Impedance: general information, dependence on other quantities and calculation formulas

Any substance, being in various states, has a certain resistance. In some cases, it becomes necessary to calculate the impedance of a circuit or a specific section. In such a situation, you should use the formulas. In addition, you need to understand the basic meaning of resistance and electrical conductivity, as well as the dependence of these concepts on certain quantities.

Physical sense

All substances by the conductivity of electric current (ET) are divided into conductors, semiconductors and dielectrics. The conductors are elements that conduct ET well. This is due to the presence of free electrons (FE). Semiconductors are a special group of substances whose conductivity depends on external factors, for example, temperature, illumination, etc. Dielectrics are all substances that do not conduct ET due to the absence or insufficient amount of SE. For the current to flow through the substance, the presence of FEs is required, the number of which depends on the electronic configuration.

The electronic configuration of any element is taken from the periodic table. The current has a thermal effect on the conductor, since the FE interacts with the crystal lattice (CR). They slow down, but over time, under the influence of the electromagnetic field, they accelerate again, after which the process of interaction is repeated many times. once.

The process of interaction of free charged particles with the CR of a substance is called the electrical resistance of the conductor. Resistance or electrical conductivity is indicated by the letter R, the unit of measurement of this quantity is Ohm.

Dependence of electrical conductivity

R depends on external environmental factors, electrical quantities, as well as the characteristics of the conductor. These dependencies are used in the calculation of circuits and the manufacture of radio components. There are several ways to find R, and sometimes they are combined for efficiency and computational accuracy.

Electrical quantities

The electrical quantities on which the value of R depends include I, U, electromotive force (EDS is denoted e) and the type of current. R in electrical circuits is calculated according to Ohm's law for a certain section of the circuit: I, flowing in a given section of the electrical circuit, is directly proportional to U in this section and inversely proportional to R of the selected section chains. In the form of a formula, it can be written as follows: I = U / R.

Based on the consequence of this law, you can get the resistance of the circuit section: R = U / I. If you want to calculate R over the entire section of the circuit, then you need to use the formula (a consequence of Ohm's law for a complete circuit), taking into account the internal R of the power source: R = (e / I) - R internal. The value of electrical conductivity is calculated not only using Ohm's laws, but also using the geometric parameters of the conductor and temperature. In addition, it is necessary to take into account the type of current (direct or alternating).

Geometric parameters and type of substance

If the main charge carriers are FE, and the conductivity properties are directly proportional to on their number and structure of the RR, then the type of substance is one of the factors affecting R conductor. Substances and their constituent elements, which have different electronic configurations, according to the periodic table, have different CRs, which is due to different R.

Material dependence is expressed by a coefficient denoted p. It characterizes the specific R index of the conductor. Its value is taken from the table (at a temperature of +20 ° C). The reciprocal of p is called the conductivity and is denoted σ. The relationship between σ and p can be expressed by the formula p = 1 / σ.

In addition, the R of the conductor also depends on the cross-sectional area (S). This dependence is due to the fact that at a small cross-section, the flux density E flows through the conductor and the interaction with the RR becomes more frequent. The cross-sectional area is fairly easy to calculate. To do this, you need to use some algorithmif the conductor (P) is a cylindrical wire:

1. Measurement of the conductor diameter using a vernier caliper (ШЦ).
2. Finding S using the formula S = 3.1416 * sqr (d) / 4.

P can be a stranded wire, therefore, for an accurate calculation, it is necessary to find S of one core, using the algorithm for finding a cylindrical shape P, and then multiply the result by the number lived.

In addition, there are square and rectangular wires, but these are rare. To do this, you need to perform the following calculations:

1. For a square shape, you need to measure the SC one of the sides and square it: S = sqr (a).
2. For a rectangular shape, two opposite sides should be measured using the SHTs, and then the calculation should be made using the formula S = a * b.

From these algorithms for finding S, a universal (abstract algorithm) can be made. It is suitable for finding or calculating values, regardless of the shape of P when it is cut strictly perpendicular to P. The algorithm looks like this:

1. Visually determine the geometric figure when cutting P.
2. Find the S formula in the reference.
3. Carry out measurements with the help of the SHTs of the required values.
4. Substitute in the formula and calculate S.

Another quantity is the length P, with an increase in which R increases. Based on these values, the following formula can be derived for the dependence on the type of substance, length (L) and S of the conductor: R = p * L / S.

However, this R-value can be determined at + 20 ° C. To obtain more accurate calculations, you need to consider the dependence on temperature.

Conductor temperature

Scientifically confirmed is the fact that p depends on temperature. This statement can be proved in a practical way. To conduct the experiment, the following elements are required, shown in the diagram: a nichrome spiral (used in heating elements), connecting copper wires, power supply, ammeter (for measuring I), voltmeter (measures U) and rheostat.

In the diagram, the heating element is shown as a resistor. When you turn it on, you should carefully observe the readings of the ammeter. The coil begins to heat up and the ammeter reading decreases as it warms up. According to Ohm's law for a section of the circuit, it is necessary to conclude that with an increase in R, the current decreases (inversely proportional dependence). Therefore, the R value is temperature dependent. When heated, there is an increase in ions in the RC of the nichrome spiral and E begin to collide with them more often.

In the formula R = p * L / S, you can use the elimination method to find an indicator that depends on temperature. The latter does not affect the length of P. According to the formula for calculating S, the dependence is also not traced, since the geometry of P does not depend on temperature. There remains p, which depends on the temperature. In physics, there is a dependence formula p = p0 * [1 + a * (t - 20)]. Letter a is the temperature coefficient:

• for metals a> 0;
• for electrolytes a <0.

The variable t is the temperature P, p0 is the resistivity taken from the reference book for a particular material. In addition, p also depends on the deformation P, since in this case the RR changes its structure. This occurs during metal processing at low temperatures and pressures. Such deformation is plastic, with it, the CR is distorted, and the R of the flow E.

In this case, p increases. The process is reversible; therefore, some of the defects are reduced (recrystalline annealing). If tensile or compressive forces act on the metal, then this deformation is elastic. The value of p decreases under the action of the compression force, at which there is a sharp decrease in thermal vibrations (TC), and E move more easily. But under the action of the tensile force, a directly proportional increase in p occurs, at which the TC amplitude increases.

The final formula can be written as R = p0 * [1 + a * (t - 20)] * L / S. However, this variant of finding R was considered in circuits with constant I, and under the influence of variable I, new quantities appear that affect the calculations.

AC circuit

Ohm's law applies only to DC circuits. For the variable U, it has been changed and, therefore, there are other formulas for finding R. Resistance in circuits with variable I (PT) is:

• active;
• inductive;
• capacitive;
• complete.

Resistance indicates that a resistor or any other non-capacitive or non-inductive load is present in the circuit. To calculate it, it is necessary to measure the values ​​of the amplitudes Um and Im. Only the actual values ​​of these quantities can be obtained with the instruments. Amplitude values ​​are calculated by the formulas Um = Ud * sqrt (2) and Im = Id * sqrt (2). To determine the active resistance (denoted by R), you need to use the formula Im = Um / R. From it you can get R = Ud * sqrt (2) / Im = Id * sqrt (2).

If there is an inductor, a choke, a circuit, etc. in the variable I circuit (CLT), then an inductive R appears, which is denoted Xl. To calculate, you must use the formula Xl = w * L, having previously measured the frequency of the FET and calculated the inductance.

The value of the cyclic frequency is found by the formula for which you need to measure the frequency of the FET (f): w = 2 * 3.1416 * f. The latter is measured using an oscilloscope or frequency meter. To calculate the inductance of a coil, you must use a physics reference or an online calculator.

If there is a capacitor (capacitor) in the CPT, a capacitive R appears, which is denoted by Xc. At flowing constant U, the capacitor does not allow I, and in the central heating circuit it passes I and has a capacity (C) and Xc. This value is calculated by the formula Xc = 1 / (w * C), where:

• w is the cyclic frequency, which is calculated similarly to the calculation of Xl;
• C is the capacitance of the capacitor, indicated on the case or measured with a suitable device.

The impedance of the circuit is indicated by Z and is the sum of the entire load of the CLT (active, inductive and capacitive resistance). To calculate, you need to use the impedance formula: Z = sqrt [sqr (R) + sqr (Xc - Xl)]. In the CPT, the value of Z depends on:

• geometry P;
• the type of substance from which the P is made;
• temperature;
• deformations of various types;
• electrical indicators I, U, f, L, C and R.

Ohm's law for a section of the chain takes the following form: I = U / Z. It is not necessary to calculate the electrical conductivity P, since there are ohmmeters for this purpose. Calculation of Xl and Xc should be done independently.

Resistance measurement

It takes some time to calculate R. This task is simplified by a device called an ohmmeter. It consists of a digital or dial indicator. Almost all modern combined instruments (multimeters) are equipped with the R measurement function. However, there are also specialized devices that are used for specific purposes, for example, to measure R insulation of cable conductors. This type of device is called a megohmmeter. An ohmmeter is used not only to measure the value of R, but also to call radio components, cables, individual loops and other elements for serviceability and open circuit.

To measure R, it is necessary to de-energize the section or radio component and monitor the discharge of circuits where capacitors are present. Before taking measurements, you need to set the required mode on the device and short-circuit the probes to check the device for serviceability. Some models are equipped with a buzzer function. After testing the device, you should start measuring.

To find the exact value of Z of the conductor, it is necessary to take into account all the quantities on which it depends. Calculating Z allows you to accurately calculate the electrical circuit of a device to avoid time-consuming measurements. You can measure with an ohmmeter only the value of the active resistance, and Xl and Xc should be calculated independently. However, with the help of online calculators, this will not be difficult.