Temperature coefficient of electrical resistance
Temperature coefficient of electrical resistance - a value equal to the relative change electrical resistance section electrical circuit or resistivity substance when changing temperatures per unit.
The temperature coefficient of resistance characterizes the dependence of electrical resistance on temperature and is measured in kelvin in the minus first degree (K -1).
Also commonly used is the term "Temperature coefficient of conductivity". It is equal to the inverse of the resistance coefficient.
The temperature dependence of the resistance of metallic alloys , gases , doped semiconductors and electrolytes is more complex.
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The resistance of the conductor (R) (resistivity) () depends on the temperature. This dependence for small changes in temperature () is represented as a function:
where is the resistivity of the conductor at a temperature of 0 ° C; - temperature coefficient of resistance.
DEFINITION
The temperature coefficient of electrical resistance () is the physical quantity equal to the relative increment (R) of the chain segment (or the resistivity of the medium ()) that occurs when the conductor is heated by 1 ° C. Mathematically, the determination of the temperature coefficient of resistance can be represented as:
The value serves as a characteristic of the electrical resistance relationship with temperature.
At temperatures belonging to the range, for most metals the coefficient under consideration remains constant. For pure metals, the temperature coefficient of resistance is often taken equal to
Sometimes they speak 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 the result of electron scattering on the crystal lattice, which enhances thermal oscillations.
At temperatures close to absolute zero (-273 o C), the resistance of a large number of metals falls sharply to zero. It is said that metals become superconducting.
Semiconductors that do not have impurities have a negative temperature coefficient of resistance. Their resistance decreases with increasing temperature. This is due to the fact that the number of electrons that go into the conduction band increases, hence, the number of holes per unit volume of the semiconductor increases.
Solutions of electrolytes have. The resistance of electrolytes decreases with increasing temperature. This is because the increase in the number of free ions due to dissociation of molecules exceeds the increase in ion scattering due to 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
The basic unit of measurement of the temperature coefficient of resistance in the SI system is:
Examples of problem solving
EXAMPLE 1
| The task | An incandescent lamp having a spiral of tungsten is included in a network with a voltage of B, current is flowing through it. What will be the temperature of the spiral, if at a temperature of o C it has an ohm resistance? Temperature coefficient of tungsten resistance  .
|
| Decision | As a basis for solving the problem, we use the formula for the dependence of the resistance on the temperature of the species: where is the resistance of the tungsten filament at a temperature of 0 ° C. We express from the expression (1.1), we have: According to Ohm's law for the chain segment, we have:
We calculate
We write the equation relating resistance and temperature: Let's carry out the calculations: |
| Answer | K |
EXAMPLE 2
| The task | At a temperature, the resistance of the rheostat is equal to, the resistance of the ammeter is equal and it shows the current strength of the rheostat, made of iron wire, it is connected in series with the ammeter (Fig. 1). How will the current flow through the ammeter, if the rheostat is heated to a temperature? Read the temperature coefficient of resistance of the iron equal. |
About the effect of superconductivity know, probably, everything. In any case, we heard about him. The essence of this effect is that at minus 273 ° C the resistance of the conductor to the flowing current disappears. Already one of this example is enough to understand that there is its dependence on temperature. A describes a special parameter - the temperature coefficient of resistance.
Any conductor prevents current flowing through it. This counteraction for each electrically conductive material is different, it is determined by many factors inherent in a particular material, but this will not be the matter further. Interest at the moment is its dependence on temperature and the nature of this relationship.
Conductors of electrical current usually are metals, they have a resistance with increasing temperature, and when it decreases, it decreases. The magnitude of this change, occurring at 1 ° C, is called the temperature coefficient of resistance, or abbreviated to TCS.
The value of TCS can be positive and negative. If it is positive, then as the temperature rises, if it increases, it decreases. For most metals, used as conductors of electric current, TCS is positive. One of the best conductors is copper, the temperature coefficient of copper resistance is not that best, but compared to other conductors, it is smaller. You just need to remember that the value of TCR determines how the resistance value will change when the environmental parameters change. Its change will be more significant than this coefficient is greater.
Such a temperature dependence of the resistance should be taken into account when designing radioelectronic equipment. The matter is that the equipment should work under any environmental conditions, the same cars are operated from minus 40 ° С to plus 80 ° С. And there are a lot of electronics in the car, and if you do not take into account the influence of the environment on the operation of the circuit elements, you can face the situation when the electronic unit works fine under normal conditions, but refuses to work under the influence of a reduced or increased temperature.
This dependence on the conditions of the environment is taken into account and the developers of the equipment take into account when designing it, using the temperature coefficient of resistance for calculating the parameters of the circuit. There are tables with TCS data for the materials used and calculation formulas for which, knowing the TCR, it is possible to determine the resistance value under any conditions and to take into account possible changes in the operation modes of the circuit. But to understand that, TCS, now no formulas, no tables are needed.
It should be noted that there are metals with a very small value of TCS, and they are used in the manufacture of resistors, the parameters of which depend little on the changes in the environment.
The temperature coefficient of resistance can be used not only to take into account the influence of fluctuations in the environmental parameters, but also for what is enough. Knowing the material that has been exposed, it is possible to determine from the tables the temperature of the measured resistance. As such a meter can be used ordinary copper wire, however, it is necessary to use it a lot and rewind in the form, for example, of a coil.
All of the above does not cover completely all the issues of using the temperature coefficient of resistance. There are very interesting possibilities of application associated with this coefficient in semiconductors, in electrolytes, but also what is presented is sufficient for understanding the concept of TCS.
|
Metal |
Specific resistance ρ at 20 ºС, Ohm * mm² / m |
Temperature coefficient of resistance α, ºС -1 |
|
Aluminum | ||
|
Iron (steel) | ||
|
Constantan | ||
|
Manganin | ||
The temperature coefficient of resistance α indicates how much the resistance of the conductor increases by 1 ohm with increasing temperature (heating of the conductor) by 1 ° C.
The conductor resistance at temperature t is calculated by the formula:
r t = r 20 + α * r 20 * (t - 20 ° C)
r t = r 20 *,
where r 20 is the resistance of the conductor at a temperature of 20 ° C, r t is the conductor resistance 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?
The current density is J = I / S = 10 A / 4 mm² = 2.5 A / mm².
[The current I = 2.5 A flows through the cross-sectional area 1 mm²; the current I = 10 A] flows through the entire cross section S.
The current I = 1000A passes through the bus of the rectangular cross-section (20x80) mm² switchgear. What is the current density in the bus?
The cross-sectional area of the tire S = 20х80 = 1600 мм². Current Density
J = I / S = 1000 A / 1600 mm² = 0.625 A / mm².
At the coil, the wire has a circular cross-section of 0.8 mm in diameter and allows a current density of 2.5 A / mm². What permissible current can be passed through the wire (heating should not exceed permissible)?
The cross-sectional area of the wire is S = π * d² / 4 = 3/14 * 0.8² / 4 ≈ 0.5 mm².
Permissible current I = J * S = 2.5 A / mm² * 0.5 mm² = 1.25 A.
The permissible current density for the transformer winding J = 2.5 A / mm². The current flows through the winding I = 4 A. What should be the cross-section (diameter) of the circular 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.
An insulated copper wire of 4 mm² crosses the maximum permissible current of 38 A (see table). What is the permissible current density? What are the permissible current densities for copper wires of 1, 10 and 16 mm²?
1). Permissible current density
J = I / S = 38 A / 4mm² = 9.5 A / mm².
2). For a cross-section of 1 mm², the allowable current density (see table)
J = I / S = 16 A / 1 mm² = 16 A / mm².
3). For a cross-section of 10 mm², the permissible current density
J = 70 A / 10 mm² = 7.0 A / mm²
4). For a cross section of 16 mm², the permissible current density
J = I / S = 85 A / 16 mm² = 5.3 A / mm².
The permissible current density decreases with increasing cross section. Table. It is valid for electrical wires with class B insulation.
Tasks for independent solutions
The 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 current of 100 mA passes through a wire 0.3 mm in diameter. What is the current density? (J = 1.415 A / mm²)
By winding an electromagnet from an insulated wire with a diameter
d = 2.26 mm (excluding insulation) is a current of 10 A. 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 of a round wire (excluding insulation)? (in mm², 2.76 mm).
|
Metal |
Specific resistance ρ at 20 ºС, Ohm * mm² / m |
Temperature coefficient of resistance α, ºС -1 |
|
Aluminum | ||
|
Iron (steel) | ||
|
Constantan | ||
|
Manganin | ||
The temperature coefficient of resistance α indicates how much the resistance of the conductor increases by 1 ohm with increasing temperature (heating of the conductor) by 1 ° C.
The conductor resistance at temperature t is calculated by the formula:
r t = r 20 + α * r 20 * (t - 20 ° C)
r t = r 20 *,
where r 20 is the resistance of the conductor at a temperature of 20 ° C, r t is the conductor resistance 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?
The current density is J = I / S = 10 A / 4 mm² = 2.5 A / mm².
[The current I = 2.5 A flows through the cross-sectional area 1 mm²; the current I = 10 A] flows through the entire cross section S.
The current I = 1000A passes through the bus of the rectangular cross-section (20x80) mm² switchgear. What is the current density in the bus?
The cross-sectional area of the tire S = 20х80 = 1600 мм². Current Density
J = I / S = 1000 A / 1600 mm² = 0.625 A / mm².
At the coil, the wire has a circular cross-section of 0.8 mm in diameter and allows a current density of 2.5 A / mm². What permissible current can be passed through the wire (heating should not exceed permissible)?
The cross-sectional area of the wire is S = π * d² / 4 = 3/14 * 0.8² / 4 ≈ 0.5 mm².
Permissible current I = J * S = 2.5 A / mm² * 0.5 mm² = 1.25 A.
The permissible current density for the transformer winding J = 2.5 A / mm². The current flows through the winding I = 4 A. What should be the cross-section (diameter) of the circular 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.
An insulated copper wire of 4 mm² crosses the maximum permissible current of 38 A (see table). What is the permissible current density? What are the permissible current densities for copper wires of 1, 10 and 16 mm²?
1). Permissible current density
J = I / S = 38 A / 4mm² = 9.5 A / mm².
2). For a cross-section of 1 mm², the allowable current density (see table)
J = I / S = 16 A / 1 mm² = 16 A / mm².
3). For a cross-section of 10 mm², the permissible current density
J = 70 A / 10 mm² = 7.0 A / mm²
4). For a cross section of 16 mm², the permissible current density
J = I / S = 85 A / 16 mm² = 5.3 A / mm².
The permissible current density decreases with increasing cross section. Table. It is valid for electrical wires with class B insulation.
Tasks for independent solutions
The 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 current of 100 mA passes through a wire 0.3 mm in diameter. What is the current density? (J = 1.415 A / mm²)
By winding an electromagnet from an insulated wire with a diameter
d = 2.26 mm (excluding insulation) is a current of 10 A. 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 of a round wire (excluding insulation)? (in mm², 2.76 mm).