It is well known that a train leaving a station cannot immediately reach the required speed.

The required speed is achieved only after a certain period of time. During this period, a significant part of the locomotive's energy is spent on overcoming the inertia of the train, i.e., on forming a reserve of kinetic energy, and a very small part on overcoming friction.

Due to the fact that a moving train has a reserve of kinetic energy, it cannot stop instantly and will continue to move by inertia for some time, that is, until the entire reserve of kinetic energy imparted to it by the locomotive at the beginning of the movement is spent on friction.

Similar phenomena occur in a closed electrical circuit when the current is turned on and off.

At the moment the direct current is turned on (Figure 1), a magnetic force field.

Picture 1. Inertia of electric current. When current is turned on, a magnetic field appears around the conductor.

In the first moments after turning on the current, a significant part of the energy of the current source is spent on creating this magnetic field and only a small part on overcoming the resistance of the conductor, or rather on heating the conductor by the current. Therefore, at the moment of circuit closure the current does not immediately reach its maximum value . The maximum current strength is set in the circuit only after the process of formation of a magnetic field around the conductor is completed (Figure 2).

Figure 2. When the current source is turned on, the current in the circuit is not immediately established.

If, without breaking a closed circuit, you turn off the current source from it, then the current in the circuit will not stop immediately, but will flow in it, gradually decreasing for some time (Figure 3) until the magnetic field around the conductor disappears, i.e. that is, until the entire supply of energy contained in the magnetic field is used up.

Figure 2. The influence of self-induction emf on the current in the circuit. When the current source is turned off, the current in the circuit does not stop immediately.

So, the magnetic field is a carrier of energy. It accumulates energy when the DC source is turned on and releases it back into the circuit after the current source is turned off. The energy of a magnetic field thus has much in common with the kinetic energy of a moving object. The magnetic field causes the “inertia” of the electric current.

We know that whenever the magnetic flux penetrating the area bounded by a closed electric circuit changes, a induced emf .

In addition, we know that any change in current in the circuit entails a change number of magnetic field lines covered by this chain. If the closed circuit is motionless, then the number of magnetic lines of force piercing a given area can change only when new lines enter from the outside within this area or when existing lines go beyond this area. In both cases, the magnetic lines of force must cross the conductor during their movement. When crossing a conductor, magnetic lines of force induce an induced emf in it. But since in this case the conductor induces an emf in itself, this emf is called Self-induced emf.

When a direct current source is connected to any closed circuit, the area limited by this circuit begins to be penetrated from the outside by magnetic lines of force. Each magnetic line of force coming from outside, crossing a conductor, induces in it Self-induced emf.

The electromotive force of self-induction, acting against the EMF of the current source, retards the increase in current in the circuit. After a few moments, when the increase in the magnetic flux around the circuit stops, the self-induction emf disappears and a current strength is established in the circuit, determined by Ohm's law:

I=U/R

When the current source is turned off from a closed circuit, the magnetic lines of force must disappear from the space limited by the conductor. Each outgoing magnetic field line, when crossing a conductor, induces a self-inductive emf in it, which has the same direction as the emf of the current source; therefore, the current in the circuit will not stop immediately, but will flow in the same direction, gradually decreasing until the magnetic flux inside the circuit completely disappears. The current flowing through a circuit after the current source is turned off is called self-induction current.

If the circuit breaks when the source is turned off, the self-induction current appears in the form of a spark at the point where the circuit opens.

The term induction in electrical engineering means the occurrence of current in an electrical closed circuit if it is in a changing state. Discovered only two hundred years ago by Michael Faraday. This could have been done much earlier by Andre Ampere, who conducted similar experiments. He inserted a metal rod into the coil, and then, bad luck, he went into another room to look at the galvanometer needle - what if it moved. And the arrow regularly did its job - it deviated, but while Ampere wandered through the rooms, it returned to the zero mark. This is how the phenomenon of self-induction waited for another ten years until the coil, the device and the researcher were simultaneously in the right place.

The main point of this experiment was that induced emf occurs only when the magnetic field passing through a closed loop changes. But you can change it in any way you like - either change the magnitude of the magnetic field itself, or simply move the source of the field relative to the same closed loop. The emf that arises in this case is called “mutual inductive emf.” But this was only the beginning of discoveries in the field of induction. Even more surprising was the phenomenon of self-induction, which was discovered around the same time. In his experiments, it was discovered that the coils not only induced a current in another coil, but also when the current in this coil changed, it induced an additional EMF in it. This is what they called the EMF of self-induction. Of great interest is the direction of the current. It turned out that in the case of self-induction EMF, its current is directed against its “parent” - the current caused by the main EMF.

Is it possible to observe the phenomenon of self-induction? As they say, nothing is simpler. Let's assemble the first two - a series-connected inductor and a light bulb, and the second - only the light bulb. Let's connect them to the battery via a common switch. When you turn it on, you can see that the light bulb in the circuit with the coil lights up “reluctantly,” and the second light bulb, which is faster “on the rise,” turns on instantly. What's happening? In both circuits, after switching on, current begins to flow, and it changes from zero to its maximum, and it is precisely the change in current that the inductor waits for, which generates the self-induction emf. There is an EMF and a closed circuit, which means there is also its current, but it is directed opposite to the main current of the circuit, which will eventually reach a maximum value determined by the parameters of the circuit and stop growing, and since there is no change in the current, there is no self-induction EMF. It's simple. A similar picture, but “exactly the opposite,” is observed when the current is turned off. True to its “bad habit” of opposing any change in current, the self-inductive emf maintains its flow in the circuit after the power is turned off.

The question immediately arose - what is the phenomenon of self-induction? It was found that the self-induction EMF is affected by the rate of change of current in the conductor, and can be written:

From this it can be seen that the self-induction EMF E is directly proportional to the rate of change of current dI/dt and the proportionality coefficient L, called inductance. For his contribution to the study of the question of what the phenomenon of self-inductance consists of, George Henry was rewarded by the fact that the unit of measurement of inductance, Henry (H), bears his name. It is the inductance of the current flow circuit that determines the phenomenon of self-induction. One can imagine that inductance is a kind of “storage” of magnetic energy. If the current in the circuit increases, electrical energy is converted into magnetic energy, delaying the increase in current, and when the current decreases, the magnetic energy of the coil is converted into electrical energy and maintains the current in the circuit.

Probably everyone has seen a spark when turning off a plug from a socket - this is the most common manifestation of self-induction EMF in real life. But in everyday life, currents open at a maximum of 10-20 A, and the opening time is about 20 ms. With an inductance of the order of 1 H, the self-induction emf in this case will be equal to 500 V. It would seem that the question of what the phenomenon of self-induction consists of is not so complicated. But in fact, self-induced emf is a big technical problem. The bottom line is that when the circuit breaks, when the contacts have already separated, self-induction maintains the flow of current, and this leads to burnout of the contacts, because In technology, circuits with currents of hundreds and even thousands of amperes are switched. Here we are often talking about a self-induction EMF of tens of thousands of volts, and this requires additional solutions to technical issues related to overvoltages in electrical circuits.

But not everything is so gloomy. It happens that this harmful EMF is very useful, for example, in internal combustion engine ignition systems. Such a system consists of an inductor in the form of an autotransformer and a chopper. A current is passed through the primary winding, which is turned off by a breaker. As a result of an open circuit, a self-inductive emf of hundreds of volts occurs (while the battery provides only 12V). Then this voltage is further transformed, and a pulse of more than 10 kV is sent to the spark plugs.

When the switch is closed in the circuit shown in Figure 1, it will occur, the direction of which is shown by single arrows. With the advent of current, a current arises, the induction lines of which cross the conductor and induce emf in it. As stated in the article “The phenomenon of electromagnetic induction”, this EMF is called self-induction EMF. Since any induced emf is directed against the cause that caused it, and this cause will be the emf of the battery of elements, the self-induction emf of the coil will be directed against the emf of the battery. The direction of self-induction EMF in Figure 1 is shown by double arrows.

Thus, the current is not established in the circuit immediately. Only when it is established, the intersection of the conductor with magnetic lines will stop and the self-induction EMF will disappear. Then the circuit will leak.

Figure 2 shows a graphical representation of direct current. The horizontal axis represents time, and the vertical axis represents current. It can be seen from the figure that if at the first moment of time the current is 6 A, then at the third, seventh and so on moments of time it will also be equal to 6 A.

Figure 3 shows how the current is established in the circuit after switching on. The self-induction emf, directed at the moment of switching on against the emf of the battery of elements, weakens the current in the circuit, and therefore at the moment of switching on the current is zero. Then, at the first moment of time, the current is 2 A, at the second moment of time - 4 A, at the third - 5 A, and only after some time a current of 6 A is established in the circuit.

Figure 3. Graph of current increase in the circuit taking into account the self-inductive emf	Figure 4. The self-induction EMF at the moment of opening the circuit is directed in the same direction as the EMF of the voltage source

When the circuit is opened (Figure 4), the disappearing current, the direction of which is shown by a single arrow, will reduce its magnetic field. This field, decreasing from a certain value to zero, will again cross the conductor and induce a self-induction emf in it.

When an electrical circuit with inductance is turned off, the self-inductive emf will be directed in the same direction as the emf of the voltage source. The direction of the self-induction EMF is shown in Figure 4 by a double arrow. As a result of the action of self-induction emf, the current in the circuit does not disappear immediately.

Thus, the self-induced emf is always directed against the cause that caused it. Noting this property, they say that the self-induction EMF is reactive in nature.

Graphically, the change in current in our circuit, taking into account the self-inductive emf when it is closed and when it is subsequently opened at the eighth moment in time, is shown in Figure 5.

Figure 5. Graph of the rise and fall of the current in the circuit, taking into account the self-induction emf	Figure 6. Induction currents when the circuit is opened

When opening circuits containing a large number of turns and massive steel cores or, as they say, having high inductance, the self-inductive emf can be many times greater than the emf of the voltage source. Then, at the moment of opening, the air gap between the knife and the fixed clamp of the switch will be broken and the resulting electric arc will melt the copper parts of the switch, and if there is no casing on the switch, it can burn a person’s hands (Figure 6).

In the circuit itself, the self-induction EMF can break through the insulation of the turns of the coils, and so on. To avoid this, some switching devices provide protection against self-induction EMF in the form of a special contact that short-circuits the electromagnet winding when switched off.

It should be taken into account that the self-induction EMF manifests itself not only at the moments when the circuit is turned on and off, but also during any changes in current.

The magnitude of the self-induction emf depends on the rate of change of current in the circuit. So, for example, if for the same circuit in one case within 1 second the current in the circuit changed from 50 to 40 A (that is, by 10 A), and in another case from 50 to 20 A (that is, by 30 A ), then in the second case a threefold greater self-induction emf will be induced in the circuit.

The magnitude of the self-inductive emf depends on the inductance of the circuit itself. Circuits with high inductance are the windings of generators, electric motors, transformers and induction coils with steel cores. Straight conductors have lower inductance. Short straight conductors, incandescent lamps and electric heating devices (stoves, stoves) have practically no inductance and the appearance of self-inductive emf in them is almost not observed.

The magnetic flux penetrating the circuit and inducing the self-induction emf in it is proportional to the current flowing through the circuit:

F = L × I ,

Where L- proportionality coefficient. It's called inductance. Let us determine the dimension of inductance:

Ohm × sec is otherwise called henry (Hn).

1 henry = 10 3 ; millihenry (mH) = 10 6 microhenry (µH).

Inductance, except Henry, is measured in centimeters:

1 henry = 10 9 cm.

For example, 1 km of telegraph line has an inductance of 0.002 H. The inductance of the windings of large electromagnets reaches several hundred henries.

If the loop current changes by Δ i, then the magnetic flux will change by the value Δ Ф:

Δ Ф = L × Δ i .

The magnitude of the self-induction EMF that appears in the circuit will be equal to (formula of the self-induction EMF):

If the current changes uniformly over time, the expression will be constant and can be replaced by the expression. Then the absolute value of the self-induction emf arising in the circuit can be found as follows:

Based on the last formula, we can define the unit of inductance - henry:

A conductor has an inductance of 1 H if, with a uniform change in current by 1 A per 1 second, a self-inductive emf of 1 V is induced in it.

As we saw above, self-induction emf occurs in a direct current circuit only at the moments of its switching on, switching off, and whenever it changes. If the circuit is unchanged, then the magnetic flux of the conductor is constant and the self-induction emf cannot arise (since . At moments of current change in the circuit, the self-induction emf interferes with changes in the current, that is, it provides a kind of resistance to it.

Often in practice there are cases when it is necessary to make a coil that does not have inductance (additional resistance to electrical measuring instruments, resistance of plug rheostats, and the like). In this case, a bifilar coil winding is used (Figure 7)

As is easy to see from the drawing, in adjacent conductors currents flow in opposite directions. Consequently, the magnetic fields of neighboring conductors cancel each other out. The total magnetic flux and inductance of the coil will be zero. To further understand the concept of inductance, let us give an example from the field of mechanics.

As is known from physics, according to Newton’s second law, the acceleration received by a body under the influence of a force is proportional to the force itself and inversely proportional to the mass of the body:

Let's compare the last formula with the formula for self-induced emf, taking the absolute value of the emf:

If in these formulas changes in speed over time are likened to changes in current over time, mechanical force is likened to the electromotive force of self-induction, then the mass of the body will correspond to the inductance of the circuit.

With uniform linear motion a= 0, so F= 0, that is, if no forces act on the body, its motion will be rectilinear and uniform (Newton’s first law).

In DC circuits, the current value does not change and therefore e L = 0.

• Inductance

Inductance

• Current I, flowing in a closed loop, creates a magnetic field around itself B .

• F ~ I.

• where is the proportionality coefficient L called circuit inductance .

Self-induction phenomenon

• When the current changes I the magnetic field it creates changes in the circuit. Consequently, an emf is induced in the circuit.

• This process is called self-induction .

• In the SI system, inductance is measured in henry: [ L] = Gn = Vb/A = V s/A.

Self-induction phenomenon

• E.m.f. induction E i created by an external magnetic field.

• E.m.f. self-induction E S is created when its own magnetic field changes.

• In general, the loop inductance L depends on

• 1) the geometric shape of the contour and its dimensions,

• 2) magnetic permeability of the medium in which the circuit is located.

• In electrostatics, an analogue of inductance is electrical capacitance WITH solitary conductor, which depends on the shape, size, dielectric constant ε environment.

• L = const, if magnetic permeability μ the environment and the geometric dimensions of the contour are constant.

Faraday's law for self-induction

• The minus sign in Faraday's law, in accordance with Lenz's rule, means that the presence of inductance L leads to a slower change in current I in the circuit.

If the current I increases, then dI/dt> 0 and, accordingly, E S < 0, т.е. ток самоиндукции IS directed towards the current I

• If the current I increases, then dI/dt> 0 and, accordingly, E S < 0, т.е. ток самоиндукции IS directed towards the current I external source and slows down its growth.

• If the current I decreases, then dI/dt< 0 и, соответственно, ES> 0, i.e. self-induction current IS has the same direction as the decreasing current I external source and slows down its decrease.

^ Faraday's law for self-induction

• If the circuit has a certain inductance L, then any change in current I the more it slows down, the more L contour, i.e. the circuit has electrical inertia .

Solenoid inductance

• Inductance L depends only on the geometric dimensions of the circuit and magnetic permeability μ environment.

• ФN– flux of magnetic induction through N turns,

• F = B.S.- magnetic flux through the pad S, limited to one turn.

Solenoid inductance

• Solenoid field:

• l– solenoid length,

• n = N/ l– number of turns per unit length of the solenoid.

• (2) (1):

• According to Lenz's rule, when turning on and off the current in a circuit containing inductance L, a self-induction current occurs IS, which is directed so as to prevent the current from changing I in the chain.

Extra currents

• Key TO pregnant 1 :

• Key TO pregnant 2 (open circuit):

• E arises S and the current caused by it

Extra currents

• constant called relaxation time – time during which the current strength I decreases in e once.

• The more L, the more τ , and the slower the current decreases I.

Extra currents

• At circuit closure in addition to the external emf. E emf arises. self-induction E S.

Extra currents

• At the moment of closure t= 0 current I= 0, variable a 0 = – I 0, at time t current strength I, variable a =I – I 0

Extra currents

• I 0 – steady current.

• The establishment of current occurs the faster, the smaller L circuit and its greater resistance R

Extra currents of closing and breaking

• Because battery resistance r is usually small, then we can assume that R R 0, where

• R 0 – circuit resistance without taking into account the resistance of the EMF source. Steady current

R 0 to R.

• ● Instant increase in circuit resistance from R 0 to R.

• The steady current was

• At turning off the source e.m.f.

• (open circuit) the current varies according to the law

• The magnitude of the e.m.f. self-induction

RR>>R 0), then E S

• If the circuit switches to very high external resistance R, for example, the chain breaks ( R>>R 0), then E S can become huge and a voltaic arc is formed between the open ends of the switch.

e.m.f. self-induction

• In a circuit with high inductance, E S there may be more emf. source E included in the circuit, which can lead to insulation breakdown and equipment failure.

• Therefore, resistance must be introduced into the circuit gradually, reducing the ratio dI /dt.

Mutual induction

• The magnetic flux formed by circuit 1 penetrates circuit 2:

• L 21 – proportionality coefficient.

• If I 1 changes, then an emf is induced in circuit 2.

Mutual induction

• Similarly, if circuit 2 changes I 2, then in the first circuit a change in the magnetic flux induces an emf:

Odds L 12 = L 21 – mutual inductance contours depends on

• 1. geometric shape,

• 2. sizes,

• 3. mutual position,

• 4. magnetic permeability of the medium μ .

For two coils on a common toroidal core

• N 1, N 2 – number of turns of the first and second circuit, respectively,

• l– length of the core (toroid) along the midline,

• S– core section.

Transformer - a device consisting of two or more coils wound on one common core.

• Serve to increase or decrease AC voltage:

• transformation ratio.

• Structurally, transformers are designed in such a way that the magnetic field is almost completely concentrated in the core.

• In most transformers, the secondary winding is wound on top of the primary winding.

Autotransformer – a transformer consisting of one winding.

• Boosting:

• 1-2 U supplied, 1-3 U removed.

• Downgrade:

• 1-3 U supplied, 1-2 U removed.

Skin effect

• When alternating current passes through a conductor, the magnetic field inside the conductor changes. A time-varying magnetic field generates in a conductor self-induction eddy currents .

Skin effect

• The planes of eddy currents pass through the axis of the conductor.

• According to Lenz's rule, eddy currents prevent the main current from changing inside the conductor and promote its change near the surface.

• For alternating current, the resistance inside the conductor is greater than the resistance at the surface R inside > R on top

Skin effect

• The alternating current density is not the same across the cross section:

• jmax on a surface, jmin inside on the axis.

• This phenomenon is called skin effect .

Consequence of the skin effect

• RF currents flow through a thin surface layer, so the conductors for them are made hollow, and part of the outer surface is coated with silver.

Application:

• a method of surface hardening of metals in which, when heated by high-frequency currents (HF), only the surface layer is heated.

Magnetic field energy. Volumetric magnetic field energy density

• The energy of a magnetic field is equal to the work expended by the current to create this field.

• Work due to induction phenomena

Magnetic field energy

• Job dA is spent on changing the magnetic flux by the amount dФ.

• Work to create magnetic flux F:

Volumetric magnetic field energy density

• We'll find ω for example a solenoid

We have already studied that a magnetic field arises near a conductor carrying current. We also studied that an alternating magnetic field generates a current (the phenomenon of electromagnetic induction). Let's consider an electrical circuit. When the current strength changes in this circuit, the magnetic field will change, as a result of which an additional induced current. This phenomenon is called self-induction, and the current arising in this case is called self-induction current.

Self-induction phenomenon- this is the occurrence of an EMF in a conducting circuit, created as a result of a change in current strength in the circuit itself.

Loop inductance depends on its shape and size, on the magnetic properties of the environment and does not depend on the current strength in the circuit.

The self-induction emf is determined by the formula:

The phenomenon of self-induction is similar to the phenomenon of inertia. Just as in mechanics it is impossible to instantly stop a moving body, so a current cannot instantly acquire a certain value due to the phenomenon of self-induction. If a coil is connected in series with the second lamp in a circuit consisting of two identical lamps connected in parallel to a current source, then when the circuit is closed, the first lamp lights up almost immediately, and the second with a noticeable delay.

When the circuit is opened, the current strength quickly decreases, and the resulting self-induction emf prevents the decrease in magnetic flux. In this case, the induced current is directed in the same way as the original one. The self-induced emf can be many times greater than the external emf. Therefore, light bulbs very often burn out when the lights are turned off.