We know that the self-induction current of the coil goes to meet the growing current of the generator. This the resistance of the self-induction current of the coil to the rising current of the generator is called inductive resistance.
To overcome this opposition, part of the generator's alternating current energy is spent. All this part of the energy is completely converted into the energy of the magnetic field of the coil. When the generator current decreases, the magnetic field of the coil will also decrease, interrupting the coil and inducing a self-induction current in the circuit. Now the self-induction current will go in the same direction as the decreasing generator current.
Thus, all the energy expended by the generator current to overcome the resistance of the self-induction current of the coil is completely returned to the circuit in the form of electric current energy. Therefore, the inductive reactance is reactive, that is, it does not cause irrecoverable energy losses.
The unit of measure for inductive reactance is Ohm
Inductive resistance is indicated by X L.
X- stands for reactance and L stands for inductive reactance.
f- frequency Hz, L- inductance of the H coil, X L- inductive resistance Ohm
The relationship between phases U and I on X L
Since the active resistance of the coil, by condition, is equal to zero (purely inductive resistance), then all the voltage applied by the generator to the coil goes to overcome e. etc. with. self-induction coil. This means that the graph of the voltage applied by the generator to the coil is equal in amplitude to the graph of e. etc. with. self-induction of the coil and is in antiphase with it.
The voltage applied by the generator to the purely inductive reactance and the current coming from the generator by the purely inductive reactance are shifted in phase by 90 0, i.e. That is, the voltage is ahead of the current by 90 0.
A real coil, in addition to inductive resistance, also has an active resistance. These resistors should be considered connected in series.
On the active resistance of the coil, the voltage applied by the generator and the current coming from the generator are in phase.
On a purely inductive reactance, the voltage applied by the generator and the current coming from the generator are out of phase by 90 0. The voltage is ahead of the current by 90 0. The resulting voltage applied by the generator to the coil is determined by the parallelogram rule.
click on the picture to enlarge
The resulting voltage applied by the generator to the coil always leads the current by an angle less than 90 0.
The value of the angle φ depends on the values \u200b\u200bof the active and inductive resistance of the coil.
About the resulting coil resistance
The resulting resistance of the coil cannot be found by summing the values \u200b\u200bof its active and reactance.
The resulting resistance of the coil Z is
An alternating current, passing through the wire, forms an alternating magnetic field around it, which induces a reverse EMF (self-induction EMF) in the conductor. Resistance to currentcaused by the self-induction EMF counteraction is called reactance inductive reactance.
The value of the reactive inductive resistance depends both on the value of the current in its own wire and on the magnitude of the currents in the adjacent wires. The farther the phase wires of the line are located, the less the influence of neighboring wires - the leakage flux and inductive reactance increase.
The value of the inductive resistance is influenced by the diameter of the wire, the magnetic permeability ( ) and the frequency of the alternating current. The linear inductive resistance is calculated by the formula:
where is the angular frequency;
- magnetic permeability;
average geometric distance between the phases of power lines;
radius of the wire.
The linear inductive reactance consists of two components 
and 
... The quantity 
called external inductive reactance. It is caused by an external magnetic field and depends only on the geometric dimensions of the transmission line. The quantity 
called internal inductive reactance. It is caused by an internal magnetic field and depends only on
, that is, from the current passing through the conductor.
The geometric mean distance between the phase wires is calculated using the formula:
.
In fig. 1.3 shows the possible arrangement of wires on the support.
When the wires are located in one plane (Figure 4.3 a, b), the formula for calculating D Wed simplifies:
If the wires are located at the vertices of an equilateral triangle, then D Wed \u003d D .
For overhead transmission lines with a voltage of 6-10 kV, the distance between the wires is 1-1.5 m; voltage 35 kV - 2-4 m; voltage 110 kV - 4-7 m; voltage of 220 kV - 7-9m.
When f\u003d 50Hz value \u003d 2 f\u003d 3.14 1 / s. Then formula (4.1) is written as follows:
For conductors made of non-ferrous metal (copper, aluminum) = 1.
On high voltage power lines (330 kV and above), phase splitting is used into several wires. At 330 kV, 2 wires per phase are usually used (inductive reactance is reduced by about 19%). At 500 kV, 3 wires per phase are usually used (inductive reactance is reduced by about 28%). At 750 kV, 4-6 wires per phase are used (inductive reactance is reduced by approximately 33%).
The linear inductive reactance with a split phase structure is calculated as:
where n- the number of wires in a phase;
R pr eq - equivalent radius of the wire.
When n= 2, 3
where and- splitting step (geometric mean distance between wires in phase);
R pr is the radius of the wire.
With a larger number of wires in phase, they are placed in a circle (see Fig.4.4). In this case, the value of the equivalent radius of the wire is:
where p is the splitting radius.
The value of the linear inductive resistance depends on the radius of the wire, and practically does not depend on the cross section (Fig. 4.5).
IN 
magnitude x 0 decreases with increasing wire radius. The smaller the average wire diameter, the more x 0, since neighboring wires are less affected, the self-induction EMF decreases. The effect of the second circuit for double-circuit transmission lines is little manifested, therefore it is neglected.
The inductive resistance of the cable is much lower than that of overhead transmission lines due to the smaller distances between the phases. In some cases, it can be neglected. Let's compare the linear inductive cable and overhead lines of different voltages:
The value of the reactance of the network section is calculated:
X= x 0 l.
1 Real and Ideal Email Sources energy. Substitution schemes... Any source of electrical energy converts other types of energy (mechanical, light, chemical, etc.) into electrical energy. The current in the source of electrical energy is directed from negative to positive due to external forces due to the type of energy that the source converts into electrical energy. A real source of electrical energy in the analysis of electrical circuits can be represented either as a voltage source or as a current source. This is shown below using an example of an ordinary battery.
Methods for representing a real source of electrical energy differ from each other in equivalent circuits (design circuits). In fig. 15, the real source is represented (replaced) by the voltage source circuit, and in Fig. 16, the real source is represented (replaced) by the current source circuit.
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Period (T) is the time (s) during which the variable is fully oscillating. Frequency - the number of periods per second. The unit of measurement for frequency is Hertz (abbreviated as Hz), 1 Hz is equal to one oscillation per second. Period and frequency are related T \u003d 1 / f. Changing over time, the sinusoidal value (voltage, current, EMF) takes on different values. The value of a quantity at a given time is called instantaneous. Amplitude - the largest value of the sinusoidal value. The amplitudes of current, voltage and EMF are designated in capital letters with an index: I m, U m, E m, and their instantaneous values \u200b\u200b- in lower case letters i, u, e... The instantaneous value of a sinusoidal value, for example a current, is determined by the formula i \u003d I m sin (ωt + ψ), where ωt + ψ is the phase-angle that determines the value of the sinusoidal value at a given time; ψ is the initial phase, that is, the angle that determines the value of the quantity at the initial moment of time. Sinusoidal quantities that have the same frequency but different initial phases are called phase-shifted.
3 In fig. 2 shows graphs of sinusoidal quantities (current, voltage), phase shifted. When the initial phases of the two quantities are equal ψ i \u003d ψ u, then the difference ψ i - ψ u \u003d 0 and, therefore, there is no phase shift φ \u003d 0 (Fig. 3). The effectiveness of the mechanical and thermal action of alternating current is estimated by its effective value. The rms value of the alternating current is equal to the value of the direct current, which, in a time equal to one period of alternating current, will release the same amount of heat in the same resistance as the alternating current. The effective value is indicated in capital letters without an index: I, U, E. Figure: 2 Phase-shifted sinusoidal current and voltage graphs. Figure: 3 Sinusoidal current and voltage plots in phase
For sinusoidal quantities, the effective and amplitude values \u200b\u200bare related by the relations:
I \u003d I M / √2; U \u003d U M / √2; E \u003d E M √2. The effective values \u200b\u200bof current and voltage are measured with ammeters and ac voltmeters, and the average power value is measured with wattmeters.
4 . Effective (effective) valuestrengthalternating current called the amount of direct current, the action of which will produce the same work (thermal or electrodynamic effect) as the considered alternating current during one period. In modern literature, the mathematical definition of this value is often used - the rms value of the alternating current. In other words, the effective value of the current can be determined by the formula:
.
For harmonic current fluctuations
5 Formula of inductive resistance:
where L is the inductance.
Capacitance formula:
where C is the capacity.
We propose to consider an alternating current circuit, in which one active resistance is included, and draw it in notebooks. After checking the figure, I tell you that in the electrical circuit (Fig. 1, a), under the action of an alternating voltage, an alternating current flows, the change of which depends on the change in voltage. If the voltage increases, the current in the circuit increases, and when the voltage is zero, there is no current in the circuit. The change in its direction will also coincide with the change in the direction of the voltage.
(Fig. 1, c).
Fig 1. AC circuit with active resistance: a - circuit; b - vector diagram; c - wave diagram
I graphically depict on the board sinusoids of current and voltage, which are in phase, explaining that although the period and frequency of oscillations, as well as the maximum and effective values, can be determined from the sinusoid, nevertheless it is rather difficult to construct a sinusoid. A simpler way to represent the values \u200b\u200bof current and voltage is vector. For this, the stress vector (to scale) should be plotted to the right from an arbitrarily chosen point. The teacher invites the students to postpone the current vector on their own, recalling that the voltage and current are in phase. After constructing a vector diagram (Fig. 1, b), it should be shown that the angle between the voltage and current vectors is zero, ie? \u003d 0. The current in such a circuit will be determined by Ohm's law: Question 2... AC circuit with inductive reactance Consider an alternating current electric circuit (Fig. 2, a), which includes an inductive reactance. Such resistance is a coil with a small number of turns of wire of large cross-section, in which the active resistance is considered to be 0.
Figure: 2. AC circuit with inductive resistance
Around the turns of the coil when the current passes, an alternating magnetic field will be created, which induces self-induction emf in the turns. According to Lenz's rule, the ede of induction is always opposed by the cause that causes it. And since the ede of self-induction is caused by changes in the alternating current, it also prevents its passage. The resistance caused by self-induction is called inductive and is denoted by the letter x L. The inductive resistance of the coil depends on the rate of change of the current in the coil and its inductance L: where X L is the inductive resistance, Ohm; - angular frequency of alternating current, rad / s; L is the inductance of the coil, G.
Angular frequency \u003d\u003d,
hence, .
Capacitive resistance in the AC circuit. Before starting the explanation, it should be recalled that there are a number of cases when in electrical circuits, in addition to active and inductive resistance, there is also a capacitive resistance. A device designed to store electrical charges is called a capacitor. The simplest capacitor is two wires separated by a layer of insulation. Therefore, stranded wires, cables, motor windings, etc. have capacitive resistance. The explanation is accompanied by showing a capacitor of various types and capacitances with their connection to an electrical circuit. I propose to consider the case when one capacitive resistance prevails in the electrical circuit, and the active and inductive ones can be neglected because of their small values \u200b\u200b(Fig. 6, a). If the capacitor is included in the DC circuit, then the current will not pass through the circuit, since there is a dielectric between the plates of the capacitor. If the capacitive resistance is connected to an alternating current circuit, then the current I will flow through the circuit, caused by the recharging of the capacitor. Recharging occurs because the alternating voltage changes its direction, and, therefore, if we connect an ammeter to this circuit, then it will show the charging and discharging current of the capacitor. In this case, the current does not pass through the capacitor either. The strength of the current passing in a circuit with a capacitive resistance depends on the capacitive resistance of the capacitor Xc and is determined by Ohm's law 
where U is the voltage of the emf source, V; Xc - capacitive resistance, Ohm; / - current strength, A.
Figure: 3. AC circuit with capacitive resistance
Capacitive resistance, in turn, is determined by the formula 
where C is the capacitive resistance of the capacitor, F. I suggest that students build a vector diagram of the current and voltage in a circuit with capacitive resistance. Let me remind you that when studying the processes in an electric circuit with a capacitive resistance, it was found that the current is ahead of the voltage by an angle φ \u003d 90 °. This phase shift of current and voltage should be shown on a wave diagram. I graphically depict a sinusoid of voltage on the blackboard (Fig. 3, b) and instruct the students to independently draw a sinusoid of current on the drawing, leading the voltage at an angle of 90 °
We know that the self-induction current of the coil goes to meet the growing current of the generator. This the resistance of the self-induction current of the coil to the rising current of the generator is called inductive resistance.
To overcome this opposition, part of the generator's alternating current energy is spent. All this part of the energy is completely converted into the energy of the magnetic field of the coil. When the generator current decreases, the magnetic field of the coil will also decrease, interrupting the coil and inducing a self-induction current in the circuit. Now the self-induction current will go in the same direction as the decreasing generator current.
Thus, all the energy expended by the generator current to overcome the resistance of the self-induction current of the coil is completely returned to the circuit in the form of electric current energy. Therefore, the inductive reactance is reactive, that is, it does not cause irrecoverable energy losses.
The unit of measure for inductive reactance is Ohm
Inductive resistance is indicated by X L.
X- stands for reactance and L stands for inductive reactance.
f- frequency Hz, L- inductance of the H coil, X L- inductive resistance Ohm
The relationship between phases U and I on X L
Since the active resistance of the coil, by condition, is equal to zero (purely inductive resistance), then all the voltage applied by the generator to the coil goes to overcome e. etc. with. self-induction coil. This means that the graph of the voltage applied by the generator to the coil is equal in amplitude to the graph of e. etc. with. self-induction of the coil and is in antiphase with it.
The voltage applied by the generator to the purely inductive reactance and the current coming from the generator by the purely inductive reactance are shifted in phase by 90 0, i.e. That is, the voltage is ahead of the current by 90 0.
A real coil, in addition to inductive resistance, also has an active resistance. These resistors should be considered connected in series.
On the active resistance of the coil, the voltage applied by the generator and the current coming from the generator are in phase.
On a purely inductive reactance, the voltage applied by the generator and the current coming from the generator are out of phase by 90 0. The voltage is ahead of the current by 90 0. The resulting voltage applied by the generator to the coil is determined by the parallelogram rule.
click on the picture to enlarge
The resulting voltage applied by the generator to the coil always leads the current by an angle less than 90 0.
The value of the angle φ depends on the values \u200b\u200bof the active and inductive resistance of the coil.
About the resulting coil resistance
The resulting resistance of the coil cannot be found by summing the values \u200b\u200bof its active and reactance.
The resulting resistance of the coil Z is
Reactance - electrical resistance to alternating current due to the transfer of energy by a magnetic field in inductors or an electric field in capacitors.
Elements with reactance are called reactive.
Inductor coil reactance.
When AC is flowing I in the coil, a magnetic field creates an EMF in its turns, which prevents the current from changing.
With an increase in the current, the EMF is negative and prevents an increase in the current, with a decrease, it is positive and prevents it from decreasing, thus providing resistance to the change in current throughout the entire period.
As a result of the created reaction, a voltage is formed at the terminals of the inductor in antiphase U, suppressing EMF, equal to it in amplitude and opposite in sign.
When the current passes through zero, the amplitude of the EMF reaches its maximum value, which forms a discrepancy in time between current and voltage in 1/4 period.
If you apply a voltage to the terminals of the inductor U, the current cannot start instantly due to the counteraction of the EMF equal to -U, therefore, the current in the inductor will always lag behind the voltage by an angle of 90 °. The lagging current shift is called positive.
We write the expression for the instantaneous voltage value u based on EMF ( ε
), which is proportional to the inductance L and the rate of change of the current: u \u003d -ε \u003d L (di / dt).
From here we express a sinusoidal current.
Integral of function sin (t) will be -cos (t), or an equal function sin (t-π / 2).
Differential dt functions sin (ωt) leaves the integral sign by a factor of 1 /ω
.
As a result, we obtain the expression for the instantaneous value of the current 
offset from the stress function by an angle π / 2 (90 °).
For rms values U and I in this case, you can write 
.
As a result, we have a dependence of sinusoidal current on voltage according to Ohm's Law, where in the denominator instead of R expression ωL, which is the reactance:
The reactance of inductors is called inductive.
Capacitor reactance.
The electric current in a capacitor is a part or a combination of the processes of its charge and discharge - the accumulation and release of energy by an electric field between its plates.
In an alternating current circuit, the capacitor will charge up to a certain maximum value until the current is reversed. Therefore, at the moments of the amplitude value of the voltage across the capacitor, the current in it will be zero. Thus, the voltage across the capacitor and the current will always have a quarter period difference in time.
As a result, the current in the circuit will be limited by the voltage drop across the capacitor, which creates an AC reactance that is inversely proportional to the rate of change of the current (frequency) and the capacitance of the capacitor.
If you apply voltage to the capacitor U, the current will instantly start from the maximum value, then decrease to zero. At this time, the voltage across its terminals will rise from zero to maximum. Consequently, the voltage on the capacitor plates in phase lags behind the current by an angle of 90 °. This phase shift is called negative.
The current in a capacitor is a derivative function of its charge i \u003d dQ / dt \u003d C (du / dt).
Derived from sin (t) will be cos (t) or an equal function sin (t + π / 2).
Then for a sinusoidal voltage u \u003d U amp sin (ωt) we write the expression for the instantaneous value of the current as follows:
i \u003d U amp ωCsin (ωt + π / 2).
From here we express the ratio of the rms values 
.
Ohm's law dictates that 1 / ωC is nothing more than a reactance for a sinusoidal current:
The reactance of a capacitor in the technical literature is often called capacitive. It can be used, for example, in the organization of capacitive dividers in AC circuits.
Online calculator for calculating reactance
It is necessary to enter the values \u200b\u200band click with the mouse in the table.
When switching the multipliers, the result is automatically recalculated.
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Capacitance reactance |
