In practice, it is often necessary to calculate the resistance of various wires. This can be done using formulas or using the data given in table. 1.
The effect of the conductor material is taken into account using the resistivity, denoted by the Greek letter? and having a length of 1 m and a cross-sectional area of 1 mm2. Lowest resistivity? = 0.016 Ohm mm2/m has silver. Let us give the average value of the resistivity of some conductors:
Silver - 0.016 , Lead - 0.21, Copper - 0.017, Nickelin - 0.42, Aluminum - 0.026, Manganin - 0.42, Tungsten - 0.055, Constantan - 0.5, Zinc - 0.06, Mercury - 0.96, Brass - 0.07, Nichrome - 1.05, Steel - 0.1, Fechral - 1.2, Phosphor bronze - 0.11, Chromal - 1.45.With different amounts of impurities and with different ratios of components included in the composition of rheostatic alloys, the resistivity may change slightly.
Resistance is calculated using the formula:
where R is resistance, Ohm; resistivity, (Ohm mm2)/m; l - wire length, m; s - cross-sectional area of the wire, mm2.
If the wire diameter d is known, then its cross-sectional area is equal to:
It is best to measure the diameter of the wire using a micrometer, but if you don’t have one, you should wind 10 or 20 turns of wire tightly onto a pencil and measure the length of the winding with a ruler. Dividing the length of the winding by the number of turns, we find the diameter of the wire.
To determine the length of a wire of a known diameter made of a given material necessary to obtain the required resistance, use the formula
Table 1.
Note. 1. Data for wires not listed in the table should be taken as some average values. For example, for a nickel wire with a diameter of 0.18 mm, we can approximately assume that the cross-sectional area is 0.025 mm2, the resistance of one meter is 18 Ohms, and the permissible current is 0.075 A.
2. For a different value of current density, the data in the last column must be changed accordingly; for example, at a current density of 6 A/mm2, they should be doubled.
Example 1. Find the resistance of 30 m of copper wire with a diameter of 0.1 mm.
Solution. We determine according to the table. 1 resistance of 1 m of copper wire, it is equal to 2.2 Ohms. Therefore, the resistance of 30 m of wire will be R = 30 2.2 = 66 Ohms.
Calculation using the formulas gives the following results: cross-sectional area of the wire: s = 0.78 0.12 = 0.0078 mm2. Since the resistivity of copper is 0.017 (Ohm mm2)/m, we get R = 0.017 30/0.0078 = 65.50 m.
Example 2. How much nickel wire with a diameter of 0.5 mm is needed to make a rheostat with a resistance of 40 Ohms?
Solution. According to the table 1, we determine the resistance of 1 m of this wire: R = 2.12 Ohm: Therefore, to make a rheostat with a resistance of 40 Ohms, you need a wire whose length is l = 40/2.12 = 18.9 m.
Let's do the same calculation using the formulas. We find the cross-sectional area of the wire s = 0.78 0.52 = 0.195 mm2. And the length of the wire will be l = 0.195 40/0.42 = 18.6 m.
Electrical resistance, expressed in ohms, is different from the concept of resistivity. To understand what resistivity is, we need to relate it to the physical properties of the material.
About conductivity and resistivity
The flow of electrons does not move unimpeded through the material. At a constant temperature, elementary particles swing around a state of rest. In addition, electrons in the conduction band interfere with each other through mutual repulsion due to similar charge. This is how resistance arises.
Conductivity is an intrinsic characteristic of materials and quantifies the ease with which charges can move when a substance is exposed to an electric field. Resistivity is the reciprocal of the material and describes the degree of difficulty electrons encounter as they move through a material, giving an indication of how good or bad a conductor is.
Important! An electrical resistivity with a high value indicates that the material is a poor conductor, while a resistivity with a low value indicates a good conductor.
Specific conductivity is designated by the letter σ and is calculated by the formula:
Resistivity ρ, as an inverse indicator, can be found as follows:
In this expression, E is the intensity of the generated electric field (V/m), and J is the electric current density (A/m²). Then the unit of measurement ρ will be:
V/m x m²/A = ohm m.
For conductivity σ, the unit in which it is measured is S/m or Siemens per meter.
Types of materials
According to the resistivity of materials, they can be classified into several types:
- Conductors. These include all metals, alloys, solutions dissociated into ions, as well as thermally excited gases, including plasma. Among non-metals, graphite can be cited as an example;
- Semiconductors, which are actually non-conducting materials, whose crystal lattices are purposefully doped with the inclusion of foreign atoms with a greater or lesser number of bound electrons. As a result, quasi-free excess electrons or holes are formed in the lattice structure, which contribute to the conductivity of the current;
- Dielectrics or dissociated insulators are all materials that under normal conditions do not have free electrons.
For the transport of electrical energy or in electrical installations for domestic and industrial purposes, a frequently used material is copper in the form of single-core or multi-core cables. An alternative metal is aluminum, although the resistivity of copper is 60% of that of aluminum. But it is much lighter than copper, which predetermined its use in high-voltage power lines. Gold is used as a conductor in special-purpose electrical circuits.
Interesting. The electrical conductivity of pure copper was adopted by the International Electrotechnical Commission in 1913 as the standard for this value. By definition, the conductivity of copper measured at 20° is 0.58108 S/m. This value is called 100% LACS, and the conductivity of the remaining materials is expressed as a certain percentage of LACS.
Most metals have a conductivity value less than 100% LACS. There are exceptions, however, such as silver or special copper with very high conductivity, designated C-103 and C-110, respectively.
Dielectrics do not conduct electricity and are used as insulators. Examples of insulators:
- glass,
- ceramics,
- plastic,
- rubber,
- mica,
- wax,
- paper,
- dry wood,
- porcelain,
- some fats for industrial and electrical use and bakelite.
Between the three groups the transitions are fluid. It is known for sure: there are no absolutely non-conducting media and materials. For example, air is an insulator at room temperature, but when exposed to a strong low-frequency signal, it can become a conductor.
Determination of conductivity
When comparing the electrical resistivity of different substances, standardized measurement conditions are required:
- In the case of liquids, poor conductors and insulators, cubic samples with an edge length of 10 mm are used;
- The resistivity values of soils and geological formations are determined on cubes with a length of each edge of 1 m;
- The conductivity of a solution depends on the concentration of its ions. A concentrated solution is less dissociated and has fewer charge carriers, which reduces conductivity. As the dilution increases, the number of ion pairs increases. The concentration of solutions is set to 10%;
- To determine the resistivity of metal conductors, wires of a meter length and a cross-section of 1 mm² are used.
If a material, such as a metal, can provide free electrons, then when a potential difference is applied, an electric current will flow through the wire. As the voltage increases, more electrons move through the substance into the time unit. If all additional parameters (temperature, cross-sectional area, length and wire material) are unchanged, then the ratio of current to applied voltage is also constant and is called conductivity:
Accordingly, the electrical resistance will be:
The result is in ohms.
In turn, the conductor can be of different lengths, cross-sectional sizes and made of different materials, which determines the value of R. Mathematically, this relationship looks like this:
The material factor takes into account the coefficient ρ.
From this we can derive the formula for resistivity:
If the values of S and l correspond to the given conditions for the comparative calculation of resistivity, i.e. 1 mm² and 1 m, then ρ = R. When the dimensions of the conductor change, the number of ohms also changes.
When an electrical circuit is closed, at the terminals of which there is a potential difference, an electric current occurs. Free electrons, under the influence of electric field forces, move along the conductor. In their movement, electrons collide with the atoms of the conductor and give them a supply of their kinetic energy. The speed of electron movement continuously changes: when electrons collide with atoms, molecules and other electrons, it decreases, then under the influence of an electric field it increases and decreases again during a new collision. As a result, a uniform flow of electrons is established in the conductor at a speed of several fractions of a centimeter per second. Consequently, electrons passing through a conductor always encounter resistance to their movement from its side. When electric current passes through a conductor, the latter heats up.
Electrical resistance
The electrical resistance of a conductor, which is denoted by a Latin letter r, is the property of a body or medium to convert electrical energy into thermal energy when an electric current passes through it.
In the diagrams, electrical resistance is indicated as shown in Figure 1, A.
Variable electrical resistance, which serves to change the current in a circuit, is called rheostat. In the diagrams, rheostats are designated as shown in Figure 1, b. In general, a rheostat is made of a wire of one resistance or another, wound on an insulating base. The slider or rheostat lever is placed in a certain position, as a result of which the required resistance is introduced into the circuit.
A long conductor with a small cross-section creates a large resistance to current. Short conductors with a large cross-section offer little resistance to current.
If you take two conductors from different materials, but the same length and cross-section, then the conductors will conduct current differently. This shows that the resistance of a conductor depends on the material of the conductor itself.
The temperature of the conductor also affects its resistance. As temperature increases, the resistance of metals increases, and the resistance of liquids and coal decreases. Only some special metal alloys (manganin, constantan, nickel and others) hardly change their resistance with increasing temperature.
So, we see that the electrical resistance of a conductor depends on: 1) the length of the conductor, 2) the cross-section of the conductor, 3) the material of the conductor, 4) the temperature of the conductor.
The unit of resistance is one ohm. Om is often represented by the Greek capital letter Ω (omega). Therefore, instead of writing “The conductor resistance is 15 ohms,” you can simply write: r= 15 Ω.
1,000 ohms is called 1 kiloohms(1kOhm, or 1kΩ),
1,000,000 ohms is called 1 megaohm(1mOhm, or 1MΩ).
When comparing the resistance of conductors from different materials, it is necessary to take a certain length and cross-section for each sample. Then we will be able to judge which material conducts electric current better or worse.
Video 1. Conductor resistance
Electrical resistivity
The resistance in ohms of a conductor 1 m long, with a cross section of 1 mm² is called resistivity and is denoted by the Greek letter ρ (ro).
Table 1 shows the resistivities of some conductors.
Table 1
Resistivities of various conductors
The table shows that an iron wire with a length of 1 m and a cross-section of 1 mm² has a resistance of 0.13 Ohm. To get 1 Ohm of resistance you need to take 7.7 m of such wire. Silver has the lowest resistivity. 1 Ohm of resistance can be obtained by taking 62.5 m of silver wire with a cross section of 1 mm². Silver is the best conductor, but the cost of silver excludes the possibility of its mass use. After silver in the table comes copper: 1 m of copper wire with a cross section of 1 mm² has a resistance of 0.0175 Ohm. To get a resistance of 1 ohm, you need to take 57 m of such wire.
Chemically pure copper, obtained by refining, has found widespread use in electrical engineering for the manufacture of wires, cables, windings of electrical machines and devices. Aluminum and iron are also widely used as conductors.
The conductor resistance can be determined by the formula:
Where r– conductor resistance in ohms; ρ – specific resistance of the conductor; l– conductor length in m; S– conductor cross-section in mm².
Example 1. Determine the resistance of 200 m of iron wire with a cross section of 5 mm².
Example 2. Calculate the resistance of 2 km of aluminum wire with a cross section of 2.5 mm².
From the resistance formula you can easily determine the length, resistivity and cross-section of the conductor.
Example 3. For a radio receiver, it is necessary to wind a 30 Ohm resistance from nickel wire with a cross section of 0.21 mm². Determine the required wire length.
Example 4. Determine the cross-section of 20 m of nichrome wire if its resistance is 25 Ohms.
Example 5. A wire with a cross section of 0.5 mm² and a length of 40 m has a resistance of 16 Ohms. Determine the wire material.
The material of the conductor characterizes its resistivity.
Based on the resistivity table, we find that lead has this resistance.
It was stated above that the resistance of conductors depends on temperature. Let's do the following experiment. Let's wind several meters of thin metal wire in the form of a spiral and connect this spiral to the battery circuit. To measure current, we connect an ammeter to the circuit. When the coil is heated in the burner flame, you will notice that the ammeter readings will decrease. This shows that the resistance of a metal wire increases with heating.
For some metals, when heated by 100°, the resistance increases by 40–50%. There are alloys that change their resistance slightly with heating. Some special alloys show virtually no change in resistance when temperature changes. The resistance of metal conductors increases with increasing temperature, while the resistance of electrolytes (liquid conductors), coal and some solids, on the contrary, decreases.
The ability of metals to change their resistance with changes in temperature is used to construct resistance thermometers. This thermometer is a platinum wire wound on a mica frame. By placing a thermometer, for example, in a furnace and measuring the resistance of the platinum wire before and after heating, the temperature in the furnace can be determined.
The change in the resistance of a conductor when it is heated per 1 ohm of initial resistance and per 1° temperature is called temperature coefficient of resistance and is denoted by the letter α.
If at temperature t 0 conductor resistance is r 0 , and at temperature t equals r t, then the temperature coefficient of resistance
Note. Calculation using this formula can only be done in a certain temperature range (up to approximately 200°C).
We present the values of the temperature coefficient of resistance α for some metals (Table 2).
table 2
Temperature coefficient values for some metals
From the formula for the temperature coefficient of resistance we determine r t:
r t = r 0 .
Example 6. Determine the resistance of an iron wire heated to 200°C if its resistance at 0°C was 100 Ohms.
r t = r 0 = 100 (1 + 0.0066 × 200) = 232 ohms.
Example 7. A resistance thermometer made of platinum wire had a resistance of 20 ohms in a room at 15°C. The thermometer was placed in the oven and after some time its resistance was measured. It turned out to be equal to 29.6 Ohms. Determine the temperature in the oven.
Electrical conductivity
So far, we have considered the resistance of a conductor as the obstacle that the conductor provides to the electric current. But still, current flows through the conductor. Therefore, in addition to resistance (obstacle), the conductor also has the ability to conduct electric current, that is, conductivity.
The more resistance a conductor has, the less conductivity it has, the worse it conducts electric current, and, conversely, the lower the resistance of a conductor, the more conductivity it has, the easier it is for current to pass through the conductor. Therefore, the resistance and conductivity of a conductor are reciprocal quantities.
From mathematics it is known that the inverse of 5 is 1/5 and, conversely, the inverse of 1/7 is 7. Therefore, if the resistance of a conductor is denoted by the letter r, then the conductivity is defined as 1/ r. Conductivity is usually symbolized by the letter g.
Electrical conductivity is measured in (1/Ohm) or in siemens.
Example 8. The conductor resistance is 20 ohms. Determine its conductivity.
If r= 20 Ohm, then
Example 9. The conductivity of the conductor is 0.1 (1/Ohm). Determine its resistance
If g = 0.1 (1/Ohm), then r= 1 / 0.1 = 10 (Ohm)
Content:
In electrical engineering, one of the main elements of electrical circuits are wires. Their task is to pass electric current with minimal losses. It has long been determined experimentally that to minimize electricity losses, wires are best made of silver. It is this metal that provides the properties of a conductor with minimal resistance in ohms. But since this noble metal is expensive, its use in industry is very limited.
Aluminum and copper became the main metals for wires. Unfortunately, the resistance of iron as a conductor of electricity is too high to make a good wire. Despite its lower cost, it is used only as a supporting base for power line wires.
Such different resistances
Resistance is measured in ohms. But for wires this value turns out to be very small. If you try to take measurements with a tester in resistance measurement mode, it will be difficult to get the correct result. Moreover, no matter what wire we take, the result on the device display will differ little. But this does not mean that in fact the electrical resistance of these wires will have the same effect on electricity losses. To verify this, you need to analyze the formula used to calculate the resistance:
This formula uses quantities such as:
It turns out that resistance determines resistance. There is a resistance calculated by a formula using another resistance. This electrical resistivity ρ (Greek letter rho) is what determines the advantage of a particular metal as an electrical conductor:
Therefore, if you use copper, iron, silver or any other material to make identical wires or conductors of a special design, the material will play the main role in its electrical properties.
But in fact, the situation with resistance is more complex than simply calculating using the formulas given above. These formulas do not take into account the temperature and shape of the conductor diameter. And with increasing temperature, the resistivity of copper, like any other metal, becomes greater. A very clear example of this would be an incandescent light bulb. You can measure the resistance of its spiral with a tester. Then, having measured the current in the circuit with this lamp, use Ohm’s law to calculate its resistance in the glow state. The result will be much greater than when measuring resistance with a tester.
Likewise, copper will not give the expected efficiency at high currents if the cross-sectional shape of the conductor is neglected. The skin effect, which occurs in direct proportion to the increase in current, makes conductors with a circular cross-section ineffective, even if silver or copper is used. For this reason, the resistance of a round copper wire at high current may be higher than that of a flat aluminum wire.
Moreover, even if their diameter areas are the same. With alternating current, the skin effect also appears, increasing as the frequency of the current increases. Skin effect means the tendency of current to flow closer to the surface of a conductor. For this reason, in some cases it is more profitable to use silver coating of wires. Even a slight reduction in the surface resistivity of a silver-plated copper conductor significantly reduces signal loss.
Generalization of the concept of resistivity
As in any other case that is associated with the display of dimensions, resistivity is expressed in different systems of units. The SI (International System of Units) uses ohm m, but it is also acceptable to use Ohm*kV mm/m (this is a non-systemic unit of resistivity). But in a real conductor, the resistivity value is not constant. Since all materials have a certain purity, which can vary from point to point, it was necessary to create a corresponding representation of the resistance in the actual material. This manifestation was Ohm’s law in differential form:
This law most likely will not apply to household payments. But during the design of various electronic components, for example, resistors, crystal elements, it is certainly used. Since it allows you to perform calculations based on a given point for which there is a current density and electric field strength. And the corresponding resistivity. The formula is used for inhomogeneous isotropic as well as anisotropic substances (crystals, gas discharge, etc.).
How to obtain pure copper
In order to minimize losses in copper wires and cable cores, it must be especially pure. This is achieved by special technological processes:
- based on electron beam and zone melting;
- repeated electrolysis cleaning.
Therefore, it is important to know the parameters of all elements and materials used. And not only electrical, but also mechanical. And have at your disposal some convenient reference materials that allow you to compare the characteristics of different materials and choose for design and work exactly what will be optimal in a particular situation.
In energy transmission lines, where the goal is to deliver energy to the consumer in the most productive way, that is, with high efficiency, both the economics of losses and the mechanics of the lines themselves are taken into account. The final economic efficiency of the line depends on the mechanics - that is, the device and arrangement of conductors, insulators, supports, step-up/step-down transformers, the weight and strength of all structures, including wires stretched over long distances, as well as the materials selected for each structural element. , its work and operating costs. In addition, in lines transmitting electricity, there are higher requirements for ensuring the safety of both the lines themselves and everything around them where they pass. And this adds costs both for providing electricity wiring and for an additional margin of safety of all structures.
For comparison, data are usually reduced to a single, comparable form. Often the epithet “specific” is added to such characteristics, and the values themselves are considered based on certain standards unified by physical parameters. For example, electrical resistivity is the resistance (ohms) of a conductor made of some metal (copper, aluminum, steel, tungsten, gold) having a unit length and a unit cross-section in the system of units of measurement used (usually SI). In addition, the temperature is specified, since when heated, the resistance of the conductors can behave differently. Normal average operating conditions are taken as a basis - at 20 degrees Celsius. And where properties are important when changing environmental parameters (temperature, pressure), coefficients are introduced and additional tables and dependency graphs are compiled.
Types of resistivity
Since resistance happens:
- active - or ohmic, resistive - resulting from the expenditure of electricity on heating the conductor (metal) when an electric current passes through it, and
- reactive - capacitive or inductive - which occurs from the inevitable losses due to the creation of any changes in the current passing through the conductor of electric fields, then the resistivity of the conductor comes in two varieties:
- Specific electrical resistance to direct current (having a resistive nature) and
- Specific electrical resistance to alternating current (having a reactive nature).
Here, type 2 resistivity is a complex value; it consists of two TC components - active and reactive, since resistive resistance always exists when current passes, regardless of its nature, and reactive resistance occurs only with any change in current in the circuits. In DC circuits, reactance occurs only during transient processes that are associated with turning on the current (change in current from 0 to nominal) or turning off (difference from nominal to 0). And they are usually taken into account only when designing overload protection.
In alternating current circuits, the phenomena associated with reactance are much more diverse. They depend not only on the actual passage of current through a certain cross section, but also on the shape of the conductor, and the dependence is not linear.
The fact is that alternating current induces an electric field both around the conductor through which it flows and in the conductor itself. And from this field, eddy currents arise, which give the effect of “pushing” the actual main movement of charges, from the depths of the entire cross-section of the conductor to its surface, the so-called “skin effect” (from skin - skin). It turns out that eddy currents seem to “steal” its cross-section from the conductor. The current flows in a certain layer close to the surface, the remaining thickness of the conductor remains unused, it does not reduce its resistance, and there is simply no point in increasing the thickness of the conductors. Especially at high frequencies. Therefore, for alternating current, resistance is measured in such sections of conductors where its entire section can be considered near-surface. Such a wire is called thin; its thickness is equal to twice the depth of this surface layer, where eddy currents displace the useful main current flowing in the conductor.
Of course, reducing the thickness of round wires does not exhaust the effective conduction of alternating current. The conductor can be thinned, but at the same time made flat in the form of a tape, then the cross-section will be higher than that of a round wire, and accordingly, the resistance will be lower. In addition, simply increasing the surface area will have the effect of increasing the effective cross-section. The same can be achieved by using stranded wire instead of single-core; moreover, stranded wire is more flexible than single-core wire, which is often valuable. On the other hand, taking into account the skin effect in wires, it is possible to make the wires composite by making the core from a metal that has good strength characteristics, for example, steel, but low electrical characteristics. In this case, an aluminum braid is made over the steel, which has a lower resistivity.
In addition to the skin effect, the flow of alternating current in conductors is affected by the excitation of eddy currents in surrounding conductors. Such currents are called induction currents, and they are induced both in metals that do not play the role of wiring (load-bearing structural elements), and in the wires of the entire conductive complex - playing the role of wires of other phases, neutral, grounding.
All of these phenomena occur in all electrical structures, making it even more important to have a comprehensive reference for a wide variety of materials.
The resistivity for conductors is measured with very sensitive and precise instruments, since metals with the lowest resistance are selected for wiring - on the order of ohms * 10 -6 per meter of length and sq. m. mm. sections. To measure insulation resistivity, you need instruments, on the contrary, that have ranges of very large resistance values - usually megohms. It is clear that conductors must conduct well, and insulators must insulate well.
Table
| Table of resistivity of conductors (metals and alloys) |
||||
| Conductor material | Composition (for alloys) | Resistivity ρ mΩ × mm 2/m |
||
| copper, zinc, tin, nickel, lead, manganese, iron, etc. | ||||
| Aluminum | ||||
| Tungsten | ||||
| Molybdenum | ||||
| copper, tin, aluminum, silicon, beryllium, lead, etc. (except zinc) | ||||
| iron, carbon | ||||
| copper, nickel, zinc | ||||
| Manganin | copper, nickel, manganese | |||
| Constantan | copper, nickel, aluminum | |||
| nickel, chromium, iron, manganese | ||||
| iron, chromium, aluminum, silicon, manganese | ||||
Iron as a conductor in electrical engineering
Iron is the most common metal in nature and technology (after hydrogen, which is also a metal). It is the cheapest and has excellent strength characteristics, therefore it is used everywhere as the basis for the strength of various structures.
In electrical engineering, iron is used as a conductor in the form of flexible steel wires where physical strength and flexibility are needed, and the required resistance can be achieved through the appropriate cross-section.
Having a table of resistivities of various metals and alloys, you can calculate the cross-sections of wires made from different conductors.
As an example, let's try to find the electrically equivalent cross-section of conductors made of different materials: copper, tungsten, nickel and iron wire. Let's take aluminum wire with a cross-section of 2.5 mm as the initial one.
We need that over a length of 1 m the resistance of the wire made of all these metals is equal to the resistance of the original one. The resistance of aluminum per 1 m length and 2.5 mm section will be equal to
Where R- resistance, ρ – resistivity of the metal from the table, S- cross-sectional area, L- length.
Substituting the original values, we get the resistance of a meter-long piece of aluminum wire in ohms.
After this, let us solve the formula for S
We will substitute the values from the table and obtain the cross-sectional areas for different metals.
Since the resistivity in the table is measured on a wire 1 m long, in microohms per 1 mm 2 section, then we got it in microohms. To get it in ohms, you need to multiply the value by 10 -6. But we don’t necessarily need to get the number ohm with 6 zeros after the decimal point, since we still find the final result in mm2.
As you can see, the resistance of the iron is quite high, the wire is thick.
But there are materials for which it is even greater, for example, nickel or constantan.