for tungsten (W) With=238, and for molybdenum (Mo) With=264J/(kg?K). Low-melting materials, on the contrary, are characterized by a high specific heat capacity. For example, aluminum (Al) With=922, and for magnesium (Mg) With=3040J/(kg? K). Copper has a specific heat capacity c = 385 J/(kg? K). For metal alloys, the specific heat capacity is in the range of 300-2000 J/(kg? K). C is an important characteristic of metal.

Thermal conductivity call the transfer of thermal energy Q in an unevenly heated medium as a result of thermal movement and interaction of its constituent particles. The transfer of heat in any environment or any body occurs from hotter parts to cold ones. As a result of heat transfer, the temperature of the environment or body is equalized. In metals, thermal energy is transferred by conduction electrons. The number of free electrons per unit volume of metal is very large. Therefore, as a rule, the thermal conductivity of metals is much greater than the thermal conductivity of dielectrics. The fewer impurities metals contain, the higher their thermal conductivity. As impurities increase, their thermal conductivity decreases.

As is known, the process of heat transfer is described by Fourier's law:

. (3.14)

Here is the heat flux density, i.e. the amount of heat passing along the coordinate x through a unit of cross-sectional area per unit of time, J/m 2?s,

Temperature gradient along the coordinate x, K/m,

The proportionality coefficient, called the thermal conductivity coefficient (previously designated), W/K?m.

Thus, the term thermal conductivity corresponds to two concepts: this is the process of heat transfer and the proportionality coefficient that characterizes this process.

So, free electrons in a metal determine both its electrical and thermal conductivity. The higher the electrical conductivity γ of a metal, the greater its thermal conductivity should be. With increasing temperature, when the mobility of electrons in the metal and, accordingly, its specific conductivity γ decrease, the ratio /γ of the thermal conductivity of the metal to its specific conductivity should increase. Mathematically this is expressed Wiedemann-Franz-Lorenz law

/γ = L 0 T, (3.15)

Where T- thermodynamic temperature, K,

L 0 - Lorentz number, equal

L 0 = . (3.16)

Substituting the values of the Boltzmann constant into this expression k= J/K and electron charge e= 3.602?30 -39 Cl we get L 0 = /

The Wiedemann-Franz-Lorentz law is satisfied in the temperature range close to normal or slightly elevated for most metals (the exceptions are manganese and beryllium). According to this law, metals that have high electrical conductivity also have high thermal conductivity.

Temperature and heat of fusion. The heat absorbed by a solid crystalline body during its transition from one phase to another is called the heat of phase transition. In particular, the heat absorbed by a crystalline solid during its transition from solid to liquid is called heat of fusion and the temperature at which melting occurs (at constant pressure) is called melting point and denote T PL.. The amount of heat that must be supplied per unit mass of a solid crystalline body at temperature T PL to convert it into a liquid state is called specific heat of fusion r PL and is measured in MJ/kg or kJ/kg. The values of the specific heat of fusion for a number of metals are given in Table 3.3.

Table.3. 3. Specific heat of fusion of some metals.

Depending on the melting point, refractory metals are distinguished, having a melting point higher than that of iron, i.e. higher than 3539 0 C and low-melting with a melting point less than 500 0 C. The temperature range from 500 0 C to 3539 0 C refers to the average melting point values.

The work function of an electron leaving a metal.

Experience shows that free electrons practically do not leave the metal at ordinary temperatures. This is due to the fact that a holding electric field is created in the surface layer of the metal. This electric field can be thought of as a potential barrier that prevents electrons from escaping from the metal into the surrounding vacuum.

A holding potential barrier is created for two reasons. Firstly, due to the attractive forces from the excess positive charge that arose in the metal as a result of electrons escaping from it, and, secondly, due to the repulsive forces from the previously emitted electrons, which formed an electron cloud near the surface of the metal. This electron cloud, together with the outer layer of positive lattice ions, forms an electric double layer, the electric field of which is similar to that of a parallel-plate capacitor. The thickness of this layer is equal to several interatomic distances (30 -30 -30 -9 m).

It does not create an electric field in external space, but creates a potential barrier that prevents free electrons from escaping from the metal. The work function of an electron leaving a metal is the work done to overcome the potential barrier at the metal-vacuum interface. In order for an electron to fly out of a metal, it must have a certain energy sufficient to overcome the attractive forces of positive charges in the metal and the repulsive forces of electrons previously emitted from the metal. This energy is denoted by the letter A and is called the work function of an electron leaving the metal. The work function is determined by the formula:

Where e- electron charge, K;

Output potential, V.

Based on the foregoing, we can assume that the entire volume of the metal for conduction electrons represents a potential well with a flat bottom, the depth of which is equal to the work function A. The work function is expressed in electron volts (eV). The electron work function values for metals are given in Table 3.3.

If you impart energy to the electrons in the metal sufficient to overcome the work function, then some of the electrons may leave the metal. This phenomenon of metal emitting electrons is called electronic emissions. To obtain free electrons in electronic devices there is a special metal electrode - cathode.

Depending on the method of transmitting energy to the electrons of the cathode, the following types of electron emission are distinguished:

- thermionic, in which additional energy is imparted to electrons as a result of heating the cathode;

- photoelectronic, in which the cathode surface is exposed to electromagnetic radiation;

- secondary electronic, which is the result of bombardment of the cathode by a stream of electrons or ions moving at high speed;

- electrostatic, in which a strong electric field at the surface of the cathode creates forces that promote the escape of electrons beyond its limits.

The phenomenon of thermionic emission is used in vacuum tubes, X-ray tubes, electron microscopes, etc.

Thermoelectromotive force (thermo-emf).

When two different metal conductors A and B (or semiconductors) come into contact (Fig. 3.5), a contact potential difference, which is due to the difference in the work function of electrons from different metals. In addition, the electron concentrations of different metals and alloys may also be different.

In this case, electrons from metal A, where their concentration is higher, will move to metal B, where their concentration is lower. As a result, metal A will have a positive charge, and metal B will have a negative charge. In accordance with the electronic theory of metals, the contact potential difference or EMF between conductors A and B is equal to (Fig. 3.5):

(3.17)

Where U A And U B— potentials of contacting metals; n A And n B- electron concentrations in metals A and B; k- Boltzmann constant, e- electron charge, T- thermodynamic temperature. If the electron concentration is greater in metal B, then the potential difference will change sign, since the logarithm of a number less than one will be negative. The contact potential difference can be measured experimentally. The first such measurements were carried out in 3797 by the Italian physicist A. Volta, who discovered this phenomenon.

Fig.3.5. The formation of a contact potential difference or EMF between two different conductors A and B.

It goes without saying that if two conductors A and B form a closed circuit (Fig. 3.6) and the temperatures of both contacts are the same, then the sum of the potential differences or the resulting emf is zero.

(3.18)

If one of the contacts or, as they are called, “junctions” of two metals has a temperature T3, and the other - temperature T2. In this case, a thermo-EMF arises between the junctions equal to

(3.19)

Where - constant thermo-EMF coefficient for a given pair of conductors, measured in μV/K. It depends on the absolute value of the temperatures of the “hot” and “cold” contacts, as well as on the nature of the contacting materials. As can be seen from formula (3.39), the thermo-EMF should be proportional to the temperature difference between the junctions.

Fig3.6. Thermocouple diagram.

The dependence of thermo-EMF on the junction temperature difference may not always be strictly linear. Therefore the coefficient with T must be adjusted according to temperature values T 3 And T 2.

A system of two wires isolated from each other, made of different metals or alloys, soldered in two places is called thermocouple. It is used to measure temperatures. The temperature of one junction (cold) is usually known, and the second junction is placed in the place whose temperature they want to measure. A measuring instrument, for example a millivoltmeter, is connected to the thermocouple mV, graduated in degrees Celsius or degrees Kelvin (Fig. 3.6).

In some cases, a control relay or solenoid coil is connected to the ends of the thermocouple (Fig. 3.7). When a certain temperature difference is reached, under the influence of thermoEMF, a current begins to flow through the relay coil P, causing the relay to operate or the valve to open using a solenoid. Examples of the most common thermocouples, their temperature ranges and applications are given below on pages 325-330.

Fig.4

Fig.3.7. Connection diagram of a thermocouple to a relay in an automatic control circuit

Thermo-EMF can be useful in some cases, but harmful in others. For example, when measuring temperature with thermocouples, it is useful. It is harmful in measuring instruments and reference resistors. Here they strive to use materials and alloys with the lowest possible thermo-EMF coefficient relative to copper.

Example 3.7. The thermocouple was calibrated at the cold junction temperature T 0 =0 o C. Calibration data is given in table 3.4

Table 3.4

Thermocouple calibration data

T, o C
Thermo-EMF, mV	0,0	0,33	0,65	3,44	2,33	3,25	4.23	5,24	6,27	7,34	8,47	9,63

This thermocouple was used to measure the temperature in the furnace. The temperature of the cold junction of the thermocouple during the measurement was 300 o C. The voltmeter during the measurement showed a voltage of 7.82 mV. Using the calibration table, determine the temperature in the oven.

Solution. If the temperature of the cold junction during measurement does not correspond to the calibration conditions, then you need to apply the law of intermediate temperatures, which is written as follows:

The junction temperatures are indicated in parentheses. The found thermo-EMF corresponds, in accordance with the calibration table, to the temperature in the furnace T= 900 o C.

Temperature coefficient of linear expansion of conductors(TCLR). This coefficient, designated shows the relative change in the linear dimensions of the conductor, and in particular its length, depending on temperature:

It is measured in K-3. Figure 3.8 shows the elongation of rods 3 m long, made of various materials, with increasing temperature,

Fig.3.8. Dependence of the elongation of a rod 1 m long on the temperature of the material.

It should be borne in mind that if the resistor is made of wire, then when it is heated, the length of the wire and its radius increase in proportion to its temperature. The cross-section increases in proportion to the square of the linear dimensions, i.e. proportional to the square of the radius. This means that as the linear dimensions of the wire increase when heated, the resistance of this wire decreases. Thus, when a wire is heated, the value of its resistance is influenced by two factors acting in opposite directions: an increase in resistivity ρ and an increase in the cross-section of the wire.

Due to the above, the temperature coefficient of the electrical resistance of the wire will be equal to:

Load expansion joints will not be able to compensate for such an extension. In this case, the adjustment of the contact network will be disrupted, the sag will increase, and the conditions for normal current collection will not be met. Under these conditions, it is impossible to ensure high train speeds and there will be a real threat of breakdown of current collectors.

In order to prevent such a development of events, the heating temperature of the wires should be limited to the value permissible under the conditions for ensuring normal operating conditions for this contact network design. If the temperature rises above this permissible value, the traction load must be limited.

In addition, the length of the anchor sections should be limited so that the length of the wire does not exceed 800 m. In this case, when the temperature of the contact wire increases by 300 0 C, the elongation will not exceed 3.4 m, which is quite acceptable under the conditions of compensation for the elongation of the traction suspension. If we take the minimum temperature as -40 0 C, then the maximum temperature of the contact wire should not exceed 60 0 C (in some designs 50 0 C).

When creating electric vacuum devices, it is necessary to select metal conductors in such a way that their TCLE is approximately the same as that of vacuum glass or vacuum ceramics. Otherwise, thermal shocks may occur, leading to the destruction of vacuum devices.

Mechanical properties of conductors characterized by tensile strength and elongation at break Δ l/l as well as fragility and hardness. These properties depend on mechanical and thermal treatment, as well as on the presence of alloying agents and impurities in the conductors. In addition, the tensile strength depends on the temperature of the metal and the duration of the tensile force.

As noted above, to compensate for the linear expansion of contact wires, their tension is carried out by temperature compensators with weights creating a tension of 30 kN (3 t). This tension ensures normal current collection conditions. The greater the tension, the more elastic the suspension will be and the better the conditions for current collection. However, the permissible tension depends on the tensile strength, which decreases with increasing temperature.

For hard-drawn copper, from which contact wires are made, a sharp decrease in tensile strength occurs at temperatures above 200 0 C. Temporary tensile strength also decreases with increasing duration of exposure to high temperatures. Time until metal fracture depending on its absolute temperature T(K) and design features and manufacturing technology are determined by the formula:

. (3.22)

Here: C 3 and C 2 are thermal resistance coefficients, depending on the design and properties of the metals. Figure 3.9 shows the dependence of the time to destruction on temperature, expressed in degrees Celsius, for wires made of different metals.

Thus, when increasing the tension of the contact wire in order to increase the elasticity of the suspension, the strength of the contact wire should also be taken into account in accordance with Fig. 3.9.

Fig.3. 9. Dependence of time before metal rupture on temperature and type of wire. 1 - aluminum and stranded steel-aluminum; 2 - copper contact; 3 - stranded steel-copper bimetallic; 4 - bronze heat-resistant contact.

Probably everyone knows. In any case, we have heard about him. The essence of this effect is that at minus 273 °C the conductor’s resistance to the flowing current disappears. This example alone is enough to understand that there is a dependence on temperature. A describes a special parameter - the temperature coefficient of resistance.

Any conductor prevents current from flowing through it. This resistance is different for each conductive material; it is determined by many factors inherent in a particular material, but this will not be discussed further. Of interest at the moment is its dependence on temperature and the nature of this dependence.

Metals usually act as conductors of electric current; their resistance increases as the temperature increases, and decreases as the temperature decreases. The magnitude of such a change per 1 °C is called the temperature coefficient of resistance, or TCR for short.

The TCS value can be positive or negative. If it is positive, then it increases with increasing temperature; if it is negative, then it decreases. For most metals used as conductors of electric current, the TCR is positive. One of the best conductors is copper; the temperature coefficient of resistance of copper is not exactly the best, but compared to other conductors, it is less. You just need to remember that the TCR value determines what the resistance value will be when the environmental parameters change. The greater this coefficient, the more significant its change will be.

This temperature dependence of resistance must be taken into account when designing electronic equipment. The fact is that the equipment must operate under any environmental conditions; the same cars are operated from minus 40 °C to plus 80 °C. But there are a lot of electronics in a car, and if you do not take into account the influence of the environment on the operation of circuit elements, you may encounter a situation where the electronic unit works perfectly under normal conditions, but refuses to work when exposed to low or high temperatures.

It is this dependence on environmental conditions that equipment developers take into account when designing it, using the temperature coefficient of resistance when calculating circuit parameters. There are tables with TCR data for the materials used and calculation formulas, according to which, knowing the TCR, you can determine the resistance value under any conditions and take into account its possible change in the operating modes of the circuit. But to understand TKS, now neither formulas nor tables are needed.

It should be noted that there are metals with a very small TCR value, and they are used in the manufacture of resistors, the parameters of which are weakly dependent on environmental changes.

The temperature coefficient of resistance can be used not only to take into account the influence of fluctuations in environmental parameters, but also for which, knowing the material that was exposed, it is enough to use the tables to determine what temperature the measured resistance corresponds to. An ordinary copper wire can be used as such a meter, although you will have to use a lot of it and wind it in the form of, for example, a coil.

All of the above does not fully cover all issues of using the temperature coefficient of resistance. There are very interesting application possibilities associated with this coefficient in semiconductors and electrolytes, but what is presented is sufficient to understand the concept of TCS.

Conductor resistance (R) (resistivity) () depends on temperature. This dependence for minor changes in temperature () is presented as a function:

where is the resistivity of the conductor at a temperature of 0 o C; - temperature coefficient of resistance.

DEFINITION

Temperature coefficient of electrical resistance() is a physical quantity equal to the relative increment (R) of a circuit section (or resistivity of the medium ()), which occurs when the conductor is heated by 1 o C. Mathematically, the definition of the temperature coefficient of resistance can be represented as:

The value characterizes the relationship between electrical resistance and temperature.

At temperatures within the range, for most metals the coefficient under consideration remains constant. For pure metals, the temperature coefficient of resistance is often taken to be

Sometimes they talk about the average temperature coefficient of resistance, defining it as:

where is the average value of the temperature coefficient in a given temperature range ().

Temperature coefficient of resistance for different substances

Most metals have a temperature coefficient of resistance greater than zero. This means that the resistance of metals increases with increasing temperature. This occurs as a result of electron scattering on the crystal lattice, which enhances thermal vibrations.

At temperatures close to absolute zero (-273 o C), the resistance of a large number of metals sharply drops to zero. Metals are said to go into a superconducting state.

Semiconductors that do not have impurities have a negative temperature coefficient of resistance. Their resistance decreases with increasing temperature. This occurs due to the fact that the number of electrons that move into the conduction band increases, which means that the number of holes per unit volume of the semiconductor increases.

Electrolyte solutions have. The resistance of electrolytes decreases with increasing temperature. This occurs because the increase in the number of free ions as a result of the dissociation of molecules exceeds the increase in the scattering of ions as a result of collisions with solvent molecules. It must be said that the temperature coefficient of resistance for electrolytes is a constant value only in a small temperature range.

Units of measurement

The basic SI unit for measuring the temperature coefficient of resistance is:

Examples of problem solving

Exercise

An incandescent lamp with a tungsten spiral is connected to a network with voltage B, current A flows through it. What will be the temperature of the spiral if at a temperature o C it has a resistance Ohm? Temperature coefficient of resistance of tungsten

Solution

As a basis for solving the problem, we use the formula for the dependence of resistance on temperature of the form:

where is the resistance of the tungsten filament at a temperature of 0 o C. Expressing from expression (1.1), we have:

According to Ohm's law, for a section of the circuit we have:

Let's calculate

Let's write the equation connecting resistance and temperature:

Let's carry out the calculations:

Answer

Metal		-1


Aluminum
Iron (steel)


Constantan
Manganin

Current Density

An insulated copper wire with a cross-section of 4 mm² carries a maximum permissible current of 38 A (see table). What is the permissible current density? What are the permissible current densities for copper wires with cross-sections of 1, 10 and 16 mm²?

1). Allowable current density

J = 70 A / 10 mm² = 7.0 A/mm²

current? (J = 2.5 A/mm²).

Temperature coefficient of electrical resistance, TKS- a value or set of values expressing the dependence of electrical resistance on temperature.

The dependence of resistance on temperature can be of a different nature, which can be expressed in the general case by some function. This function can be expressed through the dimensional constant , where is a certain specified temperature, and a dimensionless temperature-dependent coefficient of the form:

In this definition, it turns out that the coefficient depends only on the properties of the medium and does not depend on the absolute value of the resistance of the measured object (determined by its geometric dimensions).

If the temperature dependence (in a certain temperature range) is sufficiently smooth, it can be fairly well approximated by a polynomial of the form:

The coefficients at the powers of the polynomial are called temperature coefficients of resistance. Thus, the temperature dependence will have the form (for brevity we denote it as):

and, if we take into account that the coefficients depend only on the material, the resistivity can also be expressed:

Where

The coefficients have the dimensions of Kelvin, or Celsius, or another temperature unit to the same degree, but with a minus sign. The temperature coefficient of resistance of the first degree characterizes the linear dependence of electrical resistance on temperature and is measured in kelvins minus the first degree (K⁻¹). The temperature coefficient of the second degree is quadratic and is measured in kelvins minus the second degree (K⁻²). The coefficients of higher degrees are expressed similarly.

So, for example, for a platinum temperature sensor of the Pt100 type, the method for calculating resistance looks like

that is, for temperatures above 0°C the coefficients are used α₁=3.9803·10⁻³ K⁻¹, α₂=−5.775·10⁻⁷ K⁻² at T₀=0°C (273.15 K), and for temperatures below 0°C, α₃=4.183·10⁻⁹ K⁻³ and α₄=−4.183·10⁻¹² K⁻⁴ are added.

Although several powers are used for accurate calculations, in most practical cases one linear coefficient is sufficient, and this is usually what is meant by TCS. Thus, for example, a positive TCR means an increase in resistance with increasing temperature, and a negative TCR means a decrease.

The main reasons for changes in electrical resistance are changes in the concentration of charge carriers in the medium and their mobility.

Materials with high TCR are used in temperature-sensitive circuits as part of thermistors and bridge circuits made from them. For precise temperature changes, thermistors based on

Thus, as the temperature increases, more and more obstacles appear in the path of electron drift under the influence of an electric field. This leads to the fact that the average free path of an electron λ decreases, the mobility of electrons decreases and, as a consequence, the conductivity of metals decreases and the resistivity increases (Fig. 3.3). The change in the resistivity of a conductor when its temperature changes by 3K, related to the resistivity value of this conductor at a given temperature, is called the temperature coefficient of resistivity TK ρ or. The temperature coefficient of resistivity is measured in K -3. The temperature coefficient of resistivity of metals is positive. As follows from the definition given above, the differential expression for TK ρ has the form:

(3.9)

According to the conclusions of the electronic theory of metals, the values of pure metals in the solid state should be close to the temperature coefficient (TK) of expansion of ideal gases, i.e. 3: 273 =0.0037. In fact, most metals have ≈ 0.004. Some metals have higher values, including ferromagnetic metals - iron, nickel and cobalt.

Note that for each temperature there is a temperature coefficient TK ρ. In practice, for a certain temperature range, the average value is used TK ρ or:

, (3.10)

Metal	*Specific resistance ρ at 20 ºС, Ohmmm²/m**	Temperature coefficient of resistance α, ºС -1


Aluminum
Iron (steel)


Constantan
Manganin

The temperature coefficient of resistance α shows how much the resistance of a conductor of 1 ohm increases with an increase in temperature (heating of the conductor) by 1 ºС.

The conductor resistance at temperature t is calculated by the formula:

r t = r 20 + α* r 20 *(t - 20 ºС)

where r 20 is the resistance of the conductor at a temperature of 20 ºС, r t is the resistance of the conductor at temperature t.

Current Density

A current I = 10 A flows through a copper conductor with a cross-sectional area S = 4 mm². What is the current density?

Current density J = I/S = 10 A/4 mm² = 2.5 A/mm².

[A current I = 2.5 A flows through a cross-sectional area of 1 mm²; a current I = 10 A flows throughout the entire cross section S].

A switchgear bus of rectangular cross-section (20x80) mm² carries a current I = 1000 A. What is the current density in the bus?

Cross-sectional area of the tire S = 20x80 = 1600 mm². Current Density

J = I/S = 1000 A/1600 mm² = 0.625 A/mm².

The coil's wire has a circular cross-section with a diameter of 0.8 mm and allows a current density of 2.5 A/mm². What permissible current can be passed through the wire (heating should not exceed the permissible)?

Cross-sectional area of the wire S = π * d²/4 = 3/14*0.8²/4 ≈ 0.5 mm².

Allowable current I = J*S = 2.5 A/mm² * 0.5 mm² = 1.25 A.

Permissible current density for the transformer winding J = 2.5 A/mm². A current I = 4 A passes through the winding. What should be the cross-section (diameter) of the circular cross-section of the conductor so that the winding does not overheat?

Cross-sectional area S = I/J = (4 A) / (2.5 A/mm²) = 1.6 mm²

This section corresponds to a wire diameter of 1.42 mm.

1). Allowable current density

J = I/S = 38 A / 4mm² = 9.5 A/mm².

2). For a cross section of 1 mm², the permissible current density (see table)

J = I/S = 16 A / 1 mm² = 16 A/mm².

3). For a cross section of 10 mm² permissible current density

J = 70 A / 10 mm² = 7.0 A/mm²

4). For a cross section of 16 mm² permissible current density

J = I/S = 85 A / 16 mm² = 5.3 A/mm².

The permissible current density decreases with increasing cross-section. Table valid for electrical wires with class B insulation.

Problems to solve independently

A current I = 4 A should flow through the transformer winding. What should be the cross-section of the winding wire with an allowable current density of J = 2.5 A/mm²? (S = 1.6 mm²)

A wire with a diameter of 0.3 mm carries a current of 100 mA. What is the current density? (J = 1.415 A/mm²)

Along the winding of an electromagnet made of insulated wire with a diameter

d = 2.26 mm (excluding insulation) a current of 10 A passes. What is the density

current? (J = 2.5 A/mm²).

4. The transformer winding allows a current density of 2.5 A/mm². The current in the winding is 15 A. What is the smallest cross-section and diameter that a round wire can have (excluding insulation)? (in mm²; 2.76 mm).

Probably everyone knows about the effect of superconductivity. In any case, we have heard about him. The essence of this effect is that at minus 273 °C the resistance of the conductor to the flowing current disappears. This example alone is enough to understand that there is a dependence on temperature. A describes a special parameter - the temperature coefficient of resistance.

It should be noted that there are metals with a very small TCR value, and they are used in the manufacture of resistors, the parameters of which are weakly dependent on environmental changes.

The temperature coefficient of resistance can be used not only to take into account the influence of fluctuations in environmental parameters, but also for which, knowing the material that was exposed, it is enough to use the tables to determine what temperature the measured resistance corresponds to. An ordinary copper wire can be used as such a meter, however, you will have to use a lot of it and wind it in the form of, for example, a coil.

Conductor resistance (R) (resistivity) () depends on temperature. This dependence for minor changes in temperature () is presented as a function:

where is the resistivity of the conductor at a temperature of 0 o C; - temperature coefficient of resistance.

DEFINITION

The value characterizes the relationship between electrical resistance and temperature.

At temperatures within the range, for most metals the coefficient under consideration remains constant. For pure metals, the temperature coefficient of resistance is often taken to be

Sometimes they talk about the average temperature coefficient of resistance, defining it as:

where is the average value of the temperature coefficient in a given temperature range ().

Temperature coefficient of resistance for different substances

At temperatures close to absolute zero (-273 o C), the resistance of a large number of metals sharply drops to zero. Metals are said to go into a superconducting state.

Units of measurement

The basic SI unit for measuring the temperature coefficient of resistance is:

Examples of problem solving

Exercise

Solution

As a basis for solving the problem, we use the formula for the dependence of resistance on temperature of the form:

where is the resistance of the tungsten filament at a temperature of 0 o C. Expressing from expression (1.1), we have:

According to Ohm's law, for a section of the circuit we have:

Let's calculate

Let's write the equation connecting resistance and temperature:

Let's carry out the calculations:

Answer

Metal	*Specific resistance ρ at 20 ºС, Ohmmm²/m**	Temperature coefficient of resistance α, ºС -1


Aluminum
Iron (steel)


Constantan
Manganin

The temperature coefficient of resistance α shows how much the resistance of a conductor of 1 ohm increases with an increase in temperature (heating of the conductor) by 1 ºС.

The conductor resistance at temperature t is calculated by the formula:

r t = r 20 + α* r 20 *(t - 20 ºС)

where r 20 is the resistance of the conductor at a temperature of 20 ºС, r t is the resistance of the conductor at temperature t.

Current Density

A current I = 10 A flows through a copper conductor with a cross-sectional area S = 4 mm². What is the current density?

Current density J = I/S = 10 A/4 mm² = 2.5 A/mm².

[A current I = 2.5 A flows through a cross-sectional area of 1 mm²; a current I = 10 A flows throughout the entire cross section S].

A switchgear bus of rectangular cross-section (20x80) mm² carries a current I = 1000 A. What is the current density in the bus?

Cross-sectional area of the tire S = 20x80 = 1600 mm². Current Density

J = I/S = 1000 A/1600 mm² = 0.625 A/mm².

Cross-sectional area of the wire S = π * d²/4 = 3/14*0.8²/4 ≈ 0.5 mm².

Allowable current I = J*S = 2.5 A/mm² * 0.5 mm² = 1.25 A.

Cross-sectional area S = I/J = (4 A) / (2.5 A/mm²) = 1.6 mm²

This section corresponds to a wire diameter of 1.42 mm.

1). Allowable current density

J = I/S = 38 A / 4mm² = 9.5 A/mm².

2). For a cross section of 1 mm², the permissible current density (see table)

J = I/S = 16 A / 1 mm² = 16 A/mm².

3). For a cross section of 10 mm² permissible current density

J = 70 A / 10 mm² = 7.0 A/mm²

4). For a cross section of 16 mm² permissible current density

J = I/S = 85 A / 16 mm² = 5.3 A/mm².

The permissible current density decreases with increasing cross-section. Table valid for electrical wires with class B insulation.

Problems to solve independently

A current I = 4 A should flow through the transformer winding. What should be the cross-section of the winding wire with an allowable current density of J = 2.5 A/mm²? (S = 1.6 mm²)

A wire with a diameter of 0.3 mm carries a current of 100 mA. What is the current density? (J = 1.415 A/mm²)

Along the winding of an electromagnet made of insulated wire with a diameter

d = 2.26 mm (excluding insulation) a current of 10 A passes. What is the density

current? (J = 2.5 A/mm²).

Page 1

The negative temperature coefficient of resistance in intrinsic materials is used in thermistors to convert temperature changes into an electrical signal. The materials used are most often compressed powders of oxides of nickel, copper, manganese and zinc. It is also possible to use germanium or other semiconductors as a low-temperature thermometer.

The negative temperature coefficient of resistance of such semiconductors is observed in temperature regions when not all impurities are ionized or intrinsic electrical conductivity occurs. In both cases, the dependence of the resistivity of the semiconductor is determined mainly by the change in the concentration of charge carriers, since the relatively weak change in their mobility in this case can be neglected.

The negative temperature coefficient of resistance of cermet films (- 200 - 10 - b deg 1) indicates that the metallic mechanism of electrical conductivity is not predominant in them. The electrical resistance of the cermet film depends on the formulation composition and dissipation during evaporation, but can be easily adjusted by varying the temperature and holding time during final annealing. As a result of annealing, not only the resistance changes, but also its temperature coefficient.

Semiconductors have a negative temperature coefficient of resistance, which in absolute value is 10 - 20 times greater than that of metals. This property of semiconductors is used in technology for various purposes, for example, for the manufacture of thermistors, the resistance of which changes sharply with slight changes in temperature.

Semiconductors have a negative temperature coefficient of resistance, which in absolute value is 10 - 20 times greater than that of metals. This property of semiconductors is used in technology for various purposes, for example, for the manufacture of thermistors, the resistance value of which changes sharply with slight changes in temperature.

Thermistors have a negative temperature coefficient of resistance.

Semiconductors have a negative temperature coefficient of resistance, which in absolute value is 10 - 20 times greater than that of metals. This property of semiconductors is used in technology for various purposes, for example, for the manufacture of thermal resistances (thermistors), the value of which changes sharply with minor changes in temperature.

Varistors have a negative temperature coefficient of resistance. At room temperature, the value of this coefficient ranges from - 0 3 to - 0 5% X deg-1. With decreasing temperature it increases, with increasing temperature it decreases. The nonlinearity coefficient p changes little with temperature.

A thermistor has a large negative temperature coefficient of resistance, so including it in a circuit of metal resistors that has a positive temperature coefficient (see Figure 8.8) can make the circuit characteristics almost independent of temperature. Thus, with the help of thermistors it is easy to provide temperature compensation for a number of elements of the electrical circuit, thermal control of various mechanisms, and fire alarms.

The thermistor has a large negative temperature coefficient of resistance, so including it in a circuit of metallized resistors that have a positive temperature coefficient (see Fig. 8.8) can make the circuit characteristics almost independent of temperature. Thus, with the help of thermistors it is easy to provide temperature compensation for a number of elements of the electrical circuit, thermal control of various mechanisms, and fire alarms.

The results of resistivity measurements are greatly influenced by shrinkage cavities, gas bubbles, inclusions and other defects. Moreover, Fig. 155 shows that small amounts of impurity entering the solid solution also have a large effect on the measured conductivity. Therefore, it is much more difficult to produce satisfactory samples for measuring electrical resistance than for

dilatometric study. This led to another method of constructing phase diagrams, in which the temperature coefficient of resistance is measured.

Temperature coefficient of resistance

Electrical resistance at temperature

Matthiessen found that the increase in metal resistance due to the presence of a small amount of the second component in the solid solution does not depend on temperature; it follows that for such a solid solution the value does not depend on the concentration. This means that the temperature coefficient of resistance is proportional, i.e., conductivity, and the graph of the coefficient a depending on the composition is similar to the graph of the conductivity of a solid solution. There are many known exceptions to this rule, especially for transition metals, but for most cases it is approximately true.

The temperature coefficient of resistance of intermediate phases is usually of the same order of magnitude as for pure metals, even in cases where the connection itself has high resistance. There are, however, intermediate phases whose temperature coefficient in a certain temperature range is zero or negative.

Matthiessen's rule applies, strictly speaking, only to solid solutions, but there are many cases where it appears to be true also for two-phase alloys. If the temperature coefficient of resistance is plotted against composition, the curve usually has the same shape as the conductivity curve, so the phase transformation can be detected in the same way. This method is convenient to use when, due to fragility or other reasons, it is impossible to produce samples suitable for conductivity measurements.

In practice, the average temperature coefficient between two temperatures is determined by measuring the electrical resistance of the alloy at those temperatures. If no phase transformation occurs in the temperature range under consideration, then the coefficient is determined by the formula:

will have the same meaning as if the interval is small. For hardened alloys as temperatures and

It is convenient to take 0° and 100°, respectively, and the measurements will give the phase region at the quenching temperature. However, if measurements are made at high temperatures, the interval should be much less than 100°, if the phase boundary may be somewhere between the temperatures

Rice. 158. (see scan) Electrical conductivity and temperature coefficient of electrical resistance in the silver-magic system (Tamman)

The great advantage of this method is that the coefficient a depends on the relative resistance of the sample at two temperatures, and is thus not affected by pitting and other metallurgical defects in the sample. Conductivity and temperature coefficient curves

resistances in some alloy systems repeat one another. Rice. 158 is taken from Tammann's early work (the curves refer to silver-magnesium alloys); later work showed that the region of the -solid solution decreases with decreasing temperature and a superstructure exists in the region of the phase. Some other phase boundaries have also undergone changes recently, so that the diagram presented in Fig. 158 is of historical interest only and cannot be used for accurate measurements.