Active resistance, inductance and capacitance in an alternating current circuit.
Changes in current strength, voltage and e. etc. with. in an alternating current circuit occur with the same frequency, but the phases of these changes, generally speaking, are different. Therefore, if the initial phase of the current strength is conventionally taken as zero, then the initial phase of the voltage will have a certain value φ. Under this condition, the instantaneous values of the current and voltage and will be expressed by the following formulas:
i = I m sinωt
u = U m sin (ωt + φ)
a) Active resistance in the AC circuit. The resistance of the circuit, which causes irrecoverable losses of electrical energy due to the thermal effect of the current, called active ... This resistance for low frequency current can be considered equal to the resistance R the same DC conductor.
In an alternating current circuit that has only active resistance, for example, in incandescent lamps, heating devices, etc., the phase shift between voltage and current is zero, i.e. φ = 0. This means that the current and voltage in such the circuits change in the same phases, and the electric energy is completely spent on the thermal effect of the current.
We will assume that the voltage at the terminals of the circuit changes according to the harmonic law: and = U t cos ωt.
As in the case of direct current, the instantaneous value of the current strength is directly proportional to the instantaneous value of the voltage. Therefore, to find the instantaneous value of the current strength, you can apply Ohm's law:
in phase with voltage fluctuations.
b) An inductor in an alternating current circuit. The inclusion of a coil with inductance in the alternating current circuit L manifests itself as an increase in the resistance of the circuit. This is explained by the fact that with an alternating current in the coil, e is always acting. etc. with. self-induction, attenuating the current. Resistance X L, which is due to the phenomenon of self-induction, is called inductive reactance. Since e. etc. with. self-induction is the greater, the greater the inductance of the circuit and the faster the current changes, then the inductive reactance is directly proportional to the inductance of the circuit L and the circular frequency of alternating current ω: X L = ωL .
Let us determine the current strength in a circuit containing a coil, the active resistance of which can be neglected. To do this, we first find the relationship between the voltage on the coil and the EMF of self-induction in it. If the resistance of the coil is zero, then the strength of the electric field inside the conductor at any time should be zero. Otherwise, the current strength, according to Ohm's law, would be infinitely large.
The equality to zero of the field strength turns out to be possible because the strength of the vortex electric field E i, generated by an alternating magnetic field, at each point is equal in magnitude and opposite in direction of the strength of the Coulomb field E k, created in the conductor by charges located at the terminals of the source and in the wires of the circuit.
From equality E i = -E k follows that vortex field specific work(i.e. EMF of self-induction e i) is equal in magnitude and opposite in sign to the specific work of the Coulomb field... Considering that the specific work of the Coulomb field is equal to the voltage at the ends of the coil, we can write: e i = -and.
When the current strength changes according to the harmonic law i = I m sin cosωt, EMF of self-induction is equal to: e i = -Li "= -LωI m cos ωt. Because e i = -and, then the voltage at the ends of the coil turns out to be
and= LωI m cos ωt = LωI m sin (ωt + π / 2) = U m sin (ωt + π / 2)
where U m = LωI m - voltage amplitude.
Consequently, the voltage fluctuations on the coil are ahead of the phase of the current fluctuations by π / 2, or, which is the same, current fluctuations lag in phase with voltage fluctuations byπ / 2.
If we introduce the notation X L = ωL, we get 
... The value X
L, equal to the product of the cyclic frequency and inductance, is called inductive reactance. According to the formula , the current value is related to the voltage value and inductive reactance in a relationship similar to Ohm's law for a DC circuit.
The inductive reactance depends on the frequency ω. The direct current does not "notice" the inductance of the coil at all. When ω = 0, the inductive reactance is zero. The faster the voltage changes, the more self-induction EMF and the lower the amplitude of the current. It should be noted that the voltage across the inductive reactance is phase-ahead of the current.
c) Capacitor in the AC circuit. Direct current does not pass through the capacitor, since there is a dielectric between its plates. If the capacitor is included in the DC circuit, then after charging the capacitor, the current in the circuit will stop.
Let the capacitor be connected to the AC circuit. Capacitor charge (q = CU) due to voltage changes, it constantly changes, therefore, an alternating current flows in the circuit. The strength of the current will be the greater, the greater the capacitance of the capacitor and the more often it is recharged, that is, the greater the frequency of the alternating current.
Resistance due to the presence of electrical capacity in an alternating current circuit is called capacitive resistance X with. It is inversely proportional to the capacity WITH and circular frequency ω: X c = 1 / ωС.
Let us establish how the current strength in a circuit containing only a capacitor changes over time, if the resistance of the wires and capacitor plates can be neglected.
The voltage across the capacitor u = q / C is equal to the voltage at the ends of the circuit u = U m cosωt.
Therefore, q / C = U m cosωt. The capacitor charge changes according to the harmonic law:
q = CU m cosωt.
The current, which is the time derivative of the charge, is equal to:
i = q "= -U m Cω sin ωt = U m ωC cos (ωt + π / 2).
Hence, fluctuations in the current strength are ahead of the phase fluctuations in the voltage across the capacitor byπ / 2.
The value X with, the reciprocal of the product ωС of the cyclic frequency and the capacitance of the capacitor, is called capacitance. The role of this quantity is similar to the role of active resistance R in Ohm's law. The value of the current strength is related to the value of the voltage across the capacitor in the same way as the current strength and voltage for a section of a direct current circuit are related according to Ohm's law. This allows us to consider the quantity X with as the resistance of a capacitor to alternating current (capacitive resistance).
The greater the capacitance of the capacitor, the greater the recharge current. This is easily detected by the increase in the incandescence of the lamp with increasing capacitance. While the DC resistance of a capacitor is infinitely high, its AC resistance is finite. X c. With increasing capacity, it decreases. It also decreases with increasing frequency ω.
In conclusion, we note that during a quarter of the period when the capacitor is charged to its maximum voltage, energy enters the circuit and is stored in the capacitor in the form of electric field energy. In the next quarter of the period, when the capacitor is discharged, this energy is returned to the network.
From a comparison of formulas X L = ωL and X c = 1 / ωC it can be seen that the inductor. represent a very large resistance for high frequency current and small for low frequency current, and capacitors - on the contrary. Inductive X L and capacitive X C resistances are called reactive.
d) Ohm's law for an alternating current electrical circuit.
Let us now consider a more general case of an electrical circuit in which a conductor with an active resistance is connected in series R and low inductance, high inductance coil L and low active resistance and a capacitor with a capacity WITH
We have seen that when separately connected to the active resistance circuit R, capacitor capacity WITH or coils with inductors L the amplitude of the current is determined, respectively, by the formulas:
; 
; I m = U m ωC.
The amplitudes of the voltages across the active resistance, the inductor and the capacitor are related to the amplitude of the current as follows: U m = I m R; U m = I m ωL; 
In DC circuits, the voltage at the ends of the circuit is equal to the sum of the voltages at individual series-connected sections of the circuit. However, if you measure the resulting voltage on the circuit and the voltages on the individual elements of the circuit, it turns out that the voltage on the circuit (rms value) is not equal to the sum of the voltages on the individual elements. Why is this so? The fact is that harmonic voltage fluctuations in different parts of the circuit are phase-shifted relative to each other.
Indeed, the current at any time is the same in all sections of the circuit. This means that the amplitudes and phases of currents flowing through sections with capacitive, inductive and active resistances are the same. However, only on the active resistance, the voltage and current fluctuations coincide in phase. On the capacitor, the voltage fluctuations lag behind the current fluctuations by π / 2, and on the inductor, the voltage fluctuations are ahead of the current fluctuations by π / 2. If we take into account the phase shift between the added voltages, it turns out that
To obtain this equality, you need to be able to add voltage fluctuations that are phase-shifted relative to each other. The easiest way to add several harmonic vibrations is using vector diagrams. The idea behind the method is based on two fairly simple principles.
Firstly, the projection of a vector with modulus x m rotating with constant angular velocity performs harmonic oscillations: x = x m cosωt
Secondly, when adding two vectors, the projection of the total vector is equal to the sum of the projections of the vectors being added.
The vector diagram of electrical oscillations in the circuit shown in the figure will allow us to obtain the relationship between the amplitude of the current in this circuit and the amplitude of the voltage. Since the current strength is the same in all sections of the circuit, it is convenient to start building a vector diagram with the current vector I m... We represent this vector in the form of a horizontal arrow. The voltage across the resistance is in phase with the current. Therefore the vector U mR, must coincide in direction with the vector I m... Its modulus is U mR = I m R
The voltage fluctuations on the inductive reactance are ahead of the current fluctuations by π / 2, and the corresponding vector U m L must be rotated relative to the vector I m by π / 2. Its modulus is U m L = I m ωL. If we assume that the positive phase shift corresponds to the rotation of the vector counterclockwise, then the vector U m L turn left. (One could, of course, have done the opposite.)
Its modulus is U mC =I m / ωC... To find the vector of the total voltage U m you need to add three vectors: 1) U mR 2) U m L 3) U mC
At first, it is more convenient to add two vectors: U m L and U mC
The modulus of this sum is 
if ωL> 1 / ωС. This is the case shown in the figure. After that, adding the vector ( U m L + U mC) with vector U mR we get a vector U m, depicting voltage fluctuations in the network. By the Pythagorean theorem:
From the last equality, you can easily find the amplitude of the current in the circuit:
Thus, due to the phase shift between the voltages in different parts of the circuit, the impedance Z the circuit shown in the figure is expressed as follows:
From the amplitudes of the current and voltage, you can go to the effective values of these quantities:
This is Ohm's law for alternating current in the circuit shown in Figure 43. The instantaneous value of the current strength changes harmoniously over time:
i = I m cos (ωt + φ), where φ is the phase difference between the current and the voltage in the network. It depends on the frequency ω and the parameters of the circuit R, L, C.
e) Resonance in an electrical circuit. When studying forced mechanical vibrations, we got acquainted with an important phenomenon - resonance. Resonance is observed when the natural oscillation frequency of the system coincides with the frequency of the external force. At low friction, there is a sharp increase in the amplitude of steady-state forced oscillations. The coincidence of the laws of mechanical and electromagnetic oscillations immediately makes it possible to draw a conclusion about the possibility of resonance in an electric circuit, if this circuit is an oscillatory circuit with a certain natural frequency of oscillations.
The amplitude of the current during forced oscillations in the circuit, occurring under the action of an external harmonically varying voltage, is determined by the formula:
At a fixed voltage and given values of R, L and C , the current strength reaches a maximum at a frequency ω satisfying the relation
This amplitude is especially large at small R. From this equation, you can determine the value of the cyclic frequency of the alternating current at which the current is maximum:
This frequency coincides with the frequency of free oscillations in a circuit with a low active resistance.
A sharp increase in the amplitude of forced current oscillations in an oscillatory circuit with a low active resistance occurs when the frequency of the external alternating voltage coincides with the natural frequency of the oscillatory circuit. This is the phenomenon of resonance in an electric oscillatory circuit.
Simultaneously with the growth of the current at resonance, the voltages across the capacitor and the inductor increase sharply. These stresses become the same and are many times greater than the external stress.
Really,
U m, C, res = 
U m, L, res = 
The external voltage is related to the resonant current as follows:
U m = 
.
If 
then U m, C, res = U m, L, res >> U m
At resonance, the phase shift between current and voltage becomes zero.
Indeed, voltage fluctuations across the inductor and capacitor always occur in antiphase. The resonant amplitudes of these voltages are the same. As a result, the voltages on the coil and capacitor completely cancel each other out. , and the voltage drop occurs only across the active resistance.
The equality to zero of the phase shift between voltage and current at resonance provides optimal conditions for the flow of energy from the AC voltage source into the circuit. Here is a complete analogy with mechanical vibrations: at resonance, an external force (analogue of voltage in a circuit) coincides in phase with speed (analogue of current strength).
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 oscillates completely. 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 = 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 - in lower case letters i, u, e... The instantaneous value of a sinusoidal value, for example a current, is determined by the formula i = 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 = ψ u, then the difference ψ i - ψ u = 0 and, therefore, there is no phase shift φ = 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 in the same resistance the same amount of heat as the alternating current. The effective value is indicated in capital letters without an index: I, U, E. Rice. 2 Phase-shifted sinusoidal current and voltage graphs. Rice. 3 Sinusoidal current and voltage plots in phase
For sinusoidal values, the effective and amplitude values are related by the relations:
I = I M / √2; U = U M / √2; E = E M √2. The effective values of 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 more 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 reactance:
where L is inductance.
Capacitive resistance 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 - diagram; 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 build a sinusoid. A simpler way to represent the values of 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 vectors of voltage and current is zero, ie? = 0. The current in such a circuit will be determined by Ohm's law: Question 2... AC circuit with inductive resistance 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 equal to 0.
Rice. 2. AC circuit with inductive resistance
Around the turns of the coil during the passage of current, 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 ==,
hence, .
Capacitive resistance in an alternating current circuit. Before starting the explanation, it should be recalled that there are a number of cases when, in addition to active and inductive resistances, there is also a capacitive resistance in electrical circuits. 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 due to their small values (Fig. 6, a). If the capacitor is included in the DC circuit, then the current will not flow 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, 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.
Rice. 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 a 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 = 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 by an angle of 90 °
An alternating current, passing through the wire, forms an alternating magnetic field around it, which induces an EMF in the opposite direction (EMF of self-induction) in the conductor. Resistance to current caused 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 further 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 the power transmission line;
radius of the wire.
The linear inductive reactance consists of two components 
and 
... The magnitude 
called external inductive reactance. It is caused by an external magnetic field and depends only on the geometric dimensions of the transmission line. The magnitude 
called internal inductive reactance. It is due to the 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 by the formula:
.
In fig. 1.3 shows the possible arrangement of the wires on the support.
When the wires are located in one plane (Fig.4.3 a, b), the formula for calculating D cf is simplified:
If the wires are located at the vertices of an equilateral triangle, then D Wed = 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 of 110 kV - 4-7 m; voltage of 220 kV - 7-9m.
At f= 50Hz value = 2 f= 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, usually 2 wires per phase are used (inductive reactance is reduced by about 19%). At 500 kV, usually 3 wires per phase are used (inductive reactance is reduced by about 28%). At 750 kV, 4-6 wires are used per phase (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.
At n= 2, 3
where a- 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 section (Fig. 4.5).
V 
magnitude x 0 decreases with increasing wire radius. The smaller the average wire diameter, the more x 0, since neighboring wires influence to a lesser extent, 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.
In an alternating current circuit, under the influence of a continuously changing voltage, changes in this current occur. In turn, these changes cause the generation of a magnetic field, which periodically increases or decreases. Under its influence, a counter voltage is induced in the coil, which prevents changes in the current. Thus, the flow of current occurs under continuous resistance, called inductive resistance.
This value is directly related to the frequency of the applied voltage (f) and the value of the inductance (L). The inductive reactance formula will look like this: XL = 2πfL... Direct proportional dependence, if necessary, allows, by transforming the basic formula, to calculate the frequency or inductance value.
What does inductive reactance depend on?
Under the action of an alternating current passing through a conductor, an alternating magnetic field is formed around this conductor. The action of this field leads to the induction in the conductor of an electromotive force of the opposite direction, also known as the EMF of self-induction. The reaction or resistance of EMF to alternating current is called reactive inductive resistance.
This value depends on many factors. First of all, it is influenced as the value of the current, not only in its own conductor, but also in neighboring wires. That is, an increase in resistance and leakage flux occurs as the distance between the phase wires increases. At the same time, the effect of adjacent wires is reduced.
There is such a concept as linear inductive resistance, which is calculated by the formula: X0 = ω x (4.61g x (Dav / Rpr) + 0.5μ) x 10-4 = X0 '+ X0' ', in which ω is the angular frequency, μ - magnetic permeability, Dav - the geometric mean distance between the phases of the transmission line, and Rпр - the radius of the wire.
The X0 ’and X0’ ’values are two parts of the linear inductive reactance. The first of them X0 'is an external inductive reactance that depends only on the external magnetic field and the size of the transmission line. Another quantity - X0 '' is the internal resistance, which depends on the internal magnetic field and the magnetic permeability μ.
On high voltage transmission lines of 330 kV or more, the passing phases are split into several separate wires. For example, at a voltage of 330 kV, the phase is split into two wires, which reduces the inductive reactance by about 19%. Three wires are used at 500 kV - inductive reactance can be reduced by 28%. The 750 kV voltage allows phase separation into 4-6 conductors, which reduces resistance by about 33%.
The linear inductive reactance has a value depending on the radius of the wire and does not depend at all on the cross-section. If the radius of the conductor increases, then the value of the linear inductive resistance will decrease accordingly. Conductors located nearby have a significant impact.
AC inductive reactance
One of the main characteristics of electrical circuits is resistance, which can be active and reactive. Typical representatives of active resistance are ordinary consumers - lamps, incandescent, resistors, heating coils and other elements in which there is an electric one.
Reactive resistances include inductive and capacitive resistances located in intermediate power converters - inductive coils and capacitors. These parameters are necessarily taken into account when performing various calculations. For example, to determine the total resistance of a circuit section,. The addition is carried out geometrically, that is, in a vector way, by constructing a right-angled triangle. In it, both legs are both resistances, and the hypotenuse is full. The length of each leg corresponds to the effective value of one or another resistance.
As an example, we can consider the nature of the inductive resistance in the simplest AC circuit. It includes a power supply with an EMF (E), a resistor as an active component (R) and a coil with inductance (L). The emergence of inductive resistance occurs under the action of the EMF of self-induction (Eshi) in the coil turns. The inductive reactance increases in accordance with the increase in the inductance of the circuit and the value of the current flowing along the circuit.
Thus, Ohm's law for such an alternating current circuit will look like the formula: E + Esi = I x R. Further, using the same formula, you can determine the value of self-induction: Esi = -L x Ipr, where Ipr is the current derivative from time. The minus sign means the opposite direction of Esi in relation to the changing value of the current. Since such changes are constantly occurring in the alternating current circuit, there is significant opposition or resistance on the part of Esi. With a constant current, this dependence is absent and all attempts to connect the coil to such a circuit would lead to a normal short circuit.
To overcome the EMF of self-induction, such a potential difference must be created at the terminals of the coil by the power source so that it can at least minimally compensate for the resistance Esi (Ucat = -Esi). Since an increase in the alternating current in the circuit leads to an increase in the magnetic field, a vortex field is generated, which causes an increase in the opposite current in the inductance. As a result, a phase shift occurs between current and voltage.
Coil inductive resistance
Inductors are classified as passive components used in electronic circuits. It is able to store electricity by converting it into a magnetic field. This is its main function. An inductor in its characteristics and properties resembles a capacitor that stores energy in the form of an electric field.
Inductance, measured in Henry, is the appearance of a magnetic field around a current-carrying conductor. In turn, it is associated with an electromotive force, which opposes the applied alternating voltage and current in the coil. This property is the inductive reactance, which is in antiphase with the capacitive reactance of the capacitor. The inductance of the coil can be increased by increasing the number of turns.
In order to find out what the inductive resistance of the coil is, it should be remembered that it, first of all, opposes the alternating current. As practice shows, each inductive coil itself has a certain resistance.
The passage of an alternating sinusoidal current through the coil results in an alternating sinusoidal voltage or EMF. As a result, an inductive reactance arises, defined by the formula: XL = ωL = 2πFL, in which ω is the angular frequency, F is the frequency in hertz, L is the inductance in henry.
There are two types - active and reactive. Active is represented by resistors, incandescent lamps, heating coils, etc. In other words, all elements in which the flowing current directly performs useful work or, in a particular case, causes the desired heating of the conductor. In turn, reactive is an umbrella term. It is understood as capacitive and inductive resistance. In the elements of the circuit, which have reactance, various intermediate energy transformations occur during the passage of an electric current. A capacitor (capacity) accumulates charge, and then gives it to the circuit. Another example is the inductive reactance of a coil, in which some of the electrical energy is converted into a magnetic field.
In fact, there are no “pure” active or reactive resistances. The opposite component is always present. For example, when calculating wires for long-distance power lines, they take into account not only but also capacitive. And when considering inductive reactance, you need to remember that both the conductors and the power supply make their own adjustments to the calculations.
Determining the total resistance of the circuit section, it is necessary to add the active and reactive components. Moreover, it is impossible to obtain a direct sum by an ordinary mathematical action, therefore, a geometric (vector) method of addition is used. A right-angled triangle is constructed, two legs of which represent active and inductive resistance, and the hypotenuse is total. The lengths of the segments correspond to the effective values.
Consider the inductive reactance in an alternating current circuit. Imagine a simple circuit consisting of a power supply (EMF, E), a resistor (resistive component, R), and a coil (inductance, L). Since the inductive resistance arises due to the EMF of self-induction (E si) in the turns of the coil, it is obvious that it increases with an increase in the inductance of the circuit and an increase in the value of the current flowing along the circuit.
Ohm's law for such a circuit looks like:
E + E si = I * R.
Having determined the derivative of the current from time (I pr), you can calculate the self-induction:
E si = -L * I pr.
The "-" sign in the equation indicates that the action of E si is directed against the change in the current value. Lenz's rule states that with any change in current, an EMF of self-induction occurs. And since such changes in the circuits are natural (and constantly occur), then E si forms a significant opposition or, which is also true, resistance. In the case of a power source, this dependence is not fulfilled, and when trying to connect a coil (inductance) to such a circuit, a classic short-circuit would occur.
To overcome E si, the power supply must create such a potential difference at the terminals of the coil so that it is sufficient, at least, to compensate for the resistance E si. This implies:
U cat = -E si.
In other words, the voltage across the inductance is numerically equal to the electromotive force of self-induction.
Since with an increase in the current in the circuit, the generating vortex field in turn increases, causing an increase in the countercurrent in the inductance, we can say that there is a phase shift between voltage and current. Hence, one feature follows: since the EMF of self-induction prevents any change in the current, then when it increases (the first quarter of the period on the sinusoid), a countercurrent field is generated, but when it falls (the second quarter), on the contrary, the induced current is co-directed with the main one. That is, if we theoretically assume the existence of an ideal power source without internal resistance and inductance without an active component, then fluctuations in the energy "source - coil" could occur indefinitely.
