Electrical resistance of metals. Resistivity and other properties of copper
The term “resistivity” refers to a parameter possessed by copper or any other metal, and is quite often found in the specialized literature. It is worth understanding what is meant by this.
One of the types of copper cable
General information about electrical resistance
First, we should consider the concept of electrical resistance. As is known, under the influence of electric current on a conductor (and copper is one of the best conductor metals), some of the electrons in it leave their place in the crystal lattice and rush towards the positive pole of the conductor. However, not all electrons leave the crystal lattice; some of them remain in it and continue to rotate around the atomic nucleus. It is these electrons, as well as atoms located at the nodes of the crystal lattice, that create electrical resistance that prevents the movement of released particles.
This process, which we briefly outlined, is typical for any metal, including copper. Naturally, different metals, each of which has a special shape and size of the crystal lattice, resist the passage of electric current through them in different ways. It is precisely these differences that characterize resistivity - an indicator individual for each metal.
Applications of copper in electrical and electronic systems
In order to understand the reason for the popularity of copper as a material for the manufacture of elements of electrical and electronic systems, it is enough to look at the value of its resistivity in the table. For copper, this parameter is 0.0175 Ohm*mm2/meter. In this regard, copper is second only to silver.
It is the low resistivity, measured at a temperature of 20 degrees Celsius, that is the main reason that almost no electronic and electrical device can do without copper today. Copper is the main material for the production of wires and cables, printed circuit boards, electric motors and power transformer parts.
The low resistivity that copper is characterized by allows it to be used for the manufacture of electrical devices characterized by high energy-saving properties. In addition, the temperature of copper conductors increases very little when electric current passes through them.
What affects the resistivity value?
It is important to know that there is a dependence of the resistivity value on the chemical purity of the metal. When copper contains even a small amount of aluminum (0.02%), the value of this parameter can increase significantly (up to 10%).
This coefficient is also affected by the temperature of the conductor. This is explained by the fact that as the temperature increases, the vibrations of metal atoms in the nodes of its crystal lattice intensify, which leads to the fact that the resistivity coefficient increases.
That is why in all reference tables the value of this parameter is given taking into account a temperature of 20 degrees.
How to calculate the total resistance of a conductor?
Knowing what the resistivity is is important in order to carry out preliminary calculations of the parameters of electrical equipment when designing it. In such cases, the total resistance of the conductors of the designed device, having a certain size and shape, is determined. Having looked at the resistivity value of the conductor using a reference table, determining its dimensions and cross-sectional area, you can calculate the value of its total resistance using the formula:
This formula uses the following notation:
- R is the total resistance of the conductor, which must be determined;
- p is the resistivity of the metal from which the conductor is made (determined from the table);
- l is the length of the conductor;
- S is its cross-sectional area.
Specific electrical resistance, or simply the resistivity of a substance, is a physical quantity that characterizes the ability of a substance to prevent the passage of electric current.
Resistivity is denoted by the Greek letter ρ. The reciprocal of resistivity is called specific conductivity (electrical conductivity). Unlike electrical resistance, which is a property of a conductor and depends on its material, shape and size, electrical resistivity is a property of a substance only.
The electrical resistance of a homogeneous conductor with resistivity ρ, length l and cross-sectional area S can be calculated using the formula (assuming that neither the area nor the cross-sectional shape changes along the conductor). Accordingly, for ρ we have
From the last formula it follows: the physical meaning of the resistivity of a substance is that it represents the resistance of a homogeneous conductor of unit length and with unit cross-sectional area made from this substance.
The unit of resistivity in the International System of Units (SI) is Ohm m.
From the relationship it follows that the unit of measurement of resistivity in the SI system is equal to the resistivity of a substance at which a homogeneous conductor 1 m long with a cross-sectional area of 1 m², made from this substance, has a resistance equal to 1 Ohm. Accordingly, the resistivity of an arbitrary substance, expressed in SI units, is numerically equal to the resistance of a section of an electrical circuit made of a given substance with a length of 1 m and a cross-sectional area of 1 m².
In technology, the outdated non-systemic unit Ohm mm²/m is also used, equal to 10 −6 of 1 Ohm m. This unit is equal to the resistivity of a substance at which a homogeneous conductor 1 m long with a cross-sectional area of 1 mm², made from this substance, has a resistance equal to 1 Ohm. Accordingly, the resistivity of a substance, expressed in these units, is numerically equal to the resistance of a section of an electrical circuit made of this substance, 1 m long and a cross-sectional area of 1 mm².
Electromotive force (EMF) is a scalar physical quantity that characterizes the work of external forces, that is, any forces of non-electric origin acting in quasi-stationary DC or AC circuits. In a closed conducting circuit, the EMF is equal to the work of these forces to move a single positive charge along the entire circuit.
By analogy with the electric field strength, the concept of external force strength is introduced, which is understood as a vector physical quantity equal to the ratio of the external force acting on a test electric charge to the magnitude of this charge. Then in a closed loop the EMF will be equal to:
where is the contour element.
EMF, like voltage, is measured in volts in the International System of Units (SI). We can talk about electromotive force at any part of the circuit. This is the specific work of external forces not throughout the entire circuit, but only in a given area. The EMF of a galvanic cell is the work of external forces when moving a single positive charge inside the element from one pole to another. The work of external forces cannot be expressed through a potential difference, since external forces are non-potential and their work depends on the shape of the trajectory. So, for example, the work of external forces when moving a charge between the terminals of the current outside itself? source is zero.
- an electrical quantity that characterizes the property of a material to prevent the flow of electric current. Depending on the type of material, the resistance can tend to zero - be minimal (miles/micro ohms - conductors, metals), or be very large (giga ohms - insulation, dielectrics). The reciprocal of electrical resistance is .
Unit electrical resistance - Ohm. It is designated by the letter R. The dependence of resistance on current in a closed circuit is determined.
Ohmmeter- a device for direct measurement of circuit resistance. Depending on the range of the measured value, they are divided into gigaohmmeters (for large resistances - when measuring insulation), and micro/miliohmmeters (for small resistances - when measuring transition resistances of contacts, motor windings, etc.).
There is a wide variety of ohmmeters by design from different manufacturers, from electromechanical to microelectronic. It is worth noting that a classic ohmmeter measures the active part of the resistance (so-called ohms).
Any resistance (metal or semiconductor) in an alternating current circuit has an active and reactive component. The sum of active and reactive resistance is AC circuit impedance and is calculated by the formula:
where, Z is the total resistance of the alternating current circuit;
R is the active resistance of the alternating current circuit;
Xc is the capacitive reactance of the alternating current circuit; 
(C - capacitance, w - angular speed of alternating current)
Xl is the inductive reactance of the alternating current circuit; 
(L is inductance, w is the angular velocity of alternating current).
Active resistance- this is part of the total resistance of an electrical circuit, the energy of which is completely converted into other types of energy (mechanical, chemical, thermal). A distinctive property of the active component is the complete consumption of all electricity (no energy is returned to the network), and reactance returns part of the energy back to the network (a negative property of the reactive component).
The physical meaning of active resistance
Each environment where electric charges pass creates obstacles in their path (it is believed that these are nodes of the crystal lattice), into which they seem to hit and lose their energy, which is released in the form of heat.
Thus, a drop (loss of electrical energy) occurs, part of which is lost due to the internal resistance of the conducting medium.
The numerical value characterizing the ability of a material to prevent the passage of charges is called resistance. It is measured in Ohms (Ohm) and is inversely proportional to electrical conductivity.
Different elements of Mendeleev's periodic table have different electrical resistivities (p), for example, the smallest. Silver (0.016 Ohm*mm2/m), copper (0.0175 Ohm*mm2/m), gold (0.023) and aluminum (0.029) have resistance. They are used in industry as the main materials on which all electrical engineering and energy are built. Dielectrics, on the contrary, have a high shock value. resistance and are used for insulation.
The resistance of the conductive medium can vary significantly depending on the cross-section, temperature, magnitude and frequency of the current. In addition, different environments have different charge carriers (free electrons in metals, ions in electrolytes, “holes” in semiconductors), which are the determining factors of resistance.
Physical meaning of reactance
In coils and capacitors, when applied, energy accumulates in the form of magnetic and electric fields, which takes some time.
Magnetic fields in alternating current networks change following the changing direction of movement of charges, while providing additional resistance.
In addition, a stable phase and current shift occurs, and this leads to additional electricity losses.
Resistivity
How can we find out the resistance of a material if there is no flow through it and we do not have an ohmmeter? There is a special value for this - electrical resistivity of the material V
(these are tabular values that are determined empirically for most metals). Using this value and the physical quantities of the material, we can calculate the resistance using the formula:
Where, p— resistivity (units ohm*m/mm2);
l—conductor length (m);
S - cross section (mm 2).
Most laws of physics are based on experiments. The names of the experimenters are immortalized in the names of these laws. One of them was Georg Ohm.
Georg Ohm's experiments
During experiments on the interaction of electricity with various substances, including metals, he established a fundamental relationship between density, electric field strength and the property of a substance, which was called “specific conductivity”. The formula corresponding to this pattern, called “Ohm’s Law,” is as follows:
j= λE , wherein
- j— electric current density;
- λ — specific conductivity, also called “electrical conductivity”;
- E – electric field strength.
In some cases, a different letter of the Greek alphabet is used to indicate conductivity - σ . Specific conductivity depends on certain parameters of the substance. Its value is influenced by temperature, substances, pressure, if it is a gas, and most importantly, the structure of this substance. Ohm's law is observed only for homogeneous substances.
For more convenient calculations, the reciprocal of specific conductivity is used. It is called “resistivity”, which is also associated with the properties of the substance in which the electric current flows, denoted by the Greek letter ρ and has the dimension Ohm*m. But since different theoretical justifications apply to different physical phenomena, alternative formulas can be used for resistivity. They are a reflection of the classical electronic theory of metals, as well as quantum theory.
Formulas
In these formulas, which are tedious for ordinary readers, factors such as Boltzmann's constant, Avogadro's constant and Planck's constant appear. These constants are used for calculations that take into account the free path of electrons in a conductor, their speed during thermal motion, the degree of ionization, the concentration and density of the substance. In short, everything is quite complicated for a non-specialist. In order not to be unfounded, below you can familiarize yourself with how everything actually looks:
Features of metals
Since the movement of electrons depends on the homogeneity of the substance, the current in a metal conductor flows according to its structure, which affects the distribution of electrons in the conductor, taking into account its heterogeneity. It is determined not only by the presence of impurity inclusions, but also by physical defects - cracks, voids, etc. The heterogeneity of the conductor increases its resistivity, which is determined by Matthiesen's rule.
This easy-to-understand rule essentially says that several separate resistivities can be distinguished in a current-carrying conductor. And the resulting value will be their sum. The components will be the resistivity of the metal crystal lattice, impurities and conductor defects. Since this parameter depends on the nature of the substance, corresponding laws have been defined to calculate it, including for mixed substances.
Despite the fact that alloys are also metals, they are considered as solutions with a chaotic structure, and for calculating the resistivity, it matters which metals are included in the alloy. Basically, most alloys of two components that do not belong to transition metals, as well as rare earth metals, fall under the description of Nodheim's law.
The resistivity of metal thin films is considered as a separate topic. It is quite logical to assume that its value should be greater than that of a bulk conductor made of the same metal. But at the same time, a special empirical Fuchs formula is introduced for the film, which describes the interdependence of resistivity and film thickness. It turns out that metals in films exhibit semiconductor properties.
And the process of charge transfer is influenced by electrons, which move in the direction of the film thickness and interfere with the movement of “longitudinal” charges. At the same time, they are reflected from the surface of the film conductor, and thus one electron oscillates between its two surfaces for quite a long time. Another significant factor in increasing resistivity is the temperature of the conductor. The higher the temperature, the greater the resistance. Conversely, the lower the temperature, the lower the resistance.
Metals are the substances with the lowest resistivity at so-called “room” temperature. The only non-metal that justifies its use as a conductor is carbon. Graphite, which is one of its varieties, is widely used for making sliding contacts. It has a very successful combination of properties such as resistivity and sliding friction coefficient. Therefore, graphite is an indispensable material for electric motor brushes and other sliding contacts. The resistivity values of the main substances used for industrial purposes are given in the table below.
Superconductivity
At temperatures corresponding to the liquefaction of gases, that is, up to the temperature of liquid helium, which is equal to -273 degrees Celsius, the resistivity decreases almost to complete disappearance. And not just good metal conductors such as silver, copper and aluminum. Almost all metals. Under such conditions, which are called superconductivity, the structure of the metal has no inhibitory effect on the movement of charges under the influence of an electric field. Therefore, mercury and most metals become superconductors.
But, as it turned out, relatively recently in the 80s of the 20th century, some types of ceramics are also capable of superconductivity. Moreover, you do not need to use liquid helium for this. Such materials were called high-temperature superconductors. However, several decades have already passed, and the range of high-temperature conductors has expanded significantly. But mass use of such high-temperature superconducting elements has not been observed. In some countries, single installations have been made with the replacement of conventional copper conductors with high-temperature superconductors. To maintain the normal regime of high-temperature superconductivity, liquid nitrogen is required. And this turns out to be a too expensive technical solution.
Therefore, the low resistivity value given by Nature to copper and aluminum still makes them irreplaceable materials for the manufacture of various electrical conductors.
- conductors;
- dielectrics (with insulating properties);
- semiconductors.
Electrons and current
The modern concept of electric current is based on the assumption that it consists of material particles - charges. But various physical and chemical experiments give grounds to assert that these charge carriers can be of different types in the same conductor. And this heterogeneity of particles affects the current density. For calculations related to the parameters of electric current, certain physical quantities are used. Among them, conductivity and resistance occupy an important place.
- Conductivity is related to resistance in a mutually inverse relationship.
It is known that when there is a certain voltage applied to an electrical circuit, an electric current appears in it, the magnitude of which is related to the conductivity of this circuit. This fundamental discovery was made at one time by the German physicist Georg Ohm. Since then, a law called Ohm's law has been in use. It exists for different circuit options. Therefore, the formulas for them may be different from each other, since they correspond to completely different conditions.
Every electrical circuit has a conductor. If there is one type of charge carrier particle in it, the current in the conductor is similar to the flow of liquid, which has a certain density. It is determined by the following formula:
Most metals correspond to the same type of charged particles, thanks to which electric current exists. For metals, the specific electrical conductivity is calculated using the following formula:
Since conductivity can be calculated, determining electrical resistivity is now easy. It was already mentioned above that the resistivity of a conductor is the reciprocal of conductivity. Hence,
In this formula, the letter of the Greek alphabet ρ (rho) is used to represent electrical resistivity. This designation is most often used in technical literature. However, you can also find slightly different formulas that are used to calculate the resistivity of conductors. If the classical theory of metals and electronic conductivity in them is used for calculations, the resistivity is calculated using the following formula:
However, there is one “but”. The state of atoms in a metal conductor is affected by the duration of the ionization process, which is carried out by an electric field. With a single ionizing effect on a conductor, the atoms in it will receive a single ionization, which will create a balance between the concentration of atoms and free electrons. And the values of these concentrations will be equal. In this case, the following dependencies and formulas take place:
Deviations of conductivity and resistance
Next, we will consider what the specific conductivity, which is inversely related to the resistivity, depends on. The resistivity of a substance is a rather abstract physical quantity. Each conductor exists in the form of a specific sample. It is characterized by the presence of various impurities and defects in the internal structure. They are taken into account as separate terms of the expression that determines the resistivity in accordance with Matthiessen's rule. This rule also takes into account the scattering of a moving flow of electrons at the nodes of the crystal lattice of the sample that fluctuate depending on the temperature.
The presence of internal defects, such as inclusions of various impurities and microscopic voids, also increases the resistivity. To determine the amount of impurities in samples, the resistivity of materials is measured for two temperatures of the sample material. One temperature value is room temperature, and the other corresponds to liquid helium. By relating the measurement result at room temperature to the result at liquid helium temperature, a coefficient is obtained that illustrates the structural perfection of the material and its chemical purity. The coefficient is denoted by the letter β.
If a metal alloy with a solid solution structure that is disordered is considered as a conductor of electric current, the value of the residual resistivity can be significantly greater than the resistivity. This feature of metal alloys of two components that are not related to rare earth elements, as well as to transition elements, is covered by a special law. It is called Nordheim's law.
Modern technologies in electronics are increasingly moving towards miniaturization. And so much so that the word “nanocircuit” will soon appear instead of microcircuit. The conductors in such devices are so thin that it would be correct to call them metal films. It is quite clear that the film sample will differ in its resistivity to a greater extent from a larger conductor. The small thickness of the metal in the film leads to the appearance of semiconductor properties in it.
The proportionality between the thickness of the metal and the free path of electrons in this material begins to appear. There is little room left for electrons to move. Therefore, they begin to interfere with each other’s movement in an orderly manner, which leads to an increase in resistivity. For metal films, resistivity is calculated using a special formula obtained based on experiments. The formula is named after Fuchs, a scientist who studied the resistivity of films.
Films are very specific formations that are difficult to replicate so that the properties of several samples are the same. For acceptable accuracy in evaluating films, a special parameter is used - specific surface resistance.
Resistors are formed from metal films on the substrate of microcircuits. For this reason, resistivity calculations are a highly sought-after task in microelectronics. The value of resistivity is obviously influenced by temperature and is related to it by direct proportionality. For most metals, this dependence has some linear portion in a certain temperature range. In this case, the resistivity is determined by the formula:
In metals, electric current occurs due to a large number of free electrons, the concentration of which is relatively high. Moreover, electrons also determine the greater thermal conductivity of metals. For this reason, a connection has been established between electrical conductivity and thermal conductivity by a special law, which was justified experimentally. This Wiedemann-Franz law is characterized by the following formulas:
The tantalizing prospects of superconductivity
However, the most amazing processes occur at the minimum technically achievable temperature of liquid helium. Under such cooling conditions, all metals practically lose their resistivity. Copper wires, cooled to the temperature of liquid helium, are capable of conducting currents many times greater than under normal conditions. If this became possible in practice, the economic effect would be invaluable.
Even more surprising was the discovery of high-temperature conductors. Under normal conditions, these types of ceramics were very far from metals in their resistivity. But at temperatures about three tens of degrees above liquid helium, they became superconductors. The discovery of this behavior of nonmetallic materials has become a powerful stimulus for research. Due to the greatest economic consequences of the practical application of superconductivity, very significant financial resources were thrown into this direction, and large-scale research began.
But for now, as they say, “things are still there”... Ceramic materials turned out to be unsuitable for practical use. The conditions for maintaining the state of superconductivity required such large expenses that all the benefits from its use were destroyed. But experiments with superconductivity continue. There is progress. Superconductivity has already been achieved at a temperature of 165 degrees Kelvin, but this requires high pressure. Creating and maintaining such special conditions again denies the commercial use of this technical solution.
Additional influencing factors
Currently, everything continues to go its way, and for copper, aluminum and some other metals, the resistivity continues to ensure their industrial use for the manufacture of wires and cables. In conclusion, it is worth adding a little more information that not only the resistivity of the conductor material and the ambient temperature affect the losses in it during the passage of electric current. The geometry of the conductor is very important when used at high voltage frequencies and high currents.
Under these conditions, electrons tend to concentrate near the surface of the wire, and its thickness as a conductor loses its meaning. Therefore, it is possible to justifiably reduce the amount of copper in the wire by making only the outer part of the conductor from it. Another factor in increasing the resistivity of a conductor is deformation. Therefore, despite the high performance of some electrically conductive materials, under certain conditions they may not appear. The correct conductors should be selected for specific tasks. The tables shown below will help with this.
