Calculation of electrical circuits: what determines the resistance of the conductor, formulas for calculating

One of the physical properties of a substance is the ability to conduct electric current. Electrical conductivity (conductor resistance) depends on several factors: the length of the electrical circuit, structural features, the presence of free electrons, temperature, current, voltage, material and area cross section.

The physical meaning of resistance

The flow of electric current through the conductor leads to the directional movement of free electrons. The presence of free electrons depends on the substance itself and is taken from Table D. AND. Mendeleev, namely from the electronic configuration of the element. Electrons start to hit crystal lattice element and transfer energy to the latter. In this case, a thermal effect arises when the current acts on the conductor.

With this interaction, they slow down, but then, under the action of an electric field that accelerates them, they begin to move at the same speed. Electrons collide a huge number of times. This process is called the resistance of the conductor.

Consequently, the electrical resistance of a conductor is a physical quantity that characterizes the ratio of voltage to current.

What is electrical resistance: a value indicating the property of a physical body to transform electrical energy into thermal energy, due to the interaction of electron energy with the crystal lattice substances. By the nature of conductivity, they differ:

1. Conductors (capable of conducting electrical current as free electrons are present).
2. Semiconductors (can conduct electricity, but under certain conditions).
3. Dielectrics or insulators (have a huge resistance, there are no free electrons, which makes them unable to conduct current).

This characteristic is denoted by the letter R and measured in ohms (ohms). The use of these groups of substances is very important for the development of electrical schematic diagrams of devices.

To fully understand the dependence of R on something, you need to pay special attention to the calculation of this value.

Calculation of electrical conductivity

To calculate the R of the conductor, Ohm's law is applied, which states: the current (I) is directly proportional to the voltage (U) and inversely proportional to the resistance.

The formula for finding the conductivity characteristic of a material R (a consequence of Ohm's law for a section of a circuit): R = U / I.

For a complete section of the circuit, this formula takes the following form: R = (U / I) - Rvn, where Rvn is the internal R of the power source.

Dependence of material conductivity

The ability of a conductor to transmit electric current depends on many factors: voltage, current, length, cross-sectional area and conductor material, as well as the ambient temperature Wednesday.

In electrical engineering, for the calculation and manufacture of resistors, the geometric component of the conductor is also taken into account.

What the resistance depends on: on the length of the conductor - l, resistivity - p and on the cross-sectional area (with radius r) - S = Pi * r * r.

Conductor R formula: R = p * l / S.

The formula shows what determines resistivity of the conductor: R, l, S. There is no need to calculate it this way, because there is a much better way. Resistivity can be found in the appropriate reference books for each type of conductor (p is a physical quantity equal to R of a material 1 meter long and a cross-sectional area of ​​1 m2.

However, this formula is not enough for an accurate calculation of the resistor, therefore, the dependence on temperature is used.

Influence of ambient temperature

It has been proven that each substance has a resistivity that depends on temperature.

To demonstrate this, the following experiment can be performed. Take a spiral from nichrome or any conductor (indicated in the diagram as a resistor), a power source and a conventional ammeter (it can be replaced with an incandescent lamp). Assemble the chain according to diagram 1.

Scheme 1 - Electrical circuit for the experiment

It is necessary to power the consumer and carefully monitor the ammeter readings. Next, you should heat R without turning it off, and the ammeter readings will begin to drop with increasing temperature. The dependence is traced according to Ohm's law for the chain section: I = U / R. In this case, the internal resistance of the power supply can be neglected: this will not affect the demonstration of the dependence of R on temperature. It follows that there is a temperature dependence of R.

The physical meaning of an increase in the value of R is due to the effect of temperature on the vibration amplitude (increase) of ions in the crystal lattice. As a result, electrons collide more often and this causes an increase in R.

According to the formula: R = p * l / S, we find the indicator that depends on temperature (S and l are temperature independent). There remains p conductor. Based on this, the formula for the dependence on temperature is obtained: (R - Ro) / R = a * t, where Ro at a temperature of 0 degrees Celsius, t is the ambient temperature and a is the proportionality coefficient (temperature coefficient).

For metals, "a" is always greater than zero, and for electrolyte solutions, the temperature coefficient is less than 0.

The formula for finding p used in the calculations: p = (1 + a * t) * po, where po is the specific resistance value taken from the reference book for a particular conductor. In this case, the temperature coefficient can be considered constant. The dependence of power (P) on R follows from the power formula: P = U * I = U * U / R = I * I * R. The specific resistance value also depends on the deformations of the material, at which the crystal lattice is broken.

Deformation and resistivity

When metal is processed in a cold environment at a certain pressure, plastic deformation occurs. In this case, the crystal lattice is distorted and the R of the electron flow increases. In this case, the resistivity also increases. This process is reversible and is called recrystalline annealing, due to which some of the defects are reduced.

When tensile and compressive forces act on the metal, the latter undergoes deformations, which are called elastic. The resistivity decreases with compression, since the amplitude of thermal vibrations decreases. Directed charged particles it becomes easier to move. When stretched, the resistivity increases due to an increase in the amplitude of thermal vibrations.

Another factor that affects conductivity is the type of current flowing through the conductor.

AC circuits

Resistance in networks with alternating current behaves somewhat differently, because Ohm's law is applicable only for circuits with constant voltage. Therefore, the calculations should be done differently.

Impedance is denoted by the letter Z and consists of the algebraic sum of active, capacitive and inductive resistance.

When active R is connected to an alternating current circuit, a sinusoidal current begins to flow under the influence of a potential difference. In this case, the formula looks like: Im = Um / R, where Im and Um are the amplitude values ​​of the current and voltage. The resistance formula takes the following form: Im = Um / ((1 + a * t) * po * l / 2 * Pi * r * r).

Capacitive resistance (Xc) is due to the presence of capacitors in the circuits. It should be noted that an alternating current passes through the capacitors and, therefore, it acts as a conductor with a capacitance.

Calculate Xc as follows: Xc = 1 / (w * C), where w is the angular frequency and C is the capacitance of a capacitor or group of capacitors. The angular frequency is determined as follows:

1. The frequency of the alternating current is measured (typically 50 Hz).
2. Multiplied by 6.283.

Inductive resistance (Xl) - implies the presence of inductance in the circuit (choke, relay, circuit, transformer, and so on). Calculated as follows: Xl = wL, where L is inductance and w is angular frequency. To calculate inductance it is necessary to use specialized online calculators or a physics reference book. So, all the values ​​are calculated by the formulas and it only remains to write down Z: Z * Z = R * R + (Xc - Xl) * (Xc - Xl).

To determine the final value, you need to extract the square root of the expression: R * R + (Xc - Xl) * (Xc - Xl). It follows from the formulas that the frequency of the alternating current plays an important role, for example, in a circuit of the same design, with an increase in frequency, its Z also increases. It must be added that in circuits with alternating voltage Z depends on the following indicators:

1. Conductor lengths.
2. Sectional areas - S.
3. Temperatures.
4. The type of material.
5. Capacities.
6. Inductance.
7. Frequencies.

Consequently, Ohm's law for a section of the chain has a completely different form: I = U / Z. The law for the complete chain also changes.

Measurement of electrical conductivity

Calculations of resistances require a certain amount of time, therefore, to measure their values, special electrical measuring instruments are used, which are called ohmmeters. The measuring device consists of a dial indicator, to which a power supply is connected in series.

Measure R all combined appliancessuch as testers and multimeters. Separate instruments for measuring only this characteristic are used extremely rarely (a megohmmeter for checking the insulation of a power cable).

The device is used for continuity of electrical circuits for damage and serviceability of radio components, as well as for continuity of cable insulation.

When measuring R, it is necessary to completely de-energize the section of the circuit to avoid damage to the device. To do this, you must take the following precautions:

1. Unplug the mains plug.
2. Turn on the device, and the capacitors will discharge.
3. Start measuring or dialing.
4. Set the switch to resistance measurement mode.
5. Short-circuit the test leads of the device to make sure it works (it will show very low resistance).
6. Measure the required area.

In expensive multimeters there is a continuity function, duplicated by a sound signal, so there is no need to look at the instrument display.

Thus, electrical resistance plays an important role in electrical engineering. It depends in permanent circuits on temperature, current strength, length, type of material and area transverse conductor cross-section. In alternating current circuits, this dependence is supplemented by such quantities as frequency, capacitance and inductance. Thanks to this dependence, it is possible to change the characteristics of electricity: voltage and current. Ohmmeters are used to measure the resistance value, which are also used when detecting wiring problems, continuity of various circuits and radio components.