Types of reactances. Electrical resistance. Definition, units of measurement, specific, total, active, reactive. Reactive power compensation

To calculate voltages and currents through the elements of an electrical circuit, you need to know their total resistance. Energy sources are divided into two types:

direct current(batteries, rectifiers, accumulators), the electromotive force (EMF) of which does not change over time;
alternating current(domestic and industrial networks), the EMF of which changes according to a sinusoidal law with a certain frequency.

Active and reactive resistances

Load resistance can be active or reactive. Active resistance(R) does not depend on the network frequency. This means that the current in it changes synchronously with the voltage. This is the resistance that we measure with a multimeter or tester.

Reactance is divided into two types:

— inductive(transformers, chokes);

— capacitive(capacitors).

A distinctive feature of a reactive load is the presence of a leading or lagging current versus voltage. In a capacitive load, the current leads the voltage, and in an inductive load, it lags behind it. Physically, it looks like this: if a discharged capacitor is connected to a direct current source, then at the moment of switching on, the current through it is maximum, and the voltage is minimum. Over time, the current decreases and the voltage increases until the capacitor is charged. If you connect a capacitor to an AC source, it will constantly recharge at the mains frequency, and the current will increase before the voltage.

By connecting an inductance to a direct current source, we get the opposite result: the current through it will increase for some time after the voltage is connected.

The amount of reactance depends on frequency. Capacitance:

Angular frequency related to network frequency f formula:

As can be seen from the formula, as the frequency increases, the capacitance decreases.

AC circuit impedance

In an AC network there is no load that is only active or only reactive. In addition to the active element, the heating element contains inductive resistance; in an electric motor, inductive resistance prevails over active resistance.

The value of total resistance, taking into account all active and reactive components of the electrical circuit, is calculated using the formula:

Calculation of equivalent resistance of circuit elements

Several resistors can be connected to one power source. To calculate the source load current, the equivalent load resistance is calculated. Depending on how the elements are connected to each other, two methods are used.

Series connection of resistances.

In this case, their values add up:

The more resistances connected in series, the greater the equivalent resistance of this circuit. A household example: if the contact in the plug deteriorates, this is equivalent to connecting additional resistance in series with the load. The equivalent load resistance will increase, and the current through it will decrease.

Parallel connection of resistances.

The calculation formula looks much more complicated:

The case of applying this formula for two parallel-connected resistances:

Case for connection n identical resistances R:

The more resistances you connect in parallel, the lower the final resistance of the circuit. We see this in everyday life: the more consumers connected to the network, the lower the equivalent resistance and the higher the load current.

Thus, calculation of electrical circuit impedance happens in stages:

An equivalent circuit is drawn containing active and reactive resistances.
Equivalent resistances are calculated separately for the active, inductive and capacitive components of the load.
The total resistance of the electrical circuit is calculated
Currents and voltages in the power supply circuit are calculated.

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In electrical engineering, active and reactive resistance is usually called a quantity that characterizes the resistance force of a section of an electrical circuit to the directed (ordered) movement of particles or quasiparticles - electric charge carriers. This counteraction is formed by the method of converting electricity into other forms of energy. In the event of an irreversible change in the electrical energy of a chain link into other types of energy, the counteraction will be active.

Features of active and reactive resistance

An alternating current network has irreversible transformation and transfer of energy to the elements of the electrical circuit. Carrying out the exchange process of electricity with the circuit components and the power source, the resistance will be reactive.

If we consider a microwave oven as an example, electrical energy in it is irreversibly converted into heat, as a result of which the microwave oven receives active counteraction, as well as elements that transform electrical energy into light, mechanical, etc.

Alternating current passing through lumped electrical elements forms reactance, which is caused mainly by inductance and capacitance.

Active resistance is directly dependent on the number of complete cycles of change in electromotive force (EMF) occurring in one second. The larger this number, the higher the active resistance.

However, many consumers have inductive and capacitive properties when alternating current passes through them . These include:

capacitors;
chokes;
electromagnets;
transformers.

It is necessary to take into account both active and reactive resistance, which is due to the presence of capacitive and inductive characteristics in the electrical consumer. By interrupting and closing the direct current circuit passing through any of the windings, in parallel with the transformation of the current, a change in the magnetic flux inside the winding itself will occur, as a result, an electromotive force of self-induction appears in it.

A similar situation will manifest itself in the winding connected to a circuit with alternating current, with the only difference that in this case the current continuously changes both in parameter and in direction. It follows that the parameter of the magnetic flux penetrating into the winding in which the electromotive force of self-induction is induced will continuously change.

At the same time, the vector of the electromotive force is invariably such that it prevents the transformation of current. Consequently, when increasing inside the winding, the electromotive force of self-induction will aim to stop the increase in current, and when decreasing, on the contrary, it will try to maintain the decreasing current.

It turns out that the EMF appearing inside the conductor (winding) involved in the alternating current circuit will constantly oppose the current, preventing it from changing. In other words, EMF can be regarded as an auxiliary resistance, which, together with the active resistance of the coil, creates a synergistic effect of opposing the alternating current flowing through the coil.

Electrotechnical law of reactance

The formation of reactance occurs with the help of a decrease in reactive power spent on creating an electromagnetic field in an electrical circuit. The reactive power decline is formed by connecting a device with active resistance to the converter.

A two-terminal device connected to a circuit can accumulate only a limited portion of the charge before the voltage polarity changes to the diametrically opposite one. Thanks to this, the electric current does not drop to zero, as in DC circuits. The accumulation of charge by a capacitor directly depends on the frequency of the electric current.

The reactance formula determines the imaginary part of the impedance:

Z = R+jX, where Z is complex electrical resistance, R is active electrical resistance, X is reactive electrical resistance, j is an imaginary unit.

The value of electrical reactance can be expressed through the values of capacitive and inductive resistance.

Electrical impedance

The total resistance of an alternating current circuit or impedance is a reflection of the current value transforming over time. In electrical engineering literature it is denoted by the Latin letter Z. Impedance is a two-dimensional (vector) quantity that includes two independent scalar one-dimensional characteristics: active and reactive opposition to alternating electric current. Simply put, impedance is the sum of active and reactive.

The active component of impedance, denoted R, is a measure of the rate at which a material resists the flow of negatively charged particles among its atoms. Low-resistivity materials are considered to be:

gold;
silver;
copper.

High-resistance materials are called dielectrics or insulators. The list of such materials includes:

polyethylene;
mica;
plexiglass.

Substances with an intermediate degree of resistance are classified as semiconductors. This group includes:

The total resistance is calculated by the formula: Z = √ R 2 + (XL - XC) 2, where: R - active electrical resistance; XL - inductive reactance, unit of measurement Ohm; XC - capacitive reaction, unit of measurement Ohm. Full resistance is calculated step by step. First, a circuit is drawn, then the equivalent resistances are calculated individually for the active, inductive and capacitive components of the load, and the total resistance of the electrical circuit is calculated.

Active and reactive resistance

The resistance provided by passages and consumers in DC circuits is called ohmic resistance.

If any conductor is connected to an alternating current circuit, it will turn out that its resistance will be slightly greater than in a direct current circuit. This is explained by a phenomenon called the skin effect ().

Its essence is as follows. When alternating current passes through a conductor, there is an alternating magnetic field inside it that crosses the conductor. The magnetic lines of force of this field induce an emf in the conductor, but it will not be the same at different points of the conductor cross-section: more towards the center of the cross-section, and less towards the periphery.

This is explained by the fact that points located closer to the center are intersected by a large number of lines of force. Under the influence of this EMF, the alternating current will not be distributed evenly over the entire cross-section of the conductor, but closer to its surface.

This is equivalent to a decrease in the useful cross-section of the conductor, and therefore an increase in its resistance to alternating current. For example, a copper wire 1 km long and 4 mm in diameter has a resistance of: direct current - 1.86 ohms, alternating current with a frequency of 800 Hz - 1.87 ohms, alternating current with a frequency of 10,000 Hz - 2.90 ohms.

The resistance offered by a conductor to the alternating current passing through it is called active resistance.

If any consumer does not contain inductance and capacitance (an incandescent light bulb, a heating device), then it will also act as an active resistance for alternating current.

Active resistance- a physical quantity characterizing the resistance of an electrical circuit (or its section) to electric current, due to the irreversible transformations of electrical energy into other forms (mainly thermal). Expressed in ohms.

Active resistance depends on , increasing with its increase.

However, many consumers exhibit inductive and capacitive properties when alternating current passes through them. Such consumers include transformers, chokes, various kinds of wires and many others.

When passing through them, it is necessary to take into account not only the active, but also reactance, due to the presence of inductive and capacitive properties in the consumer.

It is known that if the direct current passing through any winding is interrupted and short-circuited, then simultaneously with the change in the current, the magnetic flux inside the winding will also change, as a result of which a self-inductive emf will arise in it.

The same thing will be observed in a winding connected to an alternating current circuit, with the only difference being that here the current continuously changes both in magnitude and direction. Consequently, the magnitude of the magnetic flux penetrating the winding will continuously change and will be induced in it.

But the direction of the self-induction EMF is always such that it opposes the change in current. Thus, when the current in the winding increases, the self-induction EMF will tend to delay the increase in current, and when the current decreases, on the contrary, it will tend to support the disappearing current.

It follows that the self-induction emf that occurs in a winding (conductor) connected to an alternating current circuit will always act against the current, delaying its changes. In other words, the self-induction EMF can be considered as an additional resistance that, together with the active resistance of the winding, resists the alternating current passing through the winding.

The resistance provided to alternating current by the self-inductive emf is called inductive reactance.

The greater the inductance of the consumer (circuit) and the higher the frequency of the alternating current, the greater the inductive reactance. This resistance is expressed by the formula xl = ωL, where xl is the inductive reactance in ohms; L - inductance in Henry (H); ω - angular frequency where f - current frequency).

In addition to inductive reactance, there is capacitance, caused both by the presence of capacitance in conductors and windings, and in some cases by the inclusion of capacitors in the alternating current circuit. As the capacitance C of the consumer (circuit) increases and the angular frequency of the current increases, the capacitance decreases.

The capacitive reactance is equal to xc = 1/ωC, where xc is the capacitive reactance in ohms, ω is the angular frequency, C is the consumer capacitance in farads.

Triangle of resistances

Consider a circuit whose elements have active resistance r, inductance L, and capacitance C.

Rice. 1. AC circuit with resistor, inductor and capacitor.

The total resistance of such a circuit is z = √ r 2+ (x l - xc) 2) = √r 2 + x2)

Graphically, this expression can be depicted in the form of a so-called resistance triangle.

Fig.2. Triangle of resistances

The hypotenuse of the resistance triangle represents the total resistance of the circuit, the legs represent the active and reactive resistance.

If one of the circuit resistances (active or reactive), for example, is 10 or more times less than the other, then the smaller one can be neglected, which is easy to verify by direct calculation.

In previous articles, we learned that any resistance that absorbs energy is called active, and resistance that does not absorb energy is called wattless or reactive. In addition, we have established that reactances are divided into two types - inductive and capacitive.

However, there are circuits where the resistance is not purely active or purely reactive. That is, circuits where, together with active resistance, both capacitance and inductance are included in the circuit.

Let's introduce the concept AC circuit impedance - Z, which corresponds to the vector sum of all circuit resistances (active, capacitive and inductive). We need the concept of circuit impedance for a more complete understanding of Ohm's law for alternating current

Figure 1 shows options for electrical circuits and their classification depending on which elements (active or reactive) are included in the circuit.

Picture 1. Classification of alternating current circuits.

The total resistance of a circuit with purely active elements corresponds to the sum of the active resistances of the circuit and was considered by us earlier. We also talked about the purely capacitive and inductive reactance of the circuit, and it depends, respectively, on the total capacitance and inductance of the circuit.

Let's consider more complex circuit options, where inductive and reactive reactance are connected in series with active resistance.

The total resistance of a circuit with a series connection of active and reactive resistance.

In any section of the circuit shown in Figure 2a, the instantaneous current values must be the same, since otherwise accumulations and rarefaction of electrons would be observed at some points in the circuit. In other words, the phases of the current along the entire length of the circuit must be the same. In addition, we know that the voltage phase across the inductive reactance leads the current phase by 90°, and the voltage phase across the active resistance coincides with the current phase (Figure 2b). It follows that the radius vector of voltage U L (voltage across the inductive reactance) and voltage U R (voltage across the active resistance) are shifted relative to each other by an angle of 90°.

Figure 2. The impedance of a circuit with active resistance and inductance.a) - circuit diagram; b) - phase shift of current and voltage; c) - stress triangle; e) - resistance triangle.

To obtain the radius vector of the resulting voltage at terminals A and B (Fig. 2,a), we will perform a geometric addition of the radius vectors U L and U R . This addition is performed in Fig. 2,c, from which it is clear that the resulting vector U AB is the hypotenuse of a right triangle.

From geometry it is known that the square of the hypotenuse is equal to the sum of the squares of the legs.

According to Ohm's law, voltage must equal current times resistance.

Since the current strength at all points of the circuit is the same, the square of the total resistance of the circuit (Z 2) will also be equal to the sum of the squares of the active and inductive resistance, i.e.

(1)

Taking the square root of both sides of this equality, we get,

(2)

Thus, the total resistance of the circuit shown in Fig. 2a is equal to the square root of the sum of the squares of the active and inductive resistance

The total resistance can be found not only by calculation, but also by constructing a resistance triangle, similar to the voltage triangle (Figure 2,e), i.e. the total resistance of the circuit to alternating current can be obtained by measuring the hypotenuse, a right triangle, the legs of which are active and reactance. Of course, measurements of the legs and hypotenuse must be made on the same scale. So, for example, if we agreed that 1 cm of the length of the legs corresponds to 1 ohm, then the number of ohms of total resistance will be equal to the number of centimeters that fit on the hypotenuse.

The total resistance of the circuit shown in Fig. 2a is neither purely active nor purely reactive; it contains both of these types of resistance. Therefore, the phase angle of current and voltage in this circuit will differ from both 0° and 90°, that is, it will be greater than 0°, but less than 90°. Which of these two values it is closer to will depend on which of these resistances is dominant in the circuit. If the inductive reactance is greater than the active one, then the phase angle will be closer to 90°, and vice versa, if the active resistance is predominant, then the phase angle will be closer to 0°.

In the circuit shown in Fig. 3a, active and capacitive resistances are connected in series. The total resistance of such a circuit can be determined using a resistance triangle in the same way as we determined above the total resistance of an active-inductive circuit.

Figure 3. Circuit impedance with active resistance and capacitance. .

The only difference between both cases is that the resistance triangle for the active-capacitive circuit will be turned in the other direction (Figure 3, b) due to the fact that the current in the capacitive circuit does not lag behind the voltage, but leads it.

For this case:

(3)

In the general case, when a circuit contains all three types of resistance (Fig. 4a), the reactance of this circuit is first determined, and then the total resistance of the circuit.

Figure 4. Impedance of a circuit containing R, L and C. a) - circuit diagram; b) - resistance triangle .

The reactance of this circuit consists of inductive and capacitive reactance. Since these two types of reactance are opposite in nature, the total reactance of the circuit will be equal to their difference, i.e.

(4)

The total reactance of the circuit can be inductive or capacitive, depending on which of these two resistances (X L or X C) predominates.

After we have determined the total reactance of the circuit using formula (4), determining the total resistance will not present any difficulties. The total resistance will be equal to the square root of the sum of the squares of the active and reactive resistance, i.e.

(5)

(6)

The method for constructing a resistance triangle for this case is shown in Fig. 4 b.

The total resistance of a circuit with a parallel connection of active and reactive resistance.

The total resistance of the circuit when the active and reactive elements are connected in parallel.

In order to calculate the total resistance of a circuit composed of active and inductive resistances connected to each other in parallel (Fig. 5, a), you must first calculate the conductivity of each of the parallel branches, then determine the total conductivity of the entire circuit between points A and B and then calculate the total resistance of the circuit between these points.

Figure 5. Circuit impedance when connecting active and reactive elements in parallel. a) - parallel connection R and L; b) - parallel connection R and C .

The conductivity of the active branch, as is known, is equal to 1/R, similarly, the conductivity of the inductive branch is equal to 1/ωL, and the total conductance is equal to 1/Z

Total conductivity is equal to the square root of the sum of the squares of active and reactive conductivity, i.e.

(7)

Reducing the radical expression to a common denominator, we get:

(8)

(9)

Formula (9) is used to calculate the total resistance of the circuit shown in Fig. 5a.

Finding the total resistance for this case can also be done geometrically. To do this, you need to construct a resistance triangle on the appropriate scale, and then divide the product of the lengths of the legs by the length of the hypotenuse. The result obtained will correspond to the total resistance.

Similar to the case discussed above, the total resistance with a parallel connection of R and C (Fig. 5b) will be equal to:

(10)

The total resistance can also be found in this case by constructing a resistance triangle.

In radio engineering, the most common case is the parallel connection of inductance and capacitance, for example, an oscillatory circuit for tuning receivers and transmitters. Since the inductor always has, in addition to inductive resistance, also active resistance, the equivalent (equivalent) circuit of the oscillatory circuit will contain active resistance in the inductive branch (Fig. 7).

Figure 6. Equivalent circuit of an oscillatory circuit.

The impedance formula for this case will be:

(11)

Since usually the active resistance of the coil (R) is very small compared to its inductive resistance (ωL), we have the right to rewrite formula (11) in the following form:

(12)

In an oscillatory circuit, the values of L and C are usually selected so that the inductive reactance is equal to the capacitive reactance, i.e., so that the condition is met

(13)