E. d. s. self-induction. E. d. s. e L, induction in a conductor or coil as a result of a change in the magnetic flux created by a current passing through the same conductor or coil, is called e. d.s. self-induction (Fig. 60). This e. d.s. occurs with any change in current, for example, when closing and opening electrical circuits, when the load of electric motors changes, etc. The faster the current changes in a conductor or coil, the greater the rate of change of the magnetic flux penetrating them and the greater e. d.s. self-induction is induced in them. For example, e. d.s. self-induction e L occurs in the AB conductor (see Fig. 54) when the current flowing through it i 1 changes. Therefore, a changing magnetic field induces e. d.s. in the same conductor in which the current that creates this field changes.
Direction e. d.s. self-induction is determined by Lenz's rule. E. d. s. self-induction always has such a direction in which it prevents a change in the current that caused it. Consequently, with increasing current in the conductor (coil), the e. d.s. self-induction will be directed against the current, that is, it will prevent its increase (Fig. 61, a), and vice versa, when the current decreases in the conductor (coil), e. d.s. self-induction, coinciding in direction with the current, i.e., preventing its decrease (Fig. 61, b). If the current in the coil does not change, then e. d.s. self-induction does not occur.
From the above rule for determining the direction e. d.s. self-induction it follows that this e. d.s. has a braking effect on the change in current in electrical circuits. In this respect, its action is similar to the action of the force of inertia, which prevents a change in the position of the body. In an electrical circuit (Fig. 62, a), consisting of a resistor with resistance R and a coil K, current i is created by the combined action of the source voltage U and e. d.s. self-induction e L induced in the coil. When connecting the circuit under consideration to the source of e. d.s. self-induction e L (see solid arrow) inhibits the increase in current strength. Therefore, the current i reaches a steady value I \u003d U / R (according to Ohm's law) not instantly, but over a certain period of time (Fig. 62, b). During this time, a transient process occurs in the electric circuit, during which e L and i change. Exactly
also, when the electric circuit is turned off, the current i does not instantly decrease to zero, but due to the action of e. d.s. e L (see dashed arrow) gradually decreases.
Inductance. The ability of various conductors (coils) to induce e. d.s. self-induction is estimated by the inductance L. It shows which e. d.s. self-induction occurs in a given conductor (coil) when the current changes by 1 A for 1 s. Inductance is measured in Henry (H), 1 H = 1 Ohm*s. In practice, inductance is often measured in thousandths of a henry - millihenry (mH) and in millionths of a henry - microhenry (µH).
Does the inductance of a coil depend on the number of turns of the coil? and magnetic resistance R m of its magnetic circuit, i.e. from its magnetic permeability? and geometric dimensions l and s. If a steel core is inserted into the coil, its inductance increases sharply due to the amplification of the magnetic field of the coil. In this case, a current of 1 A creates a much greater magnetic flux than in a coreless coil.
Using the concept of inductance L, one can obtain for e. d.s. self-induction the following formula:
e L = – L ?i / ?t (53)
Where? i is the change in current in the conductor (coil) over a period of time? t.
Consequently, e. d.s. self-induction is proportional to the rate of change of current.
Switching on and off DC circuits with an inductor. When connected to a DC source with a voltage U of an electrical circuit containing R and L, with a switch B1 (Fig. 63, a), the current i increases to a steady value I set \u003d U / R not instantly, since e. d.s. self-induction e L , arising in the inductance, acts against the applied voltage V and prevents the current from rising. For the process under consideration, a gradual change in current i (Fig. 63, b) and voltages u a and u L along the curves is characteristic - exhibitors. Changing i, u a and u L along the indicated curves is called aperiodic.
The rate of increase in the current strength in the circuit and the change in voltages u a and u L is characterized by circuit time constant
T=L/R (54)
It is measured in seconds, depends only on the parameters R and L of a given circuit, and allows you to estimate the duration of the current change process without plotting. This duration is theoretically infinite. In practice, it is usually considered that it is (3-4) T. During this time, the current in the circuit reaches 95-98% of the steady value. Therefore, the greater the resistance and the lower the inductance L, the faster the process of changing the current in electrical circuits with inductance. The time constant T in an aperiodic process can be defined as a segment AB, cut off by a tangent drawn from the origin to the curve in question (for example, current i) on the line corresponding to the steady value of this quantity.
The property of inductance to slow down the process of changing the current is used to create time delays when various devices are triggered (for example, when controlling the operation of sandboxes for periodically supplying portions of sand under the wheels of a locomotive). The operation of the electromagnetic time relay is also based on the use of this phenomenon (see § 94).
Switching surges. E is especially strong. d.s. self-induction when opening circuits containing coils with a large number of turns and with steel cores (for example, windings of generators, electric motors, transformers, etc.), i.e. circuits with high inductance. In this case, the resulting e. d.s. self-induction e L can many times exceed the voltage U of the source and, summing up with it, cause overvoltages in electrical circuits (Fig. 64, a), called switching(occurring when switching- switching electrical circuits). They are dangerous for the windings of electric motors, generators and transformers, as they can cause breakdown of their insulation.
Big e. d.s. self-induction also contributes to the occurrence of an electric spark or arc in electrical devices that switch electrical circuits. For example, at the moment of opening the contacts of the knife switch (Fig. 64, b), the resulting e. d.s. self-induction greatly increases the potential difference between the open contacts of the switch and breaks through the air gap. The resulting electric arc is maintained for some time e. d.s. self-induction, which thus delays the process of turning off the current in the circuit. This phenomenon is highly undesirable, since the arc melts the contacts of the disconnecting devices, which leads to their rapid failure. Therefore, in all devices used to open electrical circuits, special arc extinguishing devices are provided to ensure the acceleration of arc extinction.
In addition, in power circuits with significant inductance (for example, excitation windings of generators), a discharge resistor R p is connected in parallel with the R-L circuit (i.e., the corresponding winding) (Fig. 65, a). In this case, after switching off the switch B1, the R-L circuit is not interrupted, but is closed to the resistor R p. The current in the circuit i does not decrease instantly, but gradually - exponentially (Fig. 65.6), since e. d.s. self-induction e L , arising in the inductance L, prevents the current from decreasing. The voltage u p across the discharge resistor also changes exponentially during the current change process. It is equal to the voltage applied to the R-L circuit, i.e., to the terminals of the corresponding
current winding. At the initial moment, U p start = UR p / R, i.e., depends on the resistance of the discharge resistor; at high values of Rp, this voltage can be excessively high and dangerous for the insulation of the electrical installation. In practice, to limit the resulting overvoltages, the resistance R p of the discharge resistor is taken no more than 4-8 times the resistance R of the corresponding winding.
Conditions for the occurrence of transient processes. The processes discussed above when turning on and off the R-L circuit are called transients. They arise when turning on and off the source or individual sections of the circuit, as well as when changing the operating mode, for example, with a sudden change in load, breaks and short circuits. The same transients take place under the indicated conditions and in circuits containing capacitors with a capacity of C. In some cases, transients are dangerous for sources and receivers, since the resulting currents and voltages can many times exceed the nominal values for which these are designed. devices. However, in some elements of electrical equipment, in particular in industrial electronics devices, transients are operating modes.
Physically, the occurrence of transient processes is explained by the fact that inductors and capacitors are energy storage devices, and the process of accumulation and release of energy in these elements cannot occur instantly, therefore, the current in the inductor and the voltage across the capacitor cannot change instantly. The time of the transient process, during which, when switching on, switching off and changing the operating mode of the circuit, a gradual change in current and voltage occurs, is determined by the values of R, L and C of the circuit and can be fractions and units of seconds. After the end of the transient, the current and voltage acquire new values, which are called established.
The magnetic field of the circuit, in which the current strength changes, induces a current not only in other circuits, but also in itself. This phenomenon is called self-induction.
It has been experimentally established that the magnetic flux of the magnetic induction vector of the field created by the current flowing in the circuit is proportional to the strength of this current:
where L is the loop inductance. A constant characteristic of the circuit, which depends on its shape and size, as well as on the magnetic permeability of the medium in which the circuit is located. [L] = Hn (Henry,
1H = Wb / A).
If during the time dt the current in the circuit changes by dI, then the magnetic flux associated with this current will change by dФ \u003d LdI, as a result of which an EMF of self-induction will appear in this circuit:
The minus sign shows that the EMF of self-induction (and, consequently, the self-induction current) always prevents a change in the current strength that caused self-induction.
A good example of the phenomenon of self-induction is the extra currents of closing and opening that occur when turning on and off electrical circuits with significant inductance.
Magnetic field energy
The magnetic field has potential energy, which at the moment of its formation (or change) is replenished due to the energy of the current in the circuit, which in this case does work against the EMF of self-induction that arises as a result of a change in the field.
Work dA for an infinitely small period of time dt, during which the self-induction EMF 
and current I can be considered constant, equals:
.
(5)
The minus sign indicates that the elementary work is done by the current against the EMF of self-induction. To determine the work when the current changes from 0 to I, we integrate the right side, we get:
.
(6)
This work is numerically equal to the increase in potential energy ΔW p of the magnetic field associated with this circuit, i.e. A= -ΔW p.
Let us express the energy of the magnetic field in terms of its characteristics using the example of a solenoid. We will assume that the magnetic field of the solenoid is homogeneous and is mainly located inside it. Let us substitute in (5) the value of the inductance of the solenoid, expressed through its parameters and the value of the current I, expressed from the formula for the induction of the magnetic field of the solenoid:
, (7)
where N is the total number of turns of the solenoid; ℓ is its length; S is the cross-sectional area of the internal channel of the solenoid.
, (8)
After substitution we have:
Dividing both parts by V, we obtain the volumetric field energy density:
(10)
or, given that 
we get 
.
(11)
Alternating current
2.1 Alternating current and its main characteristics
An alternating current is a current that changes over time both in magnitude and direction. An example of alternating current is the consumed industrial current. This current is sinusoidal, i.e. the instantaneous value of its parameters change over time according to the sine (or cosine) law:
i= I 0 sinωt, u = U 0 sin(ωt + φ 0). (12)
P 
Variable sinusoidal current can be obtained by rotating the frame (circuit) at a constant speed
in a uniform magnetic field with induction B(Fig.5). In this case, the magnetic flux penetrating the circuit changes according to the law
where S is the area of the contour, α = ωt is the angle of rotation of the frame in time t. Flux change leads to induction EMF
, (17)
whose direction is determined by the Lenz rule.
E 
If the circuit is closed (Fig. 5), then current flows through it:
.
(18)
Graph of change in electromotive force 
and induction current i shown in Fig.6.
Alternating current is characterized by period T, frequency ν = 1/T, cyclic frequency 
and phase φ \u003d (ωt + φ 0) Graphically, the values \u200b\u200bof the voltage and strength of the alternating current in the circuit section will be represented by two sinusoids, generally shifted in phase by φ.
To characterize alternating current, the concepts of the effective (effective) value of current and voltage are introduced. The effective value of the alternating current strength is the strength of such a direct current that releases as much heat in a given conductor during one period as it releases heat and a given alternating current.
,
. (13)
Instruments included in the alternating current circuit (ammeter, voltmeter) show the effective values of current and voltage.
Electromagnetic induction - the generation of electric currents by magnetic fields that change over time. The discovery of this phenomenon by Faraday and Henry introduced a certain symmetry to the world of electromagnetism. Maxwell in one theory managed to collect knowledge about electricity and magnetism. His research predicted the existence of electromagnetic waves before experimental observations. Hertz proved their existence and opened the era of telecommunications to mankind.
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Faraday's experiments
Faraday and Lenz laws
Electric currents create magnetic effects. Is it possible for a magnetic field to generate an electric one? Faraday discovered that the desired effects arise due to changes in the magnetic field over time.
When a conductor is crossed by an alternating magnetic flux, an electromotive force is induced in it, causing an electric current. The system that generates the current can be a permanent magnet or an electromagnet.
The phenomenon of electromagnetic induction is governed by two laws: Faraday's and Lenz's.
Lenz's law allows you to characterize the electromotive force with respect to its direction.
Important! The direction of the induced emf is such that the current it causes tends to oppose the cause that creates it.
Faraday noticed that the intensity of the induced current increases when the number of field lines traversing the circuit changes faster. In other words, the EMF of electromagnetic induction is directly dependent on the speed of the moving magnetic flux.
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EMF induction
The induction emf formula is defined as:
E \u003d - dF / dt.
The "-" sign shows how the polarity of the induced emf is related to the sign of the flux and the changing speed.
A general formulation of the law of electromagnetic induction is obtained, from which expressions for particular cases can be derived.
The movement of a wire in a magnetic field
When a wire of length l moves in a magnetic field with induction B, an EMF will be induced inside it, proportional to its linear velocity v. To calculate the EMF, the formula is used:
- in the case of conductor movement perpendicular to the direction of the magnetic field:
E \u003d - B x l x v;
- in case of movement at a different angle α:
E \u003d - B x l x v x sin α.
The induced EMF and current will be directed in the direction we find using the right hand rule: by placing your hand perpendicular to the magnetic field lines and pointing your thumb in the direction the conductor moves, you can find out the direction of the EMF by the remaining four straightened fingers.
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Moving a wire in MP
Rotating coil
The operation of the electric power generator is based on the rotation of the circuit in the MP, which has N turns.
EMF is induced in the electrical circuit whenever the magnetic flux crosses it, in accordance with the definition of the magnetic flux Ф = B x S x cos α (magnetic induction multiplied by the surface area through which the MP passes, and the cosine of the angle formed by the vector B and the perpendicular line to the plane S).
It follows from the formula that F is subject to changes in the following cases:
- the intensity of the MF changes - the vector B;
- the area bounded by the contour varies;
- the orientation between them, given by the angle, changes.
In Faraday's first experiments, induced currents were obtained by changing the magnetic field B. However, an EMF can be induced without moving the magnet or changing the current, but simply by rotating the coil around its axis in the magnetic field. In this case, the magnetic flux changes due to a change in the angle α. The coil, during rotation, crosses the lines of the MP, an emf arises.
If the coil rotates uniformly, this periodic change results in a periodic change in magnetic flux. Or the number of MF lines of force crossed every second takes equal values with equal time intervals.
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Contour rotation in MP
Important! The induced emf changes with the orientation over time from positive to negative and vice versa. The graphical representation of the EMF is a sinusoidal line.
For the formula for the EMF of electromagnetic induction, the expression is used:
E \u003d B x ω x S x N x sin ωt, where:
- S is the area limited by one turn or frame;
- N is the number of turns;
- ω is the angular velocity with which the coil rotates;
- B – MF induction;
- angle α = ωt.
In practice, in alternators, often the coil remains stationary (stator) and the electromagnet rotates around it (rotor).
EMF self-induction
When an alternating current passes through the coil, it generates an alternating magnetic field, which has a changing magnetic flux that induces an emf. This effect is called self-induction.
Since the MP is proportional to the intensity of the current, then:
where L is the inductance (H), determined by geometric quantities: the number of turns per unit length and the dimensions of their cross section.
For the induction emf, the formula takes the form:
E \u003d - L x dI / dt.
Mutual induction
If two coils are located side by side, then an EMF of mutual induction is induced in them, depending on the geometry of both circuits and their orientation relative to each other. When the separation of the circuits increases, the mutual inductance decreases, as the magnetic flux connecting them decreases.
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Mutual induction
Let there be two coils. Through the wire of one coil with N1 turns, current I1 flows, creating an MF passing through the coil with N2 turns. Then:
- Mutual inductance of the second coil relative to the first:
M21 = (N2 x F21)/I1;
- Magnetic Flux:
F21 = (M21/N2) x I1;
- Find the induced emf:
E2 = - N2 x dФ21/dt = - M21x dI1/dt;
- EMF is induced identically in the first coil:
E1 = - M12 x dI2/dt;
Important! The electromotive force caused by mutual inductance in one coil is always proportional to the change in electric current in the other.
Mutual inductance can be considered equal to:
M12 = M21 = M.
Accordingly, E1 = - M x dI2/dt and E2 = M x dI1/dt.
M = K √ (L1 x L2),
where K is the coupling coefficient between two inductances.
The phenomenon of mutual inductance is used in transformers - electrical devices that allow you to change the value of the voltage of an alternating electric current. The device consists of two coils wound around one core. The current present in the first one creates a changing magnetic field in the magnetic circuit and an electric current in the other coil. If the number of turns of the first winding is less than the other, the voltage increases and vice versa.
SELF-INDUCTION
Each conductor through which electricity flows. current is in its own magnetic field.
When the current strength changes in the conductor, the m.field changes, i.e. the magnetic flux created by this current changes. A change in the magnetic flux leads to the emergence of a vortex el. field and induction emf appears in the circuit. 
This phenomenon is called self-induction.
Self-induction - the phenomenon of the occurrence of EMF induction in email. circuit as a result of a change in current strength.
The resulting emf is called EMF self-induction
Closing the circuit 
When closing in el. the current increases in the circuit, which causes an increase in the magnetic flux in the coil, a vortex electric arises. field directed against the current, i.e. an EMF of self-induction occurs in the coil, which prevents the current from rising in the circuit (the vortex field slows down the electrons).
As a result L1 lights up later, than L2.
Open circuit 
When the electric circuit is opened, the current decreases, there is a decrease in the m.flow in the coil, a vortex electric field appears, directed like a current (tending to maintain the same current strength), i.e. A self-inductive emf appears in the coil, which maintains the current in the circuit.
As a result, L when turned off flashes brightly.
Conclusion
in electrical engineering, the phenomenon of self-induction manifests itself when the circuit is closed (the electric current increases gradually) and when the circuit is opened (the electric current does not disappear immediately).
What does the EMF of self-induction depend on?
Email current creates its own magnetic field. The magnetic flux through the circuit is proportional to the magnetic field induction (Ф ~ B), the induction is proportional to the current strength in the conductor
(B ~ I), therefore the magnetic flux is proportional to the current strength (Ф ~ I).
The EMF of self-induction depends on the rate of change in the current strength in the email. circuits, from the properties of the conductor
(size and shape) and on the relative magnetic permeability of the medium in which the conductor is located.
A physical quantity showing the dependence of the self-induction EMF on the size and shape of the conductor and on the environment in which the conductor is located is called the self-induction coefficient or inductance. 
Inductance - physical. a value numerically equal to the EMF of self-induction that occurs in the circuit when the current strength changes by 1 ampere in 1 second.
Also, the inductance can be calculated by the formula:
where F is the magnetic flux through the circuit, I is the current strength in the circuit.
Inductance units in the SI system:
The inductance of a coil depends on:
the number of turns, the size and shape of the coil, and the relative magnetic permeability of the medium
(possible core).
EMF of self-induction prevents the increase in current strength when the circuit is turned on and the decrease in current strength when the circuit is opened.
Around a conductor with current there is a magnetic field that has energy.
Where does it come from? Current source included in el. chain, has a store of energy.
At the time of closing email. In the circuit, the current source expends part of its energy to overcome the action of the emerging EMF of self-induction. This part of the energy, called the self-energy of the current, goes to the formation of a magnetic field.
The magnetic field energy is own current energy.
The self-energy of the current is numerically equal to the work that the current source must do to overcome the self-induction EMF in order to create a current in the circuit.
The energy of the magnetic field created by the current is directly proportional to the square of the current strength.
Where does the energy of the magnetic field disappear after the current stops? - stands out (when a circuit with a sufficiently large current is opened, a spark or arc may occur)
QUESTIONS FOR THE VERIFICATION WORK
on the topic "Electromagnetic induction"
|
1. List 6 ways to obtain an induction current. |
When the current in the circuit changes, the flux of magnetic induction through the surface bounded by this circuit changes, the change in the flux of magnetic induction leads to the excitation of the EMF of self-induction. The direction of the EMF turns out to be such that when the current in the circuit increases, the emf prevents the current from increasing, and when the current decreases, it stops decreasing.
The magnitude of the EMF is proportional to the rate of change of the current strength I and loop inductance L :
.Due to the phenomenon of self-induction in an electric circuit with an EMF source, when the circuit is closed, the current is not established instantly, but after some time. Similar processes also occur when the circuit is opened, while the value of the self-induction emf can significantly exceed the source emf. Most often in ordinary life it is used in car ignition coils. Typical self-induction voltage at 12V battery voltage is 7-25kV.
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